On the Reliability of a Simple Hysteretic System

Abstract

A method to determine statistical moments of time to first passage of a simple oscillator, incorporating the modified Bouc hysteresis model, has been developed. A generalized Pontriagin‐Vitt equation is formulated from Markov process theory and solved by a Petrov‐Galerkin finite element method. The resulting moments are given as functions of initial displacement and velocity of the oscillator, as well as of the initial hysteretic force in the oscillator. The first two moments are used in conjunction with a maximum entropy distribution to estimate the probability of failure of the oscillator. The accuracy and economy of the method is demonstrated for a particular example drawn from base isolation of a simple structure. A comparison of the finite element results with those obtained by Monte Carlo simulation is then given for this example.