## 6 Numerical Analysis
### Comparing the Three SMA-LRBs  
SMA wires can be wrapped around the LRB isolators in double-cross, straight, and cross arrangements to improve the seismic performance of the isolated structure. On the one hand, it is proven that adding SMA wires to the LRBs, results in a marginal rise in the base shear and deck acceleration. As evidenced in Tables 4 and 5, this rise is the least for the bridge with cross SMA-LRBs and is the highest for the bridge with the double-cross SMA-LRBs. On the other hand, adding SMA wires reduces the maximum and residual displacements because of their re-centering and super-elasticity capabilities. According to Tables 6 and 7, amongst the three configurations, cross wires have the least reduction ratio in the maximum and residual displacements, while double-cross wires have the highest ratio. Another difference between these three isolators is the length of the used SMA wire, which can affect the cost of the project. According to Equations (8), (11), and (13), for SMA-LRBs with double-cross, straight and cross configurations, 242.71 cm, 133.12 cm, and 205.85 cm SMA wire is required to wrap around each of the LRBs, respectively.  
Figure 17: Time history curves of the mid-span displacement under synthetic#3 of the RSN8060 record.  
Figure 18: Graphical demonstration of the mid-span residual displacement for the LRB-isolated and three SMA-LRB isolated bridges.  
For a specified design perspective, in order to choose the best configuration of SMA wires to add to the LRBs, the weighted sum method (WSM) can be 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 (15). In which, \(A_{i}^{WSM-Score}\) 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.  
\[A_{i}^{WSM-Score}=\sum_{j=1}^{N}w_{j}\times a_{ij}\,,\;i=1,2,3,\ldots,M. \tag{15}\]  
\[a_{ijNormalized}=\frac{Min\;a_{ij}}{a_{ij}} \tag{16}\]  
In this multi-criteria decision-making problem, there are three alternatives, including SMA-LRB with double-cross, straight and cross wires, and five decision criteria are set as the maximum base shear, maximum deck acceleration, maximum and residual mid-span displacement and the length of the SMA wire. Moreover, the responses under the real record RSN 8161, as illustrated in Table 8, are considered here, since this record has larger PGA values. To calculate the score of each alternative, first of all, the values must be normalized. Since for all the five criteria, the lower value is desired, then normalization must be performed using Equation (16). The normalized decision matrix is shown in Table 8.  
The next step is to assign weights to the criteria. These numbers are selected based on the design perspective. Here, for example, reducing displacements is more important than a minor increase in the shear force and acceleration, and the length of the wire is presumed as the least important criteria. As a result, weights are selected as 0.15, 0.15, 0.3, 0.3, and 0.1 for base shear, acceleration, mid-span displacement, residual displacement, and wire length, respectively. The final step is to substitute the normalized values in Equation (15) and calculate the performance score for each alternative. As illustrated in Table 8, the WSM score for the SMA-LRB with double-cross, straight, and cross wires is 0.92, 0.93, and 0.74, in turn. Consequently, the straight wire arrangement has the best overall  
performance (rank 1), just above the double-cross arrangement (rank 2), but the SMA-LRB with cross wires displays by far the weakest performance (rank 3).