Probabilistic Aspects of Fatigue

Transcription

1 Probabilistic Aspects of Fatigue Introduction Professor Darrell F. Socie Department of Mechanical and Industrial Engineering Darrell Socie, All Rights Reserved Contact Information Darrell Socie Mechanical Engineering 06 West Green Urbana, Illinois 680 Office: 305 Mechanical Engineering Laboratory Tel: Fax: Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

2 Fatigue Calculations Who really believes these numbers? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 SAE Specimen Bracket Suspension Transmission Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 977 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

3 Analysis Results 99.9 % 99 % Strain-Life analysis of all test data 48 Data Points COV.7 Cumulative Probability 90 % 50 % 0 % % 0. % Non conservative 0 00 Analytical Life / Experimental Life Conservative Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Material Variability 99.9 % 99 % Material Analysis Cumulative Probability 90 % 50 % 0 % % 0. % 0 00 Analytical Life / Experimental Life Strain-Life back calculation of specimen lives Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

4 Probabilistic Models Probabilistic models are no better than the underlying deterministic models They require more work to implement Why use them? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 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? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

5 Risk Risk Acceptable risk Time, Flights etc Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Reliability 99 % Expected Failures 80 % 50 % 0 % % Fatigue Life % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

6 Risk Contribution Factors Analysis Uncertainty Speed Manufacturing Flaws Material Properties Operating Temperature Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Uncertainty and Variability customers Stress usage 00 % Failures 50% Strain Amplitude time Fatigue Life, N f materials Strength manufacturing Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

7 Deterministic versus Random Deterministic from past measurements the future position of 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 and statistical averages Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Deterministic Design Safety Factor Stress Strength Variability and uncertainty is accommodated by introducing safety factors. Larger safety factors are better, but how much better and at what cost? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

8 Probabilistic Design Stress Strength Reliability = P( Stress > Strength ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 3σ Approach 3σ contains 99.87% of the data P( s < S ) = If we use 3σ on both stress and strength P(failure) = P( Σ s I s S ) = σ The probability of the part with the lowest strength having the highest stress is very small For 3 variables, each at 3 σ: P(failure) = σ Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

9 Benefits Reduces conservatism (cost) compared to assuming the worst case for every design variable Quantifies life drivers what are the most important variables and how well are they known or controlled? Quantifies risk Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

10 Probabilistic Aspects of Fatigue Basic Probability and Statistics Professor Darrell F. Socie Department of Mechanical and Industrial Engineering 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

11 Deterministic versus Random Deterministic from past measurements the future position of 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 and statistical averages Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Random Variables Discrete - fixed number of outcomes Colors Continuous - may have any value in the sample space Strength Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

12 Descriptive Statistics Mean or Expected Value Variance / Standard Deviation Coefficient of Variation Skewness Kurtosis Correlation Coefficient Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Mean or Expected Value Central tendency of the data Mean = µ x = x = E ( X) = N i = N x i Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

13 Variance / Standard Deviation Dispersion of the data Var ( X ) = N i = ( x x ) i N Standard deviation σ x = Var( X ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Coefficient of Variation σ COV = x µ x Useful to compare different dispersions µ = 0 σ = COV = 0. µ = 00 σ = 0 COV = 0. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

14 Skewness Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data are spread out more to the left of the mean than to the right. If skewness is positive, the data are spread out more to the right. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero. Skewness N ( xi i = ( X ) = 3 Nσ x ) 3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Kurtosis Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3. Kurtosis N ( xi i = ( X ) = 4 Nσ x ) 4 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

15 Covariance A measure of the linear association between random variables [ X µ )( Y )] σxy = COV(X,Y) = E ( X µ Y Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Correlation Coefficient Y Y Y 0 X 0 X 0 X ρ = ρ = - ρ = 0 Y Y Y 0 X 0 X 0 X 0 < ρ < ρ ~ 0 ρ ~0 ρ = [ X µ )( Y µ ] E ( σ X Y ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35 X σ Y

16 Probability Basic probability Conditional probability Reliability Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 Basic Probability The probability of event A occurring: 0 P( A) P ( A) = certain ( A) 0 imposssible P = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

17 Conditional Basic Probability ( A B) = P( A) + P( B) P( A B) P U I A or B A and B union intersection P(B) given A has occurred P ( A B) P P = ( B A) = ( A IB ) P( B) P ( A IB ) P( A) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35 Independent Events If A and B are unrelated P P ( B A) = P( B) P ( A B) = P( A) ( A IB) = P( A) P( B) Suppose the probability of bar failing is 0.03 and the probability of bar failing is What is the probability of the structure failing? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

18 Failure Probability P Bar or bar fails ( A UB) = P( A) + P( B) P( A IB) P ( A) = P ( B) = P ( A IB) = P( A) P( B) = P( failure) = = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 Reliability Define P( no failure ) = Reliability P( A ) probability of not A P( A ) = P( A ) = P( A I B ) ( A I B ) = P( A) P( B) Reliability P For the bar structure ( A I B ) = P = ( failure ) = Reliability P = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35

19 Structural Reliability Collapse occurs if any member fails P P( A) = P( A I A I A3 I A4 I An ) ( A) = P( A ) P( A ) P( A ) P( A ) P( ) 3 4 An Suppose P A = ( ) 0.0 for each link ( ) = ( 0.99) 7 = P A Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 35 Reliability 6 Standard Deviations Probability Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 35

20 6 Sigma 6 sigma is in a billion ( Reliability ) Suppose a structure has 000 bolted joints: ( ) = ( ) 000 = P A in a million 3 sigma is ( Reliability ) ( ) = ( ) 000 = 0. 6 P A 74 % failures Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 35 Statistical Distributions Uniform Normal LogNormal Gumble Weibull A useful on-line reference: Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 35

21 Cumulative Distribution Function F x (X) 0 a F x b ( X) = P( X x ) F x ( ) = 0 F x ( ) = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 35 Probability Density Function f x (X) b a 0 P ( a< X b ) = fx ( X)dx F x a ( X ) f ( ξ) dξ x = b a x b Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35

22 Probability f x (X) b a ε 0 a x b P x+ε ( x ε < X x + ε ) = fx ( X)dx x ε Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Survival Function F x (X) 0 a S (X) = F x x b ( X) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 35

23 Hazard Function h x (X) a b fx(x) hx(x) = F x ( X) Instantaneous failure rate Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 35 Mean or Expected Value f x (X) 0 a b ( X) = µ = E x f ( X ) dx x x Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 35

