Journal of the Society of Materials Science, Japan
Online ISSN : 1880-7488
Print ISSN : 0514-5163
ISSN-L : 0514-5163
Evaluation of Upper Bounds of Probability of Failure and Factor of Safety Based on Means and Variances
The Case When Only Continuity and Unimodalness are Assumed
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1989 Volume 38 Issue 435 Pages 1441-1445


This paper is concerned with evaluation of the upper bound of probability of failure when the distribution forms of the strength R and the stress S are unknown and only their means μR and μS, and variances σ2R and σ2S are known. In the previous paper, assuming (a) the distribution of Z≡R-S is continuous and unimodal and (b) the mode of Z is equal to the mean of Z, the present author derived the formula PfU=(4/9)(f2cη2R2S)/(fc-1)2 using Camp-Meidell inequality, where fc≡μRS is the central factor of safety, and ηR≡σRR and ηS≡σSS are the coefficients of variation of R and S, respectively. However, the assumption (b) can be satisfied only in limited cases. Thus, it was attempted in the present paper to evaluate PfU assuming only (a) without assuming (b). The problem was formulated as an optimization problem and solved using linear programming. It was found that the above formula can still be used under the assumption (a) alone. That is, the formula gives an almost exact value of PfU under the assumption (a) alone when ηZ, the coefficient of variation of Z, is small, and gives a somewhat higher value when ηZ is large. Hence, practical applicability was given to the above formula in the present study. The upper bound of the central factor of safety can also be obtained from this formula only by assuming the condition (a). Thus, by merely assuming that the distribution of Z≡R-S is continuous and unimodal, considerably lower upper bounds can be obtained for the probability of failure and for the factor of safety as compared with the case when no assumption is made.

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