1

University of Waterloo

Department of Mechanical Engineering

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

Partial notes – 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 loading.failure of a component subjected to repeated loading.

FATIGUE FATIGUE -- What is it?What is it?

3

Metallic FatigueMetallic Fatigue• A sequence of several, very complex phenomena encompassing several

disciplines:– motion of dislocations– surface phenomena– fracture mechanics– stress analysis– probability and statistics

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

• Influenced by a component’s environment• Takes many forms:

– fatigue at notches– rolling contact fatigue– fretting fatigue– corrosion fatigue– 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 & Wanhil, ref.3)

5

σσ

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

6

Locally, the crack grows in shear;

macroscopically it grows in tension.

c

Schematic of Fatigue Crack Initiation Subsequent Gr owth Schematic of Fatigue Crack Initiation Subsequent Gr owth 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 Perspective:The Materials Science Perspective:

•• Cyclic slip,Cyclic slip,•• Fatigue crack initiation,Fatigue crack initiation,•• Stage I fatigue crack growth,Stage I fatigue crack growth,•• Stage II fatigue crack growth,Stage II fatigue crack growth,•• Brittle fracture or ductile ruptureBrittle fracture or ductile rupture

8

Features of the Fatigue Fracture Surface of a Typic al Features of the Fatigue Fracture Surface of a Typic al Ductile Metal Subjected to Variable Amplitude Cycli c Ductile Metal Subjected to Variable Amplitude Cycli c

LoadingLoading

(Collins, ref. 22 )

A – fatigue crack area

B – 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 Stresses• Stress range – The basic cause of plastic deformation and consequently the

accumulation of damage• Mean stress – Tensile mean and residual stresses aid to the formation and

growth of fatigue cracks• Stress gradients – Bending is a more favorable loading mode than axial

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

Materials• Tensile and yield strength – Higher strength materials resist plastic

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

• Quality of material – Metallurgical defects such as inclusions, seams, internal tears, and segregated elements can initiate fatigue cracks

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

• Frequency (rate of straining) – 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

StrengthStrength --Fatigue Analysis ProcedureFatigue 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

σσ

σ

=

=

σσσσn

y

dn

0

T

ρ

σpeak

Str

ess

M

b)a)

σσσσn

y

dn

0

T

ρ

σpeak

Str

ess

S

S

13

Constant and Variable Amplitude Stress Histories;Constant and Variable Amplitude Stress Histories;Definition of a Stress Cycle & Stress Reversal

One cycle

����m

����max

����min

Str

ess

Time0

Constant amplitude stress historya)

Str

ess

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

σ σσ σσ σσ σσσσσ

σσσσ

σ σ σσ σ σσ σ σσ σ σ

σ σσ σσ σσ σσσσσσσσσ

σσσσ

∆ = −∆ = −∆ = −∆ = −

−−−−∆∆∆∆====

++++====

====

====

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 “Rainflow” Counted Cycles Stress History and the “Rainflow” Counted Cycles

1 1i i i iABS ABSσ σ σ σ− +− < −

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 ABSσ σ σ σ− +− < −The stress amplitude of such a cycle is:

1

2i i

a

ABS σ σσ − −

=

The stress range of such a cycle is:

1i iABSσ σ σ−∆ = −

The mean stress of such a cycle is:

1

2i i

m

σ σσ − +=

The Mathematics of the Cycle Rainflow Counting Meth od The Mathematics of the Cycle Rainflow Counting Meth od for Fatigue Analysis of Stress/Load Historiesfor Fatigue Analysis of Stress/Load 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

ess

(MP

a)x1

02

-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

ess

(MP

a)x1

02

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

ess

(MP

a)x1

02

-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

ess

(MP

a)x1

02

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

ess

(MP

a)x1

02

-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

ess

(MP

a)x1

02

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

ess

(MP

a)x1

02

-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

ess

(MP

a)x1

02

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 SS--NN methodmethod(Nominal Stress Approach)

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

• Fatigue damage accumulation

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

•

• 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 σσσσ====Smin

Smax

WWööhler’shler’s Fatigue TestFatigue Test

24

Characteristic parameters of the S - N curve are:Se - fatigue limit corresponding to N = 1 or 2×106 cycles for

steels and N = 108 cycles for aluminum alloys,S10

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 LCF/HCF transition and the endurance limit

