Probabilistic Aspects of Fatigue
Probabilistic Aspects of Fatigue Case Studies
Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign © 2003-2011 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
1 of 62
Case Studies DARWIN Southwest Research
Bicycle Loading Histories
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
2 of 62
A Software Framework for Probabilistic Fatigue Life Assessment ASTM Symposium on Probabilistic Aspects of Life Prediction Miami Beach, Florida November 6-7, 2002 R. C. McClung, M. P. Enright, H. R. Millwater*, G. R. Leverant, and S. J. Hudak, Jr. Southwest Research Slides 6 – 27 used with permission of of Craig McClung 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
3 of 62
Motivation
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
4 of 62
UAL Flight 232 July 19,1989
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
5 of 62
Turbine Disk Failure Anomalies in titanium engine disks Hard Alpha Very rare Can cause failure Not addressed by safe life methods Enhanced life management process Requested by FAA Developed by engine industry Probabilistic damage tolerance methods Supplement to safe life approach SwRI and engine industry developed DARWIN with FAA funding
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
6 of 62
Probabilistic Damage Tolerance Anomaly Distribution
Finite Element Stress Analysis
NDE Inspection Schedule
Probabilistic Fracture Mechanics
Probability of Detection
Pf vs. Cycles
Risk Contribution Factors Material Crack Growth Data
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
7 of 62
Zone-Based Risk Assessment Define zones based on similar stress, inspection, anomaly distribution, lifetime Total probability of fracture for zone: (probability of having a defect) x (POF given a defect) Defect probability determined by anomaly distribution, zone volume POF assuming a defect computed with Monte Carlo sampling or advanced methods POF for disk obtained by summing zone probabilities As individual zones become smaller (number of zones increases), risk converges down to “exact” answer 6 Case Studies
1 2
3
5 6
4
7
m
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
8 of 62
Fracture Mechanics Model of Zone Finite Element Model
x
Fracture Mechanics Model (Not to Scale)
5
Retrieve stresses along line
4
gradient direction
7
hx
3
2
1
Defect
hy
Y
m
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
9 of 62
Stress Processing
stress gradient
Hoop Stress (ksi)
FE Stresses and plate definition
Stress gradient extraction 2.0
80 70 60 50 40 30 20 10 0
1.6
FE Analysis (z)elastic
Computed relaxed stress elastic - residual
1.2
(z)relax
/
o
0.8 0.4
(z)residual
0.0
Shakedown module
3
4
5
6 7 0 Load Step
1
2
3
-0.4 -0.8 0.0
0.2
0.4
0.6
0.8
1.0
Normalized distance from the notch tip, x/r
Rainflow stress pairing
6 Case Studies
Residual stress analysis
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
10 of 62
Anomaly Distribution # of anomalies per volume of material as function of defect size Library of default anomaly distributions for HA (developed by RISC)
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
11 of 62
Probability of Detection Curves Define probability of NDE flaw detection as function of flaw size Can specify different PODs for different zones, schedules Built-in POD library or user-defined POD
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
12 of 62
Random Inspection Time “Opportunity Inspections” during on-condition maintenance Inspection time modeled with Normal distribution or CDF table
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
13 of 62
Output: Risk vs. Flight Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
14 of 62
Output: Risk Contribution Factors Identify regions of component with highest risk
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
15 of 62
Implementation in Industry FAA Advisory Circular 33.14 requests risk assessment be performed for all new titanium rotor designs Designs must pass design target risk for rotors Risk Reduction Required
Risk
10-9 Maximum Allowable Risk
A 6 Case Studies
B
Components
C
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
