Post on 13-Apr-2018
7/27/2019 Probabilistic Fatigue
1/177
Introduction
Professor Darrell F. Socie
Department of Mechanical and
Industrial Engineering
2003-2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 352
Contact Information
Darrell Socie
Mechanical Engineering
1206 West Green
Urbana, Illinois 61801
Office: 3015 Mechanical Engineering Laboratory
dsocie@uiuc.edu
Tel: 217 333 7630
Fax: 217 333 5634
7/27/2019 Probabilistic Fatigue
2/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 352
Fatigue Calculations
Who really believesthese numbers ?
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 352
SAE Specimen
Suspension
Transmission
Bracket
Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 1977
7/27/2019 Probabilistic Fatigue
3/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 352
Analysis Results
1 10 100
48 Data Points
COV 1.27
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
CumulativeProbability
Analytical Life / Experimental Life
Strain-Life analysis of all test data
Non conservative
Conservative
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 352
Material Variability
1 10 100
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
CumulativeProbability
Analytical Life / Experimental Life
Material Analysis
Strain-Life back calculation of specimen lives
7/27/2019 Probabilistic Fatigue
4/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 352
Probabilistic Models
Probabilistic models are no better than theunderlying deterministic models
They require more work to implement
Why use them?
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 352
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 ?
7/27/2019 Probabilistic Fatigue
5/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 352
Risk
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Risk
Time, Flights etc
Acceptable risk
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 352
Reliability
99 %
80 %
50 %
10 %
1 %
0.1 %
ExpectedFailures
106
Fatigue Life
105104103
7/27/2019 Probabilistic Fatigue
6/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 352
Risk Contribution Factors
OperatingTemperature
Analysis Uncertainty
Speed
Material
Properties
ManufacturingFlaws
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 352
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
10-4StrainAmplitude
time
50%
100 %
F
ailures
Strength
Stress
7/27/2019 Probabilistic Fatigue
7/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 352
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 ofa car can only be described in terms of probability andstatistical averages
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 352
Deterministic Design
Stress Strength
SafetyFactor
Variability and uncertainty is accommodated by introducingsafety factors. Larger safety factors are better, but how muchbetter and at what cost?
7/27/2019 Probabilistic Fatigue
8/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 352
Probabilistic Design
Stress Strength
Reliability = 1 P( Stress > Strength )
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 352
3Approach3 contains 99.87% of the data
If we use 3 on both stress and strength
The probability of the part with the lowest strengthhaving the highest stress is very small
P( s < S ) = 2.3 10-3
== 5.4103.5)Sss(P)failure(P 6I
For 3 variables, each at 3 :
= 7.5102.1)failure(P 8
7/27/2019 Probabilistic Fatigue
9/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 352
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
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 352
Probabilistic Aspects of Fatigue
Introduction
Basic Probability and Statistics
Statistical Techniques
Analysis Methods
Characterizing Variability
Case Studies
FatigueCalculator.com
GlyphWorks
7/27/2019 Probabilistic Fatigue
10/177
Basic Probability and Statistics
Professor Darrell F. Socie
Department of Mechanical and
Industrial Engineering
2003-2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 352
Probabilistic Aspects of Fatigue
Introduction
Basic Probability and Statistics
Statistical Techniques
Analysis Methods
Characterizing Variability
Case Studies
FatigueCalculator.com
GlyphWorks
7/27/2019 Probabilistic Fatigue
11/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 352
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 ofa car can only be described in terms of probability andstatistical averages
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 21 of 352
Random Variables
Discrete - fixed number of outcomes
Colors
Continuous - may have any value in thesample space
Strength
7/27/2019 Probabilistic Fatigue
12/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 352
Descriptive Statistics
Mean or Expected Value Variance / Standard Deviation
Coefficient of Variation
Skewness
Kurtosis
Correlation Coefficient
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 352
Mean or Expected Value
Central tendency of the data
( )N
x
XExMean
N
1ii
x
=====
7/27/2019 Probabilistic Fatigue
13/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 352
Variance / Standard Deviation
Dispersion of the data
( )N
)xx(
XVar
N
1i
2i
=
=
)X(Varx=
Standard deviation
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 25 of 352
Coefficient of Variation
x
xCOV
=
Useful to compare different dispersions
= 10 = 1
COV = 0.1
= 100 = 10
COV = 0.1
7/27/2019 Probabilistic Fatigue
14/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 26 of 352
Skewness
Skewness is a measure of the asymmetry of the dataaround the sample mean. If skewness is negative, thedata are spread out more to the left of the mean thanto the right. If skewness is positive, the data are spreadout more to the right. The skewness of the normaldistribution (or any perfectly symmetric distribution) is zero.