Number of cycles, N

Str

ess

ampl

itude

, Sa

(ksi

)

Sy

Se

Part 1 Part 2 Part 3

Infinite life

Su

S103

10m A maS C N N= ⋅ = ⋅= ⋅ = ⋅= ⋅ = ⋅= ⋅ = ⋅

Fatigue SFatigue S --N curveN 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 = 0.90Su and Se = S106 = 0.5 Su - bending

S103 = 0.75Su and Se = S106 = 0.35 - 0.45Su - axial

S103 = 0.72Su and Se = S106 = 0.29 Su - torsion

103 104 105 106 1070.1

0.3

0.5

1.0

S103

Se

Number of cycles, Log(N)

Rel

ativ

e st

ress

am

plitu

de, S

a/S

u

Torsion

AxialBending

26

Approximate endurance limit for various materials:Approximate endurance limit for various materials:

Magnesium alloys (at 108 cycles) Se = 0.35Su

Copper alloys (at 108 cycles) 0.25Su< Se <0.50Su

Nickel alloys (at 108 cycles) 0.35Su <Se < 0.50Su

Titanium alloys (at 107 cycles) 0.45Su <Se< 0.65Su

Al alloys (at 5x108 cycles) Se = 0.45Su (if Su 48 ksi) or S e = 19 ksi (if S u> 48 ksi)

Steels (at 106 cycles) Se = 0.5Su (if Su 200 ksi) or S e = 100 ksi (if S u>200 ksi)

Irons (at 106 cycles) Se = 0.4Su (if Su 60 ksi) or S e = 24 ksi (if S u> 60 ksi)

S S –– N curveN curve

(((( )))) (((( ))))(((( ))))33

1 1 1

2

10101log lo

0

g

1

3

Am mm m

am m

e e

Aa aS C N or N C S C S

SSm and A

S S

N− −− −− −− −

= == == == =

= − == − == − == − =

= == == == =

⋅ ⋅⋅ ⋅⋅ ⋅⋅ ⋅

27

Fatigue Limit Fatigue Limit –– Modifying FactorsModifying FactorsFor many years the emphasis of most fatigue testing was to gain

an empirical understanding of the effects of variou s factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges. The variables investigated inc lude:

- Rotational bending fatigue limit, S e’,

- Surface conditions, k a,

- Size, k b,

- Mode of loading, k c, Se = ka kb kc kd ke k f·Se’

- Temperature, k d

- Reliability factor, k e

- Miscellaneous effects (notch), k f

Fatigue limit of a machine part, S e

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 “polished” or “machined”.

(from J. Bannantine, ref.1)

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:

• The size effect is seen mainly at very long lives.

• The effect is small in diameters up to 2.0 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.

' 0.097

' 0.097

1 0.3

0.869 0.3 10.0

1.0 8

1.189 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

+

-

0.95σmax

σmax

0.05d/2

0.95dThe material volume subjected to stresses σ ≥ 0.95σmax is concentrated in the ring of 0.05d/2 thick.

The surface area of such a ring is:

( )max

22 20.95 0.95 0.0766

4A d d dσ

π = − =

The effective diameter, dThe effective diameter, d ee, for members , for members with nonwith non --circular cross sectionscircular cross sections

* rectangular cross section under bending

F

0.95

t

t

Equivalent diameter

max0.95 0.95 0.05

A Ft

A Ft t Ftσ

== − =

20.0766 0.05

0.808

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 0.6 to 0.9.

The ratio of endurance limits found using torsion a nd rotating bending tests ranges from 0.5 to 0.6. A theoretical value obtaine d from von Mises-Huber-Hencky failure criterion is been used as the most po pular estimate.

( ) ( )(0.7 0.9)

0.7 0.9 ( 0.85)c

e axial e bendi

c

ngS S

suggested by Shigl y kek

≈ −≈ −≈ −≈ −

= − == − == − == − =

( ) ( )0.577

0.57( 0.59)e torsion e bendi

c

ng

c

S

suggested by Shigl kek y

ττττ ≈≈≈≈

= == == == =

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

’.

