Quadratic Limit States in Structural Reliability

TLDR

Second‑moment methods are widely used in structural reliability, and first‑order reliability methods have recently been developed to provide reliable failure‑probability estimates for arbitrary designs and uncertainty distributions. This study investigates second‑order expansions to approximate the failure surface and applies quadratic‑form theory to obtain improved failure‑probability estimates. The authors approximate nonlinear relationships with linear Taylor expansions, retaining second‑moment calculus, and compute failure probabilities by evaluating the standard normal integral over a hyperplane‑bounded domain.

Abstract

Second-moment methods are widely applied in structural reliability. Recently, so-called first-order reliability methods have been developed that are capable of producing reliable estimates of the failure probability for arbitrary design situations and distributional assumptions for the uncertainity vector. In essence, nonlinear functional relationships or probability distribution transformations are approximated by linear Taylor expansions so that the simple second-moment calculus is retained. Failure probabilities are obtained by evaluating the standard normal integral, which is the probability content of a circular normal distribution in a domain bounded by a hyperplane. In this paper second-order expansions are studied to approximate the failure surface and some results of the statistical theory of quadratic forms in normal variates are used to calculate improved estimates of the failure probability.