( )3
N
1i
3i
N
)xx(
XSkewness
=
=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 352
Kurtosis
Kurtosis is a measure of how outlier-prone a distribution is.The kurtosis of the normal distribution is 3. Distributions thatare more outlier-prone than the normal distribution havekurtosis greater than 3; distributions that are lessoutlier-prone have kurtosis less than 3.
( )4
N
1i
4i
N
)xx(
XKurtosis
=
=
7/27/2019 Probabilistic Fatigue
15/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 352
Covariance
A measure of the linear association betweenrandom variables
[ ])Y()X(E)Y,X(COV YXxy ==
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 29 of 352
Correlation Coefficient
0 X
Y
0 X
Y
0 X
Y
0 X
Y
0 X
Y
0 X
Y
= 1 = 0 = -1
0 < < 1 ~0 ~ 0
[ ]
YX
YX )Y()X(E
=
7/27/2019 Probabilistic Fatigue
16/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 352
Probability
Basic probabilityConditional probability
Reliability
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 352
Basic Probability
( ) 1AP0
( ) certain1AP =
( ) eimposssibl0AP =
The probability of event A occurring:
7/27/2019 Probabilistic Fatigue
17/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 352
Conditional Basic Probability
( ) ( ) ( ) ( )BandABorA
BAPBPAPBAP IU +=
( ) ( )
( )APBAP
ABP I=
( ) ( )
( )BP
BAPBAP
I=
P(B) given A has occurred
union intersection
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 352
Independent Events
1 2
If A and B are unrelated
( ) ( )BPABP =
( ) ( )APBAP =
( ) ( ) ( )BPAPBAP =I
Suppose the probability of bar 1 failing is 0.03and the probability of bar 2 failing is 0.04.
What is the probability of the structure failing?
7/27/2019 Probabilistic Fatigue
18/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 352
Failure Probability
Bar 1 or bar 2 fails
( ) 03.0AP =
( ) 04.0BP =
( ) ( ) ( ) 0012.0BPAPBAP ==I
P( failure) = 0.03 + 0.04 - 0.0012 = 0.0688
( ) ( ) ( ) ( )BAPBPAPBAP IU +=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 352
Reliability
P( no failure ) = Reliability
) AnotofyprobabilitAPDefine
( ) ( )A1PAP =
( )BAPliabilityRe I=
( ) ( ) ( )BPAPBAP =I
) 9312.096.097.0BAP ==I
For the 2 bar structure
( ) 0688.0liabilityRe1failureP ==
7/27/2019 Probabilistic Fatigue
19/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 352
Structural Reliability
Collapse occurs if any member fails
) )n4321 AAAAAPAP = IIII
) ) ) ) ) )n4321 APAPAPAPAPAP =
( ) linkeachfor01.0APSuppose =( ) ( ) 932.099.0AP 7 ==
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 352
Reliability
6
5
4
3
2
1
010-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1
Standard
Deviations
Probability
7/27/2019 Probabilistic Fatigue
20/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 352
6 Sigma
6 sigma is 1 in a billion ( 0.999999999 Reliability )
Suppose a structure has 1000 bolted joints:
( ) ( ) 999999.0999999999.0AP 1000 ==1 in a million
3 sigma is ( 0.99865 Reliability )
( ) ( ) 26.099865.0AP1000
==74 % failures
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 352
Statistical Distributions
Uniform
Normal
LogNormal
Gumble
Weibull
http://www.itl.nist.gov/div898/handbook/
A useful on-line reference:
7/27/2019 Probabilistic Fatigue
21/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 352
Cumulative Distribution Function
( ) ( )xXPXFx =
( ) 0Fx =
( ) 1Fx =
0
1
a b
Fx(X)
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 352
Probability Density Function
( ) ( )dxXfbXaPb
a
x= Strength )
Strength
Stress
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 352
Joint Probability Density
( )
( )
+
=
2
y
y
y
y
x
x
2
x
x2
2yx
xy
yyx2
x
12
1exp
12
1y,xf
correlation coefficient
Normal distributions
7/27/2019 Probabilistic Fatigue
34/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 352
2D - Joint Probability Density
Contours ofequal probability
y
xx
Y
-3x
3Y
12
34
Stress
Strength
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 352
Acceptable Risk
Safety
P(failure) ~ 10-6 10-9
Economic
P(failure) ~ 10-2 10-4
7/27/2019 Probabilistic Fatigue
35/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 352
Extreme Values
For most durability problems, we are not interested inthe large extremes of stress or strength. Failure is muchmore likely to come from moderately high stressescombined with moderately low strengths.