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 0.08 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 et.al.

34

(source: S. Nishijima, ref. 39)

SS--N curves for assigned probability of failure; P N curves for assigned probability of failure; P -- S S -- N curvesN curves

35

Stress concentration factor, Stress concentration factor, KK tt, and the , and the notch factor effect, notch factor effect, kk ff

Fatigue notch factor effect k f depends on the stress concentration factor K t (geometry), scale and material properties and it is expressed in terms of the Fati gue Notch Factor K f.

1f

f

kK

====

36

1

σpeak

σn

σ22

σ11

σ3

3

3

2

A D B

C

F

F

σ22

σ22

σ22

σ33

σ33

σ11

C

A, B

D

Stresses in Stresses in axisymmetricaxisymmetricnotched bodynotched body

n

peakt n

FS

Aand

K

σ

σ σ

= =

=

37

Stresses in prismatic notched bodyStresses in prismatic notched body

n

peakt n

t

FS

Aand

K

K S

σ

σ σ

= =

==

A, B, C

σ22

σ22

σ22

σ33

D

E

σ11

σ peak

σn

σ22

σ11

σ3

33

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

σσσσσσσσ

σσσσ

====

====

σn, S

x

dn

0

W

ρ

σ peak

Str

ess

S

σn

x

dn

0

W

ρ

Str

ess

M

σ peak

39

Stress concentration Stress concentration factors, Kfactors, K tt, in shafts , in shafts

Bending load

Axial load

S =

S =

40

S =

41

Similarities and differences between the stress fie ld near the nSimilarities and differences between the stress fie ld near the n otch otch and in a smooth specimenand in a smooth specimen

x

dn

0

W

ρ

σpeakS

tres

s

S

σσσσn=S σσσσpeak= σσσσn=S

P

42

The Notch Effect in Terms of the NThe Notch Effect in Terms of the N ominal Stressominal Stress

np

n1

∆∆∆∆S2

cyclesN0N2

Sesmooth

N (∆∆∆∆S)m = C

∆∆ ∆∆Sm

axS

tres

s ra

nge

S

n2

n3

n4 notchedsmoothe

ef

SS

K====

S

!!!f tK K≠≠≠≠fK −−−− Fatigue notch factor!

43

1

2

3

Nsm

ooth

22peak

smoothe

σ σσ σσ σσ σσσσσ

====

====

σpeak

Str

ess

1

2

Nnotched

notchedeσσσσ

M

notchedf notched

smoothsmoo he

e

tfo NNrKσσσσσσσσ

= == == == =

Definition of the fatigue notch factor Definition of the fatigue notch factor KK ff

1

σ22= σpeak

Str

ess

2

Nnotched

notchedeσσσσ

S

Se

44

( )11 1 1

1t

f t

KK q K

a r

−= + = + −+

1;

1q

a r=

+

PETERSON'sPETERSON's approachapproach

aaaa – constant,

r – notch tip radius;

11

1t

f

KK

rρ−= +

+

NEUBER’sNEUBER’s approachapproach

– constant,

r – notch tip radius

[[[[ ]]]]

[[[[ ]]]]

1.8

330010 .

.

u

u

a inS

for S in in

−−−− = ×= ×= ×= ×

45

The The NeuberNeuber constant constant ‘‘ ’’ for steels and aluminium alloysfor steels and aluminium alloys

46

Curves of notch sensitivity index ‘q’ versus notch radiusCurves of notch sensitivity index ‘q’ versus notch radius(McGraw Hill Book Co, from ref. 1)

47

Plate 1

W1 = 5.0 in

d1 = 0.5 in.

Su = 100 ksi

Kt = 2.7

q = 0.97

Kf1 = 2.65

Illustration of the notch/scale effectIllustration of the notch/scale effect

Plate 2

W2= 0.5 in

d2 = 0.5 i

Su = 100 ksi

Kt = 2.7

q = 0.78

Kf1 = 2.32

σσσσpeak

σσσσm1

ρρρρd1

W1

σσσσpeak

ρρρρ

d2

W2

σσσσm2

48

Procedures for construction of approximate fully re versedProcedures for construction of approximate fully re versedSS--NN curves for smooth and notched componentscurves for smooth and notched components

Nf (logartmic)

Juvinal/Shigley method

Nf (logartmic)

Collins method

Sar, σσσσar – nominal/local stress amplitude at zero mean stress σσσσm=0 (fully reversed cycle)!

