An Efficient Algorithm for Evaluation of First Excursion Probabilities of Uncertain Linear Systems Subjected to General Gaussian Dynamic Loading

In this work, an algorithm to compute the first excursion probability of a dynamic response for uncertain linear systems subjected to general stochastic Gaussian excitation, is presented. The excitation is, besides of being Gaussian, described quite generally by a finite number of deterministic functions of a Karhunen-Loève expansion [1, 2] which allows to represent any type of second moment characteristics of the loading over time, e.g. non-stationarity and evolutionary spectra. The structural uncertainties are modeled by a parametric stochastic approach, where all uncertain structural parameters are described by a probability distribution. The probability of first excursion is computed by conditional integration over the space of uncertain parameters. In this procedure, the problem reduces to the evaluation of first excursion probabilities for deterministic structural systems described by points in the parameter space for which efficient numerical methods are available. In a first step of the proposed approach, the critical domain within the parameters describing the uncertain linear system is determined, i.e. the domain for which the contribution to the first excursion probability is highest. A extended procedure of standard Line Sampling [3, 4] will be used to perform the conditional integration over the space of uncertain parameters, where the conditional integration is carried out along parallel lines in the structural parameter space. The lines are selected at random by direct Monte Carlo such that the critical domain is covered. An estimate for the excursion probability and its variance is then obtained from the statistics of the first excursion probability of all lines. It is shown that a small number in the range of two hundred of Finite element analyses are sufficient to obtain a robust estimate for the first excursion probability, making the approach very efficient when compared with available alternatives.