Variability response functions and the weighted integral method in stochastic finite element analysis.

摘要

In the field of stochastic finite element analysis, various methodologies have been developed for evaluation of response variability of stochastic structures. Few authors in this area have successfully analyzed structures with more than one stochastic material or geometric property; even fewer have been able to consider cross-correlation between material properties. In this thesis, the response variability of structures with multiple stochastic material and/or geometric properties are calculated using variability response functions, which provide a means of evaluating the effects of cross-correlation between stochastic properties.; Variability response functions allow calculation of spectral-distribution-free upper bounds on the response variability, which depend only on the mean, variance, and cross-correlation coefficient of the stochastic material/geometric properties. Such bounds are of paramount importance for the majority of real-life problems where only first and second moments of the stochastic material properties can be estimated with reasonable accuracy. Under the assumption of prespecified power spectral density functions of the material/geometric properties, it is also possible to compute the response variability and the reliability of the stochastic structure.; The concept of the variability response function is first applied to the random displacement vector of statically loaded structures. Under the assumption of a stochastic elastic modulus, a plate bending formulation is achieved, overcoming earlier computational problems associated with the large number of terms in the expression for the variability response function. Variability response functions are then successfully formulated for statically loaded plane stress/strain structures with randomly varying elastic modulus, Poisson's ratio, and thickness. General guide-lines are provided for further extension to stochastic problems involving shells, three-dimensional structural systems, etc.; The random eigenvalue problem has been of interest for many years, but few of the solutions have been able to consider multiple stochastic material/geometric properties in an elegant manner. The variability response functions presented here provide a general method of analyzing the random eigenvalues of structures with a stochastically varying elastic modulus and mass density, which may be cross-correlated to any degree. As a practical application of the random eigenvalue problem, variability response functions are calculated which consider the random maximum deflection vector of a stochastic structure under design (deterministic) earthquake loading.; All the above methods are easily extended to analyze other types of finite elements, such as three-dimensional finite elements, or to consider anisotropic materials. To formulate the stochastic finite element matrices, both a weighted integral and a local average approach are presented. Numerical examples are given to demonstrate the capabilities of the variability response functions.