Analysis of mean and mean square response of general linear stochastic finite element systems

A general finite element-based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields. A flexibility-based formulation is followed that does not involve any approximations. Closed-formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper. These integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function (MRF) and the variability response function for the mean square response (VRF1). The MRF and the VRF1 have many similarities with the classical variability response function (VRF) used to estimate the variance of the response displacement. The existence of the MRF and the VRF1 depends on a conjecture that is validated numerically using a brute force Monte Carlo simulation approach. A so-called finite element method-based fast Monte Carlo simulation procedure (FEM-FMCS) is introduced for the accurate and efficient numerical evaluation of the MRF and the VRF1. This methodology is established for the analysis of stochastic beam/frame structures, as well as for more general stochastic finite element systems (i.e., plane stress and shell problems). Numerical examples are provided including a statically indeterminate beam, a plane stress problem and a shell structure. The MRF and VRF1 are used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the mean value of the response displacement. Throughout this paper, the VRF is computed in parallel to the MRF and the VRF1. It should be noted that although the concept of the VRF has been introduced in an earlier work, this is the first time that a flexibility-based approach has been used to estimate the VRF for two-dimensional problems.