# Modified Pseudo-Dynamic Bearing Capacity of Shallow Strip Foundation near Slope on Non-Uniform Soil
## 4 Results and Analysis
### Comparisons
#### 4.2.2 Comparison Considering Seismic Forces  
Based on the results of the previous section, it can be found that the results in the static case are very reliable, but the validity of the results when incorporating seismic forces into the limit analysis framework is still unknown; therefore, this section focuses on analyzing the seismic bearing capacity of foundations under different methods. Figure 6 shows the foundation bearing capacity calculated using different methods and the comparison with the existing literature results. It should be noted that the horizontal coordinate of Figure 6 represents the \(\lambda^{\prime}/H\), where \(\lambda^{\prime}\) denotes the wavelength, which can be expressed as \(\lambda^{\prime}=V_{\rm s}T\). Four different means of seismic force quantification are used in the results given in Figure 6, including the pseudo-static method (PS), the conventional pseudo-dynamic method (CPD), the spectral pseudo-dynamic method (SPD), and the modified pseudo-dynamic method (MPD). It should be emphasized that in order to compare the results with those of other studies, the seismic bearing capacity of strip foundations on horizontal ground is calculated in this section. It can be seen from Figure 6 that the results calculated in this paper using the MPD have the same trend and are closer to the results of [PERSON] et al. (2021) [33] and [PERSON] et al. (2024) [22]. [PERSON] et al. (2021) [33] used the SPD to obtain the seismic bearing capacity, where the SPD is an improvement of the CPD to consider the \(\lambda^{\prime}/H\). [PERSON] et al. (2024) [22] also used the MPD and therefore the results are very close to those of this paper, where the slight difference may be due to the different optimization algorithms used. \begin{table}
\begin{tabular}{c c c c c} \hline  & & & & \(N_{\gamma}\) \\ \cline{3-5} \(\boldsymbol{\varphi}^{\rm(\circ)}\) & \(\boldsymbol{L}/\boldsymbol{B_{0}}\) & \(\boldsymbol{\beta}^{\rm(\circ)}\) & **[PERSON] et al. (2021) [29]** & **[PERSON] and [PERSON] (2005) [32]** & **Present Study** \\ \hline
30 & 0 & 0 & 21.84 & 21.82 & 21.84 \\  & & 10 & 13.86 & 13.86 & 13.86 \\  & & 20 & 7.77 & 7.78 & 7.79 \\  & 1 & 10 & 18.56 & 18.54 & 18.56 \\  & & 20 & 14.22 & 14.22 & 14.22 \\
40 & 0 & 0 & 120.48 & 120.27 & 120.56 \\  & & 10 & 69.86 & 69.82 & 69.87 \\  & & 20 & 37.68 & 37.69 & 37.72 \\  & 2 & 20 & 78.24 & 78.18 & 78.25 \\  & 3 & 20 & 96.61 & 96.50 & 96.63 \\ \hline \end{tabular}
\end{table} 
Table 2: The comparisons of \(N_{\gamma}\) with the other literature. Figure 6: The comparison of the seismic bearing capacity using the varying approaches [1; 22; 33; 34]. The results calculated by the pseudo-static method in this paper are also very similar to those of [PERSON] (1999) [1] and [PERSON] et al. (2022) [34], which again proves the validity of the calculation process and the results in this paper. In addition, it can be found that compared with the results obtained by the PS and the MPD, the results calculated by the MPD show a completely different trend, and it can be found that there is a significant unfavorable position, so this paper is based on the MPD to accurately assess and analyze the seismic bearing capacity of the foundations near slope. As mentioned in the analysis above, the trends of the results calculated using the pseudo-static method and the modified pseudo-dynamic method are quite different, due to the fact that these two methods use different ways to quantify the horizontal acceleration coefficients of seismic forces. The depth profiles of the horizontal acceleration coefficients of seismic forces obtained by these two methods at different \(\lambda^{\prime}/H\) are given in Figure 7. It should be noted that the same initial coefficient of seismic acceleration is assumed to be 0.1. For the pseudo-static method, regardless of the position of the soil in the soil profile, the seismic acceleration coefficient remains a constant value that will not change.