Bayesian Estimation of Weights in Weighted Residuals Techniques for Finite Element Model Updating

The problem of quantifying the uncertainty in the values of the weights in weighted residuals methods used for model updating is addressed. For this, probability distributions are used to quantify uncertainties and a Bayesian model selection framework is used to update uncertainties in the values of the weights utilising the available measured data. The probability distribution of the weight values based on the data is formulated as probability integrals over the structural model parameters. Asymptotic analysis is then used to provide approximate analytical formulas for these probability integrals and represent the probability distribution of the weights in terms of the optimal values of the model parameters, conditioned on the weight values, and the Hessian of the involved weighted residual metric. The best values of the weights are selected as the ones that maximise the probability distribution of the weights. Theoretical and computational issues arising in the estimation of the probability distribution of the weights are addressed. A formulation is also presented for estimating the uncertainty in the response predictions from the uncertainties in the weight values and the optimal structural models.