## 4 Numerical Analysis of Pseudo-Dynamic Method  
In order to further verify the reliability of the pseudo-dynamic method, a two-dimensional finite element analysis of the bedding slope under the condition of an earthquake was performed by using ANSYS software. In the first phase, a stability analysis of this slope under static working conditions was performed by the elastic-plastic finite element strength reduction method, and the slip surface could be directly obtained. The finite element model is illustrated in Figure 11. In the numerical model, three thin weak layers were used to replace the structural planes above the horizontal ground surface, and the Mohr-Coulomb constitutive model simulated the failure behaviors and deformation of the rock mass. The calculated model considered plane strain and eight-node quadrilateral elements to determine the slip surface and safety factor. The boundary conditions in the calculation program were as follows: the bottom of the slope was set as a fully fixed constraint, and both sides of the model were set as horizontal fixed hinge support, and the other sides of the model were free, as shown in Figure 11.  
Figure 12 shows the slip surface of the bedding slope obtained by the numerical simulation. The numbers on contours represent the equivalent plastic strain on the yield surface. It can be found that the plastic zone only runs through structural plane 3, so it is reasonable to determine that structural plane 3 is the slip surface and this result is consistent with the result from the pseudo-dynamic method under the static working  
Figure 11: Finite element model and boundary conditions of the bedding slope. Figure 10: Effect of cohesion \(c\) and internal friction angle \(\varphi\) on safety factor (\(t=0.1\) s, \(k_{\rm h}=0.2,f=1\)). conditions. Moreover, the safety factor calculated by the strength reduction method was \(F=1.15\), and this value was relatively close to the result of the pseudo-dynamic method \(F=1.07\), and the error was \(7.48\%\). In the second phase, seismic dynamic analysis of the bedding slope was performed. One typical sinusoidal acceleration wave used in this numerical analysis is shown in Figure 13. The seismic acceleration coefficient was set to \(k_{\text{h}}=0.2\), which was exactly the same value as in the previous pseudo-dynamic method. In this phase, the result of the static analysis was regarded as the initial stress condition of the dynamic analysis, while the static boundary conditions were modified to artificial viscous-elastic boundaries. The advantage of this is that the artificial viscous-elastic boundaries can be applied to better simulate wave propagation at the boundary. The operation process in the numerical simulation was to apply the normal and tangential spring stiffness and damping coefficients at the boundary points. The normal and tangential spring stiffness can be expressed as follows [34]:  
\[\left\{ \begin{array}{l}K_{\text{bn}}=\frac{1}{1+A}\frac{\lambda+2G}{c2R}\\ K_{\text{bt}}=\frac{1}{1+A}\frac{\lambda}{2R}\end{array} \right. \tag{15}\]  
Figure 12: Slip surface determined by the finite element method. Figure 13: Input seismic acceleration data at different positions of the bedding slope. where \(K_{\rm bn}\) and \(K_{\rm bt}\) are normal and tangential spring stiffness at the boundary points, respectively. \(G\) is the shear modulus of rock mass. \(R\) is the distance from the wave source to the viscous-elastic boundary point. \(A\) is the characteristic parameter of the input wave, \(A\) = 0.8. \(\lambda\) is the Lame coefficient, which can be expressed as  
\[\lambda=\frac{2G\mu}{(1-2\mu)} \tag{16}\]  
The normal and tangential damping coefficients can be expressed as follows:  
\[\left\{ \begin{array}{l}C_{\rm bn}=B\rho v_{\rm p}\\ C_{\rm bt}=B\rho v_{\rm s}\end{array} \right. \tag{17}\]  
where \(C_{\rm bn}\) and \(C_{\rm bt}\) are normal and tangential damping coefficients, respectively. \(B\) is the characteristic parameter of the input wave, \(B\) = 1.1.