Modeling Error Estimation and Response Prediction of a 10-Story Building Model Through a Hierarchical Bayesian Model Updating Framework

In this paper a hierarchical Bayesian model updating approach is proposed for calibration of model parameters, estimation of modeling error, and response prediction of dynamic structural systems. The approach is especially suitable for civil structural systems where modeling errors are usually significant. The proposed framework is demonstrated through a numerical case study, namely a 10-story building model. The ‘measured data’ include the numerically simulated modal parameters of a frame model which represents the true structure. A simplified shear building model with significant modeling errors is then considered for model updating with stiffness of different structural components (substructures) chosen as updating parameters. In the proposed hierarchical Bayesian framework, updating parameters are assumed to follow a known distribution model (normal distribution is considered here) and are characterized by the distribution parameters (mean vector and covariance matrix). The error function, which is defined as the misfit between model-predicted and identified modal parameters, is also assumed to follow a normal distribution with unknown parameters. The hierarchical Bayesian approach is applied to estimate the stiffness parameter distributions with mean and covariance matrix referred to as hyperparameters, as well as the modeling error which is quantified by the mean and covariance of error function. Joint posterior probability distribution of all updating parameters is derived from the likelihood function and the prior distributions. A Metropolis-Hastings within Gibbs sampler is implemented to evaluate the joint posterior distribution numerically. Two cases of model updating are studied with Case 1 assuming a zero mean for the error function, and Case 2 considering a non-zero error mean. The response time history of the building to a ground motion is predicted using the calibrated shear building model for both cases and compared with the exact response (simulated). Good agreements between predictions and measurements are observed for both cases with better accuracy in Case 2. This verifies the proposed hierarchical Bayesian approach for model calibration and response prediction and underlines the importance of considering and propagating the uncertainties of structural parameters and more importantly modeling errors.