16 of 62
DARWIN for Prognosis Studies Anomaly Distribution
NDE Inspection
Material Crack Growth Data
Finite Element Stress Analysis
Probabilistic Fracture Mechanics
Pf vs. Flights
0.6
PROBABILITY OF FAILURE AT 20,000 CYCLES
12000
8000
4000
120 Pf Cycles Per Mission
0.5
100
0.4
80
0.3
60
0.2
40
0.1
20
NUMBER OF CYCLES PER MISSION
Code Enhancements
0 0
1000
2000
3000
4000
0.0
0 0
5
10
15
20
25
30
35
40
THRESHOLD STRESS RANGE (KSI)
Sensor (RPM) Input RPM-Stress Initiation All Rights LoadReserved Spectrum Editing17 of 62 © 2003-2011 Darrell Socie,Transform University of IllinoisCrack at Urbana-Champaign,
6 Case Studies
Three Sources of Variability Anomaly size (initial crack size) FCG properties (life scatter) Mission histories (stress scatter)
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Hard Alpha Defects in Titanium Initial DARWIN focus on Hard Alpha Small brittle zone in microstructure Alpha phase stabilized by N accidentally introduced during melting Cracks initiate quickly
Extensive industry effort to develop HA distribution
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Resulting Anomaly Distributions Post 1995 Triple Melt/Cold Hearth + Vacuum Arc Remelt MZ(5-10in Billet)/#1 FBH
1.00E+02
MZ(5-10in Billet)/#3 FBH
EXCEEDENCE (per million pounds)
MZ (12-13in Billet)/#1 FBH #2/#1 FBH
1.00E+01
#2/#2 FBH #3/#1 FBH
1.00E+00
#3/#2 FBH #3/#3 FBH
1.00E-01
1.00E-02 1.00E+02
1.00E+03
1.00E+04
DEFECT INSPECTION AREA (sq mils) 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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FCG Simulations for AGARD Data Use individual fits to generate set of a vs. N curves for identical conditions
Lognormal distribution appropriate in most cases
AGARD
0.4
0.3
a, cm
Characterize resulting scatter in total propagation life
0.5
0.2
0.1 Corner Crack Specimen Ki=18.7 MPam, Kf=56.9 MPam 0.0 0
2000
4000
6000
8000
N, cycle
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Engine Usage Variability Stress/Speed: (RPM)2
10000 Air Com bat Tactics
9000
Basic Fighter M aneuvers
8000
Intercept
7000 RPM (Low Speed Spool)
Total Cyclic Life (LCF): Nf = Ni + Np Ni 3-5 Np 3-4
Air to G round & G unnery
Peace Keeping Surface to Air Tactics ( hi alt)
6000 Surface to Air Tactics ( lo alt)
5000
Suppression of Enem y Defenses Cross Country
4000 3000 2000
Life/Speed: Nf (RPM)6
1000 0 0
2500
5000
7500
10000
12500
T IM E (SEC)
Component life is very sensitive to actual usage
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Usage Variability 10000 Air C om bat Tactics
9000
Components of Usage Variability:
Air to G round & G unnery Basic Fighter M aneuvers
8000
Intercept
• Mission type • Mission-to-mission variability
RPM (Low Speed Spool)
7000
Peace Keeping Surface to Air T actics ( hi alt)
6000 Surface to Air T actics ( lo alt)
5000
Suppression of Enem y D efenses Cross Country
4000 3000 2000
• Mission mixing variability
1000 0 0
2500
5000
7500
10000
12500
TIM E (SEC )
Peace K eeping
Surface to Air Tactics ( lo alt) 9000
8000
8000
7000
7000 RPM (Low Speed Spool)
RPM (Low Speed Spool)
9000
6000 5000 4000 3000
6000 5000 4000 3000
2000
2000
1000
1000
0
0 0
6 Case Studies
2500
5000
7500
10000
12500
0
2500
5000
7500
T IM E (S EC ) © 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
TIM E (SEC)
100 00
125 00
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Variability in Mission Type Air-Ground Weapons Delivery
RPM
RPM
Air-Air Weapons Delivery
Time
Time 1
Normalized POF
Initiation and Propagation No Inspection
0.1
Air Combat Tactics Combat Air Patrol
0.01
Air-Ground Weapons Delivery Air-Air Weapons Delivery Instrument/Ferry 0.001 0
1000
2000
3000
4000
5000
6000
7000
8000
Number of Flight Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Web Site: www.darwin.swri.org
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Bicycle Assess risk in a new design
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Variability / Uncertainty Fatigue strength of fork Load history variability Load history uncertainty Analysis uncertainty
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Load-Life Data
