## 3 Three-Dimensional Numerical Model
### Finite Element Calculation Parameters
#### 3.2.2 Dynamic Blasting Loads for New Tunnels  
1. Eigenvalue analysis  
An eigenvalue is used to analyze the inherent dynamic characteristics of the structure, which is an important parameter for the design of the structure subjected to dynamic loads. Through the eigenvalue analysis [37], the dynamic characteristics of the structure such as vibration shape, self-oscillation period, and vibration parameter coefficients can be obtained. The characteristic equations for the calculation of the vibration shape and intrinsic period are as follows:  
\[K\overline{\Phi}_{n}=\omega_{n}^{2}M\overline{\Phi}_{n} \tag{3}\]  
where \(K\) is the stiffness matrix of the structure; \(M\) is the mass matrix of the structure; \"\(\omega_{n}^{2}\)\" is the eigenvalue of the nth vibration mode; and \"\(\overline{\Phi}_{n}\)\" is the eigenvector of the \(n\)th vibration mode, and the spatial iterative method is used to calculate the eigenvectors to converge [31]. 1. Boundary conditions  
According to the literature [31; 38], an elastic boundary is defined by the curved surface spring, and the spring coefficient is calculated according to the foundation reaction coefficient of the road design code. The vertical rock formation reaction force coefficient:  
\[k_{v}=k_{v0}\cdot\left(B_{v}/30\right)^{-3/4} \tag{4}\]  
The horizontal rock formation reaction factor:  
\[k_{h}=k_{h0}\cdot\left(B_{h}/30\right)^{-3/4} \tag{5}\]  
where \(k_{v0}=k_{h0}=E_{0}/30\) and \(B_{v}=A_{v}{}^{1/2}\), \(B_{h}=A_{h}{}^{1/2}\). \(A_{v}\) is the vertical cross-section area of the rock layer, \(A_{h}\) is the horizontal cross-section area of the rock layer, and \(E_{0}\) is the elastic modulus of the rock layer. After extracting the cross-sectional area in each direction based on the 3D solid model (Figure 4), the vertical and horizontal foundation reaction coefficients are calculated by Equations (4) and (5), as shown in Table 3.  
2. Analysis of results  
As shown in Figure 7, the calculation is carried out by Equation (1). A comprehensive comparison of the eigenvalue results of the mass participation coefficients and periods of different vibration modes is conducted, and the final determination of the period values of vibration modes 6 and 7, 1.085584 and 1.031460, are used as the basis data for the calculation of the damping of the vibration modes in the time-range analysis. 2. Time-course analysis  
Time-course analysis is the process of calculating the dynamic characteristics of the structure and the structural response (displacement, internal force, velocity, etc.) at any moment in time. According to the literature [39; 40; 41], the power balance equations used are as follows:  
\[M\ddot{u}(t)+C\dot{u}(t)+Ku(t)=p(t) \tag{6}\]  
where \(M\) is the total mass matrix of the finite element system; \(C\) is the total damping matrix of the finite element system; \(K\) is the total stiffness matrix of the finite element system; \(``\ddot{u}(t)\"\), \(``\ddot{u}(t)\"\) and \(``u(t)\"\) are the acceleration, velocity and displacement vectors of each node of the system; and \(``p(t)\"\) is the dynamic load. \begin{table}