## 6 Numerical Analysis
### Mid-Span Displacement  
The benchmark highway bridge in the non-isolated phase, in which the deck is connected to the central pier, experiences the smallest superstructure displacements, and the highest base shear and deck acceleration (see Sections 6.1 and 6.2). After isolating the deck from the central piers via LRBs, the acceleration, and base shear drop, but mid-span displacement rises dramatically. This rise is evident in Figure 15 and Table 6. For example, under synthetic record number 3 of the RSN 8606 record, the maximum mid-span displacement jumps from 81 to 650 mm after isolation. The extreme deck displacement resulting from the isolating process can have detrimental effects, such as the collision of the adjacent parts of the superstructure, deck unseating, or plastic deformation of the bearings. To tackle this issue, SMA wires are added to the LRBs to control displacements. As illustrated in Table 6 and Figure 15, SMA wires are quite beneficial in minimizing the mid-span displacements. For instance, the maximum mid-span displacement under synthetic record number 3 of the RSN 8606 decreases from 650 to 122 mm using double-cross SMA-LRBs. According to Table 6, SMA-LRBs with various wire configurations reduce the maximum mid-span displacements in a range from 26% to 81%. Another interesting point is that SMA-LRBs with \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Maximum Mid-Span Displacement (m)**} \\ \cline{2-7}  & \multirow{2}{*}{**Non-ISOLATED Isolated**} & \multirow{2}{*}{**LC-SMA-LRB Isolated**} & \multirow{2}{*}{**S-SMA-LRB Isolated**} & \multirow{2}{*}{**C-SMA-LRB Isolated**} \\ \cline{3-3} \cline{5-7}  & & & **Reduction** & & **Reduction** & **Reduction** \\
**Earthquake** & & & & **Ratio** & & **Ratio** \\  & & & & (\%) & & (\%) \\ \hline RSN 8161 & \(0.73\times 10^{-1}\) & \(2.98\times 10^{-1}\) & \(1.53\times 10^{-1}\) & 48.66 & \(1.62\times 10^{-1}\) & 45.64 & \(2.20\times 10^{-1}\) & 26.17 \\ Synthetic \#1 & \(0.71\times 10^{-1}\) & \(4.60\times 10^{-1}\) & \(1.54\times 10^{-1}\) & 66.52 & \(1.59\times 10^{-1}\) & 65.43 & \(1.64\times 10^{-1}\) & 64.35 \\ Synthetic \#2 & \(0.69\times 10^{-1}\) & \(3.12\times 10^{-1}\) & \(1.54\times 10^{-1}\) & 50.64 & \(1.70\times 10^{-1}\) & 45.51 & \(2.00\times 10^{-1}\) & 35.90 \\ Synthetic \#3 & \(0.73\times 10^{-1}\) & \(3.81\times 10^{-1}\) & \(1.63\times 10^{-1}\) & 57.22 & \(1.69\times 10^{-1}\) & 55.64 & \(2.44\times 10^{-1}\) & 35.96 \\ Synthetic \#4 & \(0.86\times 10^{-1}\) & \(3.87\times 10^{-1}\) & \(1.77\times 10^{-1}\) & 54.26 & \(1.78\times 10^{-1}\) & 54.01 & \(2.43\times 10^{-1}\) & 37.21 \\ Synthetic \#5 & \(0.86\times 10^{-1}\) & \(2.86\times 10^{-1}\) & \(1.73\times 10^{-1}\) & 39.51 & \(1.80\times 10^{-1}\) & 37.06 & \(1.84\times 10^{-1}\) & 35.66 \\ RSN 8606 & \(0.75\times 10^{-1}\) & \(3.92\times 10^{-1}\) & \(1.15\times 10^{-1}\) & 70.66 & \(1.22\times 10^{-1}\) & 68.88 & \(1.25\times 10^{-1}\) & 68.11 \\ Synthetic \#1 & \(0.42\times 10^{-1}\) & \(3.45\times 10^{-1}\) & \(1.31\times 10^{-1}\) & 62.03 & \(1.38\times 10^{-1}\) & 60.00 & \(1.48\times 10^{-1}\) & 57.10 \\ Synthetic \#2 & \(0.69\times 10^{-1}\) & \(4.81\times 10^{-1}\) & \(1.19\times 10^{-1}\) & 75.26 & \(1.34\times 10^{-1}\) & 72.14 & \(1.84\times 10^{-1}\) & 61.75 \\ Synthetic \#3 & \(0.81\times 10^{-1}\) & \(6.50\times 10^{-1}\) & \(1.22\times 10^{-1}\) & 81.23 & \(1.30\times 10^{-1}\) & 80.00 & \(2.17\times 10^{-1}\) & 66.62 \\ Synthetic \#4 & \(0.70\times 10^{-1}\) & \(4.06\times 10^{-1}\) & \(1.21\times 10^{-1}\) & 70.20 & \(1.30\times 10^{-1}\) & 67.98 & \(1.80\times 10^{-1}\) & 55.67 \\ Synthetic \#5 & \(0.68\times 10^{-1}\) & \(4.79\times 10^{-1}\) & \(1.00\times 10^{-1}\) & 79.12 & \(1.13\times 10^{-1}\) & 76.41 & \(1.97\times 10^{-1}\) & 58.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximum mid-span displacement for the non-isolated, LRB-isolated and three SMA-LRB isolated bridges.