Fatigue

Lecture 10Lecture 10

Engineering 473Engineering 473Machine DesignMachine Design

FatigueFatigue

Load Histories and Load Histories and Design ObjectivesDesign Objectives

t, time

Fσ,

Dynamic, Cyclic, or Unsteady

Failure

Failure

t, time

Fσ,

Monotonic, Static, or Steady

Design for StrengthDesign for Strength Design for LifeDesign for Life

Rotating Beam Fatigue TestingRotating Beam Fatigue Testing

Mott, Fig. 5-2 & 5-3

Fatigue Dynamics, Inc. rotating beam test equipment.

www.fdinc.com

SS--N CurveN Curve

Shigley, Fig. 7-6Completely reversed cyclic stress, UNS G41200 steel

Fatigue StrengthFatigue StrengthThe Fatigue Strength, Sf(N), is the stress level that a material can endure for N cycles.

The stress level at which the material can withstand an infinite number of cycles is call the Endurance Limit.

The Endurance Limit is observed as a horizontal line on the S-N curve.

Shigley, Fig. 7-6

Representative SRepresentative S--N CurvesN Curves

Mott, Fig. 5-7

Note that non-ferrous materials often exhibit no endurance limit.

Endurance LimitEndurance LimitVs Tensile StrengthVs Tensile Strength

Shigley, Fig. 7-7

SpecimenTest ofStrength TensileSSpecimenTest ofLimit EnduranceS

ut

e

≡≡′

ute 0.3SS =′

Conservative Lower Bound

for Ferrous Materials

Endurance Limit Endurance Limit Multiplying FactorsMultiplying Factors

(Marin Factors)(Marin Factors)

factor effects-ousMiscellanekfactor eTemperaturk

factor Loadkfactor Sizek

factor Surfacekspecimen test oflimit EnduranceS

part oflimit EnduranceS

SkkkkkS

e

d

c

b

a

e

e

eedcbae

≡≡≡≡≡≡′≡

′⋅⋅⋅⋅⋅= There are several factors that are known to result in differences between the endurance limits in test specimens and those found in machine elements.

See sections 7-8 & 7-9 in Shigley for a discussion on each factor.

Mean Stress EffectsMean Stress Effects

� The S-N curve obtained from a rotating beam test has completely reversed stress states.

� Many stress histories will not have completely reversed stress states.

Shigley, Fig. 7-12

DefinitionsDefinitions

2σσσ

2σσσ

σσσ

minmaxm

minmaxa

minmaxr

+=

−=

−=Stress RangeStress Range

Alternating StressAlternating Stress

Mean StressMean Stress

max

min

σσR =

m

a

σσA =

Stress RatioStress Ratio Amplitude RatioAmplitude RatioNote that R=-1 for a completely reversed stress state with zero mean stress.

Mean Stress Fatigue TestingMean Stress Fatigue Testing

Fatigue Dynamics, Inc., fluctuating fatigue stress testing equipment.

www.fdinc.com

Fluctuating Stress Failure DataFluctuating Stress Failure Data

Shigley, Fig. 7-14

This plot shows the fatigue strength of several steels as a function of mean stress for a constant number of cycles to failure.

Note that a tensile mean stress results in a significantly lower fatigue strength for a given number of cycles to failure.

Note that a curved line passes through the mean of the data.

Master Fatigue PlotMaster Fatigue Plot

Shigley, Fig. 7-15

Fluctuating Stress Failure Fluctuating Stress Failure Interaction CurvesInteraction Curves

Shigley, Fig. 7-16

Soderberg Soderberg Interaction LineInteraction Line

1SS

SSk

yt

m

e

af =+Any combination of mean and alternating stress that lies on or below the Solderberg line will have infinite life.

Factor of Safety FormatFactor of Safety Format

fyt

m

e

af

N1

SS

SSk =+

Note that the fatigue stress concentration factor is applied only to the alternating component.

Goodman Interaction LineGoodman Interaction Line

1SS

SSk

ut

m

e

af =+Any combination of mean and alternating stress that lies on or below the Goodman line will have infinite life.


fut

m

e

af

N1

SS

SSk =+


Gerber Interaction LineGerber Interaction Line

1SS

SSk

2

ut

m

e

af =��

��

�+

Any combination of mean and alternating stress that lies on or below the Gerber line will have infinite life.


1S

SNS

SNk2

ut

mf

e

aff =��

��

�+


ModifiedModified--Goodman Goodman Interaction LineInteraction Line

The Modified-Goodman Interaction Line never exceeds the yield line.

Example No. 1Example No. 1

Shigley, Example 7-5

A 1.5-inch round bar has been machined from AISI 1050 cold-drawn round bar. This part is to withstand a fluctuating tensile load varying from 0 to 16 kip. Because of the design of the ends and the fillet radius, a fatigue stress-concentration factor of 1.85 exists. The remaining Marin factors have been worked out, and are ka=0.797, kb=kd=1, and kc=0.923. Find the factor of safety using the Goodman interaction line.

Example No. 1Example No. 1(Continued)(Continued)

ksi 52.42σσσ

ksi 0σ

ksi 04.9in 1.77

kip 16σ

in 77.14dπA

ksi .50S0.50Sksi 0.10S

minmaxa

min

2max

22

ute

ut

=−=

=

==

=⋅=

=⋅≈′=

( )( )( )( )( )ksi 8.36S

ksi 501923.01797.0SkkkkS

ksi 4.522σσσ

e

edcbae

minmaxm

==

′=

=+=

Example No. 1Example No. 1(Continued)(Continued)

67.3N

N1272.0

ksi .100ksi 52.4

ksi 36.8ksi 52.485.1

N1

Sσ

Sσk

f

f

fut

m

e

af

=

==+⋅

=+

ExampleExample

1 2

1.5 in. dia. 0.875 in. dia.0.125 in. rad.

Material UNS G41200 Steel

lb 503Plb 0001P

min

max

==5 in 5 in

Notch sensitivityq=0.3

( )

( ) 4442

4441

in 088.0875.064πD

64πI

in 249.05.164πD

64πI

2

1

===

===

34

2

22

34

1

11

in 0.201in 0.438in 0.088

cIS

in 0.332in 0.75in 0.249

cIS

===

===

Will the beam have infinite life?Will the beam have infinite life?

ExampleExample(Continued)(Continued)

1 2



lb 503Plb 0001P

min

max

==5 in 5 in


( )1kq1k1k1kq

tf

t

f

−+=−−= ( )

18.1)161.1(3.011kq1k

61.1k

tf

t

=−+=−+=

=

0.143875.0125.0

dr

71.1in 0.875

in 1.5dD

==

==

Ref. Peterson


1 2



lb 503Plb 0001P

min

max

==5 in 5 in


( )( )

( )( ) ksi 5.10in 332.0

in 01lb 350SMσ

ksi 1.30in 332.0

in 10lb 1000SMσ

31

1min

31

1max

===

===

Section 1 (Base)Section 1 (Base)

ksi 3.202σσσ

ksi 8.92σσσ

minmaxm

minmaxa

=+=

=−=


1 2



lb 503Plb 0001P

min

max

==5 in 5 in


( )( )

( )( ) ksi 71.8in 201.0

in 5lb 350SMσ

ksi 9.24in 201.0

in 5lb 1000SMσ

31

1min

31

1max

===

===

Section 2 (Fillet)Section 2 (Fillet)

ksi 8.162σσσ

ksi 10.82σσσ

minmaxm

minmaxa

=+=

=−=


ee

ut

Sksi 30Sksi 116S

==′=

fult

m

e

af

N1

Sσ

Sσk =+

( )

99.1502.01N

502.0ksi 116ksi 20.3

ksi 30ksi 9.80.1

f ==

=+

Part has infinite life.Part has infinite life.

( )( )

( )( ) ksi 5.10in 332.0

in 01lb 350SMσ

ksi 1.30in 332.0

in 10lb 1000SMσ

31

1min

31

1max

===

===

Section 1 (Base)Section 1 (Base)

ksi 3.202σσσ

ksi 8.92σσσ

minmaxm

minmaxa

=+=

=−=


( )( )

( )( ) ksi 71.8in 201.0

in 5lb 350SMσ

ksi 9.24in 201.0

in 5lb 1000SMσ

31

1min

31

1max

===

===

Section 2 (Fillet)Section 2 (Fillet)

ksi 8.162σσσ

ksi 10.82σσσ

minmaxm

minmaxa

=+=

=−=

ee

ut

Sksi 30Sksi 116S

==′=

fult

m

e

af

N1

Sσ

Sσk =+

( )

16.2463.01N

463.0ksi 116ksi 16.8

ksi 30ksi 8.101.18

f ==

=+

Part has infinite life.Part has infinite life.

AssignmentAssignmentProblem 1

AssignmentAssignment(Continued)(Continued)

Problem 2

Fatigue

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