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8/10/2019 Design for Variable Loading 2 2012
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Design for Variable Loading
It has been established by experiment that
components fail when loads area repeated andreversed several million times even though the
stresses involved do not reach the elastic limit of
the material. Fatigue failure is characterised by an
absence of elongation and of reduction at the pointof failure, and is particularly dangerous for
components with discontinuities since these
always produce points of stress concentration.
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Lecture Content
Significance of the Endurance Limit.
Endurance LimitModifying Factors.
Graphical determination of fatigue strength under
fluctuating lad conditions.
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SolutionResults Summary
Ultimate Tensile Strength = 950 MPa
Endurance Limit (based on = 257 MPamaterial/loading condition)
Modified Endurance Limit = 127 MPa
(for actual conditions)
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Graphical Determination of Fatigue
Strength Under Fluctuating Load
In practice the cyclic stress applied to an elementmay be considered as a combination of an
alternating stress superimposed on a constantly
applied mean stress.
Stress
Mean Stress, m
StressAmplitude, a
Stress Range, )2( ar
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Modified Goodman Diagram
The Modified Goodman Diagram can be constructed
for any material when the ultimate tensile
strength, yield strengthand endurance limitfora completely reversed stress are known. It is
considered that if a variable stress is superimposed
on a steady stress, the plotted results will
determine a maximum and a minimum stress line
between which safe operating conditions can be
maintained.
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Modified Goodman Diagram
Known parameters: Ultimate tensile strength, Su
Yield Strength, Sy
Modified endurance limit, SeuS
uS
yS
yS
eS
Alternating
Stress,
Mean Stress, m
a
eS
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Modified Goodman Diagram
Known parameters: Ultimate tensile strength, Su
Yield Strength, Sy
Modified endurance limit, SeuS
uS
yS
yS
eS
Alternating
Stress,
Mean Stress, m
a
eS
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Modified Goodman Diagram
Known parameters: Ultimate tensile strength, Su
Yield Strength, Sy
Modified endurance limit, SeuS
uS
yS
yS
eS
Alternating
Stress,
Mean Stress, m
a
eS
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Modified Goodman Diagram
Known parameters: Ultimate tensile strength, Su
Yield Strength, Sy
Modified endurance limit, SeuS
uS
yS
yS
eS
Alternating
Stress,
Mean Stress, m
a
eS
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Modified Goodman Diagram
Known parameters: Ultimate tensile strength, Su
Yield Strength, Sy
Modified endurance limit, SeuS
uS
yS
yS
eS
Alternating
Stress,
Mean Stress, m
a
eS
maxm
maxa
maxa
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Complete Modified Goodman Diagram
Known parameters: UTS, Su
;Yield Strength, Sy
;Mod. End. limit, Se
uS
uS
eS
Alternating
Stress,
Mean Stress, m
a
eS
ytS
yc
S
ycS
ytS
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Complete Modified Goodman Diagram
Known parameters: UTS, Su
;Yield Strength, Sy
;Mod. End. limit, Se
uS
uS
e
S
Alternating
Stress,
Mean Stress, m
a
eS
ytS
yc
S
ycS
ytS
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Complete Modified Goodman Diagram
Known parameters: UTS, Su
;Yield Strength, Sy
;Mod. End. limit, Se
uS
uS
e
S
Alternating
Stress,
Mean Stress, m
a
eS
ytS
yc
S
ycS
ytS
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The diagram can be simplified considering the
symmetry about the diagonal axis and by
rotating it through 45 .
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Complete Simplified Goodman Diagram
Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se
eS
Stress
Amplitude,
Mean Stress, m
a
ycS ytS
ucS utS
yS
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Complete Simplified Goodman Diagram
Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se
eS
Stress
Amplitude,
Mean Stress, m
a
ycS ytS
ucS utS
yS
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Complete Simplified Goodman Diagram
Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se
eS
Stress
Amplitude,
Mean Stress, m
a
ycS ytS
ucS utS
yS
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Complete Modified Goodman Diagram
Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se
uS
uS
eS
Alternating
Stress,
Mean Stress, m
a
eS
ytS
yc
S
ycS
ytS
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Complete Simplified Goodman Diagram
Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se
eS
Stress
Amplitude,
Mean Stress, m
a
ycS ytS
ucS utS
yS
maxa
maxm
m
a
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Should either the yield strength, orthe
ultimate tensile strength, be unobtainable, a
further simplification can be made.
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Simplified Goodman Diagram
Known parameters: UTS, SuorYield Strength, Sy ;Mod. End. limit, Se
eS
Stress
Amplitude,
Mean Stress, m
a
ycS ytS
ucS utS
Modified Goodman Line
Soderberg Line
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Design for Variable Loading
Worked Example
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Determine the diameter of a hot drawn mild steel bar
(Sut=430MPa and Sy= 215 MPa) which is subject
to a tensile preload of 50 kN and a fluctuating
tensile load which varies between 0 and 100 kN.