24 Variance Probability Density Var ( X ) = ( X µ ) f ( X ) dx ( X ) = E( X ) Var µ σ x = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 35 x x Var( X ) x x Normal Distribution Probability Density Cumulative Probability x Quantile x f (x) ( x µ ) exp σ π σ = Quantile x tables Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 35

25 Normal Probability Plot % 99 % P = m N+ Standard Deviations Cumulative Probability 90 % 50 % 0 % % Linear Scale % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 35 Normal Plot Cumulative Probability Maximum force from 4 drivers 99.9 % Mean % COV % 50 % % % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 35

26 Probability Density LogNormal Distribution σ = 0.6 σ =.0 Cumulative Probability.0 σ = σ =.0 m= exp ( µ ) f (x) x 3 Quantile x ( ln exp xσ π σ x m = ) 3 Quantile x tables Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 35 Useful Relationships µ µ σ ln x ln x COV = ln x µ x + COV ( µ + 0. σ ) x = exp ln x 5 x ln x ( COV ) =ln + = exp( σ x ln x ) X = exp( µ x ln x ) = µ x + COV x Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

27 .0 Median / Mean ratio 0.8 X µ X COV X Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35.5 σ lnx -COV X.0 σ lnx COV X Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 35

28 LogNormal Probability Plot Standard Deviations Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Log Scale Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 35 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % LogNormal Plot Median 43 COV 0.34 Maximum force from 4 drivers Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 35

29 Gumbel Extreme Value Probability Density 0.5 Cumulative Probability a-3b a a+4b a-3b a a+4b Quantile x Quantile x ( x a) ( x a) f x (x) = exp exp exp b b b ( x a) F x (x) = exp exp b Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 35 Gumble Probability Plot % 6 -ln( -ln( F x (X) ) ) Cumulative Probability 99 % 80 % 50 % - 0 % Linear Scale - % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 35

30 Gumble Plot Cumulative Probability 99.9 % 99 % 80 % 50 % 0 % % Maximum force from 4 drivers Location Scale Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 35 Weibull Probability Density c=0.5 c=.0 c=3.0 Cumulative Probability c=3.0 c=.0 c=0.5 c x fx(x) = b 3 Quantile x c c x exp b c 3 Quantile x x F x(x) = exp b c Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 35

31 Weibull Probability Plot ln( -ln( F x (X) ) ) Cumulative Probability 99.9 % 99 % 80 % 50 % 0 % % Log Scale % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 35 Weibull Plot Cumulative Probability 99.9 % 99 % 80 % 50 % 0 % % Slope: 3.8 Scale Factor: % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35

32 99.9 % Probability Estimated maximum force from the distributions Normal LogNormal Gumbel Weibull Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Comparison Probability Density LogNormal Weibull c = 3.5 Gumble Normal Maximum Load All distributions are approximately normal around the median Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 35

33 Joint Probability Density f xy (x,y) Strength x Stress µ Y y µ X Reliability = P( Stress > Strength ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 35 f xy ( x,y ) Joint Probability Density Normal distributions = πσ exp x σ y ρ x µ x µ ρ y µ y µ + x x y y ( ) ρ σx σx σy σy ρ correlation coefficient Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 35

34 D - Joint Probability Density Stress 3σ Y y 3 4 Contours of equal probability µ Y -3σ x µ x Strength x Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 35 Acceptable Risk Safety P(failure) ~ Economic P(failure) ~ Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 35

35 Extreme Values For most durability problems, we are not interested in the large extremes of stress or strength. Failure is much more likely to come from moderately high stresses combined with moderately low strengths. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 35 Key Points Basic Nomenclature Random Variables Statistical Distribution Cumulative distribution function Mean Variance Probability Marginal Conditional Joint Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 35

36 Probabilistic Aspects of Fatigue Statistical Techniques Professor Darrell F. Socie Department of Mechanical and Industrial Engineering 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

37 Statistical Techniques Normal Distributions LogNormal Distributions Monte Carlo Sampling Distribution Fitting Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35 Failure Probability Stress, Σ Strength, S s Let Σ be the stress and S the fatigue strength Given the distributions of Σ and S find the probability of failure P( Σ s I s S ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 35

38 Normal Variables Linear Response Function Z = a o + n i= a i X X i ~ N( µ i, C i ) µ σ z = a o + n i= n z = a i i= i a µ i σ i i Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 35 Calculations Stress, Σ Strength, S S ~ N( 00, 0. ) σ S = 0 Σ ~ N( 00, 0. ) σ Σ = 0 Safety factor of Let Z be a random variable: Z = S - Σ µ z = µ S - µ Σ µ z = = 00 σ z = σ S + σσ σz = = 8. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 35

39 Failure Probability Z = S - Σ Failure will occur whenever Z <= 0 Z = µ z zσ z = 0 z = µ σ Z = Z z = 3.54 standard deviations P(failure) = x 0-4 For this case only, a safety factor of means a probability of failure of x 0-4. Other situations will require different safety factors to achieve the same reliability. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 35 Failure Distribution Stress, Σ Strength, S What is the expected distribution in fatigue lives? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 35

40 Fatigue Data 000 ' σ f Stress Amplitude 00 0 b 0 ' S σf = b (N ) f Fatigue Life S b Nf ' = σf Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 35 LogNormal Variables Z = a n o i= X ai i a s are constant and X i ~ LN( x i, C i ) COV median C Z = a n o i= X ai i n ( + CX ) i ai Z = i= Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 35

41 Calculations ' σ f Stress, Σ Strength, S ~ LN( 000, 0. ) σ = 00 S ~ LN( 50, 0. ) σ = 50 b = -0.5 S b Nf ' = σf Z = N Z = N f f S = S = 8 8 σ σ ' 8 f ' 8 f COV 8 8 ( + COV ) ( + COV ) Z = S σ f Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 35 Results S/ ' σ f N f Percentile Life µx , ,706,069 COV x ,566,63 95,363,00 µlnx ,589 X , ,676 σx 50 00,676,83 0 7,586 σlnx ,98,89 b = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35

42 Monte Carlo Methods K f S = ' σf E E b ' ' b+c ( Nf ) + σfεf ( Nf ) Given random variables for K f, S, Find the distribution of N f Z = N f =? ' ' σ f and εf Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Simple Example Probability of rolling a 3 on a die f x (x) / Uniform discrete distribution Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 35

