probabilistic aspects of fatigue -...
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Case Studies
Professor Darrell F. SocieDepartment of Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
© 2003-2012 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
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Probabilistic Aspects of Fatigue
Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing VariabilityCase Studies FatigueCalculator.comGlyphWorks
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Case Studies
DARWINSouthwest Research
Bicycle Loading Histories
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A Software Framework for Probabilistic Fatigue Life Assessment
ASTM Symposium on Probabilistic Aspects of Life Prediction
Miami Beach, FloridaNovember 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
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Motivation
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UAL Flight 232July 19,1989
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Turbine Disk FailureAnomalies in titanium engine disks
Hard AlphaVery rareCan cause failureNot addressed by safe life methods
Enhanced life management processRequested by FAADeveloped by engine industryProbabilistic damage tolerance methodsSupplement to safe life approach
SwRI and engine industry developed DARWIN with FAA funding
6 Case Studies © 2003-2012 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 62
Probabilistic Damage Tolerance
Probabilistic Fracture Mechanics
Probability of DetectionAnomaly Distribution
Finite Element Stress Analysis
Material Crack Growth Data
NDE Inspection Schedule
Pf vs. Cycles
Risk Contribution Factors
6 Case Studies © 2003-2012 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 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 volumePOF 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
1
2 3 4
m
5 6 7
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Fracture Mechanics Model of Zone
m
7
Retrieve stresses along line
Fracture Mechanics Model
hx
hy
x
Y
grad
ient
dire
ctio
n
1
2
3
4
5
Defect
Finite Element Model
(Not to Scale)
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Stress Processing
FE Stresses and plate definition
stress gradient
Stress gradient extraction
FE Analysis
0.0 0.2 0.4 0.6 0.8 1.0Normalized distance from the notch tip, x/r
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
/o
(z)relax
(z)residual
(z)elastic
Shakedown module
Computed relaxed stresselastic - residual
Residual stress analysis
3 4 5 6 7 0 1 2 3Load Step
01020304050607080
Hoo
p S
tress
(ksi
)
Rainflow stress pairing
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Anomaly Distribution# of anomalies per volume of material as function of defect size
Library of default anomaly distributions for HA (developed by RISC)
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Probability of Detection CurvesDefine probability of NDE flaw detection as function of flaw sizeCan specify different PODs for different zones, schedulesBuilt-in POD library or user-defined POD
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Random Inspection Time“Opportunity Inspections” during on-condition maintenance
Inspection time modeled with Normal distribution or CDF table
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Output: Risk vs. Flight Cycles
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Output: Risk Contribution FactorsIdentify regions of component with highest risk
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Implementation in IndustryFAA Advisory Circular 33.14 requests risk
assessment be performed for all new titanium rotor designs
Designs must pass design target risk for rotors
Components
Risk
MaximumAllowable
Risk
10-9
RiskReductionRequired
CA B
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Probabilistic Fracture Mechanics
Material Crack Growth Data
Finite Element Stress Analysis
Anomaly Distribution NDE Inspection
Pf vs. Flights
RPM-Stress Transform Crack Initiation
0
4000
8000
12000
0 1000 2000 3000 4000
Load Spectrum Editing
Code Enhancements
Sensor (RPM) Input
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40
THRESHOLD STRESS RANGE (KSI)PR
OB
AB
ILIT
Y O
F FA
ILU
RE
AT
20,0
00 C
YCLE
S
0
20
40
60
80
100
120
NU
MB
ER O
F C
YCLE
S PE
R M
ISSI
ON
PfCycles Per Mission
DARWIN for Prognosis Studies
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Three Sources of Variability
Anomaly size (initial crack size) FCG properties (life scatter) Mission histories (stress scatter)
