| Summary: | This study describes a method to reliably search and determine design points required for constructing importance sampling probability density functions (p.d.f.s) in the method of estimating the structural failure probability based on multiple importance sampling simulation. In order to search out and determine all of design points without any omission, the initial points of Rackwitz Fiessler (RF) algorithm that performs the design point search are proposed to be set on limit state surfaces within the entire quadrants of the basic random variables space. First, from the standardized normal p.d.f., several samples, referred to as real samples, are generated in the specific quadrants. Using the newly proposed "inter-quadrant relational expression", the coordinates of the real sample points are converted to the coordinates of the sample points called pseudo samples in all the quadrants except those of the real sample points. In this way, real samples or pseudo samples are generated and determined in all quadrants. Next, the coordinates of intersection points of the limit state surface and straight lines extending from the origin and passing through respective real and pseudo sample points are determined. And the intersections determined on the limit state surfaces in all quadrants are proposed to adopt as initial points of RF algorithm. By executing RF algorithm from respective initial points determined, all of design points are searched out completely, and the importance sampling p.d.f.s are constructed by making use of the respective design points determined. Numerical examples for estimating the failure probability of structural systems having multivariable nonlinear limit state functions are shown based on multiple importance sampling simulation using all of the constructed importance sampling p.d.f.s and the proposed method shows accurate estimations of structural failure probabilities.
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