Interval reliability analysis under the specification of statistical information on the input variables

Structural reliability analysis should often face the problem that there is uncertainty about the probabilistic specification of the input random variables implied in a specific problem. In the poorest information case only the knowledge of intervals of fluctuation is available, while the opposite situation is that of a full specification of the joint distribution function of the variables. Parametric or non parametric probability boxes, arising from basic statistical tools such as confidence intervals or Kolmogorov-Smirnov distance, are intermediate situations. In any case, random or fixed intervals emerge for describing eitheistribution parameters under epistemic uncertainty. A crude Monte Carlo simulation for assessing the corresponding interval of the failure probability requires the solution of a double optimization problem for each probability level of the limit state function. This, obviously, implies a prohibitive computational labor. In this paper a method for drastically simplifying this task is proposed. The method exploits the ordering statistics representation property of the reliability plot, which is shown to approximately obey an orthogonal hyperbolic pattern. Accordingly a two-level FORM approach with relaxed accuracy constraints is used to derive the polar vectors for building two plots, one for the input variable space and the second to the epistemic uncertainty space. Using a variety of examples, it is demonstrated that the extrema of the failure probability are contained amongst the samples located in specific sectors of the upper level plot as indicated by the hyperbolae. This means that, after solving the two-level FORM problem, it suffices to calculate the failure probability (or the reliability index) for a small number of mean samples thus selected. Obviously, the method yields the same reliability interval estimates as the crude two-level Monte Carlo because its samples are underlying in it.