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 μ
R and μ
S, and variances σ
2R and σ
2S 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)(
f2cη
2R+η
2S)/(
fc-1)
2 using Camp-Meidell inequality, where
fc≡μ
R/μ
S is the central factor of safety, and η
R≡σ
R/μ
R and η
S≡σ
S/μ
S 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 η
Z, the coefficient of variation of
Z, is small, and gives a somewhat higher value when η
Z 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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