Level Crossings of Random Processes

Abstract

In Section 5.2 of the lecture on basic reliability concepts, [4], the role of crossing theory in structural reliability analysis is briefly discussed. The present lecture goes into details with some fundamental topics of crossing theory that are of particular applicational interest in reliability theory.

Outcrossings of a general vector process from a safe set in the transformed formulation space are characterized. The linear interpolation mean outcrossing rate is defined. It is shown that it is an upper bound on the failure probability during a time unit. The classical formula of Rice for the mean outcrossing rate is by a heuristic argument shown to follow from the definition of the linear interpolation mean outcrossing rate. As an example, the mean outcrossing rate of a Gaussian n-dimensional vector process out of a convex polyhedral set is considered. It is demonstrated that it can be bounded by use of the elementary uncertainty algebraic methods of evaluating the generalized reliability index for a convex polyhedral safe set.

The number of linear interpolation outcrossings of the sum of two vector processes are characterized in a way leading to an upper bound on the mean outcrossing rate of the sum. It is a generalization of the so-called point crossing bound due to Cornell et al. for the sum of two mutually independent scalar processes.

Everywhere in the following the time parameter t is restricted to the interval [0,T]. In order to keep typing simple, vectors are not distinguished from scalars in the typography.