Reliability assessment of deteriorating structures using Bayesian updated probability density evolution method (PDEM)

The inspection of engineering structures is important to ensure performance; in this regard, the Bayesian updating process based on inspection data plays an essential role in the life-cycle management of engineering systems. In this paper, the probability density evolution method (PDEM), a universal approach for dynamical systems, is extended for the analysis of deteriorating structures with inspection by combining with the Bayesian updating process. Firstly, the generalized density evolution equation (GDEE) for deteriorating structures is derived. Secondly, the Bayesian updating for variables with inspection at different time instants is presented based on certain rational but simple assumptions, together with a sampling and fitting approach for the posterior PDF and the predictive one. With the Bayesian updating, the probabilistic information of variables changes, and a Bayesian updated GDEE (BU-GDEE), which is an impulsive partial differential equation and equivalent to a piecewise ordinary GDEE, is developed and proposed for deteriorating structures with inspection, including its numerical solution based on approximation via a family of δ sequences. Finally, two numerical examples are analyzed to illustrate the rationale and accuracy of the modified PDEM with Bayesian updating. These examples demonstrate the fact that inspections will highly influence the reliability of deteriorating structures.