## 2 Mapping Relationship Between Rail Deformation and Bridge Deformation
### Design of Long-Span, Cable-Stayed Bridges  
After establishing the main beam's span diameter, the primary step involves selecting the main beam material based on the span diameter. And the side span diameter, the bridge tower's height, and the spacing of the cable-stay's arrangement are provided. Furthermore, parameters such as the main beam's cross-section, the cable's diameter, and the tower's cross-section are determined. In order to ensure the stiffness of the structure under complex loads, the design considers the vertical deflection of the main beam under train loads to keep it within the range of \(L/500\). Research suggests that cable-stayed bridges with a general main span of less than 400 m are best suited for the concrete main beam. In the case of main spans ranging from 400 m to 500 m, the choice involves comparing the composite beam and the concrete beam. For spans exceeding 500 m, it is recommended that we use the steel main beam or the composite beam. In this study, concrete main beams are considered for a main span of 200 m. For a main span of 400 m, both concrete beams and steel-concrete combination beams are contemplated. Additionally, a main span of 600 m is being considered for steel-concrete combination beams and steel main beams. Following the selection of the main span and the material of the main beam, the side span's extent, the bridge tower's height, and the spacing between the stay cables are determined based on the literature [18]. These values are detailed in Tables 1 and 2. In the investigation of the large-span, cable-stayed bridges ranging from 200 m to 600 m, the common design choice is the utilization of double-tower double-side cable-stayed bridges, employing the \"H\"-type concrete bridge tower. The cable-free zone near the bridge tower and abutment supports is set to a length of two to three times the diagonal cable spacing. These values are detailed in Table 3. To reduce the horizontal displace \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Main Beam** & **Side Span/Main Span** & **Value** \\ \hline Steel Beam & 0.30–0.40 & 0.35 \\ Steel–Concrete Combination Beam & 0.40–0.50 & 0.45 \\ Concrete Beam & 0.40–0.50 & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ratio of side span to main span diameter of double-tower cable-stayed bridges. \begin{table}
\begin{tabular}{c c} \hline \hline
**Tower’s Height/Main Span** & **Value** \\ \hline
0.2–0.33 & 0.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ratio of side span to main span diameter of double tower cable-stayed bridges. ment at the mid-span of the main beam, strategically placed auxiliary piers are present at the side spans. The spanning ratio for a single auxiliary pier corresponding to the side spans is 4:6. And the ratio for double auxiliary piers corresponding to the side spans is 4:3:3. For the majority of cable-stayed bridges, a semi-floating system is employed, where dampers impose longitudinal shape constraints in the longitudinal direction. While these longitudinal dampers are active under seismic and wind loads, they have minimal impact on temperature-induced deformations. Consequently, no longitudinal restraints are introduced for the 200 m, 400 m, and 600 m cable-stayed bridges. Fifteen structural schemes for cable-stayed bridges, considering different main beam's types and auxiliary pier's arrangements, are presented in Table 4. Finite element models of cable-stayed bridges are established by the Midas 2022 software. The main beam and bridge tower are modeled by the beam element. The cable is simulated by the truss element. The connection between the main beam and the support is modeled by the rigid arm unit. The support is rigidly connected with the foundation. The model of the long-span, cable-stayed bridge is depicted in Figure 4.