Generalized variability response functions for two-dimensional elasticity problems

The ability to determine probabilistic characteristics of response quantities in structural mechanics (e.g. displacements, stresses) as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters. The concept of the variability response function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). Proofs have been provided for the existence of VRFs for linear and some nonlinear statically determinate beams. For statically indeterminate structures, the Monte Carlo based generalized variability response function (GVRF) methodology has been proposed recently as a generalization of the VRF concept to indeterminate linear and some nonlinear beams. The methodology computes GVRFs, which are analogous to VRFs for statically determinate structures, and evaluates their dependence (or lack thereof) on the PDF and SDF of the random field, thereby providing an estimate of the accuracy of the GVRF. In this paper, the GVRF methodology is extended to problems involving two-dimensional, linear continua whose stochasticity is characterized by statistically homogeneous random fields. After detailing the GVRF methodology for two-dimensional random fields, two numerical examples are provided: GVRFs are computed for the displacement response and for the effective compliance of linear plane stress systems.