Probalistic Fatigue Statistics

8/15/2019 Probalistic Fatigue Statistics

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Professor Darrell F. Socie

© 2003-2014 Darrell Socie, All Rights Reserved

Probabilistic Aspects of Fatigue


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Probabilistic Fatigue © 2003-2014 Darrell Socie, All Rights Reserved 1 of 108


Introduction

Statistical Techniques

Sources of Variability


3/110


Probabilistic Models

Probabilistic models are no better than theunderlying deterministic models

They require more work to implement

Why use them?


4/110


Quality and Cost

Taguchi

Identify factors that influence performance

Robust design – reduce sensitivity to noise

Assess economic impact of variation

Risk / Reliability

What is the increased risk from reduced testing ?


5/110


Risk

10-9

10-8

10-7

10-6

10-5

10-4

10-3

R i s k

Time, Flights etc

Acceptable risk


6/110


Reliability

99 %

80 %

50 %

10 %

1 %

0.1 %

E x p e c t e d F a

i l u r e s

106

Fatigue Life

105104103


7/110Probabilistic Fatigue © 2003-2014 Darrell Socie, All Rights Reserved 6 of 108

Risk Contribution Factors

OperatingTemperature

Analysis Uncertainty

Speed

MaterialProperties

Manufacturing

Flaws



Uncertainty and 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 S t r a i n A m p l i t u d e time

50%

100 %

F a i l u r e

s

Strength

Stress



Deterministic versus Random

Deterministic – from past measurements the future positionof a satellite can be predicted with reasonable accuracy

Random – from past measurements the future position of a car can only be described in terms of probability andstatistical averages



Deterministic Design

Stress Strength

SafetyFactor

Variability and uncertainty is accommodated by introducingsafety factors. Larger safety factors are better, but how much

better and at what cost?



Probabilistic Design

Stress Strength

Reliability = 1 – P( Stress > Strength )



3 Approach

3 contains 99.87% of the data

If we use 3 on both stress and strength

The probability of the part with the lowest strength

having the highest stress is very small

P( s < S ) = 2.3 10-3

5.4103.5)Sss(P)failure(P 6

For 3 variables, each at 3 :

7.5102.1)failure(P 8



Benefits

Reduces conservatism (cost) compared toassuming the “worst case” for every designvariable

Quantifies life drivers – what are the mostimportant variables and how well are theyknown or controlled ?

Quantifies risk



Reliability Analysis

1 32

0.2

0.4

0.6

0.8

P r o b a b i l i t y D e n s i t y

0.2

0.4

0.6

0.8


1 32

0.1

0.2

0.3

0.4

-3 -2 -1 0 1 2 3

P

r o b a b i l i t y D e n s i t y

Stressing Variables

Strength Variables0.2

0.4

0.6

0.8


1 32

P( Failure )

Analysis

?



Probabilistic Analysis Methods

Monte Carlo

Simple

Hypercube sampling

Importance sampling Analytical

First order reliability method FORM

Second order reliability method SORM



Statistical Techniques

Normal Distributions

LogNormal Distributions

Monte Carlo

Distribution Fitting



Failure Probability

)Sss(P

Let be the stress and S the fatigue strength

Given the distributions of and S find theprobability of failure

Stress, Strength, S

s



Normal Variables

Linear Response Function

n

1i

iio XaaZ

Xi ~ N( i , Ci )

n

1i

iioz aa

n

1i

2

i

2

iz a



Calculations

Z = S -

z = S -

22

Sz

~ N( 100 , 0.2 ) = 20z = 200 -100 = 100

S ~ N( 200 , 0.1 ) S = 20

Stress, Strength, S

Safety factor of 2

2.282020 22z

Let Z be a random variable:



Failure Probability

Failure will occur whenever Z



Failure Distribution

Stress, Strength, S

What is the expected distribution in fatigue lives?



