## 4 Modeling the SMA-LRB Isolator
### Optimizing the SMA-LRB Isolator
The obtained results are tabulated in Table 4.  
As can be seen in Table 4, in total, four different radii are represented as the result of the optimization. However, among these four radii, only one single radius should be selected, which is able to minimize J1 and J3 simultaneously under all earthquakes, better than the other radii. To identify the final radius, two steps need to be taken. Firstly, for each earthquake record, the performance of the isolated bridge equipped with each of the four optimized SMA diameters was collected. It should be noted that, because the main aim of this research was to minimize displacements while controlling the base shear, among different evaluation criteria, only J1 and J3 were utilized to assess the best radius of the wire. Therefore, the selected nonlinear dynamic analysis results for different radii under all records are depicted in Table 5. Secondly, in order to choose the radius with the best overall performance from the results shown in Table 5, the weighted sum method (WSM) was utilized. The WSM is the best known and commonly used multi-criteria decision-making method for evaluating a number of alternatives regarding a number of decision criteria. According to the WSM, in a decision problem with M alternatives and N criteria, a performance value can be calculated for each alternative by applying Equation (4). Then, in a minimization case, the best alternative is the one that yields the minimum total score:  
\[A_{i}^{WSM-Score}=\sum\
olimits_{j=1}^{N}w_{j}\times a_{ij},\ \ i=1,2,3,\ldots,M, \tag{4}\]  
where \(A_{i}^{WSM-Score}\) is the WSM score of the \(i^{th}\) alternative, \(w_{j}\) is the weight of importance of \(j^{th}\) criteria, and \(a_{ij}\) is the actual value of the \(i^{th}\) alternative regarding the \(j^{th}\) criterion. In this research, the main goal was to minimize displacements rather than the base shear. As a result, the weights were set as 0.8 and 0.2 for J3 and J1, respectively. Rewriting Equation (4) for the current problem leads to Equation (5):  
\[A_{i}^{WSM-Score}=\sum\
olimits_{j=1}^{6}\Big{(}0.8\times J3_{j}+0.2\times J1 _{j}\Big{)},\ \ \ i=1,3.5,4.5,6\ mm. \tag{5}\]  
By substituting the numbers of Table 5 in Equation (5), the WSM scores were calculated as 5.708, 4.658, 5.026, and 5.246 for \(r\) = 1, 3.5, 4.5, and 6 mm, in turn. It is evident that the radius with the least score has superior performance in reducing the maximum base shear and the maximum mid-span displacement. Finally, the radius of 3.5 mm with the least objective function was selected as the optimum radius. As reflected, the final optimum radius diverges from the optimized radii for some earthquake records. The reason is that these earthquakes were selected for the benchmark bridge so that they \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Earthquakes** & **NPalm Springs** & **Chi-Chi** & **El Centro** & **Northridge** & **Turkey** & **Kobe** \\ \hline optimized-SMA-wire & & & & & & \\ radius (mm) & 6 & 1 & 4.5 & 3.5 & 4.5 & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimized radii obtained from NSGA-II for each ground motion record. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Criteria** & \
\begin{tabular}{c} **Wire’s Radius** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **North Palm** \\ **Springs** \\ \end{tabular} & **El Centro** & **Chi-Chi** & **Northridge** & **Turkey** & **Kobe** \\ \hline \multirow{3}{*}{**J1 = Max Base**} & 1 & 1.08 & 0.88 & 0.99 & 1.58 & 0.96 & 1.05 \\  & 3.5 & 1.24 & 1.08 & 1.45 & 1.33 & 1.03 & 1.11 \\  & 4.5 & 1.34 & 1.24 & 2.64 & 1.63 & 1.25 & 1.27 \\  & 6 & 1.28 & 1.12 & 2.85 & 2.52 & 1.58 & 1.4 \\ \hline \multirow{3}{*}{**J3 = Max**} & 1 & 0.96 & 0.75 & 0.85 & 1.24 & 0.86 & 0.84 \\  & 3.5 & 0.74 & 0.40 & 0.89 & 0.76 & 0.59 & 0.37 \\ \cline{1-1}  & 4.5 & 0.73 & 0.38 & 0.91 & 0.74 & 0.58 & 0.31 \\ \cline{1-1}  & 6 & 0.72 & 0.31 & 0.94 & 0.71 & 0.61 & 0.29 \\ \hline \hline \end{tabular}
\end{table}