Finite Element Methods in Probabilistic Structural Analysis: A Selective Review

Abstract

This review examines the field of structural analysis where finite element methods (FEMs) are used in a probabilistic setting. The finite element method is widely used, and its application in the field of structural analysis is universally accepted as an efficient numerical solution method. The analysis of structures, whether subjected to random or deterministic external loads, has been developed mainly under the assumption that the structure’s parameters are deterministic quantities. For a significant number of circumstances, this assumption is not valid, and the probabilistic aspects of the structure need to be taken into account. We present a review of this emerging field: stochastic finite element methods. The terminology denotes the application of finite element methods with a probabilistic context. This broad definition includes two classes of methods: (i) first- and second-order second moment methods, and (ii) reliability methods. This paper addresses only the first category, leaving the second to specialists in that area. The contribution of this review is to illustrate the similarities and differences of the various methods falling in the first category. Also excluded from this review are simulation methods such as Monte Carlo and response surface, and methods that use FEM to solve deterministic equations (Fokker–Planck) governing probability densities. The essential conclusion is that the second moment methods are mathematically identical to the second order (except for the Neumann expansion). The essential distinction that can be made regarding stochastic FEM is the nature of the structure: It can be deterministic or random. By random structure is meant one with parameters that have associated uncertainties, and thus which must be modeled in a random form. Although the randomness in the structure can be of three types, random variable, random process in space, and random process in time, discussion will be limited to the first two categories. While keeping the emphasis on finite element methods, other techniques involving finite differences, which are useful in the study of multi-degree-of-freedom systems, are briefly mentioned. The present review covers only developments that are derived from the engineering literature, thus implying near-term applicability.