## 2 Establishment of Prediction Model
### Track Bed-Tunnel-Soil Coupled FE Model  
Based on metro line 3 in Tianjin, a track bed-tunnel-soil coupled FE model was established to calculate the ground vibration using MIDAS/GTS. A hybrid prediction method that combines numerical simulations and measured data was used to analyse and evaluate the vibration reduction and isolation effects of the comprehensive control method for metro-induced ground vibrations.  
The size of the model was 100 m (length) \(\times\) 40 m (width) \(\times\) 40 m (height), and the soil was divided into three layers; the on-site geological survey results are presented in Table 1. The tunnel was a double-track double-hole shield tunnel with a buried depth of 15 m, diameter of 6 m and segment thickness of 0.3 m. The reinforced-concrete tunnel lining had a Young's modulus of \(E\)= \(3.5\times 10^{10}\) Pa, Poisson's ratio of 0.25 and density of 2500 kg/m\({}^{3}\). A concrete roadbed was poured into the inverted arch of the tunnel, and the foundation backfill and roadbed plate had a dynamic elastic modulus of \(E\) = \(4.2\times 10^{10}\) Pa, Poisson's ratio of 0.3 and density of 2500 kg/m\({}^{3}\). All the parts in this FE model were considered as elastic and isotropic.  
A tetrahedral grid was adopted for the model, divided into a small grid in the near field, a large grid in the far field and a transitional grid in the middle based on the distance from the vibration source. The grid size of the roadbed and tunnel was 0.3 m, and the grid sizes of the first and second layers of soil were divided from the tunnel to the far field by 0.3\(\sim\)0.6\(\sim\)1.2 m. The grid size of the bottom layer of soil was 2 m. The final number of grids for this model was 103428. Artificial viscoelastic absorption boundaries [6; 60] were adopted to eliminate the boundary effect of vibration wave propagation. The setting of viscoelastic artificial boundaries is achieved by setting spring-damping elements on the variable interface. The stiffness and damping coefficients of the boundary layer are listed in Table 2, which could be calculated by the equations as follows [61; 62]. The final established FE analysis model is shown in Figure 4.  
\[k_{i}=\frac{2G_{i}}{r_{i}}A_{i};\ \ c_{i}=\rho_{i}C_{pi}A_{i}; \tag{18}\]  
where \(k_{i}\) and \(c_{i}\) are the stiffness coefficient and the damping coefficient of the boundary, respectively. \(G_{i}\), \(\rho_{i}\) and \(C_{pi}\) are the shear modulus, density and compressive wave velocity of soil, respectively. \(A_{i}\) is the area represented by control node \(i\). \(r_{i}\) is the distance from wave source to artificial boundary node \(i\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **Soil Type** & **Thickness/m** & \begin{tabular}{c} **Shear Wave** \\ **Velocity/(m\(\cdot\)s\({}^{-1}\))** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **/(kg\(\cdot\)m\({}^{-3}\))** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Young’s** \\ **Modulus/MPa** \\ \end{tabular} &
\begin{tabular}{c} **Shear** \\ **Modulus/MPa** \\ \end{tabular} \\ \hline
1 & Plain fill & 3.0 & 130 & 1930 & 0.35 & 88.0 & 32.6 \\
2 & Silty clay & 20.0 & 290 & 1880 & 0.45 & 458.0 & 157.9 \\
3 & Silt & 20.0 & 360 & 2100 & 0.36 & 680.0 & 250.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Stratigraphic parameters.