6 Case Studies © 2003-2012 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 62
Hard Alpha Defects in Titanium
Initial DARWIN focus on Hard AlphaSmall brittle zone in
microstructureAlpha phase stabilized by N
accidentally introduced during melting
Cracks initiate quickly
Extensive industry effort to develop HA distribution
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Resulting Anomaly DistributionsPost 1995 Triple Melt/Cold Hearth + Vacuum Arc Remelt
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+02 1.00E+03 1.00E+04
DEFECT INSPECTION AREA (sq mils)
EXC
EED
ENC
E(p
er m
illio
n po
unds
)
MZ(5-10in Billet)/#1 FBH
MZ(5-10in Billet)/#3 FBH
MZ (12-13in Billet)/#1 FBH
#2/#1 FBH
#2/#2 FBH
#3/#1 FBH
#3/#2 FBH
#3/#3 FBH
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FCG Simulations for AGARD Data
Use individual fits to generate set of a vs. N curves for identical conditions
Characterize resulting scatter in total propagation life
Lognormal distribution appropriate in most cases
N, cycle0 2000 4000 6000 8000
a, c
m
0.0
0.1
0.2
0.3
0.4
0.5
Corner Crack SpecimenKi=18.7 MPam, Kf=56.9 MPam
AGARD
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Engine Usage Variability
Stress/Speed: (RPM)2
Total Cyclic Life (LCF): Nf = Ni + Np
Ni 3-5
Np 3-4
Life/Speed: Nf (RPM)6
Component life is very sensitive to actual usage
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 2500 5000 7500 10000 12500TIM E (SEC)
RPM
(Low
Spe
ed S
pool
)
Air Combat Tactics
Air to Ground & Gunnery
Basic Fighter Maneuvers
Intercept
Peace Keeping
Surface to Air Tactics ( hi alt)
Surface to Air Tactics ( lo alt)
Suppression of Enemy Defenses
Cross Country
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Usage Variability
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 2500 5000 7500 10000 12500TIME (SEC)
RPM
(Low
Spe
ed S
pool
)
Air Combat Tactics
Air to Ground & Gunnery
Basic Fighter Maneuvers
Intercept
Peace Keeping
Surface to Air Tactics ( hi alt)
Surface to Air Tactics ( lo alt)
Suppression of Enemy Defenses
Cross Country
Surface to Air Tactics ( lo alt)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 2500 5000 7500 10000 12500TIME (SEC)
RPM
(Low
Spe
ed S
pool
)
Peace Keeping
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 2500 5000 7500 10000 12500TIM E (S EC)
RPM
(Low
Spe
ed S
pool
)
Components of Usage Variability:
• Mission type
• Mission-to-mission variability
• Mission mixing variability
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0.001
0.01
0.1
1
0 1000 2000 3000 4000 5000 6000 7000 8000
Number of Flight Cycles
Nor
mal
ized
PO
F
Air Combat Tactics
Combat Air Patrol
Air-Ground Weapons Delivery
Air-Air Weapons Delivery
Instrument/Ferry
Initiation and PropagationNo Inspection
Variability in Mission Type
Air-Air Weapons Delivery
Time
RP
M
Air-Ground Weapons Delivery
Time
RP
M
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Web Site: www.darwin.swri.org
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Bicycle
Assess risk in a new design
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Variability / Uncertainty
Fatigue strength of fork Load history variability Load history uncertainty Analysis uncertainty
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100104 105
500
Load-Life Data
PNf2
1361 19 ( ) .
Fatigue Life, Cycles
Load
Am
plitu
de, l
bs
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Variability
bf )N(2P
'P
103
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Mean 1355.67COV 0.06
104
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Loading History
0 1166.51Time (Secs)-250
250
FOR
K B
EN
D (L
BS
)
Typical Loading History
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Rainflow
0
300
150
150
0
1000
coun
ts
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Exceedance Diagram
100
Number Ranges
500
Ran
ge (L
BS
)
Range Pair Plot
101 102 103 104 105
400
300
200
100
0
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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 MeanCould be “off” by a factor of 2 LN( 1.0 , 0.25 )
Analysis Estimated from other data LN( 1.0 , 1.0 )
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Combined Variability for Loads
n
1i
a2X 1C1CCOV
2i
i
58.0125.013.01COV 2222
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Analysis
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Results
99 %80 %
50 %
10 %
1 %
0.1 %
Ris
k
102 103 105 106
Hours
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ResultsMonte Carlo Simulation Results Table
