probabilistic aspects of fatigue - illinois

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Variability

Professor Darrell F. SocieDepartment of Mechanical and

Industrial Engineering

© 2003-2005 Darrell Socie, All Rights Reserved

Probabilistic Aspects of Fatigue

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5 Variability © 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 68


n Introductionn Basic Probability and Statisticsn Statistical Techniquesn Analysis Methods nCharacterizing VariabilitynCase Studiesn FatigueCalculator.comnGlyphWorks

3


Sources of Variability

customers

materials manufacturing

usage

107

Fatigue Life, 2Nf

1

10 102 103 104 105 106

0.1

10-2

1

10-3

10-4

Str

ain

Am

plitu

de

Strength

Stress

Stress, Σ Strength, S

4


Variability and Uncertainty

Variability: Every apple on a tree has a different mass.

Uncertainty: The variety of the apple is unknown.

Variability: Fracture toughness of a material

Uncertainty: The correct stress intensity factor solution

5


Sources of Variability

n Stress VariablesnLoadingnCustomer UsagenEnvironment

n Strength VariablesnMaterialnProcessingnManufacturing TolerancenEnvironment

6


Sources of Uncertainty

n Statistical Uncertaintyn Incomplete data (small sample sizes)

nModeling ErrornAnalysis assumptions

nHuman ErrornCalculation errorsn Judgment errors

7


Modeling Variability

Products: Z = X1 • X2 • X3 • X4 • ….. Xn

Z → LogNormal as n increases

Central Limit Theorem:

COVX

0 0.5 1.0 1.5 2.0

0.5

1.0

1.5

σ lnX σlnX ~ COVX

COVX is a good measure of variability

8


Standard Deviation, lnx

COVX

1 2 3

68.3% 95.4% 99.7%

0.05

0.1

0.25

0.5

1

1.05

1.10

1.28

1.60

2.30

1.11

1.23

1.66

2.64

5.53

1.16

1.33

2.04

3.92

11.1

99.7% of the data is within a factor of ± 1.33 of the mean for a COV = 0.1

COV and LogNormal Distributions

9


Variability in Service Loading

nQuantifying Loading VariabilitynMaximum LoadnLoad RangenEquivalent Stress

10


Maximum Force

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

100 1000

Median 431COV 0.34

200 500

Maximum force from 42 automobile drivers99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

100 1000200 500

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Force

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Maximum Load Correlation

0 200 400 600 800 1000

108

107

106

105

104

103

Maximum Load

Fat

igue

Liv

es

12


Loading Variability

1 10 102 103 104 105

Cumulative Cycles

0

16

32

48

Load

Ran

ge

54 Tractors / Drivers

13


Variability in Loading

1 10

54 Tractors/Drivers

COV 0.53

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

n neq SS ∑∆=∆

Equivalent Load

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Mechanisms and Slopes

100

103

104

Stre

ss A

mpl

itude

, MP

a

100

Cycles101 102 103 104 105 106 107

110

10-1210-1110-1010-910-810-710-6

Crack Growth Rate, m/cycle

1

10

100

mM

Pa

,K

∆

1

3

Crack Nucleation

Crack Growth10

100

103 104 105 106 107 108

Total Fatigue Life, Cycles

1

51

Equ

ival

ent L

oad,

kN

Structures

A combination of nucleation and growth

n = 3

n = 5

n = 10

15


Effect of Slope on Variability

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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n = 10 5 3

Equivalent Load0.1 1 100.1 1 10

n COV3 0.535 0.4310 0.38

16


Loading History Variability

n Test TracknCustomer Service

17


Test Track Variability

0.1 1 10

COV 0.12

40 test track laps of a motorhome99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Equivalent Load

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Customer Usage Variability

0.1 1 10

COV 0.32

42 drivers / cars99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Equivalent Load

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Variability in Environment

n Inclusionsn Pit depth

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Inclusions That Initiated Cracks

Barter, S. A., Sharp, P. K., Holden, G. & Clark, G. “Initiation and early growth of fatigue cracks in an aerospacealuminium alloy”, Fatigue & Fracture of Engineering Materials & Structures 25 (2), 111-125.

COV = 0.27

21


Pit Depth Variability

Pits12 Data PointsMedian 24.37

100

COV 0.33

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

10Pit Depth, µm

Dolly, Lee, Wei, “The Effect of Pitting Corrosion on Fatigue Life”Fatigue and Fracture of Engineering Materials and Structures, Vol. 23, 2000, 555-560

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Variability in Materials

n Tensile Strengthn Fracture Toughnessn FatiguenFatigue StrengthnFatigue Life

n Strain-LifenCrack Growth

23


Tensile Strength - 1035 Steel

25

50

75

100

500 550 600 650 700

Num

ber

of h

eats

Tensile Strength, MPa

Metals Handbook, 8th Edition, Vol. 1, p64

Mean = 602COV = 0.045

24


Fracture Toughness

100

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

Median 76.7

COV 0.06

60 70 9080

26 Data Points

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KIc, Ksi√in

Kies, J.A., Smith, H.L., Romine, H.E. and Bernstein, H, “Fracture Testing of Weldments”, ASTM STP 381, 1965, 328-356

