# Train-Induced Vibration Analysis and Isolation Trench Measures in Metro Depot Structures
## 2 Environmental Vibration Model of Metro Depot
### Soil Model  
The soil depth is set to 40 m. The soil dimensions are set to 90 m in the longitudinal direction of the track and 100 m in the transverse direction. Based on the geological survey report, the soil is simplified into five layers and is considered a linear elastic material. The parameters of the soil layers are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline
**Component** & **Material** & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} \\ \hline Filling wall & C30 & 2500 & 30,000 & 0.2 \\ Structural beam & C40 & 2500 & 32,500 & 0.2 \\ Floor slab & C50 & 2500 & 34,500 & 0.2 \\ Structural column & C60 & 2500 & 36,000 & 0.2 \\ Reinforcement & HRB400 & 7800 & 200,000 & 0.3 \\ I-beam & Q345 & 7800 & 206,000 & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Material parameters of building structure. \begin{table}
\begin{tabular}{c c c c c} \hline
**Component** & **Material** & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} \\ \hline Rail 60 & Q235 & 7800 & 206,000 & 0.3 \\ Sleeper & C30 & 2500 & 30,000 & 0.2 \\ Ballast & C30 & 2500 & 30,000 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties of track structure. \begin{table}
\begin{tabular}{c c c} \hline
**Fastener Parameters** & **Numerical Value** & **Unit** \\ \hline Vertical stiffness & \(3\times 10^{7}\) & N/m \\ Lateral stiffness & \(3\times 10^{7}\) & N/m \\ Vertical damping & \(4\times 10^{4}\) & N\(\cdot\)s/m \\ Lateral damping & \(4\times 10^{4}\) & N\(\cdot\)s/m \\ Fastener spacing & 0.6 & m \\ \hline \end{tabular}
\end{table} 
Table 3: Fastener parameters. To prevent the reflection of train-induced vibration waves at the boundary of the soil medium from affecting the calculation results, a viscoelastic artificial boundary is applied at the soil boundary. Springs and dampers are added to the boundary to absorb the energy radiated outward from the vibration source during the vibration process. This eliminates the reflected waves caused by the truncation of the soil medium. As a result, the wave propagation from the near field to the far field can be accurately simulated. The parameters of the normal and tangential springs and dampers [33] at the soil model boundary in this study are shown as follows:  
\[\left\{ \begin{array}{ll}K_{N}=\frac{\lambda+2\mathbb{G}}{(1+A)r}\\ C_{N}=B\rho c_{P}\\ \end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{ll}K_{T}=\frac{G}{(1+A)r}\\ C_{T}=B\rho c_{S}\\ \end{array} \right. \tag{2}\]  
where \(\lambda\) represents the first Lame constant, and \(G\) denotes the shear modulus; \(r\) is the distance from the geometric center of the near-field structure to the boundary line or surface where the artificial boundary point is located; \(\rho\) represents the density of the medium; \(C_{P}\) and \(C_{S}\) are the wave velocities of P-waves and S-waves in the medium, respectively; and \(A\) and \(B\) are correction coefficients, taken as 0.8 and 1.1, respectively. The environmental vibration of the metro depot involves high-frequency vibration calculations, which are directly influenced by the size of the finite element mesh. If the mesh size is too large, high-frequency vibrations may be truncated, leading to distorted calculation results. Theoretically, the denser the finite element mesh, the more accurate the calculation results. However, the three-dimensional model of the depot structure covers a large area. If a uniformly dense mesh is applied to the entire soil region, the total number of elements in the model will become excessively large, resulting in prolonged computation time and higher hardware requirements. Therefore, the selection of mesh size should balance the frequency range of interest and the computation time.