1

University of Waterloo

Department of Mechanical Engineering

ME 322 - Mechanical Design 1ME 322 - Mechanical Design 1

Partial notes ndash Part 4 (Fatigue)(G Glinka)

Fall 2005

2

NNii = =

NNpp = =

NNTT = =

Metal FatigueMetal Fatigue is a process which causes premature irreversible damage or is a process which causes premature irreversible damage or failure of a component subjected to repeated loadingfailure of a component subjected to repeated loading

FATIGUE - What is itFATIGUE - What is it

3

Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several

disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics

bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component

bull Influenced by a componentrsquos environmentbull Takes many forms

ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue

Fatigue is not cause of failure per se but leads to the final fracture event

4

The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics

(from Ewalds amp Wanhil ref3)

5

Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

2

NNii = =

NNpp = =

NNTT = =

Metal FatigueMetal Fatigue is a process which causes premature irreversible damage or is a process which causes premature irreversible damage or failure of a component subjected to repeated loadingfailure of a component subjected to repeated loading

FATIGUE - What is itFATIGUE - What is it

3

Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several

disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics

bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component

bull Influenced by a componentrsquos environmentbull Takes many forms

ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue

Fatigue is not cause of failure per se but leads to the final fracture event

4

The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics

(from Ewalds amp Wanhil ref3)

5

Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

3

Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several

disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics

bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component

bull Influenced by a componentrsquos environmentbull Takes many forms

ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue

Fatigue is not cause of failure per se but leads to the final fracture event

4

The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics

(from Ewalds amp Wanhil ref3)

5

Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

4

The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics

(from Ewalds amp Wanhil ref3)

5

Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

5

Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

6

Locally the crack grows in shear

macroscopically it grows in tension

Δσ

Δσ

c

Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

7

The Process of FatigueThe Process of Fatigue

The Materials Science PerspectiveThe Materials Science Perspective

bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

8

Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic

LoadingLoading

(Collins ref 22 )

A ndash fatigue crack area

B ndash area of the final static failure

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

9

Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

10

Factors Influencing Fatigue LifeFactors Influencing Fatigue Life

Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the

accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and

growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial

loading because in bending fatigue cracks propagate into the region of lower stresses

Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic

deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives

bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks

bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)

bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

11

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Allowable Load - Fatigue Life

MaterialProperties

Information path in strength and fatigue life prediction procedures

Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

12

Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses

peak

tn

peak

K

S

n

y

dn

0

T

peak

Str

ess

M

b)a)

n

y

dn

0

T

peak

Str

ess

S

S

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

13

Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal

One cycle

m

max

min

Str

es

s

Time0

Constant amplitude stress historya)

Str

es

s

Time

0

Variable amplitude stress history

One reversal

b)

max min

max mi

max min

mi

max

n

n

2 2

2

peak peak peak

peak peak

peak peakpeak

m

peak

pe

peakpe

ak

aka

R

In the case of the peak stress history the important parameters are

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

14

Str

ess

Time

Stress history Rainflow counted cycles

i-1

i-2

i+1

i

i+2

Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles

1 1i i i iABS ABS

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

15

A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation

1 1i i i iABS ABS

The stress amplitude of such a cycle is

1

2i i

a

ABS

The stress range of such a cycle is

1i iABS

The mean stress of such a cycle is

1

2i i

m

The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

16

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

17

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

18

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

19

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

20

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str

es

s (

MP

a)x

10

2

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reversal point No

Str

es

s (

MP

a)x

10

2

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

21

Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

22

The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)

bull The principles of the S-N approach (the nominal stress method)

bull Fatigue damage accumulation

bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures

bull

bull Fatigue life prediction in the design process

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

23

AB

Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same

peakS Smin

Smax

WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

24

Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10

3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)

Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit

Number of cycles N

Str

ess

am

plitu

de

Sa (

ksi)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N

Fatigue S-N curveFatigue S-N curve

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

25

Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se

The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103

corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)

S103 = 090Su and Se = S106 = 05 Su - bending

S103 = 075Su and Se = S106 = 035 - 045Su - axial

S103 = 072Su and Se = S106 = 029 Su - torsion

103 104 105 106 10701

03

05

10

S103

Se

Number of cycles Log(N)

Rel

ativ

e st

ress

am

plitu

de

SaS

u

Torsion

AxialBending

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

26

Approximate endurance limit for various materialsApproximate endurance limit for various materials

Magnesium alloys (at 108 cycles) Se = 035Su

Copper alloys (at 108 cycles) 025Sult Se lt050Su

Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su

Titanium alloys (at 107 cycles) 045Su ltSelt 065Su

Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)

Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)

Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)

