## 3 Results Analysis of Pseudo-Dynamic Method
### Dynamic Characteristics of Bedding Slope
#### 3.2.2 The Effects of Various Parameters on Safety Factor
Accordingly, the extreme values of the safety factors calculated by the two methods are also the same. However, the change frequency of the safety factor increases with the increase in the shear wave velocity, and this response frequency is highly sensitive to the change in velocity. From Figure 5a, for \(v_{s}=1\times 10^{3}\)-\(10\times 10^{3}\) m/s, the value of the frequency increases by 901.53% (from 3.9 Hz to 39.06 Hz). It will exacerbate the risk of slope instability when considering that the rock mass is a deformable material. In order to better present the results, when the input wave velocity changes from 500 m/s to 10,000 m/s, for the slope heights \(H=50\) m, \(H=75\) m, and \(H=100\) m, the relationship between change frequency and input wave velocity is shown in Figure 5b. A Log\({}_{10}\) (\(v_{\rm s}\)) value is employed as the abscissa because the range of the velocity is too large. The change frequency of the safety factor increases significantly with the increase in the shear wave velocity, especially for the lowest-height slope. In the present study, the rock mass was considered only as a rigid body based on the limit equilibrium theory, so the change frequency did not have an influence on the safety factor. However, in the deformation mechanism of rock mass, the change frequency will affect the deformation degree and morphology of rock mass, and thereby affect the dynamic stability of rock mass. Hence, this parameter is of great importance for slope deformation. The effects of seismic acceleration coefficients \(k_{\rm h}\) and \(k_{\rm v}\) on safety factor  
The effects of the horizontal and vertical seismic coefficients on the safety factor calculated by the present method were compared with the results of the pseudo-static method, and the results are shown in Figure 6. In the whole calculations, the range and values of the parameters were \(k_{\rm h}=0\)-\(0.4\) and \(k_{\rm v}=0\)-\(0.4\), respectively. Here, the effect of seismic combination was not considered in the analysis; only the seismic action in a single horizontal or vertical direction was considered. It can be seen that the results of the two methods have a discrepancy, mainly because the shear wave velocity \(v_{\rm s}\) and \(v_{\rm p}\) are limited. For Figure 6a, the safety factor calculated by the pseudo-dynamic method is smaller than that of the pseudo-static method. This can be explained by the seismic acceleration in the pseudo-dynamic method, in which the absolute value of the acceleration varies from 0 to \(k_{\rm h}\)g, whereas the acceleration in the pseudo-static method is invariably a constant, i.e., \(k_{\rm h}\)g. It is difficult for the seismic acceleration to converge to the maximum value, due to the value of \(v_{\rm s}\) not being large enough in the pseudo-dynamic method. The smaller the input shear wave velocity, the greater the discrepancy in the results. Also, notice the directions of the seismic inertia forces associated with the calculated results and the differences between the two. Additionally, from Figure 6, it can be seen that the seismic acceleration coefficients have a direct effect on the slope stability, and that the safety factor changes slightly nonlinearly with an increase in \(k_{\rm h}\) or \(k_{\rm v}\). When \(k_{\rm h}>0\), it represents a horizontal inertia force whose direction points to the outside of the slope, which means that the rock mass is more prone to cracking. During an earthquake, this adds the unfavorably oriented seismic inertia force to the slip body, which may cause slope instability. As shown in Figure 6a, the safety factor decreases with an increase in \(k_{\rm h}\), and this seismic inertia force of outside direction will seriously attenuate the slope stability. As an illustration, for \(\varphi=40^{\circ}\), when \(k_{\rm h}\) increases 0\(\sim\)0.4, the safety factor decreases by 40.16% (from 1.9 to 1.14).