1992_system reliability using trivariate and bivariate integrals_Ramachandran

1992_system reliability using trivariate and bivariate integrals_Ramachandran

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computers & stmctwes Vol. 45, No. 5/6, pp. 959-964, 1992 Printed in Great Britaiti. 004~7949B2 S5.00 + 0.W PerpamonpragLtd SYSTEM RELIABILITY USING TRIVARIATE BIVARIATE INTEGRALS K. ~MACHANDRAN Department of Civil Engineering, Imperial College of science, Technology and London SW7 2BU, U.K. (Receiued 23 ~~re~~ 1991) AND Medicine, Abstract-Probabilistic safety analysis of civil engineering structures (system reliability analysis) requires the evaluation of multi-normal integrals and these multi-normal integrals can be estimated either from upper and lower bounds on these probabilities or by first-order second-moment approximate methods or by simulation methods. In this paper a new method based on conditional probabilities for the evaluation of series and parallel structural systems reliabilities is presented. Examples are studied to compare the accuracy of this new approach with other approximate methods. INTRODUCI’ION Reliability theory is now widely used in soil and structural engineering to assess the probability of failure of earth slopes [l-3], propped cantilever [4,5], offshore structures [6,7j and many other engineering structures [8-l 11. The technique for evaluating the reliability index of a single failure mode is well established and better statistical data are now being used in these computations. However, methods for calculating the probability of failure of a structural system with more than three failure modes still lack vigour, and occasionally give unsafe results. The calculation of system reliability usually requires the evaluation of a multi-normal integral and the current practice is to use bounds on the probability as a means of estimating the system failure prob- ability [12-141 or to use an approximate method for the evaluation of this multi-normal integral [ 151 or to use improved Monte Carlo methods [16]. Although the available bounds [12, 13, 17] are reasonably satis- factory, there is a definite need for tinding an exact or near-exact method for reliable estimation of these multi-normal integrals. In this paper, a near-exact method is presented for the determination of these failure probabilities using trivariate and bivariate integrals. PROBLEM DEACON In reliability analysis, each failure mode is rep- resented by a continuous function (safety margin) gi(x) of the statistical variables x. Usually there are a very large number of possible failure modes in a system; however, with a little experience, the number of failure modes which have significant eikct on the reliability of the system (predominant modes) can be substantially reduced. The event g,(x) < 0 is referred to as the ith failure event (mode) 4, and its prob- ability of occurrence denoted by P(,F;,). It is assumed that all P(Fj) and the correlation matrix p can be obtained by suitable transformations into an uncorre- lated standard normal space n [17]. Then the event 1% > /%I, i = 1, n is referred to as the event I$ and
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1992_system reliability using trivariate and bivariate integrals_Ramachandran

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