The exact element stiffness matrices of stochastically parametered beams

Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano's approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen-Loeve expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods.