1PSMN

Lecture 2P-S-M-N diagrams

What’s P-S-M-N?

• P = probability of failure

• S = stress amplitude

• M = mean stress

• N = number of cycles to failure of a ‘standard’plain fatigue test specimen

2PSMN

Stress cycle

sDsa

smin

smax

sm

Tid

spenningssyklussa

sD

smin

smax

sm - midtspenning- spenningsamplitude- maksimumsspenning- minimumsspenning- spenningsvidde

Spennin

g

( ) ( )

min max

a m

Stress ratio

Amplitude ratio

1 1

R

A R R

σ σ

σ σ

=

= = − +

S-N Curves

3PSMN

Olin Hanson Basquin (1869-1946), Northwestern University, Illinois (1910): Power law describes N vs. σσσσa [Dowling].

Principle S-N curve for smooth member under fully reversed loading

4PSMN

Nomenclature

Symbol Meaning

N number of cycles to failure, life(-time)

ND number of cycles at ‘knee-point’

n number of cycles

(σm ±) σA fatigue limit (at arbitrary mean stress, σm)

(σm ±) σAN fatigue strength (at N cycles and arbitrary mean stress, σm)

σW fatigue limit (at σm = 0)

σWN fatigue strength (at N cycles and σm = 0)

σa stress amplitude

σar equivalent fully reversed stress amplitude

σm mean stress

Basquin’s equation

( )

( )

a f A ; 1 W

f f m

Basquin's equation for the fully reversed fatigue strength

at cycles:

2

Fatigue strength coefficient ( 'true' fracture strength):

1

Fatigue strength exponent:

1

Basquin

b

N R N

N

N

R Z

b m

σ σ σ σ

σ σ

=−′= = =

≈

′ ≈ = −

= −

( )W D

1

a W D

's equation may be rewritten in terms of 'knee-point'

parameters and :

m

N

N N

σ

σ σ −=

5PSMN

Strain-controlled fatigue testing [Dowling]

Strain vs. life curves [Dowling]

6PSMN

ASME BPVC VIII Div 3: Design fatigue curves for non-welded machined parts made of forged carbon or low-alloy steel

MPa

7000

700

70

620 MPa

860-1200 MPa

Coffin-Manson’s equation

( )

( )( )

p

a f

e

a a f

a

Coffin-Manson's equation for the fatigue life for a fully reversed strain cycle:

2

Combining this with Basquin's equation,

2 ,

yields the Basquin-Coffin-Manson equation,

c c

b b

N

N CN

E E N BN

ε ε

ε σ σ

ε

′= =

′= = =

= ( )( ) ( )

( ) ( ) ( )

e p

a a f f

at t

a at t t

2 2 .

This may be rewritten in terms of the 'transition' parameters and :

2 .

b c b c

b c

E N N BN CN

N

N N N N

ε ε σ ε

ε

ε ε

′ ′+ = + = +

= +

7PSMN

Transition fatigue lives vs. hardnessfor a wide range of steels [Dowling].

335 1015 1725

Fatigue Limit

8PSMN

Fatigue limits of ferrous metals in rotating bending proportional to the tensile strength as long as Rm < 1400 MPa [Dowling].

Fatigue strengths in rotating bending at N = 5·108 cycles for wrought Al alloys approximately proportional to the tensile strength as long as Rm < 325 MPa [Dowling].

9PSMN

FKM fatigue ratios (and mean stress sensitivities)

W

W

W m

W W

f R

f

σ

τ

σ

τ σ

=

=

Mean Stress Effect

10PSMN

S-N curves for smooth specimens of an Al alloy under axial loading at various mean stresses [Dowling].

Haigh diagrams for smooth specimens of an Al alloy under axial loading at various lives [Dowling].

11PSMN

Normalised Haigh diagrams for smooth specimens of an Al alloy under axial loading showing the Goodman (solid) and Hempel-Morrow (long-dashed) lines and the Gerber parabola (short-dashed)[Dowling].

Mean stress effect on fatigue strength

( )

( )

a W m m

2

a W m m

a W m m

Goodman

1

Gerber

1

Gerber, generalised

1

N

N

N

R

R

Rδ

σ σ σ

σ σ σ

σ σ σ

+ =

+ =

+ =

( )

( )

a W m f

a W f m W

1

m a a W

Morrow

1

Ditto in terms of MSS

Walker (SWT, 0.5)

N

N N

N

γ γ

σ σ σ σ

σ σ σ σ σ

γ

σ σ σ σ−

′+ =

′+ =

=

+ =

12PSMN

Equivalent fully reversed stress amplitude

( )( )( )( )

( ) ( )

aar W f

m f

a f m

1

ar m a a W

Morrow

21

2

Walker

b

N

b

N

N

N

γ γ

σσ σ σ

σ σ

σ σ σ

σ σ σ σ σ−

′= = = ⇒′−

′= −

= + =

Haigh diagram at fatigue limit forSGCI EN-GJS-400-18-LT

13PSMN

Straight-line Haigh diagram at fatigue limit

according to FKM Guideline and Hempel-Morrow

Mean stress effect on fatigue limit

{ }

W W

a m W

W f m

W

Equations formally identical: fatigue strengths

are merely replaced by fatigue limits .