43 Computer Simulation. Generate random numbers between and 6, all integers. Count the number of 3 s Let X i = if 3 0 otherwise n n X i i= ( ) = P 3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 35 EXCEL =ROUNDUP( 6 * RAND(), 0 ) =IF( A = 3,, 0 ) =SUM($B$:B)/ROW(B) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 35

44 Results 0.4 probability trials Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 35 Evaluate π y P( inside circle ) - - x πr P= 4 π = 4 P x = * RAND() - y = * RAND() - IF( x + y <,, 0 ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 35

45 π 4 π trials Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 35 Monte Carlo Simulation ' σ f Stress, Σ Strength, S ~ LN( 000, 0. ) σ = 00 S ~ LN( 50, 0. ) σ = 50 b = -0.5 S b Nf ' = σf Randomly choose values of S and ' from their distributions σ f Repeat many times Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 35

46 Generating Distributions F x (X) RAND 0 x Randomly choose a value between 0 and x = F x - ( RAND ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 35 Generating Distributions in EXCEL Normal =NORMINV(RAND(),µ,σ) Log Normal =LOGINV(RAND(),lnµ,lnσ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

47 EXCEL ' σ f S N f , , , , , , , , , , , , , ,473 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35 Simulation Results 99.9 % 99 % Monte Carlo Analytical Mean.5. Std Cumulative Probability 90 % 50 % 0 % Life % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 35

48 Summary Simulation is relatively straightforward and simple Obtaining the necessary input data is difficult Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 35 Nomenclature Population: the totality of the observations Sample: a subset of the population Population mean: µ x Population variance: σ x Sample mean: X Sample variance: s ( X) = x ( s ) = σ E µ E x Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 35

49 Sample Mean and Variance Mean X = n n x in EXCEL =AVERAGE(A:A n ) Variance s = n ( x X ) n in EXCEL =STDEV(A:A n ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 35 Confidence Intervals ( X) = x E µ What is the probability that a sample X is greater than µ x? 50% P( L <= µ x <= U ) = - α There is a α chance of selecting a sample in the interval between L and U that contains the true mean of the population Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 35

50 Translation 90% confidence If we sampled a population many times to estimate the mean, 90% of the time the true population mean would lie between the computed upper and lower limit. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 35 Confidence Interval - mean For a normal distribution: Lower limit of µ s x X tα,n µ x n Upper limit of µ s x X + tα,n µ x n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 35

51 Sample Size % s x X tα,n µ x n tα, n n % Sample size, n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 00 of 35 5 Samples from Normal Distribution 40 µ x = 00 σ x = % confidence 5 samples will be outside confidence limit Lower limit above mean Upper limit below mean Sample Number Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35

52 Confidence Interval - variance For a normal distribution: Upper limit of σ (n )s x χ α,n σ x Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Sample Size 0 8 (n )s x χ α,n σ x (n ) χ α,n % 95% 50% Sample size, n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 03 of 35

53 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Maximum Load Data Median 43 COV 0.34 Maximum force from 4 drivers 90% confidence COV Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 04 of Maximum Load Data Probability Density LogNormal Normal COV = 0.34 Normal COV = Maximum Load Uncertainty in Variance is just as important, perhaps more important than the choice of the distribution Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 05 of 35

54 Choose the Best Distribution 5 samples from a Normal Distribution Probability Density LogNormal Weibull Gumble Normal Maximum Load Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 06 of 35 Goodness of Fit Tests Cumulative Probability 99.9 % 99 % 90 % 50 % % % 0. % Normal Cumulative Probability 99.9 % 99 % 80 % 50 % 0 % % Gumble Cumulative Probability 99.9 % 99 % 90 % 50 % % % LogNormal Cumulative Probability 99.9 % 99 % 80 % 50 % 0 % % Weibull % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 07 of 35

55 Analytical Tests Chi-square any univariate distribution Snedecor, George W. and Cochran, William G. (989), Statistical Methods, Eighth Edition, Iowa State University Press. Kolmogorov-Smirnov tends to be more sensitive near the center of the distribution Chakravarti, Laha, and Roy, (967). Handbook of Methods of Applied Statistics, Volume I, John Wiley and Sons, pp Anderson-Darling gives more weight to the tails Stephens, M. A. (974). EDF Statistics for Goodness of Fit and Some Comparisons, Journal of the American Statistical Association, Vol. 69, pp Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 08 of 35 Distributions Normal Strength Dimensions LogNormal Fatigue Lives Large variance in properties or loads Gumble Maximums in a population Weibull Fatigue Lives Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 09 of 35

56 Statistics Software BestFit Minitab Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Central Limit Theorem If X, X, X 3.. X n is a random sample from the population, with sample mean X, then the limiting form of X µ X Z = σ / n as n is the standard normal distribution Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

57 Translation When there are many variables affecting the outcome, The final result will be normally distributed even if the individual variable distributions are not. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Example Probability of rolling a die f x (x) / Uniform discrete distribution Let Z be the summation of six dice Z = X + X + X 3 + X 4 + X 5 + X 6 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

58 Results Frequency trials X Z =. C Z = Z Central limit theorem states that the result should be normal for large n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Central Limit Theorem Sums: Z = X ± X ± X 3 ± X 4 ±.. X n Z Normal as n increases Products: Z = X X X 3 X 4.. X n Z LogNormal as n increases Normal and LogNormal distributions are often employed for analysis even though the underlying population distribution is unknown. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

59 Key Points All variables are random and can be characterized by a statistical distribution with a mean and variance. The final result will be normally distributed even if the individual variable distributions are not. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Probabilistic Aspects of Fatigue Analysis Methods Professor Darrell F. Socie Department of Mechanical and Industrial Engineering Darrell Socie, All Rights Reserved

60 Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Reliability Analysis Stressing Variables Probability Density Strength Variables Probability Density Analysis? Probability Density P( Failure ) 3 3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

61 Probabilistic Analysis Methods Monte Carlo Simple Hypercube sampling Importance sampling Analytical First order reliability method FORM Second order reliability method SORM Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Limit States Limit States Equilibrium Strength Deformation Wear Functional Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

62 Limit State Problems Response function Z(X) = Z( Q, L, b, h ) Limit State function g = Z(X) - Z o = 0 Same as the failure function Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Limit States Strength g > 0 g < 0 failures Stress g = Strength - Stress Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

