Structural Safety, 5 (1988) 267-284 267 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands INTEGRATION FORMULAS TO EVALUATE FUNCTIONS OF RANDOM VARIABLES Jianhua Zhou and Andrzej S. Nowak Department of Civil Engineering, University of Michigan, Ann Arbor, M148109 (U.S.A.) (Received December 15, 1987; accepted in revised form September 10, 1988) Key words: reliability; numerical integration; functions; statistical moments; random variables; probability; statistical parameters. ABSTRACT Structural reliability analysis often involves extensive integration of a function of multiple basic variables. Special techniques can be used to reduce this numerical effort. This paper develops integration formulas for computing the statistical parameters of a function of a random vector, in particular calculation of the first few moments. The formula is a numerical procedure using selected weights and points to estimate integrals. The points and weights are predetermined in the indepen- dent standard normal variable space. The sample points in basic variable space are then obtained by various transformations. The formulas were developed for the basic variables with various probability information known. The procedure is convenient to implement, and allows for a direct use of the available deterministic computer programs. INTRODUCTION Recently, there have been extensive new developments in reliability theory and its application to structural engineering. Especially noteworthy has been the work of Ang and Cornell [1], Galambos and Ravindra [2], and Ellingwood et al. [3] in the United States, Lind [4] in Canada, and Ditlevsen [5] and Rackwitz [6] in Europe. A more critical review on structural reliability methods can be found in the book by Madsen et al. [7]. In structural reliability analysis, load and resistance models play a very important role. The models are derived using survey data, tests, analysis and engineering judgement. Often the resistance or load is expressed as a function of multiple basic variables such as material properties, dimensions, geometries, frequency and magnitude of load, and so on. It is then necessary to evaluate the statistical parameters, particularly the first few moments, of these 0167-4730/88/$03.50 © 1988 Elsevier Science Publishers B.V.