Hempel-Morrow in terms of FKM mean stress

sensitivity and fatigue ratio:

[MPa] 1000

N

M

M aR b

σ

σ

σ σ

σ σ σ

σ σ

σ

+ =

′= = +

=W mf Rσ

14PSMN

FKM (fatigue ratios and) mean stress sensitivities

W

m[MPa] 1000M aR b

M f M

σ

τ τ σ

= +

=

Goodman, Morrow and Gerber lines in Haigh diagram

0

100

200

300

400

500

600

700

800

900

1000

-1000 -500 0 500 1000

σmax = Rm

σmax = Re

σmin = -Re

σmin = 0

σmax = 0

Goodman

Gerber, σm > 0

Gerber, σm < 0

Morrow

σa, MPa

σm, MPa

15PSMN

Haigh diagram based on Walker’s equation

0

100

200

300

400

500

600

700

800

900

1000

-1000 -500 0 500 1000

σmax = Rm

σmax = Re

σmin = -Re

σmin = 0

σmax = 0

Walker, γ = 0

Walker, γ = 1

Walker, γ = 0,5

σm, MPa

σa, MPa

Fitting Hempel-Morrow line and Walker curve to R = -1 and R = 0 according to FKM Guideline for wrought steel

16PSMN

Hempel-Morrow parameter for wrought steel according to FKM Guideline and Walker

parameter fitted to R = -1 and R = 0

Scatter

17PSMN

Scatter in S-N data from rotating-bending specimens of an Al alloy [Dowling, Grover].

Statistical distribution (histogram) of fatigue lives for 57 smooth specimens of 7075-T6 aluminium tested at Sa = 207 MPa (30 ksi) in rotating bending [Dowling, Sinclair].

18PSMN

Rotating-bending S-N curves for various probabilities of failure for smooth specimens of 7075-T6 aluminium [Dowling, Sinclair].

Scatter in fatigue testing

Fatigue life scatter Fatigue strength scatter

19PSMN

Weibull Distribution

Waloddi Weibull (1887-1979) receives the Great Gold Medal of the Royal Swedish Academy of Engineering Sciences

Professor Weibull's proudest moment came in 1978 when he received the Great Gold medal which was

personally presented to him by King Carl XVI Gustav of Sweden. Below is the photo with King Carl XVI

Gustav of Sweden, Waloddi Weibull, and in the middle Gunar Hambræus, then President of the Royal

Swedish Academy of Engineering. When Waloddi stood in front of the King he said: "Seventy-one years ago I

stood in front of Your Majesty's grandfather's grandfather (King Oscar II) and got my officer's commission."

The King then said: "That is fantastic!"

20PSMN

Cumulative distribution function

The cumulative distribution functionof the three-parameter Weibull distribution

The cumulative distribution functionof a two-parameter Weibull distribution

λ = scale parameterk = shape parameterθ = location parameter

Two-parameter Weibull probability density function

0

2

4

6

8

10

12

0 0,5 1 1,5 2

k = 2 k = 5 k = 10 k = 20 k = 30

21PSMN

Mean and variance

Mean of the two-parameter Weibull distribution

Variance of the two-parameter Weibull distribution

Coefficient of variation, CVδ = σ/µ = √var(X)/E(X)

Weibull CV vs. Weibull exponent

22PSMN

Principle P-S-N diagram for smooth member under cyclic loading

( )( ) ( ) ( )

( )( )

( )( )( )

50

a

f a

1

a f m 50

50 a A 501

A 50 f m

Probability that failure for a stress amplitude of occurs within

cycles:

Pr 1 2

Basquin's equation:

2

2

Probability that failure at

nn N

N

m

m

Nm

N

n

P N n F n

Nn N

n

β

σ

σ

σ σ σσ σ

σ σ σ

−

−

−

= < = = −

′= − ⇒ =

′= −

( )( ) ( ) ( )a A 50 a A 50

a

f A a

a life of cycles occurs for a stress amplitude

below :

Pr 1 2 1 2m n

N N

N n

n

P n mβ βσσ σ σ σ

σ

σ

σ σ β β− −= < = − = − ⇒ =

P-S-N diagram based on Weibull life distribution

23PSMN

P-S-N diagram based on Weibull life distribution (2)

( )( )

( )( )

a A 50 m

f

1

1

a f m

f

Solving the equation for the probability of failure

1 2

for the stress amplitude yields

12 ln ln 2

1

N

m

P

NP

βσ

σ

σ σ σ

β

σ σ σ

−

−

= −

′= − −

P-S-N diagram for smooth member (ββββσσσσ = 30)under fully reversed loading

100

1000

1,00E+03 1,00E+04 1,00E+05 1,00E+06 1,00E+07 1,00E+08

Pf = 99%

Pf = 50%

Pf = 1%

Cycles to failure N

Stress amplitude σ

a, MPa

24PSMN

Safety factor based on Weibull distributed fatigue strength

( )( )

( )

a A 50 m

f

1

A 50

a f

Solving the equation for the Weibull probability of failure

1 2

for the safety factor

ln 2

ln 1 1

N

N

S

P

fP

βσ

σ

σ σ σ

β

σσ

−= −

= =

−

Safety factors based on Gaussian fatigue strength

( )

( )

a A 50

f A a

A 50

1

f

Similarly the equation for the Gaussian probability of failure

1 1Pr

1

1

coefficient of variation (= standard deviation/mean)

N S

N

N

S

fP

fP

σ σσ σ

δ σ δ

δ

δ

−

− − = < = Φ = Φ ⇒ ⋅

=+ ⋅Φ

=

25PSMN

Safety factors based on Weibull or Gauss distributed fatigue strength for coefficients of variation δδδδ = 0.025…0.1

References

� N.E. Dowling, Mechanical Behavior of Materials, 4th edition,

Prentice Hall, 2012.

� Rechnerischer Festigkeitsnachweis für Maschinenbauteile aus

Stahl, Eisenguss- und Aluminiumwerkstoffen. 6. Auflage, FKM,

Frankfurt, 2012. English version: Analytical strength assessment.