Probabilistic free vibration analysis of functionally graded beams using stochastic finite element methods

In this study, we use the stochastic finite element method for vibration analysis of functionally graded (FG) Euler-Bernoulli beams considering variability in material properties. The selected FG material consists of a mix of ceramic and metal constituents. The material properties of the FG beams studied are assumed to vary smoothly over the depth according to a power law. Constituent material properties such as the Young’s modulus, mass density and volume fraction index are modeled as random variables. For each simulation of these random parameters, finite element method is employed to estimate natural frequencies of FG beam. Several simulations need to be carried out for propagating overall inputs uncertainty to stochastic frequencies that are approximated as a series in an orthogonal space. The components of series will be determined based on both polynomial chaos expansion (PCE) and stochastic collocation (SC) methods. For PCE, the multivariate Hermite orthogonal functions are derived using Askey scheme. Their coefficients are estimated using both spectral projection, linear regression approaches. Standard tensor product is used to integrate the multi-dimensional integrals. In term of SC method, basis functions are Lagrange interpolation functions formed for known coefficients called collocation points. Postanalysis including reliability, sensitivity and distribution of uncertain frequencies are also studied. These results will also be compared with those of Monte Carlo Simulation.