## 3 Numerical Analysis
### Bridge and Track Model  
This study employs Timoshenko beam elements to model the steel truss main girder, main tower, and sleeper, which are characterized by slender to moderately stubby beam structures. The rail is treated as an infinitely long Euler-Bernoulli beam with discrete elastic point supports, and its vertical, lateral, and rotational degrees of freedom are taken into account. The ballast is distributed in mass blocks according to the actual distance between the sleepers, taking into account their vertical and lateral vibrations (as  
Figure 1: General schemata and details of ballasted railway suspension bridge: **(a)** Distribution of geological conditions and schematic view of Bridge, **(b)** Elevation of main tower, **(c)** Layout of stiffening girder, **(d)** Layout of ballasted railway, **(e)** Key sections of main tower (GM\({}_{\text{UP}}\): Ground motion in the up direction; GM\({}_{\text{FN}}\): Ground motion in the fault-normal direction). shown in Figure 2a,b). A link element is used to model the main cable and sling, with the assumption that the stress is uniformly distributed across the entire element. Shell elements are appropriate for simulating orthogonal plates, and the mesh size of the plates should correspond to the spacing of the sleepers. The transverse bounding generated by horizontal seismic motion significantly influences the dynamic response of bridge structures. Consequently, there are lateral anti-wind connections between the steel truss girders and the towers [32; 33]. Figure 3 shows the detailed FE model of the HSRDTSTSB. The present study adopts the well-established properties for the connection between track structures [34]. In addition, 300 m of ballastless tracks are constructed on each side of the ballasted tracks as boundary conditions and the parameters related to Jiang [29]. Table 1 tabulates the material properties adopted in this study. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**LS-DYNA Model** & **Component** & **Parameter** & **Value** \\ \hline \multirow{8}{*}{MAT\_ELASTIC (MAT\_001)} & \multirow{8}{*}{Main girder/Rail} & [PERSON]’s modulus & 210 GPa \\  & & Poisson’s ratio & 0.3 \\  & & Mass density & 7850 kg/m\({}^{3}\) \\  & & Young’s modulus & 34.5 GPa \\  & & Poisson’s ratio & 0.2 \\  & & Mass density & 2420 kg/m\({}^{3}\) \\  & & Young’s modulus & 33.5 GPa \\  & & Poisson’s ratio & 0.2 \\  & & Mass density & 2410 kg/m\({}^{3}\) \\  & & Young’s modulus & 31.5 GPa \\  & & Poisson’s ratio & 0.2 \\  & & Mass density & 2390 kg/m\({}^{3}\) \\ \hline \multirow{8}{*}{MAT\_CABLE\_DISCRETE\_BEAM (MAT\_071)} & \multirow{8}{*}{Main cable} & [PERSON]’s modulus & 200 GPa \\  & & Mass density & 8500 kg/m\({}^{3}\) \\  & & Young’s modulus & 200 GPa \\  & & Mass density & 8400 kg/m\({}^{3}\) \\ \hline \multirow{8}{*}{MAT\_LINEAR\_ELASTIC\_DISCRETE \_BEAM (MAT\_066)} & \multirow{8}{*}{Track} & Transversal stiffness of the fastener K\({}_{\text{ph}}\) & \(3\times 10^{7}\) N/m \\  & & Transversal damper of the fastener C\({}_{\text{ph}}\) & \(6\times 10^{4}\) N-S/m \\  & & Vertical stiffness of the fastener K\({}_{\text{pv}}\) & \(7.8\times 10^{7}\) N/m \\  & & Vertical damper of the fastener C\({}_{\text{pv}}\) & \(5\times 10^{4}\) N-S/m \\  & & Transversal stiffness of the steeper-ballast K\({}_{\text{th}}\) & \(3\times 10^{7}\) N/m \\  & & Transversal damper of the steeper-ballast C\({}_{\text{ph}}\) & \(6\times 10^{4}\) N-S/m \\  & & Vertical stiffness of the steeper-ballast K\({}_{\text{bv}}\) & \(1.2\times 10^{7}\) N/m \\  & & Vertical damper of the steeper-ballast C\({}_{\text{bv}}\) & \(7.5\times 10^{4}\) N-S/m \\  & & Stiffness of adjacent ballast K\({}_{\text{w}}\) & \(7.8\times 10^{7}\) N/m \\  & & Dumper of adjacent ballast C\({}_{\text{w}}\) & \(8\times 10^{4}\) N-S/m \\  & & Vertical stiffness of the ballast-bridge K\({}_{\text{v}}\) & \(1.7\times 10^{7}\) N/m \\  & & Vertical damper of the ballast-bridge C\({}_{\text{lv}}\) & \(3.1\times 10^{4}\) N-S/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties.