## 4 Numerical Results and Discussions
### Parametric Study
\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 & 11.13 & 11.00 & 15.15 & 14.87 & 21.33 & 20.78 & 31.33 & 30.25 \\
0.1 & 9.61 & 9.50 & 12.90 & 12.69 & 17.89 & 17.50 & 25.83 & 25.09 \\
0.2 & 8.07 & 7.96 & 10.72 & 10.54 & 14.67 & 14.37 & 20.85 & 20.32 \\
0.3 & 6.60 & 6.48 & 8.71 & 8.53 & 11.80 & 11.53 & 16.54 & 16.12 \\
0.4 & 5.22 & 5.14 & 6.85 & 6.75 & 9.21 & 9.07 & 12.78 & 12.58 \\
0.5 & 4.2  
The consideration of earthquake loads is also necessary for the foundation located in an earthquake-prone area; therefore, earthquake analysis is also conducted in this work by the classic pseudo-static method. Not only are both the horizontal and vertical seismic coefficients considered, but also inertia forces acting on the soil mass, the lateral soil of footing, and the superstructure are included. The variation in the seismic bearing capacity with a vertical seismic coefficient is illustrated in Figure 7, where it is interestingly noted that when the horizontal seismic coefficient remains in a small range, such as no more than 0.2, the increase in the vertical seismic coefficient has a negative influence on the bearing capacity of the foundation, whereas a positive influence appears as the horizontal seismic coefficient becomes relatively greater. Such a strange phenomenon somewhat may be caused by the following reasons. First, this work assumes that the failure of the foundation is a process of footing sinking and lateral soil uplifting towards a certain direction. Assuming the vertical direction to be positive, in this case, the work rate done by the horizontal inertia force of the failure blocks is always positive while that of the vertical inertia force is not always positive because the failure blocks under the foundation produce negative work rates. The combined effect of horizontal and vertical seismic loads may result in a decreasing trend of the bearing capacity when the horizontal seismic coefficient is relatively small. As illustrated in Figure 8, the equivalent surface surcharge has a favorable impact on the bearing capacity since it produces a negative work rate in the soil-foundation system and provides a resistance for preventing the foundation from failure. The improvement of the equivalent surcharge on the foundation is obvious in the absence of a horizontal seismic coefficient; then, the improvement effect gradually decreases with an increase in \(k_{\mathrm{h}}\). Figure 9 illustrates the influence of nonlinear parameters on the seismic bearing capacity of the foundations. The nonlinear coefficient \(m\) accounts for the curvature of the strength envelope, an increase in the nonlinear coefficient provokes a significant decrease in the bearing capacity, and the maximal reduction in the bearing capacity achieves 70.0% with a nonlinear coefficient ranging from 1.2 to 2.0 for \(p=25\) kPa. The same influence on the bearing capacity as the nonlinear coefficient is also observed from the uniaxial tensile strength, with a decrease of up to 82.4%. However, the initial cohesion has a beneficial effect on the bearing capacity of the foundation; an increase in the initial cohesion obtains a greater bearing capacity for the foundations. Notably, the variation in the bearing capacity with all nonlinear parameters behaves as a nonlinear tendency. Figure 7: Influence of seismic coefficients on the seismic bearing capacity.