## 3 Results and Discussion
### Objective Function Minimization  
In order to predict the structural performance under hazardous events accurately, a well-tuned baseline model is essential. With limited modeling information due to lack of design drawings and reports, an approximate model may deviate from the actual behavior of the structure. Based on the field observations, mass estimations, and fixed BCs, modal analysis results of the initial non-updated model are 8.98, 14.41, and 22.05 Hz for first, second, and third modes, respectively. Comparing these results with the actual dynamic response obtained from the identification results, one can see there is a significant mismatch in second and third modes. Therefore, such modeling discrepancy should be diminished to improve the accuracy of the baseline model. For this purpose, the FEM updating procedure explained in the previous section is adopted. The updating procedure is composed of three loops each manipulating one modeling variable to generate multiple FEM instances. These three parameters are related to the support restraints, member thicknesses, and distributed mass over the entire span. Looking at the support restraints of the bridge, there are two different types of BCs. The first type is anchored to the adjacent buildings, and the second type is bolted connections. The support details observed through visual inspection show that the bolted connections are only used for the arch restraints, and the rest of the connections are most likely anchored to the structure. To decrease the number of parameter updates, considering that anchored connections form rigid supports, the bolted connection type is considered as an updating parameter which leads to three different combinations such as fixed-fixed, fixed-pinned, and fixed-roller. For each BC case, 900 FEM instances are created ranging in stiffness (K), and mass (M) parameters. The objective function error between the FEMs and the modal identification results are computed to find the optimal parameter combination. Figure 8 shows the error surfaces of the fixed-fixed, fixed-pinned, and fixed-roller cases. According to Figure 8, the uppermost three figures of each BC case shows the error due to each individual modal frequency, whereas the three-dimensional figures show the combination of these individual components as the objective function product. For visualization purposes, the error between FEM and identification results is demonstrated with colored surfaces. The error surface ranges between  
Figure 8: Modal frequency error surfaces for fixed-fixed, fixed-pinned, fixed-roller BCs. red and blue where red corresponds to maximum dispersion from physical reality (modal frequencies from accelerometer data) and blue corresponds to minimal difference between mathematical model and identified modal parameters. Other colors (e.g., orange, yellow, green, turquoise) lay between maximum and minimum error based on the objective function calculations. The magenta spots on each subfigure points out the optimal combination of updating parameters. The overall behavior shows that the model accuracy is very sensitive to the BCs. In other words, combinatory results as well as individual modal frequency errors heavily rely on the modeling of the support restraints. Stiffness and mass domains are meshed into 30 pieces when candidate models are developed. So, each dimension consisting of 30 individual values represents the uncertainty range within the minimum and maximum values. To explain, Table 1 presents the modal frequency errors obtained from different BC cases and Figure 9 presents the modal parameters of optimal combination cases for each BC case. Stiffness parameters vary from \(1.20\times 10^{-5}\)m\({}^{4}\) to \(11.5\times 10^{-5}\)m\({}^{4}\) for moment of inertia and \(2.1\times 10^{-3}\)m\({}^{2}\) to \(36.8\times 10^{-3}\)m\({}^{2}\) for cross-sectional area of a single element. Meanwhile, mass per unit area ranges between \(5.7\times 10^{-2}\)t/m\({}^{2}\) and \(1.7\)t/m\({}^{2}\).