## 2 Numerical Model
From these geometries of the pile, the pile is defined as a long pile with a ratio of embedment length to width greater than 35, as indicated by [PERSON] [3]. The depth of the 3D FE model in the z- or out-of-plane direction is set to be 15 times the width of the pile. [PERSON] et al. [20] confirmed that this depth of  
Figure 1: Schematic view, sections, and boundaries for 2D and 3D FE models of FSPS: (**a**) top view of 3D FE model with side and out-of-plane boundaries; (**b**) transverse section at pile–wall location of 3D FE model with out-of-plane and bottom boundaries; (**c**) longitudinal section at symmetrical axis of 3D FE model and schematic view of 2D FE model with side and bottom boundaries; and (**d**) transverse section at footing location of 3D FE model with out-of-plane and bottom boundaries. Reprinted/adapted with permission from Ref. [20]. 2024, American Society of Civil Engineers. the 3D FE model enclosed by the out-of-plane boundaries has a negligible impact on the behavior of the FSPS. The above constant parameters of the FSPS are shown in Figure 1.  
The variables of the FSPS are investigated with a wide range into 180 combinations of different conditions. Three different soil types are used: kaolin, silty clay, and kaolin-sand. Four footing shape ratios (\(\mathrm{L_{f}/B_{f}}\)) are selected: 1, 2, 3, and 5/infinite. Four footing-pile distances (\(\mathrm{B_{fp}}\) face to face) are chosen: 0.914, 2.743, 4.572, and 6.401 m; three pile widths (\(\mathrm{B_{p}}\)), which are equal to the pile-to-pile spacings (\(\mathrm{S_{p}}\)), are studied: 0.254, 0.305, and 0.356 m. The variables and conditions considered for the FSPS in this study are tabulated in Table 2. [PERSON] and [PERSON] (2017) referred to the factors affecting the pile-to-pile spacing, such as soil type, installation method, layout and verticality errors, pile group efficiency, and economy. [PERSON] and [PERSON] (2018) stated a recommendation that the spacing between adjacent piles must not be less than the smallest width of the pile. Therefore, the pile spacing (\(\mathrm{S_{p}}\)) is selected to be equal to the pile width (\(\mathrm{B_{p}}\)). \begin{table}
\begin{tabular}{c c c c c} \hline
**Properties** & **Kaolin** & **Silty Clay** & **Kaolin–Sand** & **Concrete** \\ \hline Initial modulus in compression \(\mathrm{E_{i}}\) (\(\mathrm{kN/m^{2}}\)) & 19,843 & 4670 & 42,029 & 22.7 \(\times\) 10\({}^{6}\) \\ [PERSON]’s ratio \(\mathrm{\
u}\) & 0.39 & 0.28 & 0.32 & 0.2 \\ Wet unit weight \(\mathrm{\gamma}\) (\(\mathrm{kN/m^{3}}\)) & 16.7 & 18.4 & 18.5 & 24.39 \\ Unit cohesion \(\mathrm{c}\) (\(\mathrm{kN/m^{2}}\)) & 158.5 & 65.5 & 9.2 & 7235 \\ Water content \(\mathrm{\omega}\) & 23\% & 17\% & 11.8\% & - \\ Internal friction angle \(\mathrm{\phi}\) deg. & 8.0 & 13.5 & 31.0 & 20.0 \\ Friction angle at soil–concrete interface \(\mathrm{\phi_{sc}}\), (deg.) & 5.6 & 9.5 & 21.7 & - \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of backfill/ footing soils and concrete for the FSPS. Figure 2: Schematic view and sections of mesh for 2D and 3D FE models of FSPS showing the finite element sizes: (**a**) top view of 3D FE model; (**b**) transverse section at pile–wall location of 3D FE model; (**c**) longitudinal section at symmetrical axis of 3D FE model and schematic view of 2D FE model; and (**d**) transverse section at footing location of 3D FE model. Reprinted/adapted with permission from Ref. [20]. 2024, American Society of Civil Engineers. Currently, experimental tests for FSPS are not available. However, the present 3D FE models of the FSPS were validated with previous experimental tests performed for systems of footing-soil by [PERSON] [29], [PERSON] [30], and [PERSON] [31] for kaolin, silty clay, and kaolin-sand, respectively. When the footing is located outside the footing-wall critical distance, where the performance of the adjacent pile-wall is not affected by the footing, the FSPS behaves as the footing-soil system.