Stochastic dynamics of geometrically non-linear structures with random properties subject to stationary random excitation

Abstract

A non-linear stochastic finite element formulation for the stochastic response analysis of geometrically non-linear, elastic two-dimensional frames with random stiffness properties and random damping subject to stationary random excitations is derived, utilizing deterministic shape functions and random nodal displacements. Hence, a consistent weighted integral method has been applied for the discretization of the random fields of beam elements. The discretized second order non-linear stochastic differential equations with random coefficients are then solved applying the total probability theorem with a mean-centered second order perturbation method in the frequency domain to evaluate the unconditional statistics of the response. Zeroth, first and second order perturbations are computed using a spectral approach in which a system reduction scheme to the modal subspace expanded by the deterministic linear eigenmodes and equivalent linearization with Gaussian closure are applied. Sample frames are solved to illustrate the validity range of the second order perturbation random vibration analysis in terms or variability of the random damping ratios and the random bending rigidity held as well as the correlation length of the random bending rigidity field. Computed results are compared with the ones obtained from extensive Monte Carlo simulations.