Load Amplitude, lbs
500
100 104
6 Case Studies
P 1361( N f ) .19 2
Fatigue Life, Cycles © 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
105
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Variability 99.9 % Mean 1355.67 COV 0.06
99 % 90 %
50 %
103
P P' 2 b (Nf ) 104
10 % 1% 0.1 % 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Loading History Typical Loading History
FORK BEND (LBS)
250
-250 0 6 Case Studies
Time (Secs)
1166.51
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Rainflow
counts
1000
0 0
150
300
6 Case Studies
150
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Exceedance Diagram 500
Range Pair Plot
Range (LBS)
400 300 200 100 0 100
101
102
103
104
105
Number Ranges
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Random Variables Strength LN( 1356 , 0.06 ) Loading History LN( 1 , 0.3 ) Estimated from other data
Loading History Uncertainty in Mean Could be “off” by a factor of 2 LN( 1.0 , 0.25 )
Analysis Estimated from other data LN( 1.0 , 1.0 )
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Combined Variability for Loads COV C
1 C n
i1
COV
6 Case Studies
2 ai Xi
2
1
1 0.3 1 0.25 1 0.58 2 2
2 2
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Analysis
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Results 99 % 80 % 50 %
Risk
102 10 %
103
105
106
Hours
1%
0.1 %
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Results Monte Carlo Simulation Results Table mean COV r2 Si Blocks 7.778e+04 6.132 intercept 1.360e+03 0.061 0.015 5.4 slope -1.900e-01 0.000 0.000 -13.9 damage 9.856e-01 0.980 0.016 0.99 scale 9.981e-01 0.588 0.928 -5.4
6 Case Studies
i 0.10 0.31 0.94
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Correlation Coefficient Load Scale Factor
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1 10
102
103
104
105
106
Operating Hours
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Course of Action Make it stronger Run tests to reduce analysis uncertainty Field tests to reduce loading uncertainty
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Eliminate Mean Uncertainty 99 % 80 % 50 %
Risk
102 10 %
103
105
106
Hours
1%
0.1 %
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
40 of 62
Service Loading Spectra 3 2
0
4
1
Load Range,kN
From Load, kN
Lateral Force, kN
5
0 -1 -2
-5
-3
Time history
6 Case Studies
3 2 1 0
-3
-2
-1 0 1 To Load, kN
Rainflow Histogram
2
3
1
10 100 Cumulative Cycles
1000
Exceedance Diagram
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extrapolate to more users
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Probability Density 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Kernel Smoothing 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Sparse Data 3
From Load, kN
2 1 0 -1 -2
-33
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Exceedance Plot of 1 Lap
Load Range,kN
4 3
weibull distribution
2 1 0 1
10
100
1000
Cumulative Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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10X Extrapolation 6
Load Range,kN
5 4 3 2 1 0 1
10
100
1000
Cumulative Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Probability Density 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Results Test Data
3
3
2
2
From Load, kN
From Load, kN
Simulation
1 0 -1
0 -1 -2
-2 -3
1
3
2
1
0
-1
To Load, kN 6 Case Studies
-2
-3
-3
3
2
1
0
-1
-2
-3
To Load, kN
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Exceedance Diagram 6
Load Range,kN
5
Model Prediction
4 3 2
Actual 10 Laps
1 1 Lap (Input Data)
0 1
10
100
103
104
Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extropolate to more users
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
51 of 62
Extrapolated Data Sets 99.9 % Airplane
99 %
ATV Tractor
90 % 50 % 0.1 10 %
1
10 102 103 Normalized Life
104
1% 0.1 % 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Issues In the first problem the number of cycles is known but the variability is unknown and must be estimated In the second problem the variability is known but the number and location of cycles is unknown and must be estimated
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Assumption On average, more severe users tend to have more higher amplitude cycles and fewer low amplitude cycles
Load Range
48
32
16
0 1
10
102
103
104
105
Cumulative Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Translation 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Damage Regions 3 2
From Load, kN
1 0 -1 -2 -3 3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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ATV Data - Most Damaging in 19 Test Data
3
3
2
2
From Load, kN
From Load, kN
Simulation
1 0 -1
0 -1 -2
-2 -3
1
3
2
1
0
-1
To Load, kN 6 Case Studies
-2
-3
-3 3
2
1
0
-1
-2
-3
To Load, kN
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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ATV Exceedance
Load Range, kN
6 5 4 3
Simulation
2 Actual Data 1 0
6 Case Studies
1
10
100 Cycles
103
104
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Airplane Data - Most Damaging in 334 Test Data
30
30
20
20
From Stress, ksi
From Stress, ksi
Simulation
10 0 -10 -20
-30
10 0 -10 -20
-30 30
20
10
0
-10
To Stress, ksi 6 Case Studies
-20
-30
30
20
10
0
-10
-20
-30
To Stress, ksi
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Airplane Exceedance 40 Stress Range,ksi
Simulation 30 20 Actual Data 10 0 1
6 Case Studies
10
Cycles
100
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
1000
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Tractor Data - Most Damaging in 54 Test Data
Simulation 1.5
From Strain x 10-3
From Starin x10-3
1.5 1.0 0.5 0
-1.0
0
-1.0
1.0
0.5
0
-0.5 -1.0 -2.0
To Strain x 10-3 6 Case Studies
0.5
-0.5
-0.5
-1.5 1.5
1.0
-1.5
1.5
1.0
0.5
0
-0.5 -1.0 -2.0
To Strain x 10-3
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Tractor Exceedance
Strain Range x 10-3
3.0 Simulation 2.0 Actual Data 1.0
0
6 Case Studies
1
10
100
103 Cycles
104
105
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Probabilistic Aspects of Fatigue
Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign © 2003-2011 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
1 of 62
Case Studies DARWIN Southwest Research
Bicycle Loading Histories
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
2 of 62
A Software Framework for Probabilistic Fatigue Life Assessment ASTM Symposium on Probabilistic Aspects of Life Prediction Miami Beach, Florida November 6-7, 2002 R. C. McClung, M. P. Enright, H. R. Millwater*, G. R. Leverant, and S. J. Hudak, Jr. Southwest Research Slides 6 – 27 used with permission of of Craig McClung 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
3 of 62
Motivation
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
4 of 62
UAL Flight 232 July 19,1989
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
5 of 62
Turbine Disk Failure Anomalies in titanium engine disks Hard Alpha Very rare Can cause failure Not addressed by safe life methods Enhanced life management process Requested by FAA Developed by engine industry Probabilistic damage tolerance methods Supplement to safe life approach SwRI and engine industry developed DARWIN with FAA funding
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
6 of 62
Probabilistic Damage Tolerance Anomaly Distribution
Finite Element Stress Analysis
NDE Inspection Schedule
Probabilistic Fracture Mechanics
Probability of Detection
Pf vs. Cycles
Risk Contribution Factors Material Crack Growth Data
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
7 of 62
Zone-Based Risk Assessment Define zones based on similar stress, inspection, anomaly distribution, lifetime Total probability of fracture for zone: (probability of having a defect) x (POF given a defect) Defect probability determined by anomaly distribution, zone volume POF assuming a defect computed with Monte Carlo sampling or advanced methods POF for disk obtained by summing zone probabilities As individual zones become smaller (number of zones increases), risk converges down to “exact” answer 6 Case Studies
1 2
3
5 6
4
7
m
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
8 of 62
Fracture Mechanics Model of Zone Finite Element Model
x
Fracture Mechanics Model (Not to Scale)
5
