PSI - Issue 38

Robin Hauteville et al. / Procedia Structural Integrity 38 (2022) 507–518 Robin Hauteville, Xavier Hermite, Fabien Lefèbvre / Structural Integrity Procedia 00 (2021) 000 – 000 7 o For each stress level the parameter (μ for a log -normal distribution, for a Weibull distribution) is assumed to follow a S-N curve, whose parameters are ( 0 , 1 , ..., ). o For example, for a Basquin’s law (inverse power law): = 0 + 1 ln( ) (14) where "Failed" is the number of failures, "Censored" is the number of survivals, and Failed + Censored is the number of tests. Because product manipulation is complex, it is much more interesting to transpose this equation into logarithm: Λ( , 0 , 1 , … , ) = ln(L( , 0 , 1 , … , )) = ∑ ln( ( , , 0 , 1 , … , , )) =1 + ∑ ln (1 − ( , , 0 , 1 , … , , )) =1 (15) The parameters { 0 , 1 , … , } are estimated by maximising the log-likelihood, i.e. by solving the system of (k + 2) partial derivatives: Λ = 0 et Λ = 0 , = 0 … (16) Often, solving this system of partial derivatives is tedious. Therefore, generally, the estimation of parameters by maximising likelihood is performed through numerical methods (iteration by solver using Newton-Raphson or 513 The likelihood is therefore expressed as: L( , 0 , 1 , … , ) = ∏ ( , , 0 , 1 , … , , ) × ∏ [1 − ( , , 0 , 1 , … , , )] =1 =1 Cetim proposes to use the maximum likelihood method on the three models seen above to take into consideration the censored data regardless of the model studied. However, for non-asymptotic models such as Basquin, censored data can improve the parameter estimates but may not be considered because they are not relevant for that kind of behaviour. Through the model parameters calculation, Cetim proposal is also to make 3 statistical tests to assess the goodness of fit of distribution law on a dataset (lifetime at a stress level) and of the studied S-N curve on the test results (lifetime at all stress levels). 5.1. Outliers value test (Grubbs ’ test) [10] The Grubbs test (also called maximum normed residual test) is used to detect outliers in terms of means deviation. It is operated here on a dataset of lifetimes at a stress level. Grubbs' test detects one outlier at each iteration and is only valid for a sample size greater than or equal to 3 and it is not adapted for censored data. The test is defined by assuming: reduced gradient methods). 5. Cetim methodologies

H0: There are no outliers in the dataset

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• H1: There is at least one outlier in the dataset The Grubbs test statistic is defined by: = | − ̄ |