STATISTICAL ANALYSIS OF FATIGUE SIMULATION DATA J R Technical Services, LLC Julian Raphael 140 Fairway Drive Abingdon, Virginia

OUTLINE Specimen Raw Material Crack Growth Model Crack Growth Parameter Variations Simulation Results Some Fatigue Models Statistical Considerations Summary

Compact Tension Specimen

Compact Specimen Dimensions (mm) W B a0a0

Applied Loading MN

Validity Check In order that K Ic is valid the following requirement is imposed on the length of the uncracked ligament, W-a, and the thickness, B.

Raw Material Material:Aluminum Alloy 2219 Temper:T87 Orientation:L-T Yield Strength:393 MPa Ultimate Strength:476 MPa Nominal K I c :36 MPa C v of K Ic :0.06

Crack Growth Model Crack Growth Model: Paris C and m Are Normally Distributed Nominal Paris Constants –C = 6.27 x –m = 3.3 Coefficients of Variation –C = 0.05 –m = 0.02

Variations in a 0 and a c The remaining stochastic variables are the initial and final crack lengths, a 0 and a c, respectively. We assign a small coefficient of variation to a 0, because the precrack is short. However, a c is a function of K Ic or K c, so the variations in K Ic account for the uncertainty in a c.

Variations in C Normally Distributed

Variations in m Normally Distributed

Initial Crack Length (m) Normally Distributed

Critical Crack Length Not Normally Distributed

Final Crack Growth Equation The resulting equation for N f, the number of fatigue cycles to fracture, with the stochastic variables shown in red is

Histogram of Simulation Results

Some Statistical Concepts Hypothesis Test Significance Level Null Hypothesis Alternate Hypothesis P - Value Goodness-Of-Fit Confidence Intervals

Hypothesis Testing And The Significance Level A method of making decisions using scientific data A result is statistically significant if it is unlikely to have occurred by chance alone, according to a predetermined threshold probability, the significance level Significance level is the probability of incorrectly rejecting the null hypothesis when it is, in fact, true, i.e. a Type I error

We Have The Failure Data, So What Do We Do Now? State a null hypothesis, e.g., the data are Normally distributed with mean  and standard deviation  Usually denoted as H 0 State the alternate hypothesis, e.g., the data are not Normally distributed with mean  and standard deviation  Denoted H a Pick a significance level, e.g.,  Choose a goodness-of-fit test, e.g., Kolmogorov-Smirnov or Anderson-Darling

Goodness-Of-Fit Testing Compare your data to a standard statistical model such as Normal. Calculate a test statistic and compare that to a known value that depends on the significance level and sample size Goodness-of-fit tests can only tell you if your distribution can be rejected at a specific significance level

How Do We Choose The Best Distribution? Use Goodness-Of-Fit Tests Question - Which GOF Test is Best? Answer - It depends on What You Want For Example –Kolmogorov-Smirnov gives more weight to the center of the distribution –Anderson-Darling gives more weight to the tails

What Is A P-Value? A p-value is a probability Assume that your data do fit the distribution under consideration, i.e., accept H 0 temporarily The p-value is the probability that you would get a goodness-of-fit statistic as extreme or more extreme as the one you got P-values greater than  generally mean that we accept the null hypothesis

PDF Comparison

CDF Comparison

Lognormal Distribution  and  are the mean and standard deviation of ln(t)

Birnbaum-Saunders Distribution 

Confidence Intervals After we’ve determined what distributions cannot be excluded it is necessary to set the ranges over which the parameters can be expected to vary, given the sample size Most commonly we use 95% two-sided confidence intervals on each parameter The confidence intervals depend on the mathematical formulation of the distribution and the sample size The confidence interval does not mean that the parameter will lie between the calculated values, but rather that the true value will lie in the confidence interval 95% of the time

Summary Statistical techniques are available for distribution fitting of fatigue data Historically, these techniques have not been employed frequently Variable amplitude loading can be simulated by applying Miner’s Rule Sometimes the statistical parameters can be estimated as functions of the loading

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