# Simulation Study on Ground Vibration Reduction Measures of the Elevated Subway Line
## 2 Numerical Model
### Transmission Path Model  
The finite element software Workbench (v.17.0) was used to establish the pier-soil-building finite element model as a transmission path model, and the dynamic force of the pier calculated by the vibration source model was used as an input and loaded into the pier-soil-building finite element model.  
In the process of research, these special characteristics of the soil medium should be taken into account. When the soil medium is basically in the range of elastic deformation under some specific conditions, it can be treated as an elastic medium and the fluctuation law of elastic mediums can be applied directly. Because the deformation caused by subway vibration is a small deformation, the elastic principal relationship is used for simulation. According to the provided soil dynamics parameters and based on the related research, four layers of soil bodies were established, and the material parameters of each soil layer are shown in Table 3.  
The study shows that the ground vibration decreases as the distance of the building from the track centerline increases. In this paper, we only study the buildings at the nearest distance from the centerline of the track and establish the following geometric model and finite element model according to the building of the sensitive point and the location information of the corresponding bridge piers.  
The soil model is 109 m \(\times\) 60 m \(\times\) 50 m. the building has a floor height of 3 m and 6 floors. The closest distance between the building and the centerline of the track is about 18.5 m. Figure 5 shows the geometric model of the pier-soil-buildings, which is a 1/2 model with symmetric boundary conditions in order to reduce the amount of calculation.  
Figure 6 shows the corresponding pier-soil-building finite element model. When the cell length is divided \(\Delta x=\lambda_{\text{smin}}/12\), the results are more accurate; when the cell length is divided \(\Delta x=\lambda_{\text{smin}}/6\), the accuracy of the calculation can be guaranteed beyond 0.5 \(\lambda_{\text{smin}}\) of the vibration source. The grid size is divided according to Equation (3):  
\[l<\frac{C_{s}}{6f_{\text{max}}} \tag{3}\]  
where \(l\)--Size of model meshing cells (m);  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soil Layer** & **Density (kgm\({}^{-3}\))** & **Poisson’s Ratio** & \begin{tabular}{c} **Shear Wave** \\ **Velocity (m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Shear Modulus** \\ **(GPa)** \\ \end{tabular} &
\begin{tabular}{c} **Elastic Modulus** \\ **(GPa)** \\ \end{tabular} \\ \hline First layer & 1930 & 0.35 & 130 & 0.102 & 0.276 \\ Second layer & 1850 & 0.35 & 260 & 0.134 & 0.363 \\ Third layer & 1841 & 0.36 & 270 & 0.134 & 0.365 \\ Fourth layer & 1839 & 0.35 & 201 & 0.072 & 0.200 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Soil material properties.  
Figure 5: Pier–soil–building geometric modeling: (**a**) front view; (**b**) side view.  
A calculation according to Equation (3) can be obtained: \(l<\frac{C_{s}}{6f_{\max}}=\frac{201}{6\times 80}=0.41875\) m, taking the grid size of 0.3 m. If all cells in the model are divided according to 0.3 m, a large number of cells will be generated, which will seriously affect the efficiency of the calculation. Under the premise of ensuring the correct results, the grid size of the vibration source region is controlled as 0.3 m, and the grid size can be relaxed accordingly in the region farther away from the vibration source. The final overall grid number of the model is 674,397 and the number of nodes is 2,107,071.