## 3 Bearing Capacity of Foundation
### Numerical Analysis  
A selection of typical soil strata was chosen from several engineering cases [12; 38; 42; 43], and their bearing capacity of foundation (\(q_{\rm u}\)) was calculated using the method mentioned above. The detailed parameters can be found in Table 2. The soil arching of the piled embankment with the pile ends located in soil strata S1-S7 was investigated regarding vertical stresses, the earth pressure coefficient, and settlement. Figure 9 shows the vertical stress distribution along the height of the embankment above the subsoil and the pile cap. The results of the finite element analysis are in good agreement with the stress distribution predicted by [PERSON] and [PERSON] [29]. At the top of the embankment, the vertical stress above the subsoil and the pile top is equal to \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Soils** & \(\gamma\) **(kN/m3)** & \(\lambda\) & \(\kappa\) & \(M\) & \(c_{\rm u}\) & \(q_{\rm u}\) & \(q_{\rm u}\) **(Rounded) 1** \\  & & & & & **(kPa)** & **(kPa)** & **(kPa)** \\ \hline S1 & 14 & 0.760 & 0.217 & 0.898 & 17.51 & 99.98 & 100 \\ S2 & 14 & 0.308 & 0.046 & 1.226 & 21.76 & 124.25 & 125 \\ S3 & 17 & 0.326 & 0.023 & 0.898 & 26.40 & 150.74 & 150 \\ S4 & 17 & 0.15 & 0.03 & 0.95 & 30.56 & 174.50 & 175 \\ S5 & 17.5 & 0.326 & 0.078 & 0.984 & 34.85 & 198.99 & 200 \\ S6 & 19 & 0.074 & 0.011 & 0.613 & 48.93 & 279.39 & 280 \\ S7 & 20 & 0.06 & 0.12 & 1.2 & 55.14 & 314.85 & 315 \\ \hline \hline \end{tabular}  
* Note: the bearing capacity of foundation was rounded to 100-315 kPa.  
\end{table} 
Table 2: Detailed parameters of soils. the self-weight stress of the embankment. As the embankment depth increases to a certain level (2 m), the vertical stress above the subsoil begins to decrease while the vertical stress above the pile top increases. This first inflection point corresponds to the height of the external arching. With a further increase in embankment depth, another inflection point appears in the vertical stress distribution above the subsoil, corresponding to the height of the internal arching. Subsequently, the stress above the subsoil increases with depth. On the other hand, the stress above the pile top continues to increase continuously until the bottom of the embankment. It has been found that the vertical stress above the subsoil gradually decreases while the vertical stress above the pile increases as the bearing capacity of the subsoil increases. When the bearing capacity of a foundation reaches 280 kPa, the stress variation becomes less significant, indicating the complete development of soil arching. Additionally, the pile-soil stress ratio increases with the bearing capacity of the foundation. When the bearing capacity of the foundation is 100 kPa, the stress ratio of the pile soil is 1.17, which increases to 3.75 when it is 150 kPa, and finally reaches 34.14 at the bearing capacity of the foundation of 280 kPa. It can be concluded that the increase in bearing capacity of the foundation enhances the transfer of stress to the pile, thereby reflecting the strengthening of the soil arching effect with increasing bearing capacity of the foundation. The complete manifestation of the soil arching effect is achieved when the bearing capacity reaches 280 kPa. The earth pressure coefficient (\(K\)) is the most straightforward parameter to quantify the soil arching effect in a piled embankment. It is defined as the ratio of the horizontal stress to the vertical stress at a specific location within the embankment. The horizontal stress is considered to be independent of direction:  
\[K=\frac{\sigma_{\mathrm{h}}}{\sigma_{\mathrm{v}}} \tag{11}\]  
Figure 10 illustrates the variation in the coefficient (\(K\)) with the height of the embankment supported by piles on different bearing capacities of subsoil.