Probabilistic Aspects of Fatigue -...

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

6 Case Studies © 2003-2012 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 62


Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing VariabilityCase Studies FatigueCalculator.comGlyphWorks


Case Studies

DARWINSouthwest Research

Bicycle Loading Histories


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


Motivation


UAL Flight 232July 19,1989


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


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


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


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)


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


Anomaly Distribution# of anomalies per volume of material as function of defect size

Library of default anomaly distributions for HA (developed by RISC)


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


Random Inspection Time“Opportunity Inspections” during on-condition maintenance

Inspection time modeled with Normal distribution or CDF table


Output: Risk vs. Flight Cycles


Output: Risk Contribution FactorsIdentify regions of component with highest risk


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


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


Three Sources of Variability

Anomaly size (initial crack size) FCG properties (life scatter) Mission histories (stress scatter)


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


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


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


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


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)


Suppression of Enemy Defenses

Cross Country


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


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


Web Site: www.darwin.swri.org


Bicycle

Assess risk in a new design


Variability / Uncertainty

Fatigue strength of fork Load history variability Load history uncertainty Analysis uncertainty


100104 105

500

Load-Life Data

PNf2

1361 19 ( ) .

Fatigue Life, Cycles

Load

Am

plitu

de, l

bs


Variability

bf )N(2P

'P

103

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

Mean 1355.67COV 0.06

104


Loading History

0 1166.51Time (Secs)-250

250

FOR

K B

EN

D (L

BS

)

Typical Loading History


Rainflow

0

300

150

150

0

1000

coun

ts


Exceedance Diagram

100

Number Ranges

500

Ran

ge (L

BS

)

Range Pair Plot

101 102 103 104 105

400

300

200

100

0


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 )


Combined Variability for Loads

n

1i

a2X 1C1CCOV

2i

i

58.0125.013.01COV 2222


Analysis


Results

99 %80 %

50 %

10 %

1 %

0.1 %

Ris

k

102 103 105 106

Hours


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


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


Course of Action

Make it strongerRun tests to reduce analysis uncertainty Field tests to reduce loading uncertainty


Eliminate Mean Uncertainty

99 %80 %

50 %

10 %

1 %

0.1 %

Ris

k

102 103 105 106

Hours


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


Problem Statement

Given a rainflow histogram for a single user, extrapolate to longer times

Given rainflow histograms for multiple users, extrapolate to more users


Probability Density

-3

-2

-1

0

1

2

3

0 -1 -2 -33 2 1To Load, kN

From

Loa

d, k

N


Kernel Smoothing

-3

-2

-1

0

1

2

3Fr

om L

oad,

kN

0 -1 -2 -33 2 1To Load, kN


Sparse Data

-3

-2

-1

0

1

2

3

From

Loa

d, k

N

0 -1 -2 -33 2 1To Load, kN


Exceedance Plot of 1 Lap

0

1

2

3

4


Load

Ran

ge,k

N

weibull distribution


10X Extrapolation

0

1

2

3

4


Load

Ran

ge,k

N6

5


-3

-2

-1

0

1

2

3

From

Loa

d, k

N

Probability Density

0 -1 -2 -33 2 1To Load, kN


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


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


Problem Statement

Given a rainflow histogram for a single user, extrapolate to longer times

Given rainflow histograms for multiple users, extropolate to more users


Extrapolated Data Sets

0.1 1 10 102 103 104

Normalized Life

Airplane

TractorATV

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %


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


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


Translation

-3

-2

-1

0

1

2

3Fr

om L

oad,

kN

0 -1 -2 -33 2 1To Load, kN


Damage Regions

-3

-2

-1

0

1

2

3

From

Loa

d, k

N

0 -1 -2 -33 2 1To Load, kN


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


0 -1 -2 -33 2 1To Load, kN

0 -1 -2 -33 2 1To Load, kN


ATV Exceedance

0

1

2

3

4

5

6

1 10 100 103 104

Cycles

Load

Ran

ge, k

N

Actual Data

Simulation


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


0 -10 -20 -3030 20 10To Stress, ksi


Airplane Exceedance

0

10

20

30

40

1 10 100 1000Cycles

Stre

ss R

ange

,ksi

Simulation

Actual Data


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


0 -0.5 -1.0 -2.01.5 1.0 0.5To Strain x 10-3


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

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