## 3 Research Method  
### Finite Element Analysis  
The shape memory alloy used in this study was assumed to be a nitinol-based shape memory alloy comprising 50.7% Ni and 49.3% Ti. To simulate the hyperelastic behavior of the shape memory alloy, the properties were generated using the UMAT function within the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{3}{c}{**Nitinol**} \\ \cline{2-4}  & \multicolumn{2}{c}{**Austenine Phase**} & \multicolumn{2}{c}{**Martensite Phase**} & \multicolumn{1}{c}{**Structural Steel**} \\ \hline \multirow{4}{*}{Physical Properties} & Melting point & 1240–1310 \({}^{\circ}\)C & 1500 \({}^{\circ}\)C \\ \cline{2-4}  & Density & 6.45 g/cm\({}^{3}\) & 7.849 g/cm\({}^{3}\) \\ \cline{2-4}  & Thermal conductivity & 0.28 W/cm\({}^{\circ}\)C & 0.14 W/cm\({}^{\circ}\)C & 0.65 W/cm\({}^{\circ}\)C \\ \cline{2-4}  & Coeff. of thermal expansion & 11.3 \(\times\) 10\({}^{-6}\)/\({}^{\circ}\)C & 6.6 \(\times\) 10\({}^{-6}\)/\({}^{\circ}\)C & 11.7 \(\times\) 10\({}^{-6}\)/\({}^{\circ}\)C \\ \hline \multirow{4}{*}{Mechanical Properties} & Recoverable elongation & up to 8\% & 0.2\% \\ \cline{2-4}  & Modulus of elasticity & 30–83 GPa & 21–41 GPa & 200 GPa \\ \cline{2-4}  & Yield strength & 195–690 MPa & 70–140 MPa & 248–517 MPa \\ \cline{2-4}  & Ultimate tensile strength & 895–1900 MPa & 448–827 MPa \\ \cline{2-4}  & Elongation at failure & 5–50\% (typically \(\sim\)25\%) & \(\sim\)20\% \\ \cline{2-4}  & Poisson ratio & 0.33 & 0.27–0.30 \\ \cline{2-4}  & Hot workability & Quite Good & Good \\ \cline{2-4}  & Cold workability & Difficult due to rapid work hardening & Good \\ \cline{2-4}  & Machinability & Difficult, abrasive techniques preferred & Good \\ \cline{2-4}  & Hardness & 30–60 R\({}_{C}\) & Varies \\ \cline{2-4}  & Weldability & Quite Good & Very good \\ \hline Electrical Properties & Resistivity & 100 \(\mu\Omega\cdot\)cm & 80 \(\mu\Omega\cdot\)cm & 13–125 \(\mu\Omega\cdot\)cm \\ \hline Chemical Properties & Corrosion performance & Excellent (similar to stainless steel) & Fair \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Characteristics of shape memory alloys. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process Factors** & **NiTi** & **Cu-Based** & **Fe-Based** \\ \hline Maximum Recoverable Strain & 8\% & 5\% & Less than 5\% \\ \hline Cost & High & Low & Low \\ \hline Shape Memory Effect & High & Moderate & Low \\ \hline Workability & Moderate & Low & Good \\ \hline Fabrication & Low & Good & Moderate \\ \hline Processing & Demanding & Easy & Easy \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Characteristics of shape memory alloys. In the finite element analysis of the superelastic shape memory alloy expansion joints, as illustrated in Figure 7, a uniform thickness of 5 mm was maintained for all joints, irrespective of the expansion amount. To evaluate the behavior characteristics across different bridge gap distances, four distinct models were selected, as detailed in Table 8. The table presents the normal gap distance as the standard measurement, with the minimum and maximum gap distances corresponding to the least and greatest extents of expansion, respectively. This approach enables the determination of the maximum permissible expansion amount for the girders, labeled as \"Maximum Allowed\". Based on the established design concept, modeling was performed using the finite element analysis software Abaqus [13; 14; 15; 16; 17]. The expansion joint modeled through Abaqus is shown in Figure 8. During the meshing stage, all water-stop material parts were meshed in a hexagonal form based on the \"structured technique\". The hexagonal mesh enhances the speed and stability of the finite element analysis in this study, owing to its stable rectangular parallelepiped shape. The modeled expansion joints were designed to maintain the specified gap distance and to accommodate movements both along and perpendicular