Variational Principles Developed for and Applied to Analysis of Stochastic Beams

Abstract

In the present paper the deterministic governing equations and boundary conditions for mean and covariance functions of the displacement for statically determinate beams with spatially varying stochastic stiffness are derived. The corresponding variational principles for the mean and covariance functions of the displacement are established. Based on the governing equations or variational principles, Galerkin and Rayleigh-Ritz methods are proposed to find probabilistic characteristics of the response. Several problems involving stochastic stiffness are exemplified. It is suggested that the displacements corresponding to an associated deterministic beam, which possesses the same geometry and load as the original beam but has a deterministic stiffness, be adopted as the trial functions in Galerkin or Rayleigh-Ritz formulation. Examples show that statically determinate beams with stochastic stiffness can be effectively analyzed by the proposed approximate methods. The agreement between the solutions obtained by the Galerkin or Rayleigh-Ritz method and the exact solutions is extremely good.