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 352
Key Points
Basic Nomenclature
Random VariablesStatistical Distribution
Cumulative distribution function
Mean
Variance
ProbabilityMarginal
Conditional
Joint
7/27/2019 Probabilistic Fatigue
36/177
Statistical Techniques
Professor Darrell F. Socie
Department of Mechanical and
Industrial Engineering
2003-2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 352
Probabilistic Aspects of Fatigue
Introduction
Basic Probability and Statistics
Statistical Techniques
Analysis Methods
Characterizing Variability
Case Studies
FatigueCalculator.com
GlyphWorks
7/27/2019 Probabilistic Fatigue
37/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 352
Statistical Techniques
Normal Distributions LogNormal Distributions
Monte Carlo
Sampling
Distribution Fitting
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 352
Failure Probability
)Sss(P I
Let be the stress and S the fatigue strength
Given the distributions of and S find theprobability of failure
Stress, Strength, S
s
7/27/2019 Probabilistic Fatigue
38/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 352
Normal Variables
Linear Response Function
=
+=n
1iiio XaaZ
Xi ~ N( i , Ci )
=
+=n
1iiioz aa
= =n
1i
2
i
2
iz a
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 352
Calculations
Z = S -
z = S -
22
Sz +=
~ N( 100 , 0.2 ) = 20
z
= 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:
7/27/2019 Probabilistic Fatigue
39/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 352
Failure Probability
Failure will occur whenever Z
7/27/2019 Probabilistic Fatigue
40/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 352
Fatigue Data
bf
'f
)N2(2
S=
102 103 104 105 106 107
Fatigue Life101
1000
100
10
1
StressAmplitude
'
f
b
1
'f
f2
SN2
=
b1
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 352
LogNormal Variables
=
=n
1i
a
ioiXaZ
as are constant and Xi ~ LN( xi , Ci )
==
n
1i
a
io
i
XaZmedian
( )=
+=n
1i
a2
XZ 1C1CCOV2
i
i
7/27/2019 Probabilistic Fatigue
41/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 352
Calculations
Stress, Strength, S
~ LN( 250 , 0.2 ) = 50
~ LN( 1000 , 0.1 ) = 100
b
1
'f
f2
SN2
=
'f
b = -0.125
2
S8'
f
8
f2
SN2Z
==
8'f
8
f2
SN2Z
==
( ) ( ) 1COV1COV1COV2
f
2 8282
SZ ++=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 352
Results
'fS/2 2Nf Pe rce ntile Life
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 90 715,589
X 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
7/27/2019 Probabilistic Fatigue
42/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 352
Monte Carlo Methods
( ) ( )
+
=
+cbf
'f
'f
b2
f
2'ff N2N2
EE
2
SK
Given random variables for Kf, S,Find the distribution of 2Nf
'f
'fand
Z = 2Nf= ?
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 352
Simple Example
Probability of rolling a 3 on a die
1 2 3 4 5 6
fx(x) 1/6
Uniform discrete distribution
7/27/2019 Probabilistic Fatigue
43/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 352
Computer Simulation
1. Generate random numbers between 1 and 6,all integers
2. Count the number of 3s
Let Xi = 1 if 30 otherwise
( ) =
=n
1i
iXn
13P
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 352
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
7/27/2019 Probabilistic Fatigue
44/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 352
Results
0.1
0.2
0.3
0.4
1 100 200 300 400 500
trials
probability
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 352
Evaluate
x
yP( inside circle )
4
rP
2=
= 4 P
x = 2 * RAND() - 1y = 2 * RAND() - 1
IF( x2 + y2 < 1 , 1 , 0 )
1
-1
1
-1
7/27/2019 Probabilistic Fatigue
45/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 352
0
1
2
3
4
1 100 200 300 400 500
trials
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 352
Monte Carlo Simulation
Stress, Strength, S
~ LN( 250 , 0.2 ) = 50
~ LN( 1000 , 0.1 ) = 100'f
b = -0.125
2
S
b
1
'f
f2
SN2
=
Randomly choose values of S andfrom their distributions'f
Repeat many times
7/27/2019 Probabilistic Fatigue
46/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 352
Generating Distributions
0
1
x
Fx(X)
Randomly choose a value between 0 and 1
x = Fx-1( RAND )
RAND
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 352
Generating Distributions in EXCEL
=LOGINV(RAND(),ln,ln)
=NORMINV(RAND(),,)
Normal
Log Normal
7/27/2019 Probabilistic Fatigue
47/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 92 of 352
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 5,878
965 240 68,671
993 207 277,192
1191 368 11,967
831 210 59,473
'
f
2
S2N
f
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 352
Simulation Results
103
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
104 105 106 107 108
Monte Carlo Analytical
Mean 11.25 11.21
Std 1.79 1.77
CumulativeProbability
Life
7/27/2019 Probabilistic Fatigue
48/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 352
Summary
Simulation is relatively straightforward and simple
Obtaining the necessary input data is difficult
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 352
Nomenclature
Population: the totality of the observations
Sample: a subset of the population
Population mean: x
Population variance: x2
Sample mean: X
Sample variance: s2
( ) xXE =( ) 2x2sE =
7/27/2019 Probabilistic Fatigue
49/177
7/27/2019 Probabilistic Fatigue
50/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 352
Translation
90% confidence
If we sampled a population many times toestimate the mean, 90% of the time the truepopulation mean would lie between thecomputed upper and lower limit.
