## 3 Three-Dimensional Numerical Model
### Finite Element Calculation Parameters
#### 3.2.2 Dynamic Blasting Loads for New Tunnels
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Rock Layers (Top to Bottom)**} & \multicolumn{3}{c}{**Model X, Y, and Z Direction Rock Reaction Coefficient (t/m3)**} \\ \cline{2-6}  & \(k_{\text{LXh}}\) & \(k_{\text{RXh}}\) & \(k_{\text{FYh}}\) & \(k_{\text{BYh}}\) & \(k_{\text{Zv}}\) \\ \hline Mudstone & 2665.52 & 4387.63 & 3145.71 & 2432.20 \\ Malmstone & 7244.13 & 8970.20 & 7759.39 & 7153.64 \\ Mudstone & 3000.16 & 3126.16 & 2946.43 & 2907.13 \\ Malmstone & 5915.44 & 5884.53 & 5710.24 & 5710.24 \\ Mudstone & 4530.42 & 4527.31 & 4384.49 & 4384.49 \\ Malmstone & 9279.97 & 9156.92 & 8921.78 & 8921.78 \\ Mudstone & 3433.82 & 3471.77 & 3330.95 & 3330.95 \\ Malmstone & 9418.76 & 9440.04 & 9170.55 & 9170.55 \\ Mudstone & 3778.67 & 4061.60 & 3665.52 & 3665.52 \\ Malmstone & 12,121.47 & 12,114.00 & 111,728.72 & 11,728.72 \\ Mudstone & 2872.95 & 2876.63 & 2781.49 & 2781.49 \\ Malmstone & 8071.31 & 8006.92 & 7782.58 & 7782.58 \\ Mudstone & 2849.02 & 3341.86 & 2944.74 & 2944.74 \\ Malmstone & 10,695.17 & 11,704.80 & 10,214.62 & 10,214.62 \\ Mudstone & 5055.50 & 3273.70 & 3833.84 & 3833.84 \\ Malmstone & 16,257.58 & 17,026.51 & 16,334.96 & 16,334.96 \\ Mudstone & 4388.29 & 4656.22 & 4334.61 & 4334.61 \\ Malmstone & 14,776.82 & 14,146.10 & 14,016.62 & 14,016.62 \\ Mudstone & 7913.75 & 8064.80 & 8000.96 & 8000.96 \\ Malmstone & 7416.49 & 7348.26 & 7214.18 & 7214.18 \\ Mudstone & 4157.90 & 4093.52 & 3915.15 & 3915.15 \\ Malmstone & 10,895.32 & 11,533.88 & 10,782.67 & 10,782.67 \\ Mudstone & 3940.98 & 3941.80 & 3755.93 & 3755.93 \\ Malmstone & 11,277.14 & 10,721.60 & 10,633.81 & 10,633.81 \\ Mudstone & 6106.59 & 6398.02 & 6045.57 & 6045.57 \\ Malmstone & 6596.91 & 6338.03 & 6255.26 & 6255.26 \\ Mudstone & 2212.34 & 2078.98 & 2073.10 & 2073.10 & 1556.58 \\ \hline \hline \end{tabular} 
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Rock (1)**} \\ \hline Mud & 2665.52 & 4387.63 & 3145.71 & 2432.20 \\ Malmstone & 7244.13 & 8970.20 & 7759.39 & 7153.64 \\ Mudstone & 3000.16 & 3126.16 & 2946.43 & 2907.13 \\ Malmstone & 5915.44 & 5884.53 & 5710.24 & 5710.24 \\ Mudstone & 4530.42 & 4527.31 & 4384.49 & 4384.49 \\ Malmstone & 9279.97 & 9156.92 & 8921.78 & 8921.78 \\ Mudstone & 3433.82 & 3471.77 & 3330.95 & 3330.95 \\ Malmstone & 9418(1) Boundary conditions  
Define the viscous boundary through the surface spring [31; 38]. The P-wave and S-wave damping calculations in the X, Y, and Z directions of the rock layer required to establish the viscous boundary are shown in Equations (7) and (8). The P-wave damping calculation formula:  
\[C_{P}=\rho\cdot A\cdot[(\lambda+2G)/\rho]^{1/2}=\gamma\cdot A\cdot[(\lambda+2G)/( \lambda g)]^{1/2}=c_{P}\cdot A \tag{7}\]  
The S-wave damping formula:  
\[C_{S}=\rho\cdot A\cdot(G/\rho)^{1/2}=\gamma\cdot A\cdot[(G/(\lambda g)]^{1/2}=c _{s}\cdot A \tag{8}\]  
where \(\rho\) is the density, kg/m\({}^{3}\); \(\gamma\) is the bulk weight, t/m\({}^{3}\); \(\lambda=vE/[(1+v)(1-2v)]\), the bulk modulus, t/m\({}^{2}\); \(G=E/2(1+v)\), and the shear modulus, t/m\({}^{2}\); \(E\) is the modulus of elasticity, t/m\({}^{2}\); \(v\) is Poisson's ratio; \(g\) is the acceleration of gravity; \(A\) is the cross-sectional area of the boundary rock layer, m\({}^{3}\); \(c_{p}\) is the P-wave damping constant; and \(c_{s}\) is the S-wave damping constant. When inputting damping in GTS NX, only \(c_{P}\) and \(c_{s}\) can be input because the program automatically calculates the cross-sectional area of each unit. According to Equations (7) and (8), combined with the mechanical parameters of the rock formation, the results of cp and cs are shown in Table 4.  
(2) Dynamic load  
Blasting load parameters include loading waveform, peak stress, location and direction of action, loading and unloading time, total vibration time, and loading boundary. The blast load waveform adopts the triangular load waveform, the peak load is reached quickly at 10 ms, and the unloading time is 100 ms.