On the use of infinite random sets for bounding the probability of failure in the case of parameter uncertainty

This contribution presents a novel technique for estimating the bounds of the probability of failure of structural systems when there is aleatory and epistemic uncertainty in the representation of the basic variables. The proposed methodology allows the designer to model parameter uncertainty without making further suppositions that would be reflected in the estimated value of the probability of failure; since the method takes in consideration all possible variation due to uncertainty in the representation of the basic variables, it gives as an answer upper and lower bounds on the probability of failure. In particular, the methodology allows to represent parameter uncertainty as a possibility distribution, cumulative distribution function, probability box or family of intervals provided by experts. These four representations are special cases of the theory of random sets, which is a generalization of the theories of probability, possibility and interval analysis. Up to the best of the author’s knowledge, until now, all the papers that employ random sets in the case of uncertainty analysis have been confined to a finite random set representation, or to its analog a Dempster-Shafer body of evidence. This implies that the information provided by the experts must be discretized. In this paper, an infinite random set representation is introduced. With this new approach, the information provided does not have to be discretized. Also, a new geometrical representation of the space of basic variables is given, where the methods already existing for estimating the probability of failure and that only require the sign of the evaluations of the basic variables vector on the limit state function may work without additional overhead. Using this kind of methods, the computational cost required for the estimation of the bounds of the probability of failure could decrease notably compared with the discrete approach. Furthermore, the proposed method allows the analyst to model the lack of information about the dependence of the basic variables, and in consequence, provides a new methodology to avoid the misused assumption of independence between the basic variables and the myth that varying the correlation coefficients constitutes a sensitivity analysis for uncertainty about dependence. A benchmark example is used to demonstrate the usefulness of the method.