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 352
Confidence Interval - mean
Lower limit of
xx
1n,n
stX
Upper limit of
xx
1n,n
stX +
For a normal distribution:
7/27/2019 Probabilistic Fatigue
51/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 100of 352
Sample Size
0 5 10 15 20 25 30
Sample size, n
0
0.5
1.0
1.5
2.0
xx
1n,n
stX
n
t 1n,
95%
90%
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 101of 352
5 Samples from Normal Distribution
0
20
40
60
80
100
120
140
0 10 20 30 40 50 60 70 80 90 100
95 % confidence 5 samples will be outside confidence limit
Lower limit above mean
Upper limit below mean
Sample Number
x = 100 x = 10
7/27/2019 Probabilistic Fatigue
52/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 102of 352
Confidence Interval - variance
Upper limit of
2
x1n,1
2
2
xs)1n(
For a normal distribution:
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103of 352
Sample Size
0 5 10 15 20 25 300
2
4
6
8
10
Sample size, n
2
x1n,1
2
2
xs)1n(
1n,12
)1n(
95%
50%
90%
7/27/2019 Probabilistic Fatigue
53/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104of 352
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
CumulativeProbability
90% confidence COV 0.41
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105of 352
Maximum Load Data
Maximum Load
0 200 400 600 800 1000 1200
0.001
0.002
0.003
0.004
Pr
obabilityDensity
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
7/27/2019 Probabilistic Fatigue
54/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 106of 352
Choose the Best Distribution
0 200 400 600 800 1000
Maximum Load
Normal
LogNormal
Gumble
Weibull
0.001
0.002
0.003
ProbabilityDensity
15 samples from a Normal Distribution
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 107of 352
Goodness of Fit Tests
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
0 200 400 600 800 1000
CumulativeProbability
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
0 200 400 600 800 1000
CumulativeProbability
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
100 1000200 500
CumulativeProbability
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
100 1000200 500
CumulativeProbability
99.9 %
99 %
80 %
50 %
10 %
1 %
200 400 600 800CumulativeProbability
99.9 %
99 %
80 %
50 %
10 %
1 %
200 400 600 800CumulativeProbability
Normal
LogNormal
Gumble
200
99.9 %99 %
80 %
50 %
10 %
1 %
0.1 %
1000500
CumulativeProbability
200
99.9 %99 %
80 %
50 %
10 %
1 %
0.1 %
1000500
CumulativeProbability
Weibull
7/27/2019 Probabilistic Fatigue
55/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 108of 352
Analytical Tests
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons,Journal of the American Statistical Association, Vol. 69, pp. 730-737.
Kolmogorov-Smirnov
Chakravarti, Laha, and Roy, (1967). Handbook of Methods of Applied Statistics, Volume I,John Wiley and Sons, pp. 392-394.
Anderson-Darling
Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition,Iowa State University Press.
Chi-squareany univariate distribution
tends to be more sensitive near the center of the distribution
gives more weight to the tails
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 109of 352
Distributions
Normal Strength
Dimensions
LogNormal Fatigue Lives
Large variance in properties or loads
Gumble Maximums in a population
Weibull Fatigue Lives
7/27/2019 Probabilistic Fatigue
56/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 110of 352
Statistics Software
www.palisade.com
BestFit Minitab
www.minitab.com
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 111of 352
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
=
7/27/2019 Probabilistic Fatigue
57/177
7/27/2019 Probabilistic Fatigue
58/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 114of 352
Results
0
10
20
30
40
50
60
6 12 18 24 30 36
Z
Frequency
Central limit theorem states that the result should benormal for large n
500 trials
CZ = 0.20
XZ = 21.12
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 115of 352
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.