63 Monte Carlo Strength g > 0 g < 0 failures Stress Monte Carlo is simple but what if each of these calculations required a separate FEM model? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Low Probabilities of Failure Strength rare event failures Stress Need about 0 5 simulations for P(Failure) = 0-4 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

64 Transform Variables x µ u = σ x µ u = 0 σ u = x u u Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Standard Monte Carlo u trials u and u normally distributed g = 3 ( u + u ) Nf P(f) = = 0.08 N u Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

65 Multiple Simulations trials 5 simulations 5,000 calculations µ = 0.07 σ = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Stratified Sampling Methods Divide into regions of constant probability and then sample by region Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

66 Stratified Sample u Each box should have one sample drawn at random from the underlying pdf 00 trials 5 simulations,500 calculations µ = 0.0 σ = u Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 Latin Hypercube Sample u trials 5 simulations 50 calculations µ = 0.07 σ = u Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

67 Importance Sampling (continued) u β = The probability of a sample outside the circle is P S = 0.4 ( standard deviations ) 38 simulations Nf P(f) = PS = N S u Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35 Analytical Methods Strategy Develop a response function Transform the set of N variables to a set of uncorrelated u variables with µ=0 and σ= Locate the most likely failure point Approximate the integral of the joint probability distribution over the failure region Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

68 Analytical Methods Outline Response Surface Transform Variables Limit State Concept Most Probable Point Probability Integration FORM SORM Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 Response Surface Methodology Response surface methodology is a well established collection of mathematical and statistical techniques for applications where the response of interest is influenced by several variables. Z(X) = f( x,x,x, x ) + ε 3 n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35

69 Response Surface Z Solution points X X Fit a surface through the points Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 35 Mathematical Representation Z(X) = A o + Ax + A x + A 3x3 + Linear Bx + Bx + B3x3 + Incomplete Quadradic C x x + C x x + C x x + Complete Quadradic 3 3 Solutions Linear N + Incomplete Quadradic N + Complete Quadradic N(N + )/ 3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 35

70 Evaluation of Response Surface Suppose stress is affected by speed and temperature Factorial Design temperature high low speed high low X X X X 4 deterministic solutions needed n solutions Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 35 Evaluation of Response Surface Factorial Design temperature high low speed high low X X X X X X X X X 9 deterministic solutions needed 3 n solutions Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 35

71 Graphical Representation Z X X Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 35 Joint Probability Density f xy (x,y) Stress x Strength y Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35

72 D - Joint Probability Density Strength Contours of equal probability µ y µ x Stress Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 x µ u = σ µ u = 0 σ u = Transform Variables The mathematics and behavior of normal distributions are well understood x x u d P exp( 0.5d ) u Any distribution can be mapped into a normal distribution Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 35

73 X to u mapping f(x) uniform f(x) normal F(X) F(X) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 35 Limit States Limit states define probability problems For example: g = Strength(x ) Stress(x ) Prob( g<= 0 ) = Probability of failure = P f Analysis focuses on g = 0 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 35

74 Failure Probability u g = 0 u P(Failure) = fu u g< 0 (u,u )du du Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 35 Thought Experiment Suppose stress and strength had the same distribution Probability Density x What is the probability of failure? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 35

75 g function g = Strength(x ) Stress(x ) u u P f = 50% Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 35 Probable Combinations of u and u g functions are not straight lines in N dimensional space u g = 0 u Most likely failure point Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 35

76 Most Probable Point u g = 0 u β Probability is related to the minimum distance point β Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 35 Safety Index β Sometimes called reliability index β is a number measured in standard deviations Unlike safety factors, failure probability is directly related to the safety index P f = Φ(-β) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

77 Determining the MPP Requires an efficient numerical search to find the tangent point of a hypersphere (β-sphere) and the limit state function in u space Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35 Probability Integration FORM SORM Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 35

78 First Order Reliability Model FORM Linear approximation u g = 0 u exact β n () = ao + ai( ui ui *) g u i= MPP Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 35 FORM The joint probability density is rotationally symmetric so it is possible to rotate the coordinate system u u * P f Φ(-β) u β MPP u * Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 35

79 Second Order Reliability Model SORM Quadradic approximation u u g = 0 β exact n n n n ( ) = ao + ai( ui ui *) + bi( ui ui *) + cij( ui ui *)( uj uj *) g u i= i= i= j= Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 35 SORM ( ) ( βκi) P f =Φ β i= κ i surface curvature n u * u β MPP u * Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 35

80 Sensitivity Factors 99.9 % 99 % Cumulative Probability 90 % 50 % 0 % Design point Change in response with a change in mean % 0. % Change in probability with a change in mean Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 35 Sensitivity Factors 99.9 % 99 % Cumulative Probability 90 % 50 % 0 % Design point Change in response with a change in variance % 0. % Change in probability with a change in variance Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 35

81 Deterministic Sensitivity Factors Z(X) X i Change in response mean with respect to a change in the input variable mean Frequently normalized by the means to compare relative importance of variables Z(X)/ Z X / X i Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 35 Probabilistic Sensitivity Coefficient Standard deviation sensitivity P /P Sσ i = σ / σ i i Mean deviation sensitivity S µ i = P /P µ / µ i i Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35

82 Probabilistic Sensitivity Factors (X) α i X σ i Z α = i i α i Z(X) X i σ i - probabilistic sensitivity - response - input variable - standard deviation of X i Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Software General Purpose Durability Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 35

83 Probabilistic Aspects of Fatigue Variability Professor Darrell F. Socie Department of Mechanical and Industrial Engineering 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 35

84 Sources of Variability customers Stress usage Stress, Σ Strength, S Strain Amplitude Fatigue Life, N f materials Strength manufacturing Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 35 Variability and Uncertainty Variability: Every apple on a tree has a different mass. Uncertainty: The variety of the apple is unknown. Variability: Fracture toughness of a material Uncertainty: The correct stress intensity factor solution Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 35

85 Sources of Variability Stress Variables Loading Customer Usage Environment Strength Variables Material Processing Manufacturing Tolerance Environment Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 35 Sources of Uncertainty Statistical Uncertainty Incomplete data (small sample sizes) Modeling Error Analysis assumptions Human Error Calculation errors Judgment errors Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 35

86 Modeling Variability Central Limit Theorem: Products: Z = X X X 3 X 4.. X n Z LogNormal as n increases.5 σ lnx.0 σ lnx ~ COV X COV X COV X is a good measure of variability Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 35 COV and LogNormal Distributions Standard Deviation, lnx COV X 68.3% 95.4% 99.7% % of the data is within a factor of ±.33 of the mean for a COV = 0. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