Mar-M 250 Steel

25


Fatigue Variability

102 103 104 105 106 107

Fatigue Life101

1000

100

10

1

Str

ess

Am

plitu

de Fatigue life

Fat

igue

str

engt

h

26


Fatigue Life Variability

50 100 150 200 250

10

20

30

40

10

20

Num

ber

of T

ests

Life x10 350 100 150 200 250

10

20

30

40

10

20

Num

ber

of T

ests

Life x10 3

Production torsion bars5160H steelOne lot, 71 parts

25 lots, 300 parts

Metals Handbook, 8th Edition, Vol. 1, p219

µx = 123,000 cyclesCOV = 0.25

µx = 134,000 cyclesCOV = 0.27

27


Statistical Variability of Fatigue Life

50

10

2

90

98

104 105 106 107 108

Cycles to Failure

Per

cent

Sur

viva

l

∆s=210∆s=245∆s=315∆s=280∆s=440

Sinclair and Dolan, “Effect of Stress Amplitude on the Variability in Fatigue Life of 7075-T6 Aluminum Alloy”Transactions ASME, 1953

7075-T6 Specimens

28


COV vs Fatigue Life

440 14,000 0.12315 25,000 0.38280 220,000 0.70245 1,200,000 0.67210 12,000,000 1.39

∆S COVX

29


Variability in Fatigue Strength

( )∏=

−+=n

1i

a2X 1C1CCOV

2i

i

085.0b)N(S2S b

f'f −≈=

∆

( ) 088.0139.11C2

'f

)085.(2S =−+=

−

30


Strain Life Data for 950X Steel

102 103 104 105 106 107

Fatigue Life101

1

0.1

10-2

10-3

10-4

Str

ain

Am

plitu

de

378 Fatigue Tests

29 Data Sets

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29 Individual Data Sets'fσ

Median 820

300 1000

COV 0.25

2000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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b

-0.12 -0.08 -0.04

Mean -0.09COV 0.25

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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29 Individual Data Sets (continued)

0.1 1 10

Median 0.57COV 1.15

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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'fε

-0.8 -0.6 -0.4 -0.2

c

Mean -0.62COV 0.23

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Input Data Simulation

( )( ) ( ) ( )( ) ( )bb

ff

bbff ,,Ncf

'f

,,Nbf

'f N2,,LN2

E

,,L

2σµ

εεσµσσ σµε+

σµσ=

ε∆

34


Simulation Results

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 103 104 105 106 107

Stra

in A

mpl

itude

Fatigue Life

35


Correlation

0

0.25

0.5

0.75

1.0

0.01 0.1 1 10'fε

c

0

0.05

0.1

0.15

0.2

102 103 104

'fσ

b

ρ = -0.828 ρ = -0.976

36


Generating Correlated Data

z1 = ( rand() )Φz2 = ( rand() )Φ

2213 1zzz ρ−+ρ=

)zexp( 1lnln'f '

f'f σσ σ+µ=σ

3bb zb σ+µ=

z1= N(0,1)

37


Correlated Properties

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 103 104 105 106 107

Fatigue Life

Stra

in A

mpl

itude

38


Curve Fitting

102 103 104 105 106 107

Fatigue Life101

1

0.1

10-2

10-3

10-4

Str

ain

Am

plitu

de

cf

'f

bf

'f )N2()N2(

E2ε+

σ=

ε∆

Assume a constant slope to get a distribution of properties

'fσ

39


Property Distribution

0.1 1

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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365 Data Points

Median 0.26

COV 0.42

'fε

378 Data Points

Median 802 MPa

COV 0.12

'fσ

100 1000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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b = -0.086 c = -0.51

40


Correlation

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7'fε

'fσ

41


Simulation

0.001

0.01

0.1

1

10

0.00011 10 100 103 104 105 106 107

Stra

in A

mpl

itude

Fatigue Life

42


Strength Coefficient

5000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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500

0

500

1000

1500

0 500 1000 1500 2000 2500

365 Data Points

Median 1002 MPa

COV 0.14

'fσ

'K

'K

ρ = 0.863

43


Crack Growth Data

0

10

20

30

40

50

Cra

ck L

engt

h, m

m

0 50 100 150 200 250 300 350

Cycles x103

Virkler, Hillberry and Goel, “The Statistical Nature of Fatigue Crack Propagation”, Journal of Engineering Materials

and Technology, Vol. 101, 1979, 148-153

44


Crack Growth Rate Data

300,000

68 Data Points

Median 257,000COV 0.07

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Fatigue Lives Crack Growth Rate

5 10 5010-5

10-4

10-3

10-2C

rack

gro

wth

rate

, mm

/cyc

le

Stress intensity, MPa√m

45


Crack Growth Properties

Median 5 x 10-8

COV 0.44

10-7

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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10-6

Median 3.13COV 0.06

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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2 54

mKCdNda

∆=

C m

46


Beware of Correlated Variables

( )2/m1SC

aaN

2m

m

2/m1i

2/m1f

f

−π∆

−=

−−

Nf and C are linearly related and should have the same variability, but

07.0COVfN =

44.0COVC =

because C and m are correlated.