7/27/2019 Probabilistic Fatigue
59/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 116of 352
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 ifthe individual variable distributions are not.
Analysis Methods
Professor Darrell F. Socie
Department of Mechanical and
Industrial Engineering
2003-2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
7/27/2019 Probabilistic Fatigue
60/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 118of 352
Probabilistic Aspects of Fatigue
Introduction Basic Probability and Statistics
Statistical Techniques
Analysis Methods
Characterizing Variability
Case Studies
FatigueCalculator.com
GlyphWorks
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 119of 352
Reliability Analysis
1 32
0.2
0.4
0.6
0.8
ProbabilityDensity
0.2
0.4
0.6
0.8
ProbabilityDensity
1 32
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
ProbabilityDensity
Stressing Variables
Strength Variables0.2
0.4
0.6
0.8
Proba
bilityDensity
1 32
P( Failure )
Analysis
?
7/27/2019 Probabilistic Fatigue
61/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120of 352
Probabilistic Analysis Methods
Monte CarloSimple
Hypercube sampling
Importance sampling
Analytical
First order reliability method FORM
Second order reliability method SORM
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 121of 352
Limit States
Limit States
Equilibrium
Strength
Deformation
Wear Functional
7/27/2019 Probabilistic Fatigue
62/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 122of 352
Limit State Problems
Z(X) = Z( Q, L, b, h )
Response function
Limit State function
g = Z(X) - Zo = 0
Same as the failure function
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 123of 352
Limit States
Stress
Strength
g < 0
g > 0
failures
g = Strength - Stress
7/27/2019 Probabilistic Fatigue
63/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124of 352
Monte Carlo
Stress
Strength
g < 0
g > 0
failures
Monte Carlo is simple but what if each of these calculationsrequired a separate FEM model?
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125of 352
Low Probabilities of Failure
Stress
Strength
failures
rare event
Need about 105 simulations for P(Failure) = 10-4
7/27/2019 Probabilistic Fatigue
64/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 126of 352
Transform Variables
u1
u2
x
x1
xu
=
u1 = 0
u1 = 1
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 127of 352
Standard Monte Carlo
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4u1
u2
1000 trials
g = 3 ( u1 + u2 )
018.0
N
N)f(P f ==
u1 and u2 normally distributed
7/27/2019 Probabilistic Fatigue
65/177
7/27/2019 Probabilistic Fatigue
66/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 130of 352
Stratified Sample
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4u1
u2
Each box should haveone sample drawn at randomfrom the underlying pdf
100 trials
25 simulations
= 0.02
= 0.0093
2,500 calculations
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 131of 352
Latin Hypercube Sample
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4u1
u2
10 trials
25 simulations
= 0.017 = 0.038
250 calculations
7/27/2019 Probabilistic Fatigue
67/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 132of 352
Importance Sampling (continued)
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
u1
u2
= 2
The probability of a sampleoutside the circle is PS = 0.14( 2 standard deviations )
0183.0PN
N)f(P S
S
f ==
138 simulations
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 133of 352
Analytical Methods Strategy
Develop a response function
Transform the set of N variables to a set ofuncorrelated u variables with =0 and =1
Locate the most likely failure point
Approximate the integral of the jointprobability distribution over the failure region
7/27/2019 Probabilistic Fatigue
68/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 134of 352
Analytical Methods Outline
Response Surface
Transform Variables
Limit State Concept
Most Probable Point
Probability Integration
FORM
SORM
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 135of 352
Response Surface Methodology
Response surface methodology is a wellestablished collection of mathematical andstatistical techniques for applications where theresponse of interest is influenced by several
variables.