87 Variability in Service Loading Quantifying Loading Variability Maximum Load Load Range Equivalent Stress Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Maximum Force Maximum force from 4 automobile drivers Median 43 COV Force Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 35

88 Maximum Load Correlation Fatigue Lives Maximum Load Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 35 Loading Variability Tractors / Drivers Load Range Cumulative Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 35

89 Variability in Loading 99.9 % 54 Tractors/Drivers Cumulative Probability 99 % 90 % 50 % 0 % % COV Equivalent Load S eq = n S n 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 35 Mechanisms and Slopes Equivalent Load, kn 00 0 Structures n = Total Fatigue Life, Cycles A combination of nucleation and growth 5 Stress Amplitude, MPa Crack Nucleation n = Cycles Crack Growth n = Crack Growth Rate, m/cycle m K,MPa Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 35

90 Effect of Slope on Variability Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % n = n COV Equivalent Load % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 35 Loading History Variability Test Track Customer Service Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 35

91 Test Track Variability Cumulative Probability 99.9 % 40 test track laps of a motorhome 99 % COV % 50 % 0. 0 Equivalent Load 0 % % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 35 Customer Usage Variability Cumulative Probability 99.9 % 4 drivers / cars 99 % COV % 50 % 0. 0 Equivalent Load 0 % % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35

92 Variability in Environment Inclusions Pit depth Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Inclusions That Initiated Cracks COV = 0.7 Barter, S. A., Sharp, P. K., Holden, G. & Clark, G. Initiation and early growth of fatigue cracks in an aerospace aluminium alloy, Fatigue & Fracture of Engineering Materials & Structures 5 (), -5. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 35

93 Pits That Initiated Cracks 700-T765 Pre-corroded specimens 300 specimens 46 failed from pits Crawford et.al. The EIFS Distribution for Anodized and Pre-corroded 700-T765 under Constant Amplitude Loading Fatigue and Fracture of Engineering Materials and Structures, Vol. 8, No , Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 35 Pit Size Distribution 40 Frequency 30 0 Mean = 30 COV = area µm Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 35

94 Pit Depth Variability Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % 0 Pits Data Points Median 4.37 COV 0.33 Pit Depth, µm 00 % 0. % Dolly, Lee, Wei, The Effect of Pitting Corrosion on Fatigue Life Fatigue and Fracture of Engineering Materials and Structures, Vol. 3, 000, Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 35 Variability in Materials Tensile Strength Fracture Toughness Fatigue Fatigue Strength Fatigue Life Strain-Life Crack Growth Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 35

95 Tensile Strength Steel Number of heats Mean = 60 COV = Tensile Strength, MPa Metals Handbook, 8 th Edition, Vol., p64 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 35 Fracture Toughness Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % 6 Data Points Median 76.7 COV K Ic, Ksi in Mar-M 50 Steel 00 % 0. % Kies, J.A., Smith, H.L., Romine, H.E. and Bernstein, H, Fracture Testing of Weldments, ASTM STP 38, 965, Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 35

96 Fatigue Variability 000 Stress Amplitude 00 0 Fatigue life Fatigue strength Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 35 Fatigue Life Variability 0 One lot, 7 parts Production torsion bars 560H steel 0 µ x = 3,000 cycles COV = 0.5 Number of Tests lots, 300 parts µ x = 34,000 cycles COV = Life x0 3 Metals Handbook, 8 th Edition, Vol., p9 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

97 Statistical Variability of Fatigue Life s=440 s=80 s=35 98 s=45 s= T6 Specimens Percent Survival Cycles to Failure Sinclair and Dolan, Effect of Stress Amplitude on the Variability in Fatigue Life of 7075-T6 Aluminum Alloy Transactions ASME, 953 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35 COV vs Fatigue Life S X COV 440 4, , , ,00, ,000, Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 35

98 Variability in Fatigue Strength S ' = S (N ) f f b b COV C = n ( + ) i i= a i C X C (.085 ( +.39 ) ) ' = = S f Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 35 Strain Life Data for 950X Steel Strain Amplitude Fatigue Tests 9 Data Sets Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 35

99 9 Individual Data Sets ' σ f b Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Median 80 COV Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Mean COV % % 0. % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 35 9 Individual Data Sets (continued) Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Median 0.57 COV.5 ' ε f 0. 0 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % c Mean -0.6 COV Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 35

100 Input Data Simulation ε = σ ' f ( L, µ σ, σσ ) ( ) b( N, µ, σ ) c ( )( ) ( N, µ σ ) E f f b b ' b, Nf + ε f L, µ ε, σ N f εf f b Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 35 Simulation Results 00 Strain Amplitude Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 35

101 Correlation b 0. c ' σ f ' ε f ρ = ρ = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 00 of 35 Generating Correlated Data z = Φ( rand() ) z = Φ( rand() ) z = N(0,) z 3 = z ρ + z ρ σ ' f = exp( µ ln + σln z ) ' σ f ' σ f b = µ b + σ b z 3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35

102 00 Correlated Properties Strain Amplitude Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Curve Fitting Strain Amplitude ' σ f ε σ = E ' f b ' c (N f ) + ε ) f(nf Assume a constant slope to get a distribution of properties Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 03 of 35

103 Property Distribution Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % ' σ ' f ε f 378 Data Points Median 80 MPa COV % 99 % 365 Data Points Median 0.6 COV 0.4 b = % c = Cumulative Probability 50 % 0. 0 % % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 04 of 35 Correlation 500 ' σ f ' ε f Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 05 of 35

104 Simulation 0 Strain Amplitude Fatigue Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 06 of 35 Strength Coefficient Cumulative Probability 99.9 % 99 % 90 % 50 % % % 0. % 5000 K' 365 Data Points Median 00 MPa COV ' σ f K' ρ = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 07 of 35

105 Crack Growth Data 50 Crack Length, mm Cycles x0 3 Virkler, Hillberry and Goel, The Statistical Nature of Fatigue Crack Propagation, Journal of Engineering Materials and Technology, Vol. 0, 979, Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 08 of 35 Crack Growth Rate Data 0 - Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % 300, Data Points Median 57,000 COV 0.07 Fatigue Lives Crack growth rate, mm/cycle Stress intensity, MPa m Crack Growth Rate Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 09 of 35

106 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Crack Growth Properties Median 5 x 0-8 COV da dn = C K 0-6 m Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % 4 5 Median 3.3 COV 0.06 % % 0. % C 0. % m Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 Beware of Correlated Variables N f = a C S m / f m π a m m / i ( m / ) N f and C are linearly related and should have the same variability, but = N COV C = COV f 0.07 because C and m are correlated. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

107 Correlation m C Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Correlated Variables σ Z = σx + ρσxσy + σy σ Z.5 σ X = σ Y = ρ Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

108 Calculated Lives Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Computed from C and m pairs experimental N f = af ao C da ( K) m % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 Manufacturing/Processing Variability Bolt Forces Surface Finish Drilled Holes Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

109 Variability in Bolt Force Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Force 00 Data Points Median 30 COV Bolt Force, kn Preload force in bolts tightened to 350 Nm Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Surface Roughness Variability Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Surface Finish 5 Data Points Median 43 COV Surface Finish, µin 00 % 0. % Machined aluminum casting Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

110 Drilled Holes Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % Fighter Spectrum 54 Data Points Median 6,750 COV 0. in life COV 0.07 in strength 0 5 Cycles 80 drilled holes in a single plate 0. % From: J.P. Butler and D.A. Rees, "Development of Statistical Fatigue Failure Characteristics of 0.5-inch 04-T3 Aluminum Under Simulated Flight-by-Flight Loading," ADA-0030 (NTIS no.), July 974. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Analysis Uncertainty Miners Linear Damage rule Strain Life Analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

111 Miners Rule Cumulative Probability of Failure Tests COV = Computed Life / Experimental A safety factor of 0 in life would result in a 0% chance of failure From Erwin Haibach Betriebsfestigkeit, Springer-Verlag, 00 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 35 SAE Specimen Bracket Suspension Transmission Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 977 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35

112 Analysis Results 99.9 % 99 % Strain-Life analysis of all test data 48 Data Points COV.7 Cumulative Probability 90 % 50 % 0 % % 0. % 0 00 Analytical Life / Experimental Life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved of 35 Material Variability 99.9 % 99 % Material Analysis Cumulative Probability 90 % 50 % 0 % % 0. % 0 00 Analytical Life / Experimental Life Strain-Life back calculation of specimen lives Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

113 Modeling Uncertainty Analysis Uncertainty C U =? The variability in reproducing the original strain life data from the material constants is C M ~ 0.44 COV C = n ( + ) i i= + CNf + CU = + CM C U =.09 i C X 90% of the time the analysis is within a factor of 3! 99% of the time the analysis is within a factor of 0! Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 a Variability from Multiple Sources COV C = n ( + ) i i= a i C X Suppose we have 4 variables each with a COV = 0. The combined variability is COV = 0.9 Suppose we reduce the variability of one of the variables to 0.05 The combined variability is now COV = 0.7 If all of the COV s are the same, it doesn t do any good to reduce only one of them, you must reduce all of them! Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

114 Variability from Multiple Sources COV C = n ( + ) i i= a i C X Suppose we have 3 variables each with a COV = 0. and one with COV = 0.4 The combined variability is COV = 0.65 Suppose we reduce the variability of these variables to 0.05 The combined variability is now COV = 0.60 If one of the COV s is large, it doesn t do any good to reduce the others, you must reduce the largest one! Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Variability Summary Stress Strength Source COV Service Loading 0.5 Environment 0.3 Materials 0. Manufacturing 0. Surface Finish 0. Fatigue Lives Analysis Uncertainty.0.0 Strength Fatigue life Stress 5 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

115 Variability and Uncertainty Variability: Every apple on a tree has a different mass. Uncertainty: The variety of the apple is unknown. Variability: Multiple samples of the same material Uncertainty: What is the material Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Strain Life Data for 93 Steels 0 Strain Amplitude Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

116 Cumulative Probability Uncertainty for all Steels 99.9 % 55 Data Points 99 % COV % 50 % % % 0. % ' σ f Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 Cumulative Probability Uncertainty for Structural Steels 99.9 % 99 % 90 % 50 % 0 % % 0. % ' σf 9 Data Points COV 0. S u MPa Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

117 Variability and Uncertainty Fatigue Strength Coefficient Variability Uncertainty Combined All Steels Structural Steel Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35 Quiz At my last seminar everyone hit a golf ball and we recorded the maximum acceleration. What is the expected variability? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

118 Results µ σ COV Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 Probabilistic Aspects of Fatigue Case Studies Professor Darrell F. Socie Department of Mechanical and Industrial Engineering Darrell Socie, All Rights Reserved

119 Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 35 Case Studies DARWIN Southwest Research Bicycle Loading Histories Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 35

120 A Software Framework for Probabilistic Fatigue Life Assessment ASTM Symposium on Probabilistic Aspects of Life Prediction Miami Beach, Florida November 6-7, 00 R. C. McClung, M. P. Enright, H. R. Millwater *, G. R. Leverant, and S. J. Hudak, Jr. Southwest Research Slides 6 7 used with permission of of Craig McClung Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 35 Motivation Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 35

121 UAL Flight 3 July 9,989 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 35 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35

122 Probabilistic Damage Tolerance Anomaly Distribution NDE Inspection Schedule Probability of Detection Finite Element Stress Analysis Probabilistic Fracture Mechanics Pf vs. Cycles Material Crack Growth Data Risk Contribution Factors Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 35 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 m Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 35

123 Fracture Mechanics Model of Zone Finite Element Model x Fracture Mechanics Model 5 (Not to Scale) 7 Retrieve stresses along line 4 gradient direction 3 Defect h x hy Y m Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 35 Stress Processing stress gradient FE Stresses and plate definition Stress gradient extraction Hoop Stress (ksi) Load Step Rainflow stress pairing.0 FE Analysis (σz ) elastic.6 Computed relaxed stress σelastic - σresidual. (σz ) relax 0.8 σ0/σσ σ/ o 0.4 (σz ) residual 0.0 Shakedown module Normalized distance from the notch tip, x/r Residual stress analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 35

124 Anomaly Distribution # of anomalies per volume of material as function of defect size Library of default anomaly distributions for HA (developed by RISC) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 35 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 35

125 Random Inspection Time Opportunity Inspections during on-condition maintenance Inspection time modeled with Normal distribution or CDF table Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 35 Output: Risk vs. Flight Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 35

126 Output: Risk Contribution Factors Identify regions of component with highest risk Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 35 Implementation in Industry FAA Advisory Circular 33.4 requests risk assessment be performed for all new titanium rotor designs Designs must pass design target risk for rotors Risk Risk Reduction Required Maximum Allowable Risk 0-9 A B C Components Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35

127 Pf Cycles Per Mission THRESHOLD STRESS RANGE (KSI) DARWIN for Prognosis Studies Anomaly Distribution NDE Inspection Material Crack Growth Data Finite Element Stress Analysis Probabilistic Fracture Mechanics Pf vs. Flights Code Enhancements PROBABILITY OF FAILURE AT 0,000 CYCLES NUMBER OF CYCLES PER MISSION RPM-Stress Transform Crack Initiation Load Spectrum Editing Probabilistic Sensor Fatigue (RPM) Input Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 35 Three Sources of Variability Anomaly size (initial crack size) FCG properties (life scatter) Mission histories (stress scatter) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 35

128 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 35 Resulting Anomaly Distributions Post 995 Triple Melt/Cold Hearth + Vacuum Arc Remelt.00E+0 MZ(5-0in Billet)/# FBH MZ(5-0in Billet)/#3 FBH MZ (-3in Billet)/# FBH EXCEEDENCE (per million pounds).00e+0.00e+00.00e-0 #/# FBH #/# FBH #3/# FBH #3/# FBH #3/#3 FBH.00E-0.00E+0.00E+03.00E+04 DEFECT INSPECTION AREA (sq mils) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 35

129 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 a, cm AGARD Corner Crack Specimen K i =8.7 MPa m, K f =56.9 MPa m N, cycle Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 35 Engine Usage Variability Stress/Speed: σ (RPM) 0000 Air Combat Tactics Total Cyclic Life (LCF): N f = N i + N p N i σ 3-5 N p σ 3-4 Life/Speed: N f (RPM) 6 RPM (Low Speed Spool) 9000 Air to Ground & Gunnery Basic Fighter Maneuvers 8000 Intercept 7000 Peace Keeping Surface to Air Tactics ( hi alt) 6000 Surface to Air Tactics ( lo alt) 5000 Suppression of Enemy Defenses Cross Country TIME (SEC) Component life is very sensitive to actual usage Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 35

130 Usage Variability 0000 Air Combat Tactics Components of Usage Variability: Mission type Mission-to-mission variability RPM (Low Speed Spool) 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 Mission mixing variability Surface to Air Tactics ( lo alt) TIME (SEC) Peace Keeping RPM (Low Speed Spool) RPM (Low Speed Spool) TIME (SEC) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved TIME (SEC) 58 of 35 Variability in Mission Type Air-Air Weapons Delivery Air-Ground Weapons Delivery RPM RPM Time Time Initiation and Propagation No Inspection Normalized POF Air Combat Tactics Combat Air Patrol Air-Ground Weapons Delivery Air-Air Weapons Delivery Instrument/Ferry Number of Flight Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 35

131 Web Site: Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 35 Fatigue Design and Reliability in the Automotive Industry Fatigue Design and Reliability ESIS Publication 3 J-J. Thomas, G. Perroud, A. Bignonnet and D. Monnet PSA Peugeot Citroën Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35

132 Fatigue Design at PSA Fatigue design at PSA is done with a probabilistic approach that includes analysis of customer usage, production scatter, definition of the appropriate design loads and an acceptance testing criterion. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 35 Stress-Strength Interference Design Load Stress Strength Reliability = P( Stress > Strength ) How do you really get these distributions? How do you establish a design load? How do you validate the design? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 35

133 Stress Distribution Variability in loading has two components, how it is used and how it is driven. Car Usage Highway, city, fully loaded, empty etc. Driving Style passive, aggressive etc. The usage of a car is independent of the owners driving style so that the distributions of car usage and driving style can be obtained separately. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 35 Car Usage Customer surveys k l c k % r kl % Unloaded 7 Highway 0 Good Road 5 3 Mountain 40 4 City 5 Half Load 58 Highway 5 Good Road 30 3 Mountain 30 4 City 35 3 Fully Loaded 5 Highway 5 Good Road 5 3 Mountain 40 4 City 0 Customer Usage Categories Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 35

134 Owner Behavior Instrumented Vehicles From Load, kn Extensive field testing for each customer usage category produces a large number of histograms. Let the usage histogram be denoted U kl To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 35 Extrapolation U kl U* kl 3 3 From Load, kn 0 - From Load,kN To Load, kn Measured data, e.g.,000 km To Load, kn Extrapolated data, 300,000 km Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 35

135 Virtual Customers Thousands of virtual customers can now be generated by combining customer usage with driving style. [ U ] i i * [ U ] = N c r [ U ] k kl rainflow histogram for an individual car N kilometers c k car loading, % r kl car usage, % * [ U ] kl distributions of rainflow histograms for different car usage classifications Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 35 kl Design Loads Initial design is done on the basis of a single constant amplitude load, F eq Find a constant amplitude load and number of cycles that will produce the same fatigue damage as the customer operating a car for the design life. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 35

136 Equivalent Fatigue Loading n Nf = F eq F eq F = 6 0 m m Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 35 Distribution of F eq µ F ασ F F n Design Customer: F n = µ F +ασ F Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35

137 Choosing α y 3σ Y Stress 3 Contours of equal probability 4 3σ P( s < S ) = µ Y -3σ x µ x Strength x If we use 3σ on both stress and strength P(failure) = P( Σ s I s S) = σ Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 35 Distribution of Strength σ S S exp µ S 0 6 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 35

138 What Strength is Needed? βσ S µ F ασ F F n µ S Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 35 Some Statistics Z = S - F Suppose we want a probability of failure of in 50,000 Φ ( P ) f 5 Pf = x0 µ Z = 4. = = σ Z µ S σ µ S F + σ F F n = µ S ( β COV S ) F n = µ F ( + α COV F ) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 35

139 Mean Component Strength Φ ( P ) f = 4. = µ σ Z Z = µ S σ µ S F + σ F Φ ( P ) f = µ F S n µ F S n COV + αcov S F COVF + + αcov F Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 35 Reliability α = 3 COV F = 0. COV S = 0. Reliability n Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 35 µ F S

