Stochastic finite element method for free vibration characteristics of random FGM beams

The stochastic free vibration characteristics of a functionally graded material (FGM) beam with material properties graded across spatial position are examined in this paper, in which both physical parameters of each constituent material and volume fraction index denoting material distribution are treated as random variables with low variability. Integrating random variables into modelling by using the first-order perturbation technique, a stochastic finite element model is developed for the FGM beam and the direct relationship is derived between the first-derivatives of structural modal frequencies and mode shapes with respect to random variables and those of the FGM effective material properties with respect to random variables. The material graduation in FGMs is represented by power-law function. Comparing with the Monte-Carlo simulation, the influence of constituent material properties, material distribution and slenderness ratio on the statistics of free vibration characteristics for the FGM beam is systemically examined. The results show that the randomness coming from constituent material physical properties contributes much more to the scatterings of modal parameters than the randomness from material distribution while material distribution plays an important role in the stochastic study of the structure, and the slenderness ratio has significant effect on the mean values of both modal frequencies and mode shapes.