Unified Approach to Probabilistic and Possibilistic Analysis of Uncertain Systems

Various methods exist for assessing the safety of a structural or mechanical system that has uncertain parameters. These methods are either statistical (probabilistic), in which case the probability of failure is sought, or deterministic (possibilistic), in which case bounds on the response are sought. Well-known statistical methods include the first-order reliability method (FORM) and the second-order reliability method (SORM), while deterministic methods include interval analysis, convex modeling, and fuzzy set theory (although the categorization of the latter approach as a deterministic method is debatable). The development of probabilistic and possibilistic methods has tended to occur independently, with specialized algorithms being developed for the implementation of each technique. It is shown here that a wide range of probabilistic and possibilistic methods can be encompassed by a single mathematical algorithm, so that, for example, existing codes for FORM and SORM can potentially be employed for other methods, thus allowing the designer to readily choose the method most suited to the available data. A second common algorithm is also derived, and the analysis is illustrated by application to a simple system composed of N structural components in series.