182 R. Astroza et al. 20.3.3 Sensitivity Analysis A simplified one-at-a-time sensitivity analysis is conducted for the Marga-Marga bridge and the results are shown in tornado diagrams [13]. Local sensitivities of 36 different responses with respect to the 95 parameters described above are analyzed. The considered responses include: longitudinal and transverse absolute acceleration responses at the top of piers #2, #4, and #6 and at the deck above the same piers (i.e., 12 absolute acceleration responses, y1 to y12), relative displacements collocated with the acceleration responses (i.e., 12 relative displacement responses, y13 toy24), strain responses on one steel and one concrete fiber at the bottom section of the same piers (i.e., 6 strain responses, y25 to y30), and longitudinal and transverse shear strains on isolators located above piers #2, #4, and #6 (i.e., 6 isolators’ shear strain responses, y31 to y36). A perturbation of ±5% around the true value of each model parameter (Table 20.1) is considered. Then, the relative root mean square error (RRMSE) between the true responses (i.e., computed using the true parameter values) and those obtained with the perturbed parameter values is computed and plotted in tornado diagrams. Figure 20.5 shows the sensitivity results of one global response, the absolute acceleration response on top of pier #2 in the longitudinal direction (y1), and one local response, the shear deformation of the isolator on top of pier #6 in the longitudinal direction (y35). The parameter ID in Fig. 20.5 refers to the model parameters presented in Table 20.1. From Fig. 20.5 it is observed that response y1 and y35 are mostly sensitive to parameters defining the response of the isolators in the interior piers (#2 to #6), elastic modulus of steel girders and deck (EG and ED), the stiffness-proportional parameter of the Rayleigh damping (β), and some concrete parameters (chiefly Ec, fc , and εc) of the interior piers (#2 to #6). As expected, the predominant effect of the isolation layer in the overall structural response is clearly captured by the simple sensitivity analysis conducted. Moreover, it is observed that global acceleration and displacement responses are more sensitive to concrete model parameters than shear deformation response of the isolators. This shows the importance of considering heterogeneous response quantities when calibrating large and complex nonlinear FE models. By comparing the magnitude of the RRMSEs related to the different responses analyzed, it is observed that shear deformation of the isolators experienced the highest variations (y31 to y36), followed by global acceleration (y1 to y12) and displacement (y13 to y24) responses, while fiber level responses (y25 to y30) show the lowest variations. Then, the model parameters to be estimated are selected as those having a RRMSE equal or higher than 20% of the RRMSE for the most sensitive parameter, considering all the measured responses. A threshold value of 20% is chosen in this application because of the abrupt change in the swings of the tornado diagram. The twenty-seven (27) parameters satisfying this criterion are highlighted in grey in Table 20.1 and are chosen for the estimation and model updating phase. 20.3.4 Estimation of Model Parameters The twenty-seven model parameters defined above define the vector θ. The true responses (ytrue) obtained using the true model parameters (Table 20.1) are polluted by white Gaussian noise to define the measured response (y) employed for the estimation. For acceleration and displacement-related responses, levels of noise of 0.7%gand2.0mmRMS, are respectively considered. Note that displacement-related responses includes relative displacement responses of the pier and deck, and also shear strain responses of the isolators. For the fiber strain responses, an absolute noise level of 0.005% is assumed. Then, the updating process employs the three components of seismic base acceleration (¨ug) and the measured response (y) to estimate the twenty-seven FE model parameters. To investigate the effects of the number and type of the measured responses and of the updating step (D) considered in the estimation process, different instrumentation setups are considered to update the FE model. The twelve cases analyzed are summarized in Table 20.2 and include values of D=5, 10, and 20 and measured responses considering only accelerations (cases B01 to B03), accelerations +displacements (cases B04 to B06), accelerations +displacements +strain in fibers (cases B07 to B09), and accelerations +displacements +strain in fibers +shear strain in isolators (cases B10 to B12). wk is assumed zero-mean with diagonal covariance matrix Q, whose diagonal entries are computed as q × θ i 0|0 2 , i =1, . . . ,27 andq=1×10−5.MatrixRwas constructed assuming standard deviations of 0.5% for acceleration responses, 1.3 mm for global displacement responses, 1.5 mm for shear deformation of isolator responses, and 0.003% for strain in steel and concrete fiber responses. Initial estimates of the parameters are randomly chosen from the ranges [−20%, −10%] and [10%, 20%] of the true parameter values. Pθθ 0|0 is taken as diagonal with entries equal to p× θi 0|0 2 , with i =1, . . . ,27 and p =5%. Figure 20.6 shows the time history of the mean estimate of the 27 parameters normalized with respect to the corresponding true values for cases B01and B12. In cases B01, all the isolator model parameters, the elastic modulus of steel girders and deck (EG and ED), and most of the elastic modulus of concrete of the piers converge to the true values, while
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