+= )x,x,x,x(f)X(Z n321
7/27/2019 Probabilistic Fatigue
69/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 136of 352
Response Surface
X1 X2
Z
Fit a surface through the points
Solution points
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 137of 352
Mathematical Representation
+++
+++
++++=
323312211
2
33
2
22
2
11
332211o
xxCxxCxxC
xBxBxB
xAxAxAA)X(Z Linear
Incomplete Quadradic
Complete Quadradic
Solutions
Linear N + 1
Complete Quadradic N(N + 1)/2
Incomplete Quadradic 2N + 1
7/27/2019 Probabilistic Fatigue
70/177
7/27/2019 Probabilistic Fatigue
71/177
7/27/2019 Probabilistic Fatigue
72/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 142of 352
2D - Joint Probability Density
Contours ofequal probability
x
y
Stress
Strength
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 143of 352
Transform Variables
x
x1
xu
=
u = 0u = 1
Any distribution can be mapped into a normal distribution
u1
u2
d ( )2d5.0expP
The mathematics and behavior of normal distributionsare well understood
7/27/2019 Probabilistic Fatigue
73/177
7/27/2019 Probabilistic Fatigue
74/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 146of 352
Failure Probability
g = 0u1
u2
Strength )
How do you really get these distributions ?
Design Load
How do you establish a design load ?
How do you validate the design ?
7/27/2019 Probabilistic Fatigue
133/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 264of 352
Stress Distribution
Variability in loading has two components,how it is used and how it is driven.
Car UsageHighway, city, fully loaded, empty etc.
Driving Stylepassive, aggressive etc.
The usage of a car is independent of the owners driving styleso that the distributions of car usage and driving style canbe obtained separately.
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 265of 352
Car Usage
Customer surveys
k l c % r kl%
1 Unloaded 27
1 Highway 10
2 Good Road 25
3 Mountain 40
4 City 25
2 Half Load 581 Highway 5
2 Good Road 30
3 Mountain 30
4 City 35
3 Fully Loaded 15
1 Highway 15
2 Good Road 25
3 Mountain 40
4 City 20
12 Customer UsageCategories
7/27/2019 Probabilistic Fatigue
134/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 266of 352
Owner Behavior
Instrumented Vehicles
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN0 -1 -2 -33 2 1
To Load, kN
FromL
oad,
kN
Extensive field testing foreach customer usagecategory produces a largenumber of histograms.
Let the usage histogrambe denoted Ukl
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 267of 352
Extrapolation
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN0 -1 -2 -33 2 1
To Load, kN
FromL
oad,kN
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN
FromLoad,
kN
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN
FromLoad,
kN
Measured data, e.g. 1,000 km Extrapolated data, 300,000 km
Ukl U*kl
7/27/2019 Probabilistic Fatigue
135/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 268of 352
Virtual Customers
Thousands of virtual customers can now be generatedby combining customer usage with driving style.
= *klklki UrcNU
[ ]iU rainflow histogram for an individual carN kilometers
ck car loading, %
rkl car usage, %
*klU
distributions of rainflow histograms fordifferent car usage classifications
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 269of 352
Design Loads
Initial design is done on the basis of a singleconstant amplitude load, Feq
Find a constant amplitude load and number of cyclesthat will produce the same fatigue damage as the customeroperating a car for the design life.
7/27/2019 Probabilistic Fatigue
136/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 270of 352
Equivalent Fatigue Loading
101
102
103
104
105
106
Feq
m
1
6
m
eq10
FF
=
1Nn
f
=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 271of 352
Distribution of Feq
FF
Fn
Fn= F+ FDesign Customer:
7/27/2019 Probabilistic Fatigue
137/177
7/27/2019 Probabilistic Fatigue
138/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 274of 352
What Strength is Needed?
F
F
Fn S
S
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 275of 352
Some Statistics
Z = S - F
Suppose we want a probability of failure of 1 in 50,000
5f 10x2P
=
( ) 2F
2S
FS
Z
Zf1 1.4P +
=
==
Fn = S( 1 COVS )
Fn = F( 1 + COVF )
7/27/2019 Probabilistic Fatigue
139/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 276of 352
Mean Component Strength
( ) 2F
2S
FS
Z
Zf1 1.4P +
=
==
( )2
F
F
2
S
n
S
Fn
S
f1
COV1
COVCOV
F
COV1
1
FP
++
+
=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 277of 352
Reliability
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
Relia
bility
n
S
F
= 3COVF = 0.2COVS = 0.1
7/27/2019 Probabilistic Fatigue
140/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 278of 352
Mean Strength
FF
Fn SS
S = 1.4 Fn for a reliability of 2 x 10-5
85.2COV
F1
S
S
n
=
=
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 279of 352
Design
Design calculations are made with loads Fn Large database relates Fn to vehicle parameters
so that a new vehicle can be designed fromhistorical measurements
Fatigue calculations are made with materialproperties standard deviations from themean
7/27/2019 Probabilistic Fatigue
141/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 280of 352
Component Testing
Test load is frequently higher than Fn to getfailures near the design life
Component strength not life !
