Two-Stage Hierarchical Bayesian Framework for Finite Element Model Updating

A hierarchical Bayesian modeling (HBM) framework is presented for updating finite element (FE) models. A two stage approach is proposed for which in the first stage the modal data properties (modal frequencies, damping ratios and mode shapes) are estimated using response time histories recorded from multiple independent experiments. In the second stage, the proposed framework provides a reliable approach to account for the uncertainty of the FE model parameters due to the variability in the values of modal data estimated from multiple data sets. This variability arises due to model errors, measurement errors, as well as data processing procedures used to estimate modal data from response time histories. In the proposed framework, the uncertainties are embedded into the FE model parameters by assigning a probability model involving a set of hyper parameters. A formulation is presented for quantifying the uncertainties in the hyper parameter, model parameters and output quantities of interest (QoI) using an efficient asymptotic approximation to process independently the modal data sets. In particular, the posterior distribution of the hyper parameters is analytically formulated as a product of multi-dimensional Gaussian probability distributions. Samples of this distribution are used to estimate model parameter uncertainties as well as uncertainties in response QoI. This combined asymptotic-sampling approach is computationally more efficient than available full sampling approaches. Simulated data from a spring-mass chain model are used to demonstrate that the proposed framework provides reliable and reasonable uncertainty bounds as compared to conventional Bayesian framework that considerably underestimate uncertainties and results in unrealistic predictions of thin uncertainty bounds for response QoI.