# A Finite Element Analysis of Tunnel Lining Demolition by Blasting for Subway Tunnel Expansion
## 4 Blasting Mechanical Characteristics  
### Numerical Calculation Model and Parameters  
A three-dimensional model is established for simulation using LS-DYNA. In order to reduce the boundary effect and ensure the accuracy of the calculation, the dimensions of the numerical model are as follows: the boundaries in the X, Y, and Z directions are 120 m, 70 m, and 1.5 m, respectively, as shown in Figure 4.  
The lateral boundary of the numerical model is subject to normal constraints in the horizontal direction, the bottom is subject to fixed constraints in the vertical direction, and the non-reflection boundary conditions of all boundaries are set to simulate the semi-infinite boundary of the blasting [23]. In order to approach the actual cutting seam of the existing pre-cut subway tunnel lining in the simulation, the element strength reduction method is adopted to set the cutting seam element material to 0.1 times that of the tunnel lining material [24]. For a more convenient dimensional analysis and calculation, the unit is cm-g-ms in the modeling process [19].  
The in situ expansion excavation of a subway tunnel to a subway station is carried out using the CD excavation method. The existing subway tunnel lining structure of the left drift heading is demolished after the left drift heading excavation step (i.e., the first step of the expansion excavation). Therefore, the excavation free face during the right drift heading excavation is more than that during the left drift heading excavation step. Considering the excavation free face can attenuate the propagation of blasting waves; the most dangerous blasting step occurs in the left drift heading excavation step. This paper only calculates and analyzes the left drift heading excavation of the CD excavation method [1; 2; 3; 4].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Hole Category**} & **The Segments** & **Number of Holes** & **Charge Collection (kg/m)** & **Charge Quantity** & **Dosage Per** \\  & **of Detonator(s)** & & & **Per Hole (kg)** & **Dose (kg)** \\ \hline \multirow{6}{*}{Relief holes} & 1 & 16 & 0.67 & 1 & 16.0 \\  & 3 & 17 & 0.67 & 1 & 17.0 \\  & 5 & 14 & 0.33 & 0.5 & 7.0 \\  & 7 & 7 & 0.33 & 0.5 & 3.5 \\  & 9 & 4 & 0.33 & 0.5 & 2.0 \\  & 11 & 24 & 0.33 & 0.5 & 12.0 \\ \hline Trim holes & 13 & 30 & 0.23 & 0.35 & 10.5 \\ \hline Bottom holes & 15 & 15 & 0.33 & 0.5 & 7.5 \\ \hline Total & - & 127 & - & - & 75.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The explosive charge for the left drift heading excavated by the CD excavation method.  
Figure 4: Numerical model of subway station using the CD excavation method.  
Plastic mechanical models are adopted for the rock mass and concrete, and the C30 concrete material is defined as MAT_PLASTIC_KINEMATIC in the numerical model. The material parameters of the calculation model are shown in Table 3, and the physical parameters of the No.2 rock ammonium nitrate explosive are shown in Table 4. The relationship between the unit pressure and volume (P-V) of the detonation products of high-energy explosives is described by Equation (2), which can simulate the explosive explode state accurately.  
\[P=A\left(1-\frac{\omega}{R_{1}V}\right)e^{R_{1}V}+B\left(1-\frac{\omega}{R_{2 }V}\right)e^{-R_{2}V}+\frac{\omega E}{V} \tag{2}\]  
where \(V\) is the relative volume; \(E\) is the internal energy constant; and \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are state equation coefficients.