## 4 Results and Discussions  
### Comparisons  
[PERSON] and [PERSON] [12] investigated the stability for 3D single-step slopes without considering the soil's nonlinear features and seismic effects, and came up with the stability number \(N_{s}\) (\(N_{s}=\gamma H_{cr}/c_{t}\)), from which the critical height for 3D single-step slopes can be derived. Where the slope with the critical height is about to collapse, the safety factor is \(FS=1\) at this time. To demonstrate the applicability of the current approach, the cohesion coefficient \(\lambda_{c}\) deduced from the stability number \(N_{s}\) (\(N_{s}=1/\lambda_{c}\)) is introduced in Equation (77) to calculate the safety factor \(FS\), corresponding to \(\lambda_{a}=0.0\), \(m=1\), \(k_{h}=0.0\), where the double steps effect, non-linearity effect, and seismicity effect are eliminated. The detailed data are shown in Table 1, and it is evident that the result is pretty near to one, which verifies the effectiveness of the present method in this scenario.  
In the research of seismic effects, the fundamental distinction between the modified pseudo-dynamic method and the existing pseudo-dynamic method lies in the various assumptions on seismic acceleration. The horizontal seismic acceleration in the original pseudo-dynamic analysis is written as  
\[a_{h}=\Big{[}1+\frac{z}{H}\big{(}f-1\big{)}\Big{]}\cdot k_{h}g\cdot\sin\omega \bigg{(}t-\frac{z}{V_{s}}\bigg{)} \tag{79}\]  
where \(f\) is the amplification factor of the seismic wave. According to the above equation, after eliminating the time and position variables and setting \(f=1.0\), the horizontal seismic acceleration will be simplified to \(k_{h}g\), namely, the seismic acceleration in the pseudo-static analysis. Similarly, the seismic acceleration in the modified pseudo-dynamic analysis represented by Equation (6) will also be simplified to \(k_{h}g\) by removing the time and position variables and setting the damping ratio \(D=0.0\). Thus, with the same input parameters set, the results of the modified pseudo-dynamic method and the original pseudo-dynamic method should be theoretically consistent, which can demonstrate the modified pseudo-dynamic method's efficiency in the seismic analysis of 3D slopes. Table 2 shows the outcomes of the two pseudo-dynamic approaches, and they are strikingly similar, which verifies the applicability of the proposed method once more. The fundamental input  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**B/H**} & \multicolumn{1}{c}{**Results**} & \multicolumn{3}{c}{\(\beta_{1}\)=\(\beta_{2}\)} \\ \hline  & & 45\({}^{\circ}\) & 60\({}^{\circ}\) & 75\({}^{\circ}\) & 90\({}^{\circ}\) \\ \hline
1.0 & \(N_{s}\) by [PERSON] & & & & \\  & and [PERSON] & 54.850 & 23.835 & 14.701 & 11.028 \\  & \(FS\) by the present method & 0.9893 & 0.9823 & 0.9993 & 0.9849 \\
1.5 & \(N_{s}\) by [PERSON] & 46.845 & 20.773 & 12.976 & 8.935 \\  & and [PERSON] & 0.9922 & 0.9743 & 0.9754 & 0.9924 \\
2.0 & \(N_{s}\) by [PERSON] & 42.732 & 19.103 & 12.109 & 8.604 \\  & \(FS\) by the present method & 0.9994 & 0.9919 & 0.9796 & 0.9598 \\
3.0 & \(N_{s}\) by [PERSON] & 39.956 & 17.873 & 11.184 & 7.974 \\  & \(FS\) by the present method & 1.0019 & 1.0010 & 0.9955 & 0.9925 \\
5.0 & \(N_{s}\) by [PERSON] & 37.994 & 17.063 & 10.628 & 7.266 \\  & \(FS\) by the present method & 1.0063 & 1.0033 & 0.9986 & 0.9939 \\
10.0 & \(N_{s}\) by [PERSON] & 36.703 & 16.527 & 10.265 & 6.944 \\  & \(FS\) by the present method & 1.0085 & 1.0026 & 1.0012 & 1.0004 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons of both the present outcomes and the solutions of [PERSON] and [PERSON] [12], corresponding to \(\lambda_{a}=0.0\), \(m=1\), \(k_{h}=0.0\).  
parameters are as follows: \(H=15\) m, \(\beta_{1}=\beta_{2}=45^{\circ}\), \(\lambda_{a}=0.0\), \(m=1.6\), \(c_{0}=25\) kPa, \(\sigma_{t}=30\) kPa, \(T=0.1\) s, \(V_{\rm s}=\infty\).