## 5 Numerical Analysis
### Base Shear  
The deck acceleration and base shear of the two-span continuous bridge are larger without a base isolation system (Phase I). By employing base isolators in the central pier of the bridge, its stiffness and the load transmitted from the base to the superstructure decreases (phase II). In Table 8, the responses of the uncontrolled bridge in both phases are presented, and the significant reduction in deck acceleration and base shear can be seen for all earthquakes after isolation ([PERSON] et al. [18], [PERSON] et al. [26]). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Chi-Chi**} & \multicolumn{3}{c}{**Northridge**} & \multicolumn{3}{c}{**Turkey**} & \multicolumn{3}{c}{**Kobe**} \\ \cline{2-13}  & **SRB** & **NSD** & **N.D** & **Cu.D** & **SRB** & **NSD** & **N.D** & **Cu.D** & **SRB** & **NSD** & **N.D** & **Cu.D** & **SRB** & **NSD** & **N.D** & **Cu.D** \\ \hline
**J1** & 1.45 & 0.92 & 0.91 & 0.90 & 1.33 & 0.98 & 1.07 & 1.55 & 1.03 & 0.65 & 0.78 & 1.07 & 1.11 & 0.82 & 0.91 & 1.20 \\
**J2** & 1.68 & 0.72 & 0.95 & 1.03 & 1.41 & 0.93 & 1.04 & 1.60 & 1.08 & 0.58 & 0.70 & 1.06 & 1.28 & 0.83 & 0.90 & 1.34 \\
**J3** & 0.89 & 0.61 & 0.76 & 0.94 & 0.76 & 0.81 & 0.90 & 0.83 & 0.59 & 0.62 & 0.59 & 0.68 & 0.39 & 0.63 & 0.83 & 0.39 \\
**J4** & 2.97 & 1.24 & 1.13 & 1.57 & 2.03 & 1.00 & 1.05 & 1.70 & 1.76 & 1.01 & 0.92 & 1.65 & 2.22 & 1.34 & 1.16 & 2.57 \\
**J5** & 0.85 & 0.6 & 0.76 & 0.94 & 0.77 & 0.81 & 0.90 & 0.82 & 0.55 & 0.63 & 0.56 & 0.64 & 0.35 & 0.43 & 0.83 & 0.37 \\
**J6** & 1.68 & 0.72 & 0.95 & 1.03 & 1.41 & 0.93 & 1.04 & 1.60 & 1.08 & 0.58 & 0.70 & 1.06 & 1.28 & 0.83 & 0.90 & 1.34 \\
**J7** & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
**J8** & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
**J9** & 1.53 & 0.58 & 0.98 & 0.86 & 1.33 & 0.74 & 0.84 & 1.56 & 0.59 & 0.41 & 0.58 & 0.84 & 1.08 & 0.9 & 0.97 & 1.11 \\
**J10** & 1.54 & 0.56 & 0.98 & 0.86 & 1.31 & 0.71 & 0.82 & 1.54 & 0.59 & 0.41 & 0.58 & 0.84 & 1.08 & 0.89 & 0.96 & 1.11 \\
**J11** & 0.73 & 0.71 & 0.98 & 0.82 & 0.81 & 0.56 & 0.76 & 1.03 & 0.26 & 0.41 & 0.52 & 0.23 & 0.35 & 0.36 & 0.96 & 0.33 \\
**J12** & 2.30 & 1.45 & 1.19 & 1.23 & 1.94 & 1.03 & 0.98 & 2.02 & 1.43 & 0.96 & 0.87 & 1.84 & 1.36 & 1.04 & 0.99 & 1.64 \\
**J13** & 0.75 & 0.53 & 0.98 & 0.82 & 0.79 & 0.54 & 0.76 & 1.02 & 0.26 & 0.16 & 0.52 & 0.24 & 0.34 & 0.29 & 0.96 & 0.32 \\
**J14** & 1.54 & 0.64 & 0.98 & 0.86 & 1.31 & 0.79 & 0.82 & 1.54 & 0.59 & 0.32 & 0.58 & 0.84 & 1.09 & 0.49 & 0.96 & 1.11 \\
**J15** & 0.01 & 0.00 & 0.00 & 0.01 & 0.01 & 0.00 & 0.00 & 0.01 & 0.01 & 0.00 & 0.00 & 0.01 & 0.01 & 0.00 & 0.00 & 0.00 \\
**J16** & 0.85 & 0.68 & 0.76 & 0.94 & 0.77 & 0.75 & 0.90 & 0.82 & 0.55 & 0.5 & 0.56 & 0.64 & 0.40 & 0.31 & 0.83 & 0.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Nonlinear dynamic analysis results of the benchmark bridge equipped with SMA isolators under near-field earthquake records. Figure 9: Comparison of (**a**) base shear, (**b**) mid-span acceleration, (**c**) mid-span displacement, and (**d**) bearing deformation for four passive control systems. For example, under the North Palm Springs earthquake, the use of base isolation leads to an 88.24% and 82.37% reduction in the base shear and deck acceleration of the bridge, respectively. However, after isolating the superstructure, its displacements increase. For controlling these displacements, the SMA wires were employed. It is expected that adding the wires to the LRB of the bridge increases the horizontal stiffness and, consequently, the deck acceleration and base shear. The evaluation parameters related to the base shear are denoted by J1 and J9, which are the maximum base shear and the norm one normalized based on their uncontrolled quantities. According to Table 6, under far-field earthquakes, J1 goes up by 8% to 24%, and J9 increases by 8% to 36%.