Improved Formula for Upper Bound of Probability of Failure Based on Means and Variances

Abstract

In relation to practical structural reliability analysis, this paper is concemed with the formula $P_{fu} = (4/9)(f_C^2 \eta _R^2 + \eta _S^2 )/(f_C - 1)^2$ for the upper bound of the probability of failure based on means and variances of strength R and stress S, where f _C is the central factor of safety, and η _R and η _S are coefficients of variation of R and S, respectively. This formula was previously derived using Camp-Meidell inequality under the assumptions (a) the distribution of Z(=R-S) is continuous and unimodal, and (b) the mode of Z is equal to the mean of Z. By formulating and solving an optimization problem, the present paper reveals that this formula holds even when only (a) is assumed. This finding extends largely the applicability of this formula in practical analysis of structural reliability.