Retrieve stresses along line
4
gradient direction
7
hx
3
2
1
Defect
hy
Y
m
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
9 of 62
Stress Processing
stress gradient
Hoop Stress (ksi)
FE Stresses and plate definition
Stress gradient extraction 2.0
80 70 60 50 40 30 20 10 0
1.6
FE Analysis (z)elastic
Computed relaxed stress elastic - residual
1.2
(z)relax
/
o
0.8 0.4
(z)residual
0.0
Shakedown module
3
4
5
6 7 0 Load Step
1
2
3
-0.4 -0.8 0.0
0.2
0.4
0.6
0.8
1.0
Normalized distance from the notch tip, x/r
Rainflow stress pairing
6 Case Studies
Residual stress analysis
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
10 of 62
Anomaly Distribution # of anomalies per volume of material as function of defect size Library of default anomaly distributions for HA (developed by RISC)
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
11 of 62
Probability of Detection Curves Define probability of NDE flaw detection as function of flaw size Can specify different PODs for different zones, schedules Built-in POD library or user-defined POD
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
12 of 62
Random Inspection Time “Opportunity Inspections” during on-condition maintenance Inspection time modeled with Normal distribution or CDF table
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
13 of 62
Output: Risk vs. Flight Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
14 of 62
Output: Risk Contribution Factors Identify regions of component with highest risk
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
15 of 62
Implementation in Industry FAA Advisory Circular 33.14 requests risk assessment be performed for all new titanium rotor designs Designs must pass design target risk for rotors Risk Reduction Required
Risk
10-9 Maximum Allowable Risk
A 6 Case Studies
B
Components
C
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
16 of 62
DARWIN for Prognosis Studies Anomaly Distribution
NDE Inspection
Material Crack Growth Data
Finite Element Stress Analysis
Probabilistic Fracture Mechanics
Pf vs. Flights
0.6
PROBABILITY OF FAILURE AT 20,000 CYCLES
12000
8000
4000
120 Pf Cycles Per Mission
0.5
100
0.4
80
0.3
60
0.2
40
0.1
20
NUMBER OF CYCLES PER MISSION
Code Enhancements
0 0
1000
2000
3000
4000
0.0
0 0
5
10
15
20
25
30
35
40
THRESHOLD STRESS RANGE (KSI)
Sensor (RPM) Input RPM-Stress Initiation All Rights LoadReserved Spectrum Editing17 of 62 © 2003-2011 Darrell Socie,Transform University of IllinoisCrack at Urbana-Champaign,
6 Case Studies
Three Sources of Variability Anomaly size (initial crack size) FCG properties (life scatter) Mission histories (stress scatter)
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
18 of 62
Hard Alpha Defects in Titanium Initial DARWIN focus on Hard Alpha Small brittle zone in microstructure Alpha phase stabilized by N accidentally introduced during melting Cracks initiate quickly
Extensive industry effort to develop HA distribution
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
19 of 62
Resulting Anomaly Distributions Post 1995 Triple Melt/Cold Hearth + Vacuum Arc Remelt MZ(5-10in Billet)/#1 FBH
1.00E+02
MZ(5-10in Billet)/#3 FBH
EXCEEDENCE (per million pounds)
MZ (12-13in Billet)/#1 FBH #2/#1 FBH
1.00E+01
#2/#2 FBH #3/#1 FBH
1.00E+00
#3/#2 FBH #3/#3 FBH
1.00E-01
1.00E-02 1.00E+02
1.00E+03
1.00E+04
DEFECT INSPECTION AREA (sq mils) 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
20 of 62
FCG Simulations for AGARD Data Use individual fits to generate set of a vs. N curves for identical conditions
Lognormal distribution appropriate in most cases
AGARD
0.4
0.3
a, cm
Characterize resulting scatter in total propagation life
0.5
0.2
0.1 Corner Crack Specimen Ki=18.7 MPam, Kf=56.9 MPam 0.0 0
2000
4000
6000
8000
N, cycle
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
21 of 62
Engine Usage Variability Stress/Speed: (RPM)2
10000 Air Com bat Tactics
9000
Basic Fighter M aneuvers
8000
Intercept
7000 RPM (Low Speed Spool)
Total Cyclic Life (LCF): Nf = Ni + Np Ni 3-5 Np 3-4
Air to G round & G unnery
Peace Keeping Surface to Air Tactics ( hi alt)
6000 Surface to Air Tactics ( lo alt)
5000
