# Study on Vibration Effects and Optimal Delay Time for Tunnel Cut-Blasting Beneath Existing Railways
## 2 Numerical Simulation of Blasting Vibration of Tunnel Crossing Under the Existing Railway
### Numerical Model and Parameters  
For the situation involving blasting around railways, the Chinese railway specification, Safety Technical Specification for Blasting Vibration of Railway Engineering (TB 10313-2019) [29], specifies the maximum permissible vibration velocity at different vibration frequencies. To analyze the effect of blasting on the ground vibration, a model containing the tunnel is established (Figure 2). The model size is 100 \(\times\) 75 \(\times\) 70 (m). The tunnel excavation was carried out using the two-step method with upper and lower benches, featuring a total diameter of 12 m and with the crown positioned 42 m below the ground surface. Six cutting holes were drilled at a spacing of 1.2 m, perpendicular to the tunnel heading  
Figure 1: The location of the Zhongliangshan Tunnel (red line) and the Chongqing-Guiyang Railway. face, with each hole measuring 80 mm in diameter and 2.2 m in depth. Moreover, three observation points were selected at the surface for subsequent analysis. The monitoring points consist of three key locations: (1) point A, positioned vertically above the vault; (2) point B\({}_{1}\), located 40 m horizontally from the unexcavated side of the tunnel face; and (3) point B\({}_{2}\), situated 40 m horizontally from the excavated side of the tunnel face. The initiation sequence of blastholes is indicated by numerical labels in Figure 2, with sequential detonation progressing according to the assigned serial numbers. All models employ 8-node SOLID164 elements. In the dynamic finite element simulations, Lagrange grids are adopted for rock and lining, whereas Eulerian grids are utilized for explosives and air. The base mesh size is set at 2 m, with refinement to 20 cm around the tunnel periphery and further refinement to 2 cm near the excavation face. The model's upper surface incorporates a free boundary condition, while all other surfaces are assigned non-reflective boundaries. The millisecond blasting process in the tunnel excavation was simulated using LS-DYNA with a coupled Lagrange-Eulerian formulation. The Lagrange algorithm is used for rock and lining, and the Euler algorithm is used for explosives and air. The top surface of the model and the tunnel contour are set as free boundaries, and the rest are non-reflective boundaries. The computational model includes explosives, slurry, geotechnical surroundings, and lining materials. The parameters of the surrounding rock are shown in Table 1, and the material model is MAT_PLASTIC_KINEMATIC. The parameters of the lining are shown in Table 2, and the material model is MAT_JOHNSON_HOLMQUIST_CONCRETE. The blast explosives are simulated using high-explosive materials, which are controlled by the \(*\)MAT_HIGH_EXPLOSIVE_BURN keyword. The Jones-Wilkins-Lee (JWL) equation is used to simulate the relationship between pressure and specific volume in the detonation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Density** & **Elastic Modulus** & **Poisson’s Ratio** & **Yield Stress** \\
**(g/cm3)** & **(GPa)** & **Poisson’s Ratio** & **(GPa)** \\ \hline
2.4 & 13 & 0.25 & \(8\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the rock model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Density** & **Elastic Modulus** & **Shear Modulus** & **Poisson’s Ratio** & **Tensile** \\
**(g/cm3)** & **(GPa)** & **(GPa)** & **Ratio** & **Strength(MPa)** \\ \hline
2.6 & 0.003 & 0.015 & 0.3 & 1.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of the lining model. Figure 2: Tunnel modeling diagram. process.