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Introduction

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

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Contact Information

Darrell Socie

Mechanical Science and Engineering

1206 West Green

Urbana, Illinois 61801

Office: 3015 Mechanical Engineering Laboratory

dsocie@uiuc.edu 

 Tel: 217 333 7630

Fax: 217 333 5634

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Fatigue Calculations

Who really believesthese numbers ?

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SAE Specimen

Suspension

 Transmission

Bracket

Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 1977

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Analysis Results

1 10 100

48 Data Points

COV 1.27

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

   C  u  m  u   l  a   t   i  v  e   P  r  o   b  a

   b   i   l   i   t  y

Analytical Life / Experimental Life

Strain-Life analysis of all test data

Non conservative

Conservative

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Material Variability

1 10 100

99.9 %

99 %

90 %

50 %

10 %

1 %

0.1 %

   C  u  m  u   l  a   t   i  v  e   P  r  o   b  a

   b   i   l   i   t  y

Analytical Life / Experimental Life

Material Analysis

Strain-Life back calculation of specimen lives

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Probabilistic Models

Probabilistic models are no better than theunderlying deterministic models

 They require more work to implement

Why use them?

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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 ?

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Risk

10-9

10-8

10-7

10-6

10-5

10-4

10-3

   R   i  s   k

 Time, Flights etc

Acceptable risk

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Reliability

99 %

80 %

50 %

10 %

1 %

0.1 %

   E  x  p  e  c   t  e   d   F  a

   i   l  u  r  e  s

106

Fatigue Life

105104

103

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Risk Contribution Factors

Operating Temperature

Analysis Uncertainty

Speed

MaterialProperties

Manufacturing

Flaws

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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

10-4

   S   t  r  a   i  n   A  m  p   l   i   t  u   d  e time

50%

100 %

   F  a   i   l  u  r  e

  s

Strength

Stress

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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 of a car can only be described in terms of probability andstatistical averages

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Deterministic Design

Stress Strength

SafetyFactor

Variability and uncertainty is accommodated by introducingsafety factors. Larger safety factors are better, but how much

better and at what cost?

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Probabilistic Design

Stress Strength

Reliability = 1 – P( Stress > Strength )

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3σ Approach

3σ contains 99.87% of the data

If we use 3σ on both stress and strength

 The probability of the part with the lowest strength

having the highest stress is very small

P( s < S ) = 2.3 10-3

σ≈=≤≥Σ=− 5.4103.5)Sss(P)failure(P 6

For 3 variables, each at 3 σ:

σ≈=− 7.5102.1)failure(P 8

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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

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Probabilistic Aspects of Fatigue

Introduction

Basic Probability and Statistics

Statistical Techniques

Analysis Methods

Characterizing Variability

Case Studies

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