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

•SM analysis is enhanced to determine the third and fourth central statistical moments.•SM analysis is also implemented for the structures having a local variable parameter.•As an application, SMA is used to determine probability density function of eigenvalues of cracked structures.•Mathematical model of the cracked structure is constructed by Rayleigh-Ritz method.•The results are compared with Monte Carlo simulations.•The results yield that the SM analysis is a promising method for the analysis of the structures having input parameter variation.