Excursion Probabilities for Linear and Non-linear Systems

This paper suggests a procedure for estimating excursion probabilities for linear and nonlinear systems subjected to Gaussian excitation processes. The stochastic excitation process is modeled most generally by the Karhunen-Loève expansion. The approach is based on the so called “averaged excursion probability flow” which allows for a simple solution for the interaction in excursion problems. Simplifying, the dynamic reliability problem can be reduced to a simpler “static” problem by considering the probability flow at fixed time instances. The proposed approach is very general and can be applied to both linear and non-linear systems of which the response can be determined by deterministic methods. Hence the procedure applies to arbitrary structures and any suitable mathematical model including large FE-models solved by deterministic FE-codes. INTRODUCTION The excursion probability is one of the most basic reliability measures in stochastic dynamics. Although this problem received considerable attention in the past (see e.g. [9, 4, 10, 3, 6] ), no general applicable analytical procedures emerged, not even for the most simple one degree of freedom linear oscillator. It seems that the problem is too complex to be solved analytically without stringent assumptions. Simulation offers an alternative approach to study the complexity related with the dynamics of the structural model. Deterministic structural analysis is nowadays well developed and complex structural models with linear and nonlinear behavior can be computed straight