Reliability of hysteretic systems subjected to white noise excitation

Abstract

A single-degree-of-freedom bilinear hysteretic system driven by filtered white noise is analysed by means of stochastic differential equations. Failure of the system takes place by first passages at critical levels of so-called damage indicators, which are non-decreasing response quantities measuring at a macroscopic scale the strength and stiffness deterioration. For illustration the cumulative plastic deformation ratio has been used as damage indicator. The white noise excitation is filtered through a time-invariant rational second-order filter. The analytical technique applied is based on a replacement of the original system with non-linear and non-analytical right hand sides for the constitutive equation and damage indicator differential equation by an equivalent system, where these equations are given by a cubic polynomial expansion. The expectations appearing in the equations for the coefficients of the equivalent polynomials are calculated by a multi-dimensional Gram-Charlier expansion. In order to close the infinite hierarchy of moment equations the conventional cumulant-neglect closure scheme and a modification with due consideration to the finite probability of yielding are considered. The analytical results obtained are compared to those obtained by Monte Carlo simulation, from which it is concluded that substantial improvements are obtained by applying the modified closure scheme.