PSI - Issue 28

C. Mallor et al. / Procedia Structural Integrity 28 (2020) 619–626 C. Mallor et. al. / Structural Integrity Procedia 00 (2020) 000–000

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Now, to couple the FSOA with the fatigue crack growth life prediction, the NASGRO model is considered as the multivariable function of random input variables. In the probabilistic analysis, the distribution of the number of load cycles to reach a given crack depth is desired. For that, the original NASGRO crack propagation rate , is adapted isolating and thus the discretised version for every �� crack growth increment leads to Eq. (6). � � � � 1 � � 1 � � � � � �1 � �� �� � � � �1 � � � � � � � (4) where, is the stress ratio, ∆ is the stress intensity factor (SIF) range from the maximum and minimum , i.e. ��� and ��� , is the crack opening function, ∆ �� is the threshold stress intensity factor range, � is the critical stress intensity factor and , , , and are material empirically derived constants. For a more detailed description of the previous equation refer to [13,14]. Afterwards, for each �� crack growth increment, the first to fourth moments of the fatigue lifetime increment � are calculated by applying the full second-order approach method on the discretised version of the NASGRO equation. Then, the first to fourth moments of the fatigue lifetime are obtained by applying the probabilistic NASGRO equations (Pr. Eq.) in [14,15], providing a continuous result along the crack depth . Finally, the expected value of , its variance, the skewness and the kurtosis are calculated based on the first to fourth predicted moments. 2.2. Pearson distribution family fit using the method of moments Commonly, the normal distribution is assumed when there is not much information available about the underlying probability distribution, notwithstanding that this assumption might not reflect the reality in some scenarios. Among the different distributions that can be considered, the Pearson distribution family is used in the methodology as it is a versatile family that covers a broad range of distribution shapes. Additionally, it enables the expression of the parameters of the distribution as a function of the first four moments of the distribution without a priori hypotheses. The Pearson distribution is a family of continuous probability distributions [16]. A Pearson probability density function � � is defined to be any valid solution to the first-order linear differential equation presented in Eq. (5). � � � � � � � � � � � � � � � � � � � � � � � 0 (5) with: � � 4 � � � � 10 � � 12 � � 1� � , � � � � � � � � � 10 � � 12 � � 1� , � � 2 � � � � � � 10 � � 12 � � 1� where, is a location parameter, � is the square of the skewness, � is the kurtosis, and � is the second central moment of the distribution. The solution to the above differential equation is shown in Eq. (6). � � � ��� ��� � � � � � � � � (6) The variety of solutions differ in the values of the parameters , � , � and � . Depending on these quantities, different common probability distributions arise, for instance, the beta, symmetrical beta, gamma, Cauchy, inverse gamma distribution, beta prime, Student's t and the normal distribution. The two, three or four parameters of the particular distribution type can be calculated as a function of the expected value, variance, skewness and kurtosis, i.e. from the first four moments. The formulas to calculate the parameters for each type of Pearson distribution are enclosed in [17]. 3. Results and discussion This example shows the propagation of uncertainty in terms of expected value, variance, skewness and kurtosis of the fatigue lifetime based on NASGRO model. Then it presents the parameters of a particular Pearson distribution