Finite element reliability methods for geometrically nonlinear stochastic structures

A general framework for first- and second-order reliability analysis of structures with geometrical nonlinearity is presented. The structure is either linear or nonlinear elastic, and is subjected to static loads. The material properties, geometry, and external loads of the structure are considered as random variables or random fields. The failure criteria of the structure is expressed in terms of limit-state functions.; Four major steps are involved in the first- and second-order reliability methods: (1) selection of probability models for random variables and random fields, and representation of the latter in terms of random variables; (2) transformation of the random variables into a set of independent, standard normal variates; (3) iterative solution of a constrained optimization problem to find the nearest point on the limit-state surface to the origin in the standard normal space; and (4) integration of the failure probability using a first- or second-order approximation of the limit-state surface. In the course of the optimization programming, the structural response and its gradient with respect to the basic random variables are required at each iteration step. The finite element method is used to compute these two quantities. Analytical formulas for the gradient of the response are derived to improve the efficiency and accuracy of the reliability computation. The formulas are in terms of the tangent stiffness matrix and the initial load stiffness matrix, which are readily available if Newton's method is used to solve for the structural responses. No iterations are involved in the computation of the gradient.; A general-purpose reliability code, CALREL-FEAP, is developed to perform the finite-element reliabilities analysis. The reliabilities of a built-up column and a stochastic plate with a random hole are studied using this code. Sensitivities of the failure probabilities with respect to parameters in the probability distribution functions or in the limit-state functions are examined. The usefulness of these sensitivity measures in structural design process is demonstrated.