A higher order statistical moment based approach for the distribution of eigenvalues of beams with variable cracked depth

Abstract

Statistical moment analysis proved its accuracy on the determination of mean and variance of free and forced vibration response for the structures having normally distributed global input and output parameters. Apart from the mean and variance, higher order standardized central statistical moments (CSMs) i.e., skewness and kurtosis, must also be evaluated for the nonnormal distributions to obtain probability density function of random variables. In this study, statistical moment analysis is enhanced to calculate higher order CSMs in order to evaluate the distribution of eigenvalues of beams. The method is applied for intact and cracked beams having variable parameters. For the intact beam case, higher order CSMs of eigenvalues are determined corresponding to the normally distributed global variable parameter, i.e., Young’s modulus. The latter application of the method is on the cracked beams having a nonnormal variable local parameter, i.e., crack depth. For this case, the method is tested by modelling the crack depth by two different distributions. In this regard, firstly, expressions of CSMs are analytically derived for the mathematical operations (summation and multiplication) of two statistical variables. Then, these expressions are fed to mathematical model (constructed via Rayleigh Ritz method) of cracked beams to calculate statistical moments of eigenvalues. Next, distributions of the eigenvalues corresponding to the variable cracked depth are obtained by utilizing CMSs in Pearson distribution. The results are compared with Monte Carlo simulation and present unique advantages in the sense of computational cost for the structures having variable parameters.