PSI - Issue 28

C. Mallor et al. / Procedia Structural Integrity 28 (2020) 619–626

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C. Mallor et. al. / Structural Integrity Procedia 00 (2020) 000–000

1. Introduction The structural integrity and fatigue assessment of a damage–tolerant component, such as a railway axle, relies on a proper lifespan prediction to define a maintenance program for crack detection [1,2]. This lifespan prediction is governed by the fatigue crack growth (FCG) process which is affected by many uncertainties. For example, the statistical nature of the mechanical properties of materials observed when repeating experiments [3,4], the scattering of loads during the component operation [5,6], and the uncertainties inherent to geometrical parameters that may not be exactly as designed [7]. The existence of these uncertainties causes variability in the axle performance leading to different results that can only be predicted in a statistical sense. For this reason, the use of deterministic approaches cannot predict properly the fatigue lifespan and that is why many researchers propose probabilistic analyses [8–12]. Generally, the propagation of uncertainty is the quantification of uncertainties in the system output propagated from uncertain inputs. Commonly, it is focused on the calculation of the probability density function of a model response as a result of the randomness of the input sources. For that purpose, the Pearson distribution family is widely used as it is a good way to construct probability distributions based on prescribed statistical moments. Its cornerstone is the calculation of the moments of the output lifespan , i.e. the expected value, the variance, the skewness and the kurtosis. This moments can be estimated by applying the full second order approach (FSOA) to the fatigue crack growth NASGRO model [13] as thoroughly described in [14,15]. The previous probabilistic procedure is considered to be a rigorous approach to uncertainty analysis in fatigue crack growth life. This paper focuses on a novel propagation of uncertainty methodology for efficiently estimating the parameters of the probability distribution of fatigue lifespan considering the Pearson distribution family. For that, the Pearson distribution type is determined and its parameters are calculated based on the first four prescribed moments of the underlying lifespan distribution according to the NASGRO equation. Due to the fact that probabilistic approaches often introduce complex mathematical calculations and, in fact, they are sometimes intractable when handled in detail, the target here is to promote a deeper understanding of the methods mentioned above rather than to provide a collection of equations. The methodology is discussed in the context of a practical numerical example specifically addressing the fatigue crack growth in a metal railway axle, providing very useful results. 2. Propagation of uncertainty methods Propagation of uncertainty deals with the quantification of the effect of the input variables randomness on the uncertainty of functions based on them. For example, when using random inputs to calculate something else, the input variability propagates through the function and thus has an effect on the output calculation. To take this propagation into account, the objective of the methodology is to efficiently estimate the parameters of the probability density distribution of a random output variable of interest. To establish a general methodology, the Pearson distribution family is considered because it is a rich family that is able to adjust a broad range of distribution shapes and it is easy to use in practice because the different distribution parameters can be expressed as function of the first four moments without a priori hypotheses and therefore it is an efficient and objective procedure. The first four moments of the fatigue crack growth life distribution are obtained by using the full second-order approach (FSOA). 2.1. Full second-order approach for the first four moments of the probabilistic fatigue crack growth life The full second-order approach (FSOA) for the moments of functions of random variables presented in [14] enables the prediction of the expected value and the variance of the fatigue lifespan of interest. Further extensions developed in [15] enable the prediction of the skewness and the kurtosis of the probabilistic fatigue crack growth life. The first order second-moment (FOSM) method is the theoretical foundation of the FSOA. For simplicity, this section focuses on the most general equations for the expected value and covariance matrix for the FOSM method in matrix form and then illustrates how the FSOA is applied to the fatigue crack growth NASGRO model for propagating the first to fourth moments of the fatigue life .