Fatigue Data

b

f

'

f )N2(2

S

102 103 104 105 106 107Fatigue Life

101

1000

100

10

1

S t r e s s

A m p l i t u d e

'

f

b

1

'

f

f 2

SN2

b

1


23/110


LogNormal Variables

n

1i

a

ioiXaZ

a’s are constant and Xi ~ LN( xi , Ci )

n

1i

a

ioiXaZmedian

n

1i

a2

XZ 1C1CCOV

2i

i


24/110


Calculations

Stress, Strength, S

~ LN( 250 , 0.2 ) = 50

~ LN( 1000 , 0.1 ) = 100

b1

'

f

f 2

SN2

'

f

b = -0.125

2S

8'

f

8

f 2

S

N2Z

8'

f

8

f 2

SN2Z

1COV1COV1COV2

f

2 8282

SZ


25/110


Results

'

f S/2 2Nf Percenti l e Li fe

x 250 1000 355,368 99.9 17,706,069

COVx 0.2 0.1 4.72 99 4,566,613

95 1,363,200

lnx 5.50 6.90 11.21 90 715,589X 245 995 73,676 50 73,676

x 50 100 1,676,831 10 7,586

lnx 0.198 0.100 1.774 5 3,982

1 1,189

b = -0.125 0.1 307


26/110


Monte Carlo Methods

cbf

'

f

'

f

b2

f

2'

f f N2N2E

E2

SK

Given random variables for Kf , S,Find the distribution of 2Nf

'

f

'

f and

Z = 2Nf = ?


27/110


Simple Example

Probability of rolling a 3 on a die

1 2 3 4 5 6

f x(x) 1/6

Uniform discrete distribution


28/110


Computer Simulation

1. Generate n random numbers between 1 and 6,all integers

2. Count the number of 3’sLet Xi = 1 if 3

0 otherwise

n

1i

iXn13P


29/110


EXCEL

=ROUNDUP( 6 * RAND() , 0 )

=IF( A1 = 3 , 1 , 0 )

=SUM($B$1:B1)/ROW(B1)

5 0 0

3 1 0.5

4 0 0.333333

4 0 0.25

5 0 0.2

6 0 0.166667

1 0 0.142857

3 1 0.25

3 1 0.333333

6 0 0.3


30/110


Results

0.1

0.2

0.3

0.4

1 100 200 300 400 500

trials

p r o b a

b i l i t y


31/110


Evaluate

x

yP( inside circle )

4

r P

2

= 4 P

x = 2 * RAND() - 1

y = 2 * RAND() - 1IF( x2 + y2 < 1 , 1 , 0 )

1

-1

1

-1


32/110


0

1

2

3

4

1 100 200 300 400 500

trials


33/110


Monte Carlo Simulation

Stress, Strength, S

~ LN( 250 , 0.2 ) = 50

~ LN( 1000 , 0.1 ) = 100'f

b = -0.125

2S

b1

'

f

f 2

SN2

Randomly choose values of S andfrom their distributions'

f

Repeat many times


34/110


Generating Distributions

0

1

x

Fx(X)

Randomly choose a value between 0 and 1

x = Fx-1( RAND )

RAND


35/110


Generating Distributions in EXCEL

=LOGINV(RAND(),ln,ln)

=NORMINV(RAND(),,)

Normal

Log Normal


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EXCEL

893 204 134,677

1102 301 32,180

852 285 6,355

963 173 929,249

1050 283 35,565

1080 265 77,057

965 313 8,227

1073 213 420,456

1052 226 224,000

954 322 5,878965 240 68,671

993 207 277,192

1191 368 11,967

831 210 59,473

'f 2S

2Nf


37/110


Simulation Results

103

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

104 105 106 107 108

Monte Carlo AnalyticalMean 11.25 11.21

Std 1.79 1.77

C u

m u l a t i v e P r o b a b i l i t y

Life


38/110


Summary

Simulation is relatively straightforward and simple

Obtaining the necessary input data and distributionsis difficult


39/110


Maximum Load Data

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

100 1000

Median 431

COV 0.34

200 500

Maximum force from 42 drivers

C u m u l a t i v e P r o b

a b i l i t y

90% confidence COV 0.41


40/110


Maximum Load Data

Maximum Load

0 200 400 600 800 1000 1200

0.001

0.002

0.003

0.004

P r o b a b

i l i t y D e n s i t y

Normal COV = 0.34

LogNormal

Normal COV = 0.41

Uncertainty in Variance is just as important,perhaps more important than the choice of the distribution


41/110


Choose the “Best” Distribution

0 200 400 600 800 1000

Maximum Load

Normal

LogNormal

Gumble

Weibull

0.001

0.002

0.003

P r o b a b i l i t y D e

n s i t y

15 samples from a Normal Distribution


42/110


Distributions

Normal Strength

Dimensions

LogNormal

Fatigue Lives Large variance in properties or loads

Gumble Maximums in a population

Weibull Fatigue Lives


43/110


Central Limit Theorem

as n is the standard normal distribution

If X1, X2, X3 ….. Xn is a random sample from the population,with sample mean X, then the limiting form of

n/

XZ X


44/110


Translation

When there are many variables affecting the outcome,The final result will be normally distributed even if theindividual variable distributions are not.

