## 5 Case Study
### Blasting Load and Its Application Method  
The purpose of tunnel blasting numerical simulation in this paper is to obtain the damage range of surrounding rock caused by blasting construction, and it does not discuss the crushing mechanism and fracture effect of rock near blast-holes. Therefore, it is reasonable and feasible to simulate the blasting effect by applying simplified equivalent loads. In tunnel blasting numerical simulation, the shapes of blast-holes are not reflected, and blasting pressure load is equivalently applied to the axis plane of blast-holes through conversion [30]. And explosion pressure load is approximated by triangular explosion load, as shown in Figure 7. In Figure 7, \(P_{\text{max}}\) is peak pressure; \(t_{\tau}\) is the time corresponding to the rise of load; \(t_{\text{s}}\) is total load time. \(P_{\text{max}}\) can be calculated by Formula (39) [34]:  
\[P_{\text{max}}=\frac{1}{8}\rho_{\text{e}}V^{2}k_{\text{d}}^{6}\eta \tag{39}\]  
where \(\rho_{\text{e}}\) is charge density; \(V\) is explosive detonation velocity; \(k_{\text{d}}\) is borehole charge decoupling coefficient, \(k_{\text{d}}=\frac{d_{\text{b}}}{d_{\text{c}}}\), \(d_{\text{b}}\) and \(d_{\text{c}}\) are diameter of blast-hole and diameter of equivalent explosive coil, respectively; \(\eta\) is the multiple of pressure increase value when detonating gas hits blast-hole wall, \(\eta=8\sim 11\). Based on the Saint-Venant principle in mechanics, the equivalent blasting load time-history curve is applied to the plane determined by the central line of blast-hole and the axis of blast-holes in the same row (see Figure 8). Equivalent pressure is determined according to the above principle, and is calculated by Formula (40) [34]. \[P_{\text{e}}=\frac{2\tau_{0}}{a}P_{0} \tag{40}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\rho\)/kg/m\({}^{3}\) & \(E\)/GPa & \(\
u\) & \(\sigma_{0}\)/MPa & \(E_{\text{tan}}\)/GPa & \(\frac{\text{C}}{/\text{g}-1}\) & \(\text{P}\) & \(\alpha\) & \(\beta\) & \(\delta\) \\ \hline
2600 & 30 & 0.27 & 100 & 50 & 2.5 & 4.0 & \(3.15\times 10^{6}\) & 1.0 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main calculation parameters of rock [7; 9]. Figure 7: Schematic diagram of triangular blasting load. where \(P_{\rm e}\) is equivalent pressure; \(P_{0}\) is blasting force acting on single blast-hole wall, \(P_{0}=P_{\rm max}\); \(r_{0}\) is blast-hole radius; and \(a\) is blast-hole spacing. \(t_{\tau}\) can be calculated by Formula (41):  
\[t_{\tau}=\frac{L_{1}}{V} \tag{41}\]  
where \(L_{1}\) is charge length. In the paper, \(t_{\tau}\) = 170 \(\upmu\)s, \(t_{\rm s}\) = 1000 \(\upmu\)s [35]. Explosive parameters are listed in Table 3. The diameter of blast-holes is 42 mm; the diameter of the charges is 32 mm. According to the acoustic wave velocity of rock, \(D_{0}\) = 0.375 is obtained by Formula (7) in this study. Combined with our experience and considering the safety of the box culvert, the threshold value of rock damage corresponding to the edge of the blasting damage impact range can be determined by Formula (42):  
\[D_{\rm lim} =0.20+0.80D_{0} \tag{42}\]  
where \(D_{\rm lim}\) is the threshold value of rock blasting damage; \(D_{0}\) is initial damage of rock. Formula (42) can be used to determine the damage accumulation range of rock caused by blasting. According to Formula (42), the blasting damage threshold of surrounding rock around the box culvert structure is 0.50. In the process of tunnel blasting construction, the bottom of the box culvert is obviously the most dangerous. Therefore, the surrounding rock elements close to the bottom of the box culvert structure were selected for the blasting damage analysis, as shown in Figure 9.  
Figure 8: Blast-hole layout diagram of drilling and blasting pattern. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Explosive** & \(\rho_{\rm e}\) (\(\upmu\)s/\(\upmu\)s) & \(V\) (m/s) \\ \hline
2\# rock emulsion explosive & 950 & 3500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Explosive parameters.