140 Mean Strength ασ F βσ S µ F Fn µ S µ S =.4 F n for a reliability of x 0-5 Fn µ S β = COV S =.85 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 35 Design Design calculations are made with loads F n Large database relates F n to vehicle parameters so that a new vehicle can be designed from historical measurements Fatigue calculations are made with material properties β standard deviations from the mean Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 35

141 Component Testing Component strength not life! Test load is frequently higher than F n to get failures near the design life Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 35 Interpreting the Test Data Component tests are done with small sample sizes Confidence limits (N )s x χ α,n σ x Φ n F ( Pf ) = µ F S n COV µ F S S + αcov N χ α,n COVF + + αcov F Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35

142 Validation Full scale vehicle simulation done at the end for design final validation Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 35 Extrapolation 3 U kl 3 U* kl From Load, kn 0 - From Load, kn To Load, kn Measured data, e.g.,000 km To Load, kn Extrapolated data, 300,000 km How does this process work? Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 35

143 Service Loading Spectra Lateral Force, kn From Load, kn To Load, kn Load Range,kN Cumulative Cycles Time history Rainflow Histogram Exceedance Diagram Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 35 Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extrapolate to more users Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 35

144 Probability Density 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 35 Kernel Smoothing 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 35

145 Sparse Data 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 35 Exceedance Plot of Lap 4 Load Range,kN 3 weibull distribution Cumulative Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 35

146 0X Extrapolation 6 5 Load Range,kN Cumulative Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 35 Probability Density 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35

147 Results 3 Simulation 3 Test Data From Load, kn 0 - From Load, kn To Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 35 Exceedance Diagram 6 Load Range,kN 5 Model Prediction 4 3 Actual 0 Laps Lap (Input Data) Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 35

148 Problem Statement Given a rainflow histogram for a single user, extrapolate to longer times Given rainflow histograms for multiple users, extropolate to more users Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 35 Extrapolated Data Sets 99.9 % 99 % 90 % ATV Airplane Tractor 50 % 0 % Normalized Life % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 35

149 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 35 Assumption 48 On average, more severe users tend to have more higher amplitude cycles and fewer low amplitude cycles Load Range Cumulative Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 35

150 Translation 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 35 Damage Regions 3 From Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 35

151 ATV Data - Most Damaging in 9 3 Simulation 3 Test Data From Load, kn 0 - From Load, kn To Load, kn To Load, kn Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 300 of 35 ATV Exceedance 6 Load Range, kn Actual Data Simulation Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35

152 Airplane Data - Most Damaging in Simulation 30 Test Data 0 0 From Stress, ksi From Stress, ksi To Stress, ksi To Stress, ksi Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 Airplane Exceedance Stress Range,ksi Simulation Actual Data Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 303 of 35

153 Tractor Data - Most Damaging in 54.5 Simulation.5 Test Data From Starin x From Strain x To Strain x To Strain x 0-3 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 304 of 35 Tractor Exceedance 3.0 Strain Range x Actual Data Simulation Cycles Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 305 of 35

154 Probabilistic Aspects of Fatigue Professor Darrell F. Socie Department of Mechanical and Industrial Engineering 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 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 307 of 35

155 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 308 of 35 Probabilistic Fatigue Analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 309 of 35

156 Modeling Variability Central Limit Theorem: Products: Z = X X X 3 X 4.. X n Z LogNormal as n increases.5 σ lnx.0 σ lnx ~ COV X COV X COV X is a good measure of variability Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 COV and LogNormal Distributions Standard Deviation, lnx COV X 68.3% 95.4% 99.7% % of the data is within a factor of ±.33 of the mean for a COV = 0. Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

157 Strain Life Analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35 Material Properties Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

158 Stress Concentration Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 Results Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35

159 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 35 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 35

160 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 35 Ten Simulations Life COV Mean COV Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 35

161 9 Individual Data Sets ' σ f b Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Median 80 COV Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % Mean COV % % 0. % 0. % Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 35 9 Individual Data Sets (continued) Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % Median 0.57 COV.5 ' ε f 0. 0 Cumulative Probability 99.9 % 99 % 90 % 50 % 0 % % 0. % c Mean -0.6 COV Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35

162 Correlation b 0. c ' σ f ' ε f ρ = ρ = Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 35 Correlated Variables Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

163 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35

164 Uncorrelated Variables Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 35 Results (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 35

165 Strain Amplitude ± 000 ± 50 Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 35 Deterministic Analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 35

166 Deterministic Analysis (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 330 of 35 Deterministic Analysis (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35

167 Deterministic Analysis Results Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 35 Probabilistic Analysis Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 333 of 35

168 Probabilistic Analysis (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 334 of 35 Probabilistic Analysis (continued) Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 335 of 35

169 Probabilistic Analysis Results Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 336 of 35 Probabilistic Aspects of Fatigue GlyphWorks Professor Darrell F. Socie Department of Mechanical and Industrial Engineering 005 Darrell Socie, All Rights Reserved

170 Probabilistic Aspects of Fatigue Introduction Basic Probability and Statistics Statistical Techniques Analysis Methods Characterizing Variability Case Studies FatigueCalculator.com GlyphWorks Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 338 of 35 GlyphWorks / Rainflow Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 339 of 35

171 StatStrain.flo Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 340 of 35 StatStrain Glyph Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35

172 StatOutput.xls 99.9 % Life Distribution, Channel 6 99 % 90 % 50 % % % 0% Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 35 StatOutput.xls (continued) Probabilistic Sensitivity, Channel 6 Analysis Variables 4% Material Properties 6% Load History Variables 57% Stress Concentrators 3% Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 343 of 35

173 StatOutput.xls (continued) Channel # 6 Total Channels Life Distribution Sensitivity Variable Probabilistic Deterministic NumCases Load History Variables Median ScaleFactor COV.9 Offset Stress Concentrators Probability (%) Life Kf Material Properties E Sfp b efp c np Kp Analysis Variables Uncertainty Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 344 of 35 StatOutput.xls (continued) Inputs Outputs Scale Scale Variable Distribution Value Parameter Value Parameter ScaleFactor Normal Offset Normal Kf Uniform E None Sfp Log-Normal b None efp None c None np None Kp Log-Normal Uncertainty Log-Normal Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 345 of 35

174 StatStress.flo Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 346 of 35 Rainflow Extrapolation Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 347 of 35

175 Duration Extrapolation Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 348 of 35 Results Files Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 349 of 35

176 Rainflow Duration Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 350 of 35 Percentile Extrapolation Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35

177 Results Probabilistic Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 35 Probabilistic Aspects of Fatigue