As a result, normal distributions are frequentlyassumed for all of the input variables


45/110


Example

Probability of rolling a die

1 2 3 4 5 6

f x(x) 1/6

Uniform discrete distribution

Let Z be the summation of six dice

Z = X1 + X2 + X3 + X4 + X5 + X6


46/110


Results

0

10

20

30

40

50

60

6 12 18 24 30 36

Z

F r e q u e

n c y

Central limit theorem states that the result should benormal for large n

500 trials

CZ = 0.20

XZ = 21.12


47/110


Central Limit Theorem

Sums: Z = X1 X2 X3 X4 ….. Xn

Z Normal as n increases

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

Z LogNormal as n increases

Normal and LogNormal distributions are often employedfor analysis even though the underlying populationdistribution is unknown.


48/110


Key Points

All variables are random and can be characterizedby a statistical distribution with a mean and variance.

The final result will be normally distributed even if

the individual variable distributions are not.


49/110



customers

materials manufacturing

usage

107

Fatigue Life, 2Nf

1

10 102 103 104 105 106

0.1

10-2

1

10-3

10-4 S t r a i n A m p l i t u d

e

Strength

Stress

Stress, Strength, S


50/110


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


51/110



Stress Variables

Loading

Customer Usage

Environment

Strength Variables

Material

Processing

Manufacturing Tolerance

Environment


52/110


Sources of Uncertainty

Statistical Uncertainty

Incomplete data (small sample sizes)

Modeling Error

Analysis assumptions Human Error

Calculation errors

Judgment errors


53/110


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

l n X lnX ~ COVX

COVX

is a good measure of variability


54/110


Standard Deviation, lnx

COVX1 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


55/110


Variability in Service Loading

Quantifying Loading Variability

Maximum Load

Load Range

Equivalent Stress


56/110


Maximum Force

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

100 1000

Median 431

COV 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

C u

m u l a t i v e P r o b a b i l i t y

Force


57/110


Maximum Load Correlation

0 200 400 600 800 1000

108

107

106

105

104

103

Maximum Load

F a t i g u e L i v e s


58/110


Loading Variability

1 10 102 103 104 105

Cumulative Cycles

0

16

32

48

L o a d R a

n g e

54 Tractors / Drivers


59/110


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 C u m u l a t i v e P r o b a b i l i t y


60/110


Mechanisms and Slopes

100

103

104

S t r e s s A m p l i t u d e ,

M P a

100Cycles

101 102 103 104 105 106 107

1

10

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

Crack Growth Rate, m/cycle

1

10

100

m

M P a

, K

1

3

Crack Nucleation

Crack Growth

10

100

103 104 105 106 107 108

Total Fatigue Life, Cycles

1

51

E q u i v a l e n t L o a d ,

k N

Structures

A combination of nucleation and growth

n = 3

n = 5

n = 10


61/110


Effect of Slope on Variability

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C u m u l a t i v e P r o b a b i l i t y

n = 10 5 3

Equivalent Load

0.1 1 100.1 1 10

n COV

3 0.53

5 0.43

10 0.38


62/110


Loading History Variability

Test Track

Customer Service


63/110


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 %

C

u m u l a t i v e P r o

b a b i l i t y

Equivalent Load


64/110


Customer Usage Variability

0.1 1 10

COV 0.32

42 drivers / cars99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C


b a b i l i t y

Equivalent Load


65/110


Variability in Environment

Inclusions

Pit depth


66/110


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


67/110


Pits That Initiated Cracks

7010-T7651

Pre-corroded specimens

300 specimens

246 failed from pits

Crawford et.al.”The EIFS Distribution for Anodized and Pre-corroded 7010-T7651 under Constant Amplitude Loading”Fatigue and Fracture of Engineering Materials and Structures, Vol. 28, No. 9 2005, 795-808


68/110


Pit Size Distribution

40

30

20

10

100 200 400 500300

F r e q u e n c y

area m

Mean = 230COV = 0.32


69/110


Pit Depth Variability

Pits12 Data PointsMedian 24.37

100

COV 0.33

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

10Pit Depth, m

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



70/110


Variability in Materials

Tensile Strength

Fracture Toughness

Fatigue

Fatigue Strength Fatigue Life

Strain-Life

Crack Growth


71/110


Tensile Strength - 1035 Steel

25

50

75

100

500 550 600 650 700

N u m b e r o f h e a t s

Tensile Strength, MPa

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

Mean = 602COV = 0.045


72/110


Fracture Toughness

100

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

Median 76.7

COV 0.06

60 70 9080

26 Data Points

C u m u l a t i v e P

r o b a b i l i t y

KIc, Ksiin

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


73/110


Fatigue Variability

102 103 104 105 106 107

Fatigue Life101

1000

100

10

1

S t r e s s A m p l i t u d e

Fatigue life

F a t i g u e s t r

e n g t h


74/110


Fatigue Life Variability

50 100 150 200 250

10

20

30

40

10

20

N u m b e r o f T e

s t s

Life x10 350 100 150 200 250

10

20

30

40

10

20

N u m b e r o f T e

s t s

Life x10 3

Production torsion bars5160H steelOne lot, 71 parts

25 lots, 300 parts

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

x = 123,000 cyclesCOV = 0.25

x = 134,000 cyclesCOV = 0.27


75/110


Statistical Variability of Fatigue Life

50

10

2

90

98

104 105 106 107 108

Cycles to Failure

P e r c e n t S u

r v i v a l

s=210s=245s=315s=280s=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


76/110


COV vs Fatigue Life

440 14,000 0.12

315 25,000 0.38

280 220,000 0.70245 1,200,000 0.67

210 12,000,000 1.39

S COVX


77/110


Variability in Fatigue Strength

n

1i

a2

X 1C1CCOV

2i

i

085.0b)N(S2S b

f

'

f

088.0139.11C2

'f

)085.(2

S


78/110


Strain Life Data for 950X Steel

102 103 104 105 106 107

Fatigue Life101

1

0.1

10-2

10-3

10-4

S t r a i n A m p

l i t u d e

378 Fatigue Tests

29 Data Sets


79/110


29 Individual Data Sets

'

f Median 820

300 1000

COV 0.25

2000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C u m u l a t i v e P r o

b a b i l i t y

b

-0.12 -0.08 -0.04

Mean -0.09

COV 0.2599.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


b a b i l i t y


80/110


29 Individual Data Sets (continued)

0.1 1 10

Median 0.57

COV 1.1599.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


b a b i l i t y

'

f

-0.8 -0.6 -0.4 -0.2

c

Mean -0.62

COV 0.23

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


b a b i l i t y


81/110


Input Data Simulation

bb

f f

bbf f ,,Nc

f

'

f

,,Nb

f

'

f N2,,LN2

E

,,L

2


82/110


Simulation Results

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 103 104 105 106 107

S t r a i n A m p l i t u d e

Fatigue Life


83/110


Correlation

0

0.25

0.5

0.75

1.0

0.01 0.1 1 10'

f

c

0

0.05

0.1

0.15

0.2

102 103 104'

f

b

= -0.828 = -0.976


84/110


Generating Correlated Data

z1 = ( rand() )

z2 = ( rand() )2

213 1zzz

)zexp( 1lnln'

f 'f

'f

3bb zb

z1= N(0,1)


85/110


Correlated Properties

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 103 104 105 106 107

Fatigue Life

S t r a i n A m

p l i t u d e


86/110


Curve Fitting

102 103 104 105 106 107

Fatigue Life101

1

0.1

10-2

10-3

10-4

S t r a i n A m p

l i t u d e

c

f

'

f

b

f

'f )N2()N2(

E2

Assume a constant slope to geta distribution of properties

'

f


87/110


Property Distribution

0.1 1

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


r o b a b i l i t y

365 Data Points

Median 0.26

COV 0.42

'

f 378 Data PointsMedian 802 MPa

COV 0.12

'

f

100 1000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


r o b a b i l i t y

b = -0.086 c = -0.51

C


88/110


Correlation

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7'

f

'

f

Si l i


89/110


Simulation

0.001

0.01

0.1

1

10

0.00011 10 100 103 104 105 106 107

S t r a i n A m

p l i t u d e

Fatigue Life

St th C ffi i t


90/110


Strength Coefficient

5000

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


500

0

500

1000

1500

0 500 1000 1500 2000 2500

365 Data Points

Median 1002 MPa

COV 0.14

'

