Towards Efficient Ways of Estimating Failure Probability of Mechanical Structures under Interval Uncertainty
Whether a structure is stable depends on the values of the parameters θ = (θ1, . . . , θn) which describe the structure and its environment. Usually, we know the limit function g(θ) describing stability: a structure is stable if and only if g(θ) > 0. If we also know the probability distribution on the set of all possible combinations θ, then we can estimate the failure probability P. In practice, we often know that the probability distribution belongs to the known family of distributions (e.g., normal), but we only know the approximate values p˜i of the parameters pi characterizing the actual distribution. Similarly, we know the family of possible limit functions, but we have only approximate estimates of the parameters corresponding to the actual limit function. In many such situations, we know the accuracy of the corresponding approximations; i.e., we know an upper bound Δi for which |p˜i - pi| ≤ Δi. In this case, the only information that we have about the actual (unknown) values of the corresponding parameters pi is that pi is in the interval [p˜i - Δi, p˜i + Δi]. Different values pi from the corresponding intervals lead, in general, to different values of the failure probability P. So, under such interval uncertainty, it is desirable to find the range [P_, P– ]. In this paper, we describe efficient algorithms for computing this range. We also show how to take into account the model inaccuracy, i.e., the fact that the finite-parametric models of the distribution and of the limit function provide only an approximate descriptions of the actual ones.