## 3 Numerical Model and Validation
### Track-Soil-Building Finite Element Model  
As shown in Figure 8, the numerical model consists of the track structure, foundation soil, and over-track building. All materials were modeled using a linear elastic principal model. The floor slabs, walls, and platform were assumed to be conventional shell elements (S4R). The structural columns and beams were constructed using two-node linear beam elements (B31). The foundation soil dimensions were 110 m \(\times\) 170 m \(\times\) 70 m (X \(\times\) Y \(\times\) Z) and were constructed using hexahedral solid units (C3D8R). The track structure consists of rails, fasteners, sleepers, and ballasts, and the fasteners were modeled using spring-damping units, while the rest of the track structure was constructed using hexahedral solid units (C3D8R). To address the interference caused by vibration wave reflection at the far field boundary, which can lead to energy superposition and result in inaccuracies, an infinite element approach was employed to simulate this boundary issue within the soil domain. The absorbing boundary was constructed using a three-dimensional solid eight-node infinite unit (CIN3D8). Figure 7: Wheel–rail contact force. The specific structural parameters of the over-track building as well as the pile foundation are shown in Table 1. For detailed information regarding the soil properties in each layer, please refer to Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Location** & **Structural Component** & \begin{tabular}{c} **Concrete** \\ **Material** \\ \end{tabular} & \begin{tabular}{c} **Thickness** \\ **(m)** \\ \end{tabular} & \begin{tabular}{c} **Dimension of** \\ **Cross-Section (m)** \\ \end{tabular} \\ \hline \multirow{4}{*}{Building} & Slab & C35 & 0.15 & - \\  & Column & C35 & - & \(0.9\times 0.7\) \\  & Beam & C35 & - & \(0.7\times 0.7\) \\  & Wall & C30 & 0.35 & - \\ \hline \multirow{4}{*}{Platform} & Platform & C35 & 0.2 & - \\  & Column & C35 & - & \(0.8\times 0.8\) \\  & Column & C35 & - & \(0.8\times 1\) \\  & \multirow{4}{*}{\begin{tabular}{c} Transversal partition \\ Longitudinal beam \\ Secondary beam \\ \end{tabular} } & C35 & - & \(1.5\times 1.5\) \\  & \multirow{2}{*}{\
\begin{tabular}{c} Transversal partition \\ Longitudinal beam \\ Secondary beam \\ \end{tabular} } & C35 & - & \(1\times 0.6\) \\  & \multirow{2}{*}{
\begin{tabular}{c} C35 \\ Secondary beam \\ \end{tabular} } & C35 & - & \(2\times 1\) \\  & & C35 & - & \(0.8\times 0.6\) \\ \hline \multirow{2}{*}{Pile foundation} & Pile & C30 & - & R = 0.4 \\  & Pile cap & C30 & 1 & \(3\times 1.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of structural components. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soil Layer** & **Plain Fill** & **Clay** & **Silt** & **Weathered Rock** \\ \hline
**Thickness (m)** & 1.5 & 18.1 & 25.4 & \(\infty\) \\ \hline
**Density (kg/m\({}^{3}\))** & 1650 & 1840 & 2010 & 3200 \\ \hline
**Elastic modulus (MPa)** & 133.88 & 328.5 & 308.8 & 5706 \\ \hline
**Shear wave** & 174 & 259 & 242 & 833 \\
**velocity (m/s)** & & & & \\ \hline
**Compressional** & 367 & 418.8 & 399.57 & 653.2 \\
**velocity (m/s)** & & & & \\ \hline
**[PERSON]’s ratio** & 0.341 & 0.331 & 0.312 & 0.285 \\ \hline
**Damping ratio** & 0.03 & 0.03 & 0.03 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of the soil layer. Figure 8: Track–foundation–soil–building finite element model. The track structure was situated at ground level, with a three-story building erected above it. The soil was represented in distinct layers, exhibiting a damping ratio of 0.03. Both the dimensions and parameters of the building and track structure strictly adhere to realistic values. Under small strain conditions, the soil and pile demonstrate reasonable elastic properties; however, their dynamic behavior exhibits strong coupling. Therefore, an embedded regional constraint was incorporated into the pile-soil interaction model to accurately describe their contact behavior.