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STATISTICAL ANALYSIS OF FATIGUE SIMULATION DATA J R Technical Services, LLC 140 Fairway Drive Abingdon, Virginia Julian Raphael 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) B W a0 Applied Loading 0.012 MN 0.012 MN Validity Check • In order that KIc is valid the following requirement is imposed on the length of the uncracked ligament, W-a, and the thickness, B. 2 K Ic B, W a 2.5 21 mm ys Raw Material • • • • • • • Material: Temper: Orientation: Yield Strength: Ultimate Strength: Nominal KIc: Cv of KIc: Aluminum Alloy 2219 T87 L-T 393 MPa 476 MPa 36 MPa m 0.06 Crack Growth Model C K • Crack Growth Model: Paris • C and m Are Normally Distributed • Nominal Paris Constants da dN – C = 6.27 x 10-11 – m = 3.3 • Coefficients of Variation – C = 0.05 – m = 0.02 m Variations in a0 and ac • The remaining stochastic variables are the initial and final crack lengths, a0 and ac, respectively. • We assign a small coefficient of variation to a0, because the precrack is short. • However, ac is a function of KIc or Kc, so the variations in KIc account for the uncertainty in ac. 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 Nf, the number of fatigue cycles to fracture, with the stochastic variables shown in red is ac K Ic Nf a0 da C K m 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 m and standard deviation . Usually denoted as H0 • State the alternate hypothesis, e.g., the data are not Normally distributed with mean m and standard deviation . Denoted Ha • Pick a significance level, e.g., a 0.05 • 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 H0 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 a generally mean that we accept the null hypothesis PDF Comparison CDF Comparison Lognormal Distribution 1 f t e t 2 2 ln( t ) m x 1 e F ( x) 2 0 2 ln( t ) m 2 t dt m and are the mean and standard deviation of ln(t) Birnbaum-Saunders Distribution x F x a x 1 z z ) e 2 t2 2 dt a is the shape factor is the scale factor 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 Thank You!