Distribution of Quadratic Forms in Normal Space—Application to Structural Reliability

Abstract

In second‐order reliability methods the failure surface in the standard normal space is approximated by a parabolic surface at the design point. The corresponding probability is computed by asymptotic formulas and by approximation formulas. In this paper the probability content of the parabolic failure domain is computed exactly by inversion of the characteristic function for the parabolic quadratic form. Also, the exact result for the probability content of the failure domain obtained from the full second‐order Taylor expansion of the failure function at the design point is presented. The approximating parabola does not depend on the formulation of the failure function as long as this preserves the original failure surface. This invariance characteristic is in general not shared by the approximation obtained using the full second‐order Taylor expansion of the failure function at the design point. The exact results for the probability content of the approximating quadratic domains significantly extend the class of problems that can be treated by approximate methods.