# Safety Assessment of the Cover-and-Cut Method Under Blasting Vibration Induced by Tunnel Excavation
## 2 Numerical Modeling and Parameter Selection
### Determination of the Blast Load  
The blast-induced load impacts both the tunnel wall and its supporting structures. To model the stress propagation caused by blasting, the dynamic load is simulated using FLAC3D software. For simplicity, the time-history curve of the blast load is represented as a triangular wave, serving as an approximation of the equivalent blast load. The corresponding calculation formula is provided below:  
\[P_{D}=\frac{D^{2}\rho_{0}}{2(\gamma+1)} \tag{1}\]  
The parameter \(P_{D}\) denotes the average detonation pressure of the explosive, while \(D\) corresponds to its detonation velocity. The symbol \(\rho\) represents the density of the explosive, and \(\gamma\) refers to its isentropic coefficient, given as 3.0. When the blast hole is partially filled with a decoupled charge, the resulting blasting gas pressure, \(P\), can be expressed as:  
\[P=A\rho^{v_{0}} \tag{2}\]  
In this context, \(P\) represents the gas pressure of the explosive under specific conditions, A is a constant, \(\rho\) denotes the gas density in the same state, and \(v_{0}\) corresponds to the isentropic index of the explosive gas. When the uncoupling coefficient of the charge is low, the conditions \(P\geq P_{k}\) and \(v_{0}=\gamma=3\). Also, when \(P<P_{k}\), and \(v_{0}=v=4\), \(v_{0}=v=1.4\). Here, \(P_{k}\) indicates the critical blasting pressure, and the initial average pressure generated by the blast holes is given by:  
\[P_{0}=\frac{1}{8}\rho_{0}D^{2}k_{d}^{-6}\eta \tag{3}\]  
When the charge uncoupling coefficient is high, the explosive gas expansion proceeds only through the stage where \(P\geq P_{k}\), leading to the derivation of Formula (4).  
Figure 4: Finite element model of the tunnel structure.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Materials** & **Density** & **Elastic** & **Poisson’s** & **Tensile Stress** \\  & **/g/cm\({}^{3}\)** & **Modulus/MPa** & **Ratio** & **/MPa** \\ \hline Dolomite & 2.8 & 60 & 0.25 & 80,250 \\ Clay & 1.93 & 26 & 0.33 & - \\ RC & 3.0 & 25.5 & 0.13 & 1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material parameters.  
\[P_{0}=P_{k}^{\frac{7-\eta}{7}}(\frac{\varrho_{e}D^{2}}{2(1+\lambda)})^{\frac{2}{3}}k _{d}^{-6}\eta \tag{4}\]  
The function \(p(t)\) is typically regarded as an exponential time-delay function, characterized by a generally decreasing exponential trend. Calculations involving this function can be addressed using the triangle approximation. The relationship between load and time is expressed as follows: [31]:  
\[P(t)=P_{0}\frac{t}{t_{r}}(0\leq t\leq t_{r}) \tag{5}\]  
\[P(t)=P_{0}\frac{t_{r}+t_{b}-t}{t_{r}}(t_{r}\leq t\leq t_{r}+t_{b}) \tag{6}\]  
Based on the treatment approach for explosive parameters and the relevant literature, the rise time of the explosion load is established at 100 us, while the duration of the blasting seismic wave is designated as 0.6 s. Figure 5 illustrates the triangular shock wave load, representing the peak value of the blasting load. Here, \(t_{r}\) denotes the rise time, and it indicates the duration of the positive pressure action [32].