A single-layer dimension-reduced cubature method for sequential updating of the first two moments of the posterior failure probability

In the context of random inputs with uncertain distribution parameters and the gradual collection of new observations during the service life of a structure, the structure’s safety level can be tracked via the sequential update of the first two moments of the posterior failure probability. However, this requires repeated double-layer analysis for the estimation of the first two moments of the posterior failure probability as observations are gradually collected. To address the high cost of calculating the first two moments of the posterior failure probability, this study proposes a single-layer method featuring two main innovations. The first innovation is to equivalently derive the dimension-reduction integration expression of the first two moments of the posterior failure probability. This derivation improves the integrand behaviour and enables the efficient estimation of the first two moments of the posterior failure probability using a cubature formula. The second innovation is to construct a unified probability density function, on which the efficiency of estimating the first two moments of the posterior failure probability is enhanced by de-coupling the above-mentioned double-layer framework. The efficiency improvement resulted from both innovations of the proposed method is fully verified via numerical and engineering examples.