Probabilistic Aspects of Fatigue Case Studies

Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign © 2003-2012 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

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Case Studies  DARWIN  Southwest Research

 Bicycle  Loading Histories

6 Case Studies

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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

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Motivation

6 Case Studies

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UAL Flight 232 July 19,1989

6 Case Studies

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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

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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

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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

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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

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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

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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 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

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Random Inspection Time “Opportunity Inspections” during on-condition maintenance Inspection time modeled with Normal distribution or CDF table

6 Case Studies

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Output: Risk vs. Flight Cycles

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Output: Risk Contribution Factors Identify regions of component with highest risk

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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

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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 Crack Initiation All Load Spectrum Editing17 of 62 © 2003-2012 Darrell Socie,Transform University of Illinois at Urbana-Champaign, Rights Reserved

6 Case Studies

Three Sources of Variability  Anomaly size (initial crack size)  FCG properties (life scatter)  Mission histories (stress scatter)

6 Case Studies

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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

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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

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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 MPam, Kf=56.9 MPam 0.0 0

2000

4000

6000

8000

N, cycle

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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

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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

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2500

5000

7500

10000

12500

0

2500

5000

7500

T IM E (S EC ) © 2003-2012 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

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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

6 Case Studies

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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 © 2003-2012 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

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Loading History Typical Loading History

FORK BEND (LBS)

250

-250 0 6 Case Studies

Time (Secs)

1166.51

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Rainflow

counts

1000

0 0

150

300

6 Case Studies

150

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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

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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 )

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Combined Variability for Loads COV C 

 1 C n

i1

COV 

6 Case Studies



2 ai Xi

2

1

1 0.3  1 0.25  1  0.58 2 2

2 2

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Analysis

6 Case Studies

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Results 99 % 80 % 50 %

Risk

102 10 %

103

105

106

Hours

1%

0.1 %

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Results Monte Carlo Simulation Results Table mean COV r2 Si 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

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i 0.10 0.31 0.94

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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

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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

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Eliminate Mean Uncertainty 99 % 80 % 50 %

Risk

102 10 %

103

105

106

Hours

1%

0.1 %

6 Case Studies

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Airplane Exceedance 40 Stress Range,ksi

Simulation 30 20 Actual Data 10 0 1

6 Case Studies

10

Cycles

100

© 2003-2012 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

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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

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Probabilistic Aspects of Fatigue