264 M. K. Ramancha et al. Fig. 29.6 Time histories of (a) First invariant of the stress tensor, and (b) Frobenius norm of the deviatoric stress tensor at heel of the dam 29.4.3 Validation For validation purposes, the posterior mean estimates of the material parameters at different time steps (tk =0 s, 3 s, 10 s, 20 s) are used to predict the seismic response of the dam to an earthquake ground motion (the North-South component of the 1940 El Centro earthquake recorded at the El Centro station scaled by a factor of two) significantly different from the one (1994 Northridge earthquake recorded at the Sylmar station) used in the parameter estimation stage of this study. The predicted response and true response histories of the horizontal displacement at the top of the dam are compared in Fig. 29.7. It is observed (see Fig. 29.7a) that the predicted response using θk|k at tk =0 s (initial mean estimate of the unknown parameter vector) is in bad agreement with the true response. The response predicted using θk|k at tk =3 s matches the true response in the linear elastic range (in the time window 0–4 s, see Fig. 29.7b) since the estimation of the parameters governing the linear elastic response (Gand K) is converged by tk =3 s (see Fig. 29.4). However, the match between this predicted response and the true response is found to degrade when the dam enters its nonlinear range of behavior (after 4 s). The predicted response histories obtained using θk|k at tk =10 sandtk =20 s are found to follow very closely the entire true response histories (see Fig. 29.7c, d). In fact, it was observed that the response predicted using any set of parameter estimates after tk =7 s agrees with the true response equally well and is confirmed by the low RRMS error in Fig. 29.5: Evolution of RRMS error for all sensors during filtering after tk =7 s. This confirms that the predicted response converges to the true response even though the parameter estimates do not converge to their true values, due to parameter non-identifiability issues. 29.5 Conclusions This paper studies the application of the unscented Kalman filter, an advanced nonlinear Bayesian filtering technique, to recursively estimate, using earthquake input and output response data, the time-invariant material parameters of a multiaxial multi-surface plasticity model characterizing the concrete behavior of a nonlinear FE model of a dam. This study is based on numerically simulated dam seismic response data and does not consider the effects of modeling error/uncertainty. The input earthquake ground motion record is appropriately scaled to drive the dam into adequate levels of constitutive nonlinearity and a local sensitivity analysis is performed to ensure that the resulting output measured responses (using the scaled ground motion as input) are sufficiently sensitive to all the material parameters. It is observed that, although the estimates of some parameters do not converge to their corresponding true values, the predicted responses (for both measured and unmeasured response quantities) match the respective true responses extremely well (i.e., with a low relative root-mean-square error). In other words, the filter finds different sets of parameter values (non-true) that yield a very good match between the FE
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