Suppression of Enem y Defenses Cross Country
4000 3000 2000
Life/Speed: Nf (RPM)6
1000 0 0
2500
5000
7500
10000
12500
T IM E (SEC)
Component life is very sensitive to actual usage
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
22 of 62
Usage Variability 10000 Air C om bat Tactics
9000
Components of Usage Variability:
Air to G round & G unnery Basic Fighter M aneuvers
8000
Intercept
• Mission type • Mission-to-mission variability
RPM (Low Speed Spool)
7000
Peace Keeping Surface to Air T actics ( hi alt)
6000 Surface to Air T actics ( lo alt)
5000
Suppression of Enem y D efenses Cross Country
4000 3000 2000
• Mission mixing variability
1000 0 0
2500
5000
7500
10000
12500
TIM E (SEC )
Peace K eeping
Surface to Air Tactics ( lo alt) 9000
8000
8000
7000
7000 RPM (Low Speed Spool)
RPM (Low Speed Spool)
9000
6000 5000 4000 3000
6000 5000 4000 3000
2000
2000
1000
1000
0
0 0
6 Case Studies
2500
5000
7500
10000
12500
0
2500
5000
7500
T IM E (S EC ) © 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
TIM E (SEC)
100 00
125 00
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Variability in Mission Type Air-Ground Weapons Delivery
RPM
RPM
Air-Air Weapons Delivery
Time
Time 1
Normalized POF
Initiation and Propagation No Inspection
0.1
Air Combat Tactics Combat Air Patrol
0.01
Air-Ground Weapons Delivery Air-Air Weapons Delivery Instrument/Ferry 0.001 0
1000
2000
3000
4000
5000
6000
7000
8000
Number of Flight Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
24 of 62
Web Site: www.darwin.swri.org
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
25 of 62
Bicycle Assess risk in a new design
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
26 of 62
Variability / Uncertainty Fatigue strength of fork Load history variability Load history uncertainty Analysis uncertainty
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
27 of 62
Load-Life Data
Load Amplitude, lbs
500
100 104
6 Case Studies
P 1361( N f ) .19 2
Fatigue Life, Cycles © 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
105
28 of 62
Variability 99.9 % Mean 1355.67 COV 0.06
99 % 90 %
50 %
103
P P' 2 b (Nf ) 104
10 % 1% 0.1 % 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
29 of 62
Loading History Typical Loading History
FORK BEND (LBS)
250
-250 0 6 Case Studies
Time (Secs)
1166.51
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
30 of 62
Rainflow
counts
1000
0 0
150
300
6 Case Studies
150
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
31 of 62
Exceedance Diagram 500
Range Pair Plot
Range (LBS)
400 300 200 100 0 100
101
102
103
104
105
Number Ranges
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
32 of 62
Random Variables Strength LN( 1356 , 0.06 ) Loading History LN( 1 , 0.3 ) Estimated from other data
Loading History Uncertainty in Mean Could be “off” by a factor of 2 LN( 1.0 , 0.25 )
Analysis Estimated from other data LN( 1.0 , 1.0 )
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
33 of 62
Combined Variability for Loads COV C
1 C n
i1
COV
6 Case Studies
2 ai Xi
2
1
1 0.3 1 0.25 1 0.58 2 2
2 2
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Analysis
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Results 99 % 80 % 50 %
Risk
102 10 %
103
105
106
Hours
1%
0.1 %
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
36 of 62
Results Monte Carlo Simulation Results Table mean COV r2 Si Blocks 7.778e+04 6.132 intercept 1.360e+03 0.061 0.015 5.4 slope -1.900e-01 0.000 0.000 -13.9 damage 9.856e-01 0.980 0.016 0.99 scale 9.981e-01 0.588 0.928 -5.4
6 Case Studies
i 0.10 0.31 0.94
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
37 of 62
Correlation Coefficient Load Scale Factor
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1 10
102
103
104
105
106
Operating Hours
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
38 of 62
Course of Action Make it stronger Run tests to reduce analysis uncertainty Field tests to reduce loading uncertainty
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
39 of 62
Eliminate Mean Uncertainty 99 % 80 % 50 %
Risk
102 10 %
103
105
106
Hours
1%
0.1 %
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
40 of 62
