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Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com A Probabilistic Approach To Modeling Fatigue.

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Presentation on theme: "Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com A Probabilistic Approach To Modeling Fatigue."— Presentation transcript:

1 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com A Probabilistic Approach To Modeling Fatigue Life Variation Julian Raphael 1, Bart McPheeters 2, Ray DelDin 2 1. J R Technical Services, LLC, Abingdon, Virginia 24211, USA 2. NEiSoftware, Inc, Westminster, California 92683, USA

2 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Presentation Outline Objective Rationale Simulation Algorithm Probabilistics Finite Element Results Fatigue Model Stochastic Results Correlated Random Variables Crack Growth Models Summary

3 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Objective To develop and implement a numerical procedure that can reasonably estimate both fatigue life and fatigue life variation. Output is the Cumulative Distribution Function (CDF) that predicts life expectancy and its variation.

4 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Rationale For The Approach “ Factors of 100 in life are not uncommon for very low stress level fatigue tests ” –Stephens, R. I., Fatemi, A., Stephens, R. R., Fuchs, H.O., Metal Fatigue in Engineering, 2nd edition, 2001 “ Variability in test conditions,  and  p, will be much smaller than the variability in material properties; all of the variability in the fatigue lives can be attributed to the material constants ” –Socie, D., Reemsnyder, H., Downing, S., Tipton, S., et al, Fatigue Life Prediction, SAE Fatigue Design Handbook, 3rd edition, 1997 “ The $119 billion cost of fracture and its prevention, expressed in 1982 dollars, amounts to about 4% of the gross national product.” –Duga, J. J., Fisher, W. H., Buxbaum, R.W., Rosenfield, A. R., Burh, A. R., Honton, E.J., McMillan, S. C., The Economic Effects of Fracture in the United States, NBS Special Publication, 647-2, United States Department of Commerce, Washington, DC, March 1983

5 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Fatigue Model Simulation Algorithm Stochastics Simulation of Material Properties Fatigue Model Damage Parameter Material Properties Data Analysis Failure CDF Stress Analysis Stress State Strain State

6 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Monte Carlo Simulation Generate Random Numbers Between Zero and Unity Using Mean and Std Dev Convert the RNs to Material Constants Solve Fatigue Life Equation for Cycles to Failure, N f Analyze Failure Data to Compute CDF for Life Get Another Set of Material Constants Damage Parameter Comes From Stress And Strain States

7 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Probabilistics

8 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Uniform Distribution Simulation N=100

9 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Uniform Distribution Simulation N=1000

10 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Uniform Distribution Simulation N=100000

11 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Finite Element Results

12 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com The Solid Model

13 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Bending Stress Distribution t = 500  sec (1000 Hz Sine Wave) Units = MPa

14 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Location Of Maximum Bending Stress Crack Nucleation Is Expected To Occur At This Point. Max Principal Stress Is 1729 MPa Units = MPa

15 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Stress & Strain States at Expected Crack Initiation Site

16 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Fatemi-Socie Fatigue Model

17 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Damage Parameter:

18 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Plane of Maximum Damage Plane on which maximum damage occurs is not known a priori It must be calculated from the stress state, the strain state, & normal stress sensitivity The value of the damage parameter must be evaluated on every possible plane Non-proportional loading complicates the calculation

19 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Fatemi-Socie Crack Nucleation Plane Units = MPa Damage Parameter = 0.008713

20 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Simplified Fatemi-Socie Fatigue Model

21 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Cycles to Crack Nucleation vs Fatemi-Socie Damage Parameter  ’ f = 1758 MPa  ’ f = 2.12 b = -0.0977 c = -0.774 G = 79615 MPa

22 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Stochastic Results

23 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Simulation Results of Cycles to Failure With D = 0.008713, 0.0043565 0.384780E+04 0.331892E+04 0.375395E+04 0.420746E+04 0.317882E+04 0.459811E+04 0.391109E+04 0.667456E+04 0.376841E+04 0.298723E+04 Cycles to Failure D = 0.008713 0.689154E+05 0.510689E+05 0.603324E+05 0.854142E+05 0.498819E+05 0.890546E+05 0.712504E+05 0.135121E+06 0.620238E+05 0.341762E+05 Cycles to Failure D=0.0043565

24 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Weibull Data Analysis Linear Least Square Estimates D=0.008713 N=25000  =6.744  =4499 r=0.966 F -1 (0.01)=2274 F -1 (0.10)=3222 F -1 (0.50)=4261 F -1 (0.90)=5091 D=0.0043565 N=25000  =3.575  =87896 r=0.959 F -1 (0.01)=24273 F -1 (0.10)=46837 F -1 (0.50)=79331 F -1 (0.90)=110991

25 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Maximum Likelihood Estimates Can Be Obtained For The Weibull Parameters Abernathy, R., The New Weibull Handbook, 4th edition, North Palm Beach, Florida, 2000

26 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Some Useful Statistical Distributions Weibull Lognormal Birnbaum-Saunders (Fatigue Life) General Extreme Value Gumbel Frechet

27 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Good Statistical Fits (Compact Specimen) Lognormal Birnbaum-Saunders (Fatigue Life)

28 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Not So Good Statistical Fits Both Distributions Excluded At Significance Levels Between 0.01 And 0.20 NormalWeibull

29 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Goodness-of-Fit Tests Goodness-of-fit tests won’t tell you what the distribution function is However, they will tell you that a candidate distribution is unsuitable at a particular significance level Some general goodness-of-fit tests –Kolmogorov-Smirnov –Anderson-Darling –Chi Square (   )

30 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Cumulative Distribution Function  = 6.744  = 4499

31 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Probability Density Function  =6.744  =4499

32 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Reliability Function (Survival Probability)  =6.744  =4499

33 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Hazard Function  =6.744  =4499

34 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Correlated Random Variables  f and b are correlated  f and c are correlated  Failure to account for these correlations will overestimate actual fatigue life variation

35 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Other Fatigue Models

36 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Crack Propagation And Other Fatigue Models The approach is applicable to any fatigue model or crack propagation model Variations in load can be considered Variations in initial and final crack lengths can be modeled The only requirement is that the necessary CDFs be known or estimated

37 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Compact Specimen

38 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Basic Equations Of Crack Growth For A Compact Specimen Anderson, T. L., Fracture Mechanics: Fundamentals and Applications, 1 st edition, CRC Press, 1991

39 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Summary Monte Carlo methods are well known and appropriate for problems of this type. Solution is based on assuming the CDFs for Material Properties are Normally Distributed and known a priori. These assumptions should be replaced with experimental verification. Correlation between paired fatigue variables must be accounted for - otherwise too much variation.

40 Reducing Uncertainty in Fatigue Life Estimates Design, Analysis, and Simulation 1-77-Nastran www.NEiSoftware.com Thank You!

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