Optimal Inequalities to Bound a Performance Probability

Abstract

A challenging problem encountered in engineering applications is the estimation of a probability-of-failure based on incomplete knowledge of the sources of uncertainty and/or limited sampling. Theories formulated to derive upper probability bounds offer an attractive alternative because first, they avoid postulating the probability laws that are often unknown and second, they substitute numerical optimization for statistical sampling. A critical assessment of one such technique is presented. It derives upper probability bounds from the McDiarmid concentration-of-measure theory, which postulates that fluctuations of a function are more-or-less concentrated about its mean value. Two applications of this theory are presented. The first application analyzes a “toy” polynomial function defined in two dimensions. The upper bounds of probability are calculated and compared to sampling-based estimates of the true-but-unknown probabilities. For this function, the upper bounds obtained are too broad to be useful. These results are confirmed by conducting a similar analysis on a real engineering system, where upper bounds of probability associated with resonant frequencies of a structural system are estimated. A high-fidelity finite element model, previously validated using vibration measurements, is used to predict the frequencies. In this application, the uncertainty is introduced by way of material properties and the effective preload of a beam-to-column connection, modeled explicitly. These applications suggest that the theory not only leads to upper bounds that are inefficient but that can also be sub-optimal if their numerical estimation is based on too few model runs. It is concluded that this particular theory, while mathematically attractive, may not be well suited for engineering applications.