Stochastic Finite Element Method: Response Statistics

Abstract

An important objective of a stochastic finite element analysis of an engineering system should be the determination of a set of design criteria which can be implemented in a probabilistic context. In other words, it is important that any method that is used for the numerical treatment of stochastic systems be compatible with some rationale that permits a reliability analysis. Traditionally, this problem has been addressed by relying on the second order statistical moments of the response process. These moments offer useful statistical information regarding the response of the system and equations will be developed in an ensuing section for their determination. However, second order moments do not suffice for a complete reliability analysis. For this, higher order moments and related information are required. In this context, the response of any system to a certain excitation may be viewed as a point in the space defined by the parameters describing the system. In other words, for every set of such parameters, the response is uniquely determined as a function of the independent variables. When these parameters are regarded as random variables, the response function may be described as a point in the space spanned by these variables, defining a surface that corresponds to all possible realizations of the random parameters.