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 281of 352
Interpreting the Test Data
Component tests are done with small sample sizes
2
x1N,1
2
2
xs)1N(
Confidence limits
( )2
F
F2
1N,1
2
S
n
S
Fn
S
f1
COV1
COV1NCOV
F
COV11
FP
++
+
=
7/27/2019 Probabilistic Fatigue
142/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 282of 352
Validation
Full scale vehicle simulation done at the end for
design final validation
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 283of 352
Extrapolation
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1To Load, kN
0 -1 -2 -33 2 1To Load, kN
FromL
oad,
kN
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1To Load, kN
FromL
oad,
kN
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1To Load, kN
FromL
oad,
kN
Measured data, e.g. 1,000 km Extrapolated data, 300,000 km
Ukl U*kl
How does this process work?
7/27/2019 Probabilistic Fatigue
143/177
7/27/2019 Probabilistic Fatigue
144/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 286of 352
Probability Density
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN
FromLoad,kN
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 287of 352
Kernel Smoothing
-3
-2
-1
0
1
2
3
From
Load,kN
0 -1 -2 -33 2 1
To Load, kN
7/27/2019 Probabilistic Fatigue
145/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 288of 352
Sparse Data
-3
-2
-1
0
1
2
3
FromLoad,kN
0 -1 -2 -33 2 1
To Load, kN
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 289of 352
Exceedance Plot of 1 Lap
0
1
2
3
4
1 10 100 1000
Cumulative Cycles
LoadR
ange,kN
weibull distribution
7/27/2019 Probabilistic Fatigue
146/177
7/27/2019 Probabilistic Fatigue
147/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 292of 352
Results
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1
To Load, kN
FromL
oad,
kN
-3
-2
-1
0
1
2
3
FromL
oad,
kN
Simulation Test Data
0 -1 -2 -33 2 1
To Load, kN
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 293of 352
Exceedance Diagram
0
1
2
3
4
5
6
1 10 100 103 104
Cycles
1 Lap (Input Data)
Model Prediction
Actual 10 LapsLoadRa
nge,kN
7/27/2019 Probabilistic Fatigue
148/177
7/27/2019 Probabilistic Fatigue
149/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 296of 352
Issues
In the first problem the number of cycles isknown but the variability is unknown andmust be estimated
In the second problem the variability is knownbut the number and location of cycles isunknown and must be estimated
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 297of 352
Assumption
0
16
32
48
1 10 102 103 104 105
Cumulative Cycles
LoadR
ange
0
16
32
48
0
16
32
48
1 10 102 103 104 105
On average, more severe users tendto have more higher amplitude cyclesand fewer low amplitude cycles
7/27/2019 Probabilistic Fatigue
150/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 298of 352
Translation
-3
-2
-1
0
1
2
3
FromLoad,kN
0 -1 -2 -33 2 1
To Load, kN
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 299of 352
Damage Regions
-3
-2
-1
0
1
2
3
FromLoad,kN
0 -1 -2 -33 2 1
To Load, kN
7/27/2019 Probabilistic Fatigue
151/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 300of 352
ATV Data - Most Damaging in 19
-3
-2
-1
0
1
2
3
FromL
oad,
kN
-3
-2
-1
0
1
2
3
FromL
oad,
kN
Simulation Test Data
0 -1 -2 -33 2 1
To Load, kN0 -1 -2 -33 2 1
To Load, kN
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 301of 352
ATV Exceedance
0
1
23
4
5
6
1 10 100 103 104
Cycles
LoadR
ange,kN
Actual Data
Simulation
7/27/2019 Probabilistic Fatigue
152/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 302of 352
Airplane Data - Most Damaging in 334
-30
-20
-10
0
10
20
30
0 -10 -20 -3030 20 10
To Stress, ksi
FromS
tress,
ksi
-30
-20
-10
0
10
20
30
FromS
tress,
ksi
Simulation Test Data
0 -10 -20 -3030 20 10
To Stress, ksi
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 303of 352
Airplane Exceedance
0
10
20
30
40
1 10 100 1000Cycles
StressRange,ksi
Simulation
Actual Data
7/27/2019 Probabilistic Fatigue
153/177
7/27/2019 Probabilistic Fatigue
154/177
7/27/2019 Probabilistic Fatigue
155/177
7/27/2019 Probabilistic Fatigue
156/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 310of 352
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
lnX lnX ~ COVX
COVX is a good measure of variability
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 311of 352
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
7/27/2019 Probabilistic Fatigue
157/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 312of 352
Strain Life Analysis