The design of the bar ends is such that a stress
concentration factor of 2 is appropriate for acorresponding fillet radius of 5 mm. The bar
should have an infinite life and is subject to a
factor of safety of 2.
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Solution
1. Strength values from test specimen data:
Ratio (Se/Su)
Material Cycle
s
U.T.S.
(MPa)
Reversed
Bending
Reversed Axial Loading Reversed
Torsion
Mild Steel 107 380 +/-0.6 +/-0.55 +/-0.36
Medium Carbon
Steel (annealed)
107 620 +/-0.5 +/-0.45 +/-0.3
Low alloy Steel 107 950 +/-0.45 +/-0.4 +/-0.27
High Strength
Steel
107 1540 +/-0.38 +/-0.32 +/-0.2
High Strength
Alloy
108 500 +/-0.3 +/-0.24 +/-0.16
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Solution
Sut= 430 MPa
Un-modified endurance limit for reversed torsion:
MPaSe 236)430(55.0'
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SolutionSize Effect, kb
For Axial Loading:
Assuming
This is the book value endurance limitincluding size factor.
utuc SS
ucuce SSxS 51068.9566.0'
MPaxSe 2254304301068.9566.0 5'
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Solution - Stress Concentration, ke
Applicable to both ductile and brittle materialswhen subject to fatigue loading.
Where q = notch sensitivity
Kt= stress concentration factor (from
charts, calculation etc.)
(If q unknown, err on the safe side and make equal tounity)
111
t
eKqk
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Notch Sensitivity Chart For Steel and Aluminium Alloys
0 1.0 4.03.02.0
0.4
0.6
0.8
1.0
0.2
Steel: Sut= 1.4GPa
Sut= 1.0GPa
Sut= 0.7GPa
Sut= 0.4GPaAluminium Alloy
Notch
Sensitivity
q
Notch Radius, r (mm)
X
Notch radius = 5 mm. Extrapolate to
determine suitable value for q
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Solution - Stress Concentration, ke
56.0)12(8.01
1
)1(1
1
t
eKq
k
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SolutionOther Factors
All other modifying factors are assumed to have no
effect and hence equal unity.
1 fdc kkk
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SolutionModified Endurance Limit, Se
MPaxxkkSS caee 8668.056.0225'
Remember Se already includes a size factor, kb
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SolutionApplied Stress
Static stress
Mean stress
Stress amplitudeStress range
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SolutionApplied Stress
Static Stress
Stress Range
Mean Stress
2
3
2
3107.63
4
1050
d
x
d
x
A
Fs
static
2
3
2
3 103.127
4
10100
d
x
d
x
A
Frange
range
2
3107.63
2 d
xrangeamplitude
2
3103.127dx
amplitudestaticmean
5.0mean
amplitude
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Solution
Determine the limiting values of mean stress and
stress amplitude by constructing a Goodman
diagram based on the strength of the component
material and the modified endurance limit.
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SolutionGoodman Diagram
Mean Stress, MPa
Stress
Amplitude,
MPa
)86(eS
)215(yS
)215(yS )430(utS
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SolutionGoodman Diagram
Mean Stress, MPa
Stress
Amplitude,
MPa
)86(eS
)215(yS
)215(yS )430(utS
Safe
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SolutionGoodman Diagram
Mean Stress, MPa
Stress
Amplitude,
MPa
)86(eS
)215(yS
)215(yS )430(utS
5.0
m
a
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SolutionGoodman Diagram
Mean Stress, MPa
Stress
Amplitude,
MPa
)86(eS
)215(yS
)215(yS )430(utS
5.0
m
a
critialm
critiala
MPaMPacriticalcritial ma
125;63
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Solution
From the Goodman Diagram:
Including Factor of Safety:
Relating to strength calculation:
MPacriticalm 125
MPacriticalm
5.622
125
MPad
xcriticalm
5.62103.127
2
3
mmdx
xd 1.45,
105.62
103.1276
32
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Design For Variable Loading
15. For a design application, explain why theendurance limit of the material is
modified form the book value. What factors
should be taken into account when making
this adjustment.
16. Construct (i) a Complete Modified
Goodman Diagram and (ii) a CompleteSimplified Goodman Diagram. List the
parameters required for each.
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Design For Variable Loading
17. A cylindrical component is to bemanufactured form mild steel (Sut= 430 MPaand Sy= 215 MPa) and a modified endurancelimit has been established (Se= 100 MPa). If
the component is subject to both an axialtensile preload of 50 kN and a fluctuatingload in which the ratio of the stress amplitudeto the mean stress is 0.8. Determine thediameter of the cylinder if it is subject to asafety factor of 1.5.