A new numerical method for general failure probability with fuzzy failure region

For general reliability analysis with fuzzy failure region F̃, the general failure probability P_F̃, is defined as the integral of product of μ_F̃[g(y)], the membership of performance function g(y) to F̃, and joint Probability Density Function (PDF) f(y) over Ω, the total variable space, i.e. P_F̃ = ∫⋯∫ μ_F̃p[g(y)]f(y)dy. On the basis of line sampling, an efficient method for random failure probability analysis with clear failure region, a new numerical method is presented to calculate P_F̃ . In the presented method, the total integral region Ω is split into m clear sub-regions F_i in a way that the value of g(y) in F_i can be approximately viewed as g_i, a constant independent of y, and the value of μ_F̃[g(y)] in F_i can be viewed as a constant μ_F̃(g_i) subsequently. Due to the closely invariant property of μ_F̃[g(x)] in F_i and Ω = F₁ ∪ F₂ ∪⋯∪ F_m, P _F̃ is transformed into the sum of μ_F̃(g _i) ∫⋯∫_Fi f(y)dy , where ∫⋯∫ _Fi f(y)dy is the random failure probability with the clear failure region F_i and can be obtained by line sampling. The high efficiency of the presented method resulted from that of the line sampling is demonstrated by the illustration.