Service Loading Spectra 3 2
0
4
1
Load Range,kN
From Load, kN
Lateral Force, kN
5
0 -1 -2
-5
-3
Time history
6 Case Studies
3 2 1 0
-3
-2
-1 0 1 To Load, kN
Rainflow Histogram
2
3
1
10 100 Cumulative Cycles
1000
Exceedance Diagram
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
41 of 62
Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extrapolate to more users
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Probability Density 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
43 of 62
Kernel Smoothing 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
44 of 62
Sparse Data 3
From Load, kN
2 1 0 -1 -2
-33
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Exceedance Plot of 1 Lap
Load Range,kN
4 3
weibull distribution
2 1 0 1
10
100
1000
Cumulative Cycles
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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10X Extrapolation 6
Load Range,kN
5 4 3 2 1 0 1
10
100
1000
Cumulative Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Probability Density 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Results Test Data
3
3
2
2
From Load, kN
From Load, kN
Simulation
1 0 -1
0 -1 -2
-2 -3
1
3
2
1
0
-1
To Load, kN 6 Case Studies
-2
-3
-3
3
2
1
0
-1
-2
-3
To Load, kN
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
49 of 62
Exceedance Diagram 6
Load Range,kN
5
Model Prediction
4 3 2
Actual 10 Laps
1 1 Lap (Input Data)
0 1
10
100
103
104
Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
50 of 62
Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extropolate to more users
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
51 of 62
Extrapolated Data Sets 99.9 % Airplane
99 %
ATV Tractor
90 % 50 % 0.1 10 %
1
10 102 103 Normalized Life
104
1% 0.1 % 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
52 of 62
Issues In the first problem the number of cycles is known but the variability is unknown and must be estimated In the second problem the variability is known but the number and location of cycles is unknown and must be estimated
6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Assumption On average, more severe users tend to have more higher amplitude cycles and fewer low amplitude cycles
Load Range
48
32
16
0 1
10
102
103
104
105
Cumulative Cycles 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
54 of 62
Translation 3
From Load, kN
2 1 0 -1 -2 -3
3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
55 of 62
Damage Regions 3 2
From Load, kN
1 0 -1 -2 -3 3
2
1
0
-1
-2
-3
To Load, kN 6 Case Studies
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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ATV Data - Most Damaging in 19 Test Data
3
3
2
2
From Load, kN
From Load, kN
Simulation
1 0 -1
0 -1 -2
-2 -3
1
3
2
1
0
-1
To Load, kN 6 Case Studies
-2
-3
-3 3
2
1
0
-1
-2
-3
To Load, kN
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
57 of 62
ATV Exceedance
Load Range, kN
6 5 4 3
Simulation
2 Actual Data 1 0
6 Case Studies
1
10
100 Cycles
103
104
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
58 of 62
Airplane Data - Most Damaging in 334 Test Data
30
30
20
20
From Stress, ksi
From Stress, ksi
Simulation
10 0 -10 -20
-30
10 0 -10 -20
-30 30
20
10
0
-10
To Stress, ksi 6 Case Studies
-20
-30
30
20
10
0
-10
-20
-30
To Stress, ksi
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
59 of 62
Airplane Exceedance 40 Stress Range,ksi
Simulation 30 20 Actual Data 10 0 1
6 Case Studies
10
Cycles
100
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
1000
60 of 62
Tractor Data - Most Damaging in 54 Test Data
Simulation 1.5
From Strain x 10-3
From Starin x10-3
1.5 1.0 0.5 0
-1.0
0
-1.0
1.0
0.5
0
-0.5 -1.0 -2.0
To Strain x 10-3 6 Case Studies
0.5
-0.5
-0.5
-1.5 1.5
1.0
-1.5
1.5
1.0
0.5
0
-0.5 -1.0 -2.0
To Strain x 10-3
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
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Tractor Exceedance
Strain Range x 10-3
3.0 Simulation 2.0 Actual Data 1.0
0
6 Case Studies
1
10
100
103 Cycles
104
105
© 2003-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved
62 of 62
Probabilistic Aspects of Fatigue
