## 6 Validation  
Since no theoretical models have estimated rock slopes' critical seismic acceleration coefficient in the MHB failure criterion, a numerical simulation using finite element limit analysis (FELA) was adopted here to validate the proposed method. A recent study showed that FELA can be adopted to accurately evaluate the seismic stability of slopes by combining with the pseudo-static approach [21]. It should be noted that FELA was a 2D numerical simulation software, which had been widely used to study the stability issues of various engineering structures [18; 21; 22]. Due to the 2D analysis neglecting the influence of the energy dissipation rate done by horn failure part (see Figure 4), the results obtained from 2D analysis are more conservative when compared with those from 3D analysis [7; 9]. Recent works also showed that the results for \(B/H=+\infty\) can be regarded as the 2D solutions [7; 23]. Figure 5 shows the critical seismic acceleration coefficients \(k_{hc}\) calculated in this work compared with those calculated by FELA for \(\beta=45^{\circ}\), \(GSI=30\), \(D_{i}=0.5\), and \(m_{i}=10\). It is worth noting that, in Figure 5, \(SR\) is a defined dimensionless parameter, and can be calculated by using Equation (30).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(SR\) & \(m_{i}\) & \(GSI\) & \(D_{i}\) & \(\beta\) (\({}^{\circ}\)) & \(\sigma_{ci}\) (kPa) & \(\gamma\) (kPa) & \(H\) (m) & \(k_{hc}\) (g) \\ \hline \multirow{3}{*}{4} & \multirow{3}{*}{10} & \multirow{3}{*}{10} & \multirow{3}{*}{0} & \multirow{3}{*}{35} & 160 & 20 & 2 & 0.147 \\  & & & & & 252 & 21 & 3 & 0.147 \\  & & & & & 352 & 22 & 4 & 0.147 \\  & & & & & 460 & 23 & 5 & 0.147 \\ \hline \multirow{3}{*}{6} & \multirow{3}{*}{5} & \multirow{3}{*}{20} & \multirow{3}{*}{0.3} & \multirow{3}{*}{40} & 240 & 20 & 2 & 0.220 \\  & & & & & 378 & 21 & 3 & 0.220 \\  & & & & & 528 & 22 & 4 & 0.220 \\  & & & & & 690 & 23 & 5 & 0.220 \\ \hline \multirow{3}{*}{8} & \multirow{3}{*}{15} & \multirow{3}{*}{20} & \multirow{3}{*}{0.5} & \multirow{3}{*}{45} & 320 & 20 & 2 & 0.185 \\  & & & & & 504 & 21 & 3 & 0.185 \\  & & & & & 704 & 22 & 4 & 0.185 \\  & & & & & 920 & 23 & 5 & 0.185 \\ \hline \multirow{3}{*}{12} & \multirow{3}{*}{20} & \multirow{3}{*}{40} & \multirow{3}{*}{1.0} & \multirow{3}{*}{50} & 480 & 20 & 2 & 0.282 \\  & & & & & 756 & 21 & 3 & 0.282 \\ \cline{1-1}  & & & & & 1056 & 22 & 4 & 0.282 \\ \cline{1-1}  & & & & & 1380 & 23 & 5 & 0.282 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the \(k_{hc}\) of given rock slopes with the same \(SR\). \(SR\) is a dimensionless parameter defined in Equation (30); \(m_{i}\) is the Hoek–Brown constant; \(GSI\) is the geological strength index; \(D_{i}\) is the disturbance factor of the intact rock; \(\beta\) is the slope angle; \(\sigma_{ci}\) is the uniaxial compressive stress of the intact rock; \(\gamma\) is the unit rock weight; \(H\) is the slope height; \(k_{hc}\) is the critical seismic acceleration coefficient.  
From Figure 5, it could be seen that both \(SR\) and \(B/H\) had a significant influence on the critical seismic acceleration coefficient \(k_{hc}\). As also shown in Figure 5, one could conclude that the values of the critical seismic acceleration coefficient \(k_{hc}\) calculated in this work for \(B/H=+\infty\) were slightly higher than those from FELA, with the most significant ratio less than 3.5%, which illustrates the validity of the proposed method. It can also be seen in Figure 5 that the gap between the results from this study and FELA increased with the \(B/H\) decrease, due to the FELA is a 2D numerical simulation software, which indicated that it would lead to conservative estimations for stability of three-dimensional slope, especially for narrow slopes. The similar conclusions can be found in [PERSON] et al. (2019) and [PERSON] (2018). These observations illustrated that the proposed method could be used to accurately determine the critical seismic acceleration coefficient \(k_{hc}\) for 3D rock slopes.