## 4 Numerical Results and Discussions  
### Comparisons  
The closed-form analytical solution of Equation (31) is formulated as a multivariate optimization problem where the number of optimized variables is \(2n+1\). An increase in the number of blocks indeed results in a better accuracy of the solution; however, this is directly accompanied by the burden of the computation time, thus a compromise between the efficiency and accuracy should be accomplished. Table 1 calculates the values of seismic bearing capacity factors versus the number of blocks without considering the seismic loading. It is obvious that the ultimate bearing capacity of the foundation decreases with the number of blocks gradually increasing, and the corresponding bearing capacity factors show the same tendency as the bearing capacity. Notably, the reduction effect is significant with \(n\) varying from 2 to 5 and then slowly disappearing with a reduction of no more than 0.06% when \(n\) = 14, which indicates at this time that a further increase in the number of blocks leads to a neglectable positive effect on the upper bound solution and results in great time consumption. Therefore, a mechanism of 14 blocks is adopted in the later development. The existing work of [PERSON] [43] investigated the bearing capacity problem by the same failure mechanism; however, the optimization is implemented separately about three independent components, that is \(N_{\gamma\text{e}}\), \(N_{\text{pe}}\), and \(N_{c_{\gamma\text{e}}}\), and the ultimate bearing capacity is obtained by the superposition method. Instead, an alternative that directly optimizes Equation (31) is also prevalent among many scholars. It turns out that the latter tends to estimate the solution on the conservative side. Consequently, in this work, the direct optimization of Equation (31) is implemented. In the linear M-C failure criterion with \(n=14\), \(B=1\) m, \(\zeta=1\), \(\varphi=30^{\circ}\), \(c=15\) kPa, \(p=10\) kPa, and \(\gamma=18\) kN/m\({}^{3}\), Tables 2-5 list the results of the bearing capacity and the corresponding bearing capacity factors of [PERSON] [43] for comparison. The results of the present paper show a good consistency with those of [PERSON] [43]; the maximum difference is approximately 5% in the case of non-earthquake. However, the difference would be up to 30% in the presence of an earthquake, which states clearly that the discrepancy between direct optimization and the superposition method may be significantly amplified by the earthquake loads. The difference in the dimensionless coefficient \(N_{\gamma\text{pe}}\) is the maximum among the three components while the rest is not significant. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(k_{\text{h}}\)} & \multicolumn{4}{c}{\(\varphi\)\({}^{c}\)} \\ \cline{2-7}  & \multicolumn{2}{c}{**15**} & \multicolumn{2}{c}{**20**} & \multicolumn{2}{c}{**25**} & \multicolumn{2}{c}{**30**} \\ \cline{2-7}  & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** \\  & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** \\ \hline
0 & 176.34 & 168.4 & 268.60 & 254.83 & 429.00 & 405.24 & 726.41 & 684.02 \\
0.1 & 140.90 & 134.79 & 210.81 & 200.59 & 330.20 & 312.86 & 546.95 & 516.61 \\
0.2 & 108.75 & 102.64 & 160.22 & 151.77 & 246.73 & 233.66 & 400.75 & 378.93 \\
0.3 & 81.43 & 64.80 & 118.43 & 110.14 & 179.79 & 169.02 & 287.17 & 270.8 \\
0.4 & 59.83 & 51.40 & 86.27 & 67.50 & 129.98 & 117.92 & 206.30 & 189.09 \\
0.5 & 45.80 & 39.80 & 65.63 & 52.30 & 98.57 & 70.20 & 156.32 & 128.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of seismic bearing capacity \(p_{\text{oe}}/\text{kPa}\).