Reexamination of Convex Polyhedron Outcrossing Problem

Abstract

Structural series‐system reliability analyses formulated solely by use of a finite number of random variables can often be done with sufficient accuracy in terms of first‐ or second‐order upper and lower bounds on the system failure probability. The bounds are of the first order if they are defined solely by the single‐element‐failure probabilities, and of the second order if they also include the probabilities of the intersections of the failure events of any two single elements. A reexamination of the proof of the most well‐known pairs of second‐order bounds shows that almost the same reasoning can be used to obtain similar formulas for upper and lower bounds on the outcrossing rate of a general vector process out of the safe set of a series system. For polyhedral safe sets and smooth Gaussian vector processes, the needed outcrossing formulas are given explicitly in a form that is well suited to be put on computer.