258 M. K. Ramancha et al. Bayesian approach to model updating is attractive because it allows combining prior knowledge with a noisy and incomplete measurement data set to update the unknown state and parameter vector [4]. In this paper, the Bayesian FE model updating framework is applied to the Pine Flat dam, a concrete gravity dam on the King’s river near Fresno, California. In this respect, a 2D plane strain nonlinear FE model of the dam is developed in an FE analysis software framework (OpenSees) with the cap plasticity model, a classical 3D non-smooth multi-surface plasticity model [5–7], used to represent the behavior of plain concrete. The FE model characterized by a set of realistic material parameter values is subjected to seismic input excitation to numerically simulate the output response data. The simulated input and output response data with added Gaussian white noise (to mimic the measurement noise) are then used to update only the time-invariant material parameters of the multiaxial material model using unscented Kalman filtering (a nonlinear Bayes filter). This paper investigates the issue of convergence in FE response prediction (to the true response) in the absence of convergence of the parameter estimates to their true values due to parameter non-identifiability issues. Note that this study accounts only for the sources of uncertainty related to the input-output measurement noise and the material parameters. Moreover, the same FE model is used to simulate the response and to perform the model updating, thereby disregarding the effects of modeling uncertainty. 29.2 FE Model Updating as Parameter-Only Estimation Problem In general, FE model updating aims at jointly estimating the unknown system state and parameter vector of the nonlinear FE model using sparse and noisy input-output measurement data [8]. The state vector x for an FE model includes the displacement and velocity at every degree of freedom of the model. In addition, for nonlinear FE models, the state vector also contains all history-dependent variables (material history variables) at each integration point of the model [3]. The parameter vector θcomprises of all unknown FE model parameters such as geometric, damping, constraint and material (time-invariant) parameters. The input measurement data can consist of an earthquake recorded by a seismograph (or seismometer) in the vicinity of the structure while the output measurement data are typically provided by accelerometers mounted at various locations on the structure. Note that the input and output measurement data are noisy and often insufficient to completely estimate the joint state and parameter vector of the nonlinear FE model. In this study, the FE model with realistic values of model parameters is assumed to predict realistically the actual response of the structure. In other words, the selected FE model class has the capability to represent reasonably well the actual nonlinear behavior of the real structure. With this assumption, the FE model updating problem (joint state and parameter estimation) boils down to a parameter-only estimation problem. Therefore, the input and output measurement data are used to estimate the unknown parameter vector θ only as the updated nonlinear FE model is relied upon to provide satisfactory estimates of the state of the system. 29.3 Bayesian Parameter Estimation At time tk, let yk ∈ Rny be the measurement vector and yFE k = hk (u1:k; θ) ∈ Rny denote the response predicted by the FEmodel hk parameterized by vector θ ∈ Rnθ when subjected to input time history u1:k. The error between the measured response and the FE estimated response at time tk can be written as ek =yk −hk(u1:k; θ). The goal of parameter estimation is to estimate the parameter vector θ by minimizing the error ek at time tk (k =0, 1, 2, . . . ) while accounting for the pertinent sources of uncertainty. The Bayesian parameter estimation framework can be used to solve this mathematical inverse problem. This framework can be employed to recursively, over the time steps tk (k =0, 1, 2, . . . ), estimate the parameter vector θ by using the following discrete-time state-space representation of the system dynamics: State Equation : θk =θk−1 +wk−1 Measurement Equation : yk =hk (u1:k; θk) +vk (29.1) In this framework, the unknown parameter vector θ at each time steptk is modeled as random vector denoted by k. The state equation governing the parameter vector θk ∈ Rnθ is driven by the artificial process noise wk ∈ Rnθ. The measurement noise vk ∈ Rny is assumed to be additive to the FE predicted response hk(u1:k; θk) in the measurement equation.
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