47


Correlation

2.0

2.5

3.0

3.5

4.0

10-8 10-610-7

C

m

48


Correlated Variables

2YYX

2XZ 2 σ+σρσ+σ=σ

-1 -0.5 0 0.5 1

0.5

1

1.5

2

ρ

σZ

σX = σY = 1

49


Calculated Lives

105 106

Computed fromC and m pairs experimental

105 106

0.1 %0.1 %

99.9 %

99 %

90 %

50 %

10 %

1 %

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99.9 %

99 %

90 %

50 %

10 %

1 %

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experimental

( )∫ ∆=

f

o

a

amf

KCda

N

50


Manufacturing/Processing Variability

n Bolt Forcesn Surface FinishnDrilled Holes

51


Variability in Bolt Force

100 1000

Force200 Data PointsMedian 130COV 0.14

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

Bolt Force, kN

Preload force in bolts tightened to 350 Nm

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Surface Roughness Variability

100

125 Data PointsMedian 43COV 0.10

Machined aluminum casting

10Surface Finish, µin

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Surface Finish

53


Drilled HolesFighter Spectrum

154 Data Points

Median 126,750

COV 0.22 in life

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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105 Cycles

From: J.P. Butler and D.A. Rees, "Development of Statistical Fatigue Failure Characteristics of 0.125-inch 2024-T3 Aluminum Under Simulated Flight-by-Flight Loading," ADA-002310 (NTIS no.), July 1974.

180 drilled holes in a single plate

COV 0.07 in strength

54


Analysis Uncertainty

nMiners Linear Damage rulen Strain Life Analysis

55


50

90

99.9

99

10

1

0.01

0.01 0.1 1 10 100

Computed Life / Experimental

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babi

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of F

ailu

re 964 Tests

COV = 1.02

Miners Rule

From Erwin Haibach “ Betriebsfestigkeit”, Springer -Verlag, 2002

A safety factor of 10 in life would result in a 10% chance of failure

56


SAE Specimen

Suspension

Transmission

Bracket

Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 1977

57


Analysis Results

1 10 100

48 Data Points

COV 1.27

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Analytical Life / Experimental Life

Strain-Life analysis of all test data

58


Material Variability

1 10 100

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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Analytical Life / Experimental Life

Material Analysis

Strain-Life back calculation of specimen lives

59


Modeling Uncertainty

( )∏=

−+=n

1i

a2X 1C1CCOV

2i

i

CU = 1.09

Analysis Uncertainty CU = ?

The variability in reproducing the original strain life data from the material constants is CM ~ 0.44

2M

2N2

U C1

C1C1 f

+

+=+

90% of the time the analysis is within a factor of 3 !99% of the time the analysis is within a factor of 10 !

60


Variability from Multiple Sources

( )∏=

−+=n

1i

a2X 1C1CCOV

2i

i

Suppose we have 4 variables each with a COV = 0.1

The combined variability is COV = 0.29

Suppose we reduce the variability of one of the variables to 0.05

The combined variability is now COV = 0.27

If all of the COV’s are the same, it doesn’t do any good to reduce only one of them, you must reduce all of them !

61


Variability from Multiple Sources

( )∏=

−+=n

1i

a2X 1C1CCOV

2i

i

Suppose we have 3 variables each with a COV = 0.1and one with COV = 0.4

The combined variability is COV = 0.65

Suppose we reduce the variability of these variables to 0.05

The combined variability is now COV = 0.60

If one of the COV’s is large, it doesn’t do any good to reduce the others, you must reduce the largest one !

62


Variability Summary

Source COVService Loading 0.5

Materials 0.1Manufacturing 0.1Surface Finish 0.1

Environment 0.3

Strength

Stress

Fatigue LivesAnalysis Uncertainty

1.01.0

5

StressStrength

lifeFatigue

∝

63



Variability: Every apple on a tree has a different mass.

Uncertainty: The variety of the apple is unknown.

Variability: Multiple samples of the same material

Uncertainty: What is the material

64


Strain Life Data for 93 Steels

1 10 102 103 106 10710510410-4

10-2

10-3

0.1

10

1

Str

ain

Am

plitu

de

1 10 102 103 106 10710510410-4

10-2

10-3

0.1

10

1

Str

ain

Am

plitu

de

65


Uncertainty for all Steels

100 1000 10000

55 Data Points

COV 0.48

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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'fσ

66


Uncertainty for Structural Steels

100 1000 10000

COV 0.12

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

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'fσ Su 600 – 900 MPa

19 Data Points

67



Fatigue Strength Coefficient

Variability Uncertainty CombinedAll Steels 0.12 0.48 0.75

Structural Steel 0.12 0.12 0.24

68


Quiz

What is the expected variability ?

At my last seminar everyone hit a golf ball and we recorded the maximum acceleration.

69


Results

12.97.7

5.8811.115.510.318.111.644.260.37

µσ

COV

70


probabilistic aspects of fatigue - illinois

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