mean COV r2 Si iBlocks 7.778e+04 6.132intercept 1.360e+03 0.061 0.015 5.4 0.10slope -1.900e-01 0.000 0.000 -13.9damage 9.856e-01 0.980 0.016 0.99 0.31scale 9.981e-01 0.588 0.928 -5.4 0.94
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Correlation Coefficient
101 102 103 104 105 106
Operating Hours
0.0
0.5
1.0
1.5
2.0
2.5
Load
Sca
le F
acto
r
3.5
3.0
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Course of Action
Make it strongerRun tests to reduce analysis uncertainty Field tests to reduce loading uncertainty
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Eliminate Mean Uncertainty
99 %80 %
50 %
10 %
1 %
0.1 %
Ris
k
102 103 105 106
Hours
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Service Loading Spectra
-5
5
0
Late
ral F
orce
, kN
Time history
0 1 2 3-3 -2 -1-3
-2
-1
0
1
2
3
To Load, kN
From
Loa
d, k
N
0
1
2
3
4
1 10 100 1000Cumulative Cycles
Load
Ran
ge,k
N
RainflowHistogram
ExceedanceDiagram
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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
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Probability Density
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1To Load, kN
From
Loa
d, k
N
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Kernel Smoothing
-3
-2
-1
0
1
2
3Fr
om L
oad,
kN
0 -1 -2 -33 2 1To Load, kN
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Sparse Data
-3
-2
-1
0
1
2
3
From
Loa
d, k
N
0 -1 -2 -33 2 1To Load, kN
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Exceedance Plot of 1 Lap
0
1
2
3
4
1 10 100 1000Cumulative Cycles
Load
Ran
ge,k
N
weibull distribution
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10X Extrapolation
0
1
2
3
4
1 10 100 1000Cumulative Cycles
Load
Ran
ge,k
N6
5
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-3
-2
-1
0
1
2
3
From
Loa
d, k
N
Probability Density
0 -1 -2 -33 2 1To Load, kN
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Results
-3
-2
-1
0
1
2
3
0 -1 -2 -33 2 1To Load, kN
From
Loa
d, k
N
-3
-2
-1
0
1
2
3
From
Loa
d, k
N
Simulation Test Data
0 -1 -2 -33 2 1To Load, kN
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Exceedance Diagram
0
1
2
3
4
5
6
1 10 100 103 104
Cycles
1 Lap (Input Data)
Model Prediction
Actual 10 LapsLoad
Ran
ge,k
N
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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
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Extrapolated Data Sets
0.1 1 10 102 103 104
Normalized Life
Airplane
TractorATV
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
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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
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Assumption
0
16
32
48
1 10 102 103 104 105
Cumulative Cycles
Load
Ran
ge
0
16
32
48
0
16
32
48
1 10 102 103 104 105
On average, more severe users tend to have more higher amplitude cycles and fewer low amplitude cycles
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Translation
-3
-2
-1
0
1
2
3Fr
om L
oad,
kN
0 -1 -2 -33 2 1To Load, kN
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Damage Regions
-3
-2
-1
0
1
2
3
From
Loa
d, k
N
0 -1 -2 -33 2 1To Load, kN
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ATV Data - Most Damaging in 19
-3
-2
-1
0
1
2
3
From
Loa
d, k
N
-3
-2
-1
0
1
2
3
From
Loa
d, k
N
Simulation Test Data
0 -1 -2 -33 2 1To Load, kN
0 -1 -2 -33 2 1To Load, kN
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ATV Exceedance
0
1
2
3
4
5
6
1 10 100 103 104
Cycles
Load
Ran
ge, k
N
Actual Data
Simulation
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Airplane Data - Most Damaging in 334
-30
-20
-10
0
10
20
30
0 -10 -20 -3030 20 10To Stress, ksi
From
Stre
ss, k
si
-30
-20
-10
0
10
20
30
From
Stre
ss, k
si
Simulation Test Data
0 -10 -20 -3030 20 10To Stress, ksi
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Airplane Exceedance
0
10
20
30
40
1 10 100 1000Cycles
Stre
ss R
ange
,ksi
Simulation
Actual Data
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Tractor Data - Most Damaging in 54
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
0 -0.5 -1.0 -2.01.5 1.0 0.5To Strain x 10-3
From
Sta
rin x
10-3
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
From
Stra
in x
10-
3
Simulation Test Data
0 -0.5 -1.0 -2.01.5 1.0 0.5To Strain x 10-3
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Tractor Exceedance
0
1.0
2.0
3.0
1 10 100 103 104 105
Cycles
Stra
in R
ange
x 1
0-3 Simulation
Actual Data