Bayesian Structural Model Updating and Model Selection with Modal Data using Gibbs Sampler

Abstract

This paper presents a new Bayesian model updating approach for linear structural models based on the Gibbs sampler, a stochastic simulation method. We show that with incomplete modal data (modal frequencies and incomplete modeshapes of some lower modes), and with appropriate choices of conjugate priors, the uncertain stiffness and mass parameters of the linear structural model can be decomposed into three groups so that the sampling from any one group is possible when conditional on the other groups and the modal data. Such decomposition provides a major advantage for the Gibbs sampler: even if the number of uncertain parameters is large, the effective dimension for the Gibbs sampler is always three. Therefore, high dimensional parameter spaces that are fatal to most sampling techniques, are handled by the Gibbs sampler. At the same time, this approach inherits the advantages of Bayesian techniques: it not only updates the optimal estimate of the structural parameters but also updates the associated uncertainties. Moreover, the Gibbs sampler also estimates the complete modeshape of each contributing mode from the incomplete modal data. The approach is illustrated by applying it to two examples of structural health monitoring problems, where the goal is to detect and quantify any damage using modal data obtained from the undamaged and possibly damaged structure.