## 2 Research Methodology
### Independent Model Verification  
In addition to the database in [5], extra simulations were also performed in this study to provide an independent evaluation dataset. All the simulations were performed using the FE software Abaqus 6.14 [35]. The FE mesh in Figure 6 (Case 2, \(Z_{D}/2R\) = 1.5, \(T/2R\) = 0.5) is typical of the meshes employed in all analyses. Considering the symmetry of the problem, only half of the pile-soil system was modeled. Just as in [5], the footing has a radius of \(R\) and a thickness of \(R/50\). The soil domain was chosen as 200\(R\) (\(R\) is the radius of the foundation) in diameter and 200\(R\) in depth. Both the soil and foundation were simulated with eight-node linear strain brick elements (i.e., 'C3D8'). A sensitivity analysis with the model doubling the mesh density and geometric dimensions showed a difference of less than 3% of the stiffness coefficients. The interface behavior between the pile and soil was assumed to be fully rough and modeled using the \"tie\" constraint in Abaqus. The soil was modeled by an elastic model, which is fully characterized by two parameters: the shear modulus and Poisson's ratio. In this study, Poisson's ratio was assumed to be constant values of 0.2 and 0.49 for sand and clay, respectively, while the ratio of shear stiffness between clay and sand (\(G_{\text{sand}}/G_{\text{clay}}\))varied from 1 to 40. In the end, a total of 108 extra elastic simulations were performed in this study, with 12 of them dedicated to numerical model validation and the other 96 cases dedicated to the independent evaluation of the trained model. Details about the simulations conducted in this study can be found in Table 1.  
Table 3 shows the comparison of the elastic stiffness coefficients between the computed values using the numerical model in this study and those from [5]. All the FE simulations were performed on a computer, the CPU and RAM of which were Intel (R) Xeon (R) W-2223 CPU @ 3.60 GHz and 16 GB, respectively. In total, it required more than 100 h to finish all the simulations. As shown in the table, for all 12 simulation cases, the numerical model in this study can predict the elastic stiffness coefficients of the foundation under various loading conditions well. The maximum difference between this study and [5] is less than 1%.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**NO.**} & **Embedded** & \multirow{2}{*}{**Condition**} & \multirow{2}{*}{\(Z_{\text{D}}/2R\)} & \multirow{2}{*}{\(T/2R\)} & \multirow{2}{*}{\(G_{\text{sand}}/G_{\text{clay}}\)} & \multirow{2}{*}{\(K_{\text{V}}\)} & \multirow{2}{*}{\(K_{\text{H}}\)} & \multirow{2}{*}{\(K_{\text{M}}\)} & \multirow{2}{*}{\(K_{\text{C}}\)} \\  & & & & & & & & & \\ \hline \multirow{2}{*}{1} & Case 1 & 0 & 0 & 1 & \(5.273\) & \(4.578\) & \(3.634\) & \(-0.567\) \\  & & & & & (0.04\%) & (0.15\%) & (0.80\%) & (0.60\%) \\ \multirow{2}{*}{2} & Case 1 & 0 & 0.5 & 5 & \(16.963\) & \(7.030\) & \(6.623\) & \(0.408\) \\  & & & & & (0.30\%) & (0.35\%) & (0.8\%) & (0.75\%) \\ \multirow{2}{*}{3} & Case 1 & 0 & \(\infty\) & 1 & \(7.954\) & \(5.353\) & \(5.217\) & \(-0.030\) \\  & & & & & (0.14\%) & (0.24\%) & (0.60\%) & (1.00\%) \\ \hline \hline \end{tabular} Note: the value in the bracket represents the percentage of the difference between this study and [5].  
\end{table}
Table 3: Comparison of the elastic coefficients between this study and [5].  
Figure 6: Typical FEM mesh employed in this study.