f 'K

'K

= 0.863

C k G th D t


91/110


Crack Growth Data

0

10

20

30

40

50

C r a c k L e n

g t 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

C k G th R t D t


92/110


Crack Growth Rate Data

300,000

68 Data Points

Median 257,000

COV 0.07

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


r o b a b i l i t y

Fatigue Lives Crack Growth Rate

5 10 5010-5

10-4

10-3

10-2

C r a c k g r o w t h r a

t e , m m / c y c l e

Stress intensity, MPam

C k G th P ti


93/110


Crack Growth Properties

Median 5 x 10-8

COV 0.44

10-7

99.9 %

99 %

90 %

50 %

10 %

1 %0.1 %


10-6

Median 3.13

COV 0.06

99.9 %

99 %

90 %

50 %

10 %

1 %0.1 %

C u m u l a t i v e P r

o b a b i l i t y

2 54

mKC

dN

da

C m

Si l t d D t


94/110


Simulated Data

0.1 %0.1 %

99.9 %

99 %

90 %

50 %

10 %

1 %


r o b a b i l i t y

99.9 %

99 %

90 %

50 %

10 %

1 %


r o b a b i l i t y

105 107106

data

simulation

B f C l t d V i bl


95/110


Beware of Correlated Variables

2/m1SC

aaN

2

m

m

2/m1i

2/m1f

f

Nf and C are linearly related and should havethe same variability, but

07.0COVf N

44.0COVC

because C and m are correlated.

C l ti


96/110


Correlation

2.0

2.5

3.0

3.5

4.0

10-8 10-610-7

C

m

= -0.99

C l l t d Li


97/110


Calculated Lives

105 106

Computed fromC and m pairs experimental

105 106

0.1 %0.1 %

99.9 %

99 %

90 %

50 %

10 %

1 %


b a b i l i t y

99.9 %

99 %

90 %

50 %

10 %

1 %


b a b i l i t y

experimental

f

o

a

amf KC

daN

Manufacturing/Processing Variability


98/110


Manufacturing/Processing Variability

Bolt Forces Surface Finish

Drilled Holes

Variability in Bolt Force


99/110


Variability in Bolt Force

100 1000

Force200 Data Points

Median 130

COV 0.14

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

Bolt Force, kN

Preload force in bolts tightened to 350 Nm


r o b a b i l i t y

Surface Roughness Variability


100/110


Surface Roughness Variability

100

125 Data Points

Median 43

COV 0.10

Machined aluminum casting

10Surface Finish, in

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


r o b a b i l i t y

Surface Finish

Drilled Holes


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Drilled Holes

Fighter Spectrum

154 Data Points

Median 126,750

COV 0.22 in life

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C


b a b i l i t y

105 Cycles

From: J.P. Butler and D.A. Rees, "Development of Statistical Fatigue Failure Characteristics of 0.125-inch2024-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



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Miners Linear Damage rule Strain Life Analysis

Miners Rule


103/110


50

90

99.9

99

10

1

0.01

0.01 0.1 1 10 100

Computed Life / Experimental

C u

m u l a t i v e P r o b a b i l i t y o f F a i l u r

e 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

SAE Specimen


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SAE Specimen

Suspension

Transmission

Bracket

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

Analysis Results


105/110


Analysis Results

1 10 100

48 Data Points

COV 1.27

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C

u m u l a t i v e P r o b a b i l i t y

Analytical Life / Experimental Life

Strain-Life analysis of all test data

Material Variability


106/110


Material Variability

1 10 100

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

C


b a b i l i t y

Analytical Life / Experimental Life

Material Analysis

Strain-Life back calculation of specimen lives

Modeling Uncertainty


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Modeling Uncertainty

n

1i

a2

X 1C1CCOV

2i

i

CU = 1.09

Analysis Uncertainty CU = ?The variability in reproducing the original strain life datafrom the material constants is CM ~ 0.44

2

M

2

N2

UC1

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 !

Variability from Multiple Sources


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n

1i

a2

X 1C1CCOV2i

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 toreduce only one of them, you must reduce all of them !



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n

1i

a2

X 1C1CCOV2i

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 toreduce the others, you must reduce the largest one !


110/110


Documents

Probalistic Fatigue Statistics