## 4 Numerical Results and Discussions
### Parametric Study  
Parametric analysis is conducted in non-associative soils encompassing the seismic coefficients and the nonlinear parameters. For convenience, the minor factors that have no impact on the final result are set as constant in the whole text: \(B=1\) m, \(\gamma=18\) kN/m\({}^{3}\). The other basic inputs parameters are set as \(k_{\text{h}}=k_{\text{v}}=0.2\), \(p=15\) kPa, \(\zeta=0.8\), \(m=1.6\), \(c_{0}=15\) kPa, and \(\sigma_{\text{t}}=15\) kPa. The normality of the plastic flow rule is one of the basic assumptions in the limit analysis, whereas the dilatancy angle of the soil is not necessarily equal to the internal friction angle for the natural geomaterials, which indicates that the plastic flow is not associative. Therefore, the illustrations in Figure 4 present the influence of the nonlinear coefficient on the seismic bearing capacity of foundations concerning different seismic coefficients under associative and non-associative soils, respectively. It appears that the bearing capacity of foundations under non-associative soils is significantly smaller than that under associative soils, which implies that the previous assumption of the normality of plastic flow tends to overestimate the upper-bound of the bearing capacity. In addition, the influence of the vertical seismic coefficient is so slight that can be neglected \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{\(\varphi\)\({}^{c}\)} \\ \cline{2-9} \(k_{\text{h}}\) & \multicolumn{2}{c}{**15**} & \multicolumn{2}{c}{**20**} & \multicolumn{2}{c}{**25**} & \multicolumn{2}{c}{**30**} \\ \cline{2-9}  & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** \\  & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** & **[**43**]** \\ \hline
0 & 3.98 & 3.95 & 6.51 & 6.41 & 10.95 & 10.69 & 19.09 & 18.46 \\
0.1 & 3.10 & 3.07 & 5.07 & 5.02 & 8.49 & 8.35 & 14.70 & 14.34 \\
0.2 & 2.22 & 2.07 & 3.71 & 3.62 & 6.27 & 6.17 & 10.87 & 10.67 \\
0.3 & 1.42 & - & 2.50 & 2.25 & 4.37 & 4.22 & 7.70 & 7.54 \\
0.4 & 0.82 & - & 1.59 & - & 2.91 & 2.47 & 5.27 & 4.97 \\
0.5 & 0.50 & - & 1.03 & - & 1.99 & - & 3.71 & 2.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of seismic bearing capacity factor \(N_{\text{pe}}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{\(\varphi\)\({}^{c}\)} \\ \cline{2-9} \(k_{\text{h}}\) & \multicolumn{2}{c}{**15**} & \multicolumn{2}{c}{**20**} & \multicolumn{2}{c}{**25**} & \multicolumn{2}{c}{**30**} \\ \cline{2-9}  & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** & **This** & **Soubra** \\  & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** & **[**43**]** & **Paper** & **[**43**]** \\ \hline
0 & 2.80 & 2.10 & 5.77 & 4.67 & 11.80 & 10.06 & 24.70 & 21.88 \\
0.1 & 1.54 & 1.01 & 3.46 & 2.61 & 7.37 & 6.04 & 15.75 & 13.59 \\
0.2 & 0.66 & 0.26 & 1.78 & 1.13 & 4.14 & 3.14 & 9.28 & 7.67 \\
0.3 & 0.14 & - & 0.71 & 0.26 & 2.02 & 1.28 & 4.97 & 3.80 \\
0.4 & - & - & 0.21 & - & 0.97 & 0.28 & 2.87 & 1.51 \\
0.5 & - & - & - & - & 0.47 & - & 1.78 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of seismic bearing capacity factor \(N_{\text{pe}}\).