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 313of 352
Material Properties
7/27/2019 Probabilistic Fatigue
158/177
7/27/2019 Probabilistic Fatigue
159/177
7/27/2019 Probabilistic Fatigue
160/177
7/27/2019 Probabilistic Fatigue
161/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 320of 352
29 Individual Data Sets
'
fMedian 820
300 1000
COV 0.25
2000
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
CumulativeProbability
b
-0.12 -0.08 -0.04
Mean -0.09
COV 0.2599.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
CumulativeProbability
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 321of 352
29 Individual Data Sets (continued)
0.1 1 10
Median 0.57
COV 1.1599.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
CumulativeProbability
'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 %
CumulativeProbability
7/27/2019 Probabilistic Fatigue
162/177
7/27/2019 Probabilistic Fatigue
163/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 324of 352
Results (continued)
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 325of 352
Results (continued)
7/27/2019 Probabilistic Fatigue
164/177
7/27/2019 Probabilistic Fatigue
165/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 328of 352
Strain Amplitude 1000 250
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 329of 352
Deterministic Analysis
7/27/2019 Probabilistic Fatigue
166/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 330of 352
Deterministic Analysis (continued)
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 331of 352
Deterministic Analysis (continued)
7/27/2019 Probabilistic Fatigue
167/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 332of 352
Deterministic Analysis Results
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 333of 352
Probabilistic Analysis
7/27/2019 Probabilistic Fatigue
168/177
7/27/2019 Probabilistic Fatigue
169/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 336of 352
Probabilistic Analysis Results
GlyphWorks
Professor Darrell F. Socie
Department of Mechanical and
Industrial Engineering
2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
7/27/2019 Probabilistic Fatigue
170/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 338of 352
Probabilistic Aspects of Fatigue
Introduction Basic Probability and Statistics
Statistical Techniques
Analysis Methods
Characterizing Variability
Case Studies
FatigueCalculator.com
GlyphWorks
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 339of 352
GlyphWorks / Rainflow
7/27/2019 Probabilistic Fatigue
171/177
7/27/2019 Probabilistic Fatigue
172/177
7/27/2019 Probabilistic Fatigue
173/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 344of 352
StatOutput.xls (continued)
Channel # 6 Total Channels 1
Variable Probabilistic Deterministic
NumCases 11 Load History Variables 0.89 -5.42
Median 45634 ScaleFactor 0.89 -5.42
COV 1.91 Offset 0.00 0.00
Stress Concentrators 0.05 -5.51
Probability (%) Life Kf 0.05 -5.51
99 817179 Material Properties 0.25 -14.54
90 223648 E 0.00 -4.37
50 45634 Sfp 0.23 3.42
10 9311 b 0.00 -4.39
1 2548 efp 0.00 1.42
c 0.00 -9.86
np 0.00 -2.32Kp 0.10 1.56
Analysis Variables 0.37 1.00
Uncertainty 0.37 1.00
Life Distribution Sensitivity
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 345of 352
StatOutput.xls (continued)
Variable Distribution Value
Scale
Parameter Value
Scale
Parameter
ScaleFactor Normal 100 0.25 103 0.18
Offset Normal 0 0.10 -0.01 0.08
Kf Uniform 3 0.05 3.0 0.04
E None 208000 0.00 208000 0.00
Sfp Log-Normal 1000 0.10 1054 0.11
b None -0.1 0.00 -0.10 0.00
efp None 1 0.20 1.0 0.00
c None -0.5 0.00 -0.50 0.00
np None 0.2 0.00 0.20 0.00
Kp Log-Normal 1200 0.10 1194 0.10
Uncertainty Log-Normal 1 0.50 0.93 0.48
Inputs Outputs
7/27/2019 Probabilistic Fatigue
174/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 346of 352
StatStress.flo
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 347of 352
Rainflow Extrapolation
7/27/2019 Probabilistic Fatigue
175/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 348of 352
Duration Extrapolation
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 349of 352
Results Files
7/27/2019 Probabilistic Fatigue
176/177
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 350of 352
Rainflow Duration
Probabilistic Fatigue 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 351of 352
Percentile Extrapolation
7/27/2019 Probabilistic Fatigue
177/177
Probabilistic Fatigue 2003 2005 Darrell Socie Universityof Illinois at Urbana Champaign All Rights Reserved 352of 352
Results