# Dynamic Response Analysis of Parallel Twin Tunnels Under Different Train Loads
## 3 DEM Model  
Particle flow method (PFC5.0) is a mesoscopic analysis software based on the general discrete element model (DEM) framework, applicable for both the three-dimensional model (PFC3D) and two-dimensional model (PFC2D). PFC has been successfully applied in a series of cutting-edge scientific research applications in mining, civil engineering, machinery, materials science, and other fields [19; 20; 21; 22]. There are many studies on the establishment of large-scale models using DEM. In this study, PFC2D is used because it is recommended in the user manual for its convenience. Before establishing the super-close twin-tunnel model with \(D\) = 6.2 m using DEM, it was necessary to determine the meso-parameters of silty sand and C60 concrete, which can be influenced by particle size in DEM. This parameter determination followed the  
Figure 1: The load vibration curves of type A trains under different loads. methodology established in our previous research [23; 24]. To enhance the accuracy and reduce calculation time, the model used tiny particles (silty sand (1)) near the tunnel and larger particles (silty sand (2)) at greater distances from the tunnel. The calibration process for silty sand (1) is described as follows. First, the shear stress-strain curve of silty sand was obtained through direct shear tests in the laboratory. Second, a numerical direct shear test model was established in DEM. The normal stiffness, tangential stiffness, and friction coefficient of particles in the model were adjusted to obtain the shear force-displacement curve. Under varying loading pressures (100, 200, 300, and 400 kPa), when the curve aligned with that obtained from the direct shear tests, the resulting meso-parameters of the silty sand were essential for the model, as shown in Figure 2. The meso-parameters for silty sand and C60 concrete are presented in Table 1.  
The specific modeling steps of the super-close twin-tunnel model under different loading conditions are as follows:  
1. Generate walls and particles. Specifically, particles with \(r_{1}\) = 0.06-0.1 m are placed far from the tunnel area, particles with \(r_{2}\) = 0.03-0.05 m are positioned closer to the tunnel area, and particles with \(r_{3}\) = 0.009-0.012 m are set near the tunnel area. The overall model is shown in Figure 3.
2. The gravity is set to g = 9.81 m/s\({}^{2}\), and the damping is set to 0.7 during the model equilibrium phase and 0.1 during the subway train operation phase. 3. Set up monitoring points. The monitoring points are arranged in the twin-tunnel structure and its surroundings, with surface monitoring points established at intervals of 0.5 m. The detailed layout of the monitoring points within the tunnel structure and its surroundings is illustrated in Figure 4. Due to the proximity of T1 and T2, a single monitoring point is placed at 0\({}^{\circ}\) of T1 and 180\({}^{\circ}\) of T2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Silty Sand (1)** & **Silty Sand (2)** & **C60 Concrete** \\ \hline Radius of particles (m) & \(r_{1}\) = 0.06–0.1 & \(r_{2}\) = 0.03–0.05 & \(r_{3}\) = 0.009–0.012 \\ Density (kg/m\({}^{3}\)) & 2000 & 2000 & 2500 \\ Contact Young’s modulus (Pa) & \(2.19\times 10^{7}\) & \(2.27\times 10^{7}\) & \(3.65\times 10^{10}\) \\ The ratio of contact normal to shear stiffness & 1.0 & 1.0 & 1.0 \\ Coefficient of friction & 0.63 & 0.63 & 0.84 \\ The normal strength of the parallel bond (Pa) & \(-\) & \(-\) & \(1.0\times 10^{100}\) \\ Tangential strength of parallel bond (Pa) & \(-\) & \(-\) & \(1.0\times 10^{100}\) \\ Contact Young’s modulus of parallel bond (Pa) & \(-\) & \(-\) & \(3.65\times 10^{10}\) \\ The ratio of parallel bond contact normal to shear stiffness & \(-\) & \(-\) & 1.0 \\ Parallel bonding radius & \(-\) & \(-\) & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Micro-parameters of silty sand in the DEM model.