S ndash N curveS ndash N curve

33

1 1 1

2

10101log lo

0

g

1

3

A

m mm ma

m m

e e

Aa aS C N or N C S C S

SSm and A

S S

N

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

27

Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include

- Rotational bending fatigue limit Sersquo

- Surface conditions ka

- Size kb

- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo

- Temperature kd

- Reliability factor ke

- Miscellaneous effects (notch) kf

Fatigue limit of a machine part Se

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

28

Ca

Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens

Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]

Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo

(from J Bannantine ref1)

k a

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

29

Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress

There are many empirical fits to the size effect data A fairly conservative one is

bull The size effect is seen mainly at very long lives

bull The effect is small in diameters up to 20 in (even in bending and torsion)

Stress effects in non-circular cross section members

In the case of non-circular members the approach is based on so called effective diameter de

The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen

0097

0097

1 03

0869 03 100

10 8

1189 8 250

b

b

e

e

e

e

if d inS

S d if in d in

or

if d mmS

S d if d mm

k

k

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

30

+

-

095max

max

005d2

095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick

The surface area of such a ring is

max

22 2095 095 00766

4A d d d

The effective diameter dThe effective diameter dee for members for members

with non-circular cross sectionswith non-circular cross sections

rectangular cross section under bending

F

095

t

t

Equivalent diameter

max095 095 005

A Ft

A Ft t Ft

200766 005

0808

e

e

d Ft

d Ft

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

31

Loading Effects on Endurance LimitLoading Effects on Endurance Limit

The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09

The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate

( ) ( )(07 09)

07 09 ( 085)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

( ) ( )0577

057( 059)

e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

32

From Shigley and Mischke Mechanical Engineering Design 2001

u T u T

e RT e RTu RT u R

dT

e T dSS S

S SS

kS

k

Temperature EffectTemperature Effect

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

33

Reliability factor Reliability factor kkee

The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se

rsquo

The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution

1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

34

(source S Nishijima ref 39)

S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

35

Stress concentration factor Stress concentration factor KKtt and the and the

notch factor effect notch factor effect kkff

Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf

1f

f

kK

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

36

1

peak

n

22

11

33

3

2

A D B

C

F

F

22

22

22

33

33

11

C

A B

D

Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body

n

peakt n

FS

Aand

K

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

A B C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1

E

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

38

Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis

peak

tn

peak

K

S

n S

x

dn

0

W

peak

Str

ess

S

n

x

dn

0

W

Str

ess

M

peak

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

39

Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

40

S =

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

41

Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen

x

dn

0

W

peakS

tres

s

S

n=Speak= n=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress

np

n1

S2

cyclesN0N2

Sesmooth

N (S)m = C

Sm

ax

Str

ess

ran

ge

S

n2

n3

n4 notchedsmoothe

ef

SS

K

S

f tK KfK Fatigue notch factor

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

43

1

2

3

Nsm

ooth

22peak

smoothe

peak

Str

ess

1

2

Nnotched

notchede

M

notchedf notched

smoothsmoo he

e

tfo NNrK

Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff

1

22= peak

Str

ess

2

Nnotched

notchede

S

Se

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

44

11 1 1

1t

f t

KK q K

a r

1

1

qa r

PETERSONs approachPETERSONs approach

a ndash constant

r ndash notch tip radius

11

1t

f

KK

r

NEUBERrsquos approachNEUBERrsquos approach

ρ ndash constant

r ndash notch tip radius

18

330010

u

u

a inS

for S in in

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

45

The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

46

Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

47

Plate 1

W1 = 50 in

d1 = 05 in

Su = 100 ksi

Kt = 27

q = 097

Kf1 = 265

Illustration of the notchscale effectIllustration of the notchscale effect

Plate 2

W2= 05 in

d2 = 05 i

Su = 100 ksi

Kt = 27

q = 078

Kf1 = 232

peak

m1

d1

W1

peak

d2

W2

m2

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

48

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

JuvinalShigley method

Nf (logartmic)

Collins method

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

frsquo

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

49

Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components

Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)

Sa

r (lo

gar

tmic

)

Nf (logartmic)

Manson method09Su

100 101 102 103 104 105 106 2106

Sersquokakckbkdke

Sersquokakckbkdkekf

Se

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

50

NOTENOTE

bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary

bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy

bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

mm = 0 R = -1 = 0 R = -1

PulsatingPulsating

mm = = aa R = 0 R = 0

CyclicCyclic

mm gt 0 R gt 0 gt 0 R gt 0

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

52

Mean Stress EffectMean Stress Effect

smlt 0

sm= 0

smgt 0

No of cycles logN 2106

Str

ess

ampl

itude

log

Sa

time

smlt 0

sm= 0

smgt 0

Str

ess

ampl

itude

Sa

0

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

53

The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability

Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life

There are several empirical methods used in practice

The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives

Steel AISI 4340 Steel AISI 4340

SSyy = 147 ksi = 147 ksi (Collins)

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

54

MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit

Gereber (1874)S

S

S

Sa

e

m

u

2

1

Goodman (1899)S

S

S

Sa

e

m

u

1

Soderberg (1930)S

S

S

Sa

e

m

y

1

Morrow (1960)S

S

Sa

e

m

f

1

Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles

Sm- mean stress

Se- fatigue limit at Sm=0

Su- ultimate strength

frsquo- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress

Gereber (1874)

2

1a m

ar u

S S

S S

Goodman (1899) 1a m

ar u

S S

S S

Soderberg (1930) 1a m

ar y

S S

S S

Morrow (1960)

1a m

ar f

S S

S

Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles

Sm- mean stress

Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles

Su- ultimate strength

frsquo- true stress at fracture

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

56

Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect

Most of the experimental data lies between the Goodman and the yield line

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

57

Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

58

The following generalisations can be made when discussing mean stress effects

1 The Soumlderberg method is very conservative and seldom used

3 Actual test data tend to fall between the Goodman and Gerber curves

3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress

4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories

5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits

6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress

For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

e

NoN2

e

Ni(i)m = a

N1 N3

Str

ess

rang

e

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cycles applied atn cycles applied at n cycles applied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

1 1 2 2

1 1

Ri i

LD n N n N n N

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

60

n1 - number of cycles of stress range 1

n2 - number of cycles of stress range 2 ni - number of cycles of stress range i

11

1D

N - damage induced by one cycle of stress range 1

11

1n

nD

N - damage induced by n1 cycles of stress range 1

22

1D

N - damage induced by one cycle of stress range 2

22

2n

nD

N - damage induced by n2 cycles of stress range 2

1i

i

DN

- damage induced by one cycle of stress range i

ini

i

nD

N - damage induced by ni cycles of stress range i

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

61

Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History

1 1 2 2

1 1 2 2

1 21 2

11 2

i i

i i

ii i

n n nii i

n cyclesapplied atn cyclesapplied at n cyclesapplied atD

N cycles to failure at N cycles to failure at N cycles to failure at

n nn nD D D D

N N N N

It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1

ie if D gt 1 - fatigue failure occurs

For D lt 1 we can determine the remaining fatigue life

1 1 2 2

1 1

Ri i

LD n N n N n N

LR - number of repetitions of the stress history to failure

1 2 3 R iN L n n n n N - total number of cycles to failure

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

62

jN = am

j

if j gt e

It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to

1

j

jj

D D

It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as

BL = 1D

Hence the fatigue life of a component in cycles can be calculated as

N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history

NT = (NOP - 1)2

If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as

T = BL Tr

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

63

Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure

bull Analysis of external forces acting on the structure and the component in question

bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-

N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the

S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in

the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history

repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress

concentrations or critical locations are found in a component or structure

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

64

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

65

Stress History

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Reversal point No

Str

ess

(ksi

)

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

66

411108861

log60log6

log110log3

log60log10log

log110log1000log

logloglog

26

6

mC

Cm

Cm

Cm

Cm

CSmN

CSN

a

ma

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

67

S-N Curve

10

100

1000

1000 10000 100000 1000000 10000000

No of cycles

Str

es

s a

mp

litu

de

(k

si)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Str

es

s a

mp

litu

de

(k

si)

SY

Suts

1

0

0

1 1

11

m

m

a m maa r at S

e uts uts

a m aa r at S

ma r uts

uts

S S Sfor fatigue endurance S S

S S S

S S Sfor any stress amplitude S

SS SS

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

69

Calculations of Fatigue Damage a) Cycle No11

Sar=8793 ksi

N11= C( SaN)-m= 186610268793-114=12805 cycles

D11=0000078093 b) Cycle No 14

Sar=750 ksi

N14= C( SaN)-m= 18661026750-114=78561 cycles

D14=0000012729 c) Cycle no 15

Sar=9828 ksi

N14= C( SaN)-m= 186610269828-114=3606 cycles

D14=000027732

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70

70

Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa

-m

1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277

n0=15 D = 000036873 3677E-04

D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961

N=n0LR=15271203=40680 N=n0LR=15271961=40794

• Slide 1
• Slide 2
• Metallic Fatigue
• The Broad Field of Fracture Mechanics
• Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
• Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
• The Process of Fatigue
• Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
• Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
• Factors Influencing Fatigue Life
• Strength-Fatigue Analysis Procedure
• Stress Parameters Used in Static Strength and Fatigue Analyses
• Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
• Stress History and the ldquoRainflowrdquo Counted Cycles
• The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• The Fatigue S-N method (Nominal Stress Approach)
• Woumlhlerrsquos Fatigue Test
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70