σσσσf’

49

Procedures for construction of approximate fully re versedProcedures for construction of approximate fully re versedSS--NN curves for smooth and notched componentscurves for smooth and notched components

Sar, σσσσar – nominal/local stress amplitude at zero mean stress σσσσm=0 (fully reversed cycle)!

Sar

(loga

rtm

ic)

Nf (logartmic)

Manson method0.9Su

100 101 102 103 104 105 106 2 106

Se’kakckbkdke

Se’kakckbkdkek fSe

50

NOTE!NOTE!

• The empirical relationships concerning the S –N curve data are only estimates! Depending on the acceptable level of uncertaintyin the fatigue design, actual test data may be necessary.

• The most useful concept of the S - N method is the endurance limit , which is used in “infinite-life”, or “safe stress ” design philosophy.

• In general, the S – N approach should not be used to estimate lives below 1000 cycles (N < 1000).

51

Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories

Fully reversedFully reversed

σσσσσσσσmm = 0, R = = 0, R = --11

PulsatingPulsating

σσσσσσσσmm = = σσσσσσσσaa R = 0R = 0

CyclicCyclic

σσσσσσσσmm > 0 R > 0> 0 R > 0

52

Mean Stress EffectMean Stress Effect

sm< 0

sm= 0

sm> 0

No. of cycles, logN 2*106

Str

ess

ampl

itude

, log

Sa

time

sm< 0

sm= 0

sm> 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 – 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 – Sm curves obtained for various fatigue lives.

Steel AISI 4340, Steel AISI 4340,

SSyy = 147 = 147 ksiksi (Collins)

54

MeanMean Stress Correction for Endurance LimitStress 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

+ =σ '

1

Sa – stress amplitude applied at the mean stress S m 0 and fatigue life N = 1-2x10 6cycles.

Sm- mean stress

Se- fatigue limit at S m=0

Su- ultimate strength

σσσσf’- true stress at fracture

55

MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at nonamplitude applied at non --zero mean stresszero 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 σσσσ+ =+ =+ =+ =

Sa – stress amplitude applied at the mean stress S m 0 and resulting in fatigue life of N cycles.

Sm- mean stress

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

Su- ultimate strength

σσσσf’- 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 Good man and the yield line!

57

Approximate Goodman’s diagrams for ductile Approximate Goodman’s diagrams for ductile and brittle materialsand brittle materials

Kf

Kf

Kf

58

The following generalisations can be made when disc ussing mean stress effects:

1. The Söderberg method is very conservative and seldom used.

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

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

4. For most fatigue design situations, R < 1 (i.e., 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 (i.e., 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 finiteFor finite --life calculations the endurance limit in any of the equations calife calculations the endurance limit in any of the equations ca n be n be replaced with a fully reversed alternating stress l evel corresporeplaced with a fully reversed alternating stress l evel correspo nding to that nding to that finitefinite --life value!life value!

59

Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation

NT

n1

n2

n3

n4

cycles

σσσσe

NoN2

σσσσe'

Ni(�σi)m = a�σ2

N1 N3

�σ1

�σ3

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

σσσσσ σσ σσ σσ σσ σ σσ σ σσ σ σσ σ σ

∆∆∆∆∆ ∆∆ ∆∆ ∆∆ ∆= + += + += + += + +

∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆

= + + + = + + + == + + + = + + + == + + + = + + + == + + + = + + + =∑∑∑∑

1 1 2 2

1 1....R

i i

LD n N n N n N

= == == == =+ ++ ++ ++ +

60

n1 - number of cycles of stress range ∆σ∆σ∆σ∆σ1 n2 - number of cycles of stress range ∆σ∆σ∆σ∆σ2n i - number of cycles of stress range ∆σ∆σ∆σ∆σi,

