A stochastic finite element method (SFEM) is developed to study the reliability of structures with material nonlinearities. Bilinear stress-strain relationships that involve two correlated random fields are used to model the material nonlinearities. Random fields are represented as continuous functions by using general orthogonal series expansions, and are incorporated directly into the nonlinear finite element analysis and first-order reliability formulation. The proposed method is illustrated by two examples involving a fixed-end beam and a frame. Random spatial fluctuations in the material properties or distributed loads may have a significant effect on reliability if correlation lengths of random fields are less than the lengths of members, meaning that local fluctuations in the fields are relatively high.

A reduction in the number of random variables carried in a structural-reliability analysis can be achieved by an orthogonal transformation of the basic random vector to a vector of independent standard normal variables, followed by a reduction procedure similar to that employed in modal analysis in structural dynamics. This paper investigates the error introduced by this reduction in the representation of random fields through orthogonal-series expansions. It is found that the error in the second-moment representation of a random field is bounded by the reduction error, which is easily computed, when the correlation length of the random field is short; this often is the case in stochastic finite-element analysis.

The effects of uncertain material properties on the elastic stability of structural members and frames are analyzed using a stochastic finite-element method. The uncertain material properties are modeled as continuous random fields by a series expansion involving orthogonal functions. The expanded random fields are incorporated into the finite-element formulation leading to an eigenvalue problem involving random parameters. A mean-centered second-order perturbation technique is used to find the probabilistic characteristics of the buckling load. Illustrations involving a simply supported beam, a simply supported beam on an elastic foundation, and a frame are presented. These illustrations demonstrate the impact on instability of characteristics of the random fields, including correlation length and coefficients of variation of random flexural rigidity and/or random elastic foundation modulus, finite-element mesh selection, and the interrelation between statistical and finite-element modeling. A comparison of the perturbation solutions with the results from Monte Carlo simulation serves to validate the solutions and identify some of their limitations.

A new approach for first-order reliability analysis of structures with material parameters modeled as random fields is presented. The random field is represented by a series of orthogonal functions, and is incorporated directly in the finite-element formulation and first-order reliability analysis. This method avoids the difficulty of selecting a suitable mesh for discretizing the random field. A general continuous orthogonal series expansion of the random field is derived, and its relationship with the Karhunen-Loeve expansion used in recent stochastic finite-element studies is examined. The method is illustrated for a fixed-end beam with bending rigidity modeled as a random field. A set of Legendre polynomials is used as the orthogonal base to represent the random field. Two types of correlation models are considered. The Karhunen-Loeve expansion leads to a lower truncation error than does the Legendre expansion for a given number of terms, but one or two additional terms in the Legendre expansion yields almost the same results and avoids some of the computational difficulties associated with the use of the Karhunen-Loeve expansion.