Application of Spectral Stochastic Finite Element via Galerkin Method in Beam Bending Theories

In stochastic structural mechanics it is possible to associate uncertainties with material properties, geometry, and external loading, whereas response estimates are found in the displacement, stress, strain, frequency, and phase difference fields, among others. This paper presents the stochastic formulation of elliptic differential equations associated with the beam bending problem and explores the propagation of uncertainty from the perspective of the Neumann-Monte Carlo (NMC) asymptotic complexity methodology and the Spectral Stochastic Finite Element Method (SSFEM). The variational solution is studied for the classical Euler-Bernoulli theory (EBT) and for theories involving Timoshenko shear deformation (TBT) and Levinson-Bickford (LBT). The numerical results present the expected value, variance and covariance function for the displacement field of those three beam theories. Observations on the relationship between beam length and height are included in the comments for cases in which the initial variability is associated with the random coefficients of the stiffness matrix.