11

1D

N= - damage induced by one cycle of stress range ∆σ1,

11

1n

nD

N= - damage induced by n1 cycles of stress range ∆σ1,

22

1D

N= - damage induced by one cycle of stress range ∆σ2,

22

2n

nD

N= - damage induced by n2 cycles of stress range ∆σ2,

1i

i

DN

= - damage induced by one cycle of stress range ∆σi,

ini

i

nD

N= - damage induced by ni cycles of stress range ∆σi,

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

σσσσσ σσ σσ σσ σσ σ σσ σ σσ σ σσ σ σ

∆∆∆∆∆ ∆∆ ∆∆ ∆∆ ∆= = == = == = == = =

∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆

= + + + = + + + == + + + = + + + == + + + = + + + == + + + = + + + =∑∑∑∑

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

i.e. if D > 1 - fatigue failure occurs!

For D < 1 we can determine the remaining fatigue li fe:

1 1 2 2

1 1....R

i 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

jσσσσ−−−−

∆∆∆∆ if ∆∆∆∆σσσσj > σσσσe .

It is assumed that stress cycles lower than the fatigue limit, ∆σj < σe, 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 ‘m+2” 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, i.e. when D ≥ 1. This may happen after a certain number of repetitions, BL (blocks), of the stress spectrum, which can be calculated as:

BL = 1/D.

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

N = BL××××NT, 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 SS--NN Fatigue Life Estimation ProcedureFatigue Life Estimation Procedure

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

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

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

S-N curve (nominal/reference stress),• Determination of analogous stress parameter for the actual element in

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

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

concentrations or critical locations are found in a component orstructure.

64

Example #2

An unnotched machine component undergoes a variable amplitudestress history Si given below. The component is made from a steelwith the ultimate strength Suts=150 ksi, the endurance limitSe=60 ksi and the fully reversed stress amplitude at N1000=1000cycles given as S1000=110 ksi.Determine the expected fatigue life of the component.

Data: Kt=1, SY=100 ksi Suts=150 ksi, Se=60 ksi, S1000=110 ksi

The stress history:

Si = 0, 20, -10, 50, 10, 60, 30, 100, -70, -20, -60, -40, -80, 70, -30,20, -10, 90, -40, 10, -30, -10, -70, -40, -90, 80, -20, 10, -20, 10, 0

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

( )

4.1110886.1

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

Stre

ss a

mpl

itude

(ksi

)

68

Goodman Diagram

0

10

20

30

40

50

60

70

80

90

100

-100 -50 0 50 100 150

Mean stress (ksi)

Stre

ss a

mpl

itude

(ksi

)

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 No.11

Sa,r=87.93 ksi

N11= C×( Sa,N)-m= 1.866×1026×87.93-11.4=12805 cycles

D11=0.000078093 b) Cycle No. 14

Sa,r=75.0 ksi

N14= C×( Sa,N)-m= 1.866×1026×75.0-11.4=78561 cycles

D14=0.000012729 c) Cycle no. 15

Sa,r=98.28 ksi

N14= C×( Sa,N)-m= 1.866×1026×98.28-11.4=3606 cycles

D14=0.00027732

70

Results of "rainflow" counting D a m a g eNo. ∆∆∆∆S Sm Sa Sa,r (Sm=0) Di=1/Ni=1/C*Sa

-m

1 30 5 15 15.52 2.0155E-13 02 40 30 20 25.00 4.6303E-11 03 30 45 15 21.43 7.9875E-12 04 20 -50 10 10.00 1.3461E-15 05 50 -45 25 25.00 4.6303E-11 06 30 5 15 15.52 2.0155E-13 07 100 20 50 57.69 6.3949E-07 08 20 -20 10 10.00 1.3461E-15 09 30 -25 15 15.00 1.3694E-13 010 30 -55 15 15.00 1.3694E-13 011 170 5 85 87.93 7.8039E-05 7.80E-0512 30 -5 15 15.00 1.3694E-13 013 30 -5 15 15.00 1.3694E-13 014 140 10 70 75.00 1.2729E-05 1.27E-0515 190 5 95 98.28 0.00027732 0.000277

n0=15 D = 0.00036873 3.677E-04

D=0.000370 D=3.677E-04LR = 1/D =2712.03 LR = 1/D =2719.61

N=n0*LR=15*2712.03=40680 N=n0*LR=15*2719.61=40794