## 3 Comparison of Two- and Three-Dimensional Finite Element Analyses  
Two-dimensional (plane strain) finite element analysis (2D FEA) may not provide an accurate response of a FSPS, particulary for a footing with a small shape ratio (square footing) adjacent to a wall of spaced piles (pile-wall), because of the limitations in considering the square footing and pile-wall as a continuous (infinite shape ratio) footing and wall. The effects of out-of-plane geometries (footing length and pile spacing) of this FSPS can be modeled more appropriately using three-dimensional finite element analysis (3D FEA).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Description/Value** & **No. of Conditions** \\ \hline Analysis Type & 2D plane strain and 3D FEA & 2 \\ Soil Type & Kaolin, silty clay, and kaolin–sand & 3 \\ L\({}_{\text{f}}\)/B\({}_{\text{f}}\) & 1, 2, 3, and 5/infinite & 1 and 4 for 2D and 3D FEA, respectively \\ Bi\({}_{\text{p}}\)/B\({}_{\text{f}}\) & 1, 3, 5, and 7 & 4 \\ B\({}_{\text{p}}\) and S\({}_{\text{p}}\) (m) & 0.254, 0.305, and 0.356 & 3 \\ Total No. of Conditions & & 180 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variables and conditions of FSPS.  
Figure 3: Validation of P-\(\delta\)v curves resulting from 3D FE models with previous experimental tests for surface strip footings without adjacent pile–walls resting on (**a**) kaolin [29]; (**b**) silty clay [30]; and (**c**) kaolin–sand [31]. Reprinted/adapted with permission from Ref. [20]. 2024, American Society of Civil Engineers.  
The profiles of pile lateral deflection (\(\delta_{\rm L}\)) versus depth for a pile-wall adjacent to a square footing, which is under a pressure (P) of 200 kPa and resting on kaolin-sand soil, are shown in Figure 4 using 2D and 3D FEA. Pile deflections are measured along the middle pile of the pile-wall and at its center shown as point 1 in Figure 1c. Figure 4 shows that the pile is shifted and deflected, and the pile deflections in 2D and 3D FEA have a similar trend below a depth of 8.0 m, but they diverge above a depth of 8.0 m. In addition, Figure 4 shows that the lateral deflection of the pile head (\(\delta_{\rm LH}\)) using 2D FEA is 0.0315 m, which is 450% greater than the 0.007 m of \(\delta_{\rm LH}\) using 3D FEA. The divergence between the pile deflections is mainly because the square footing in 2D FEA is treated as a continuous/infinite footing, which induces greater lateral pressure on the pile-wall. Due to the soil support in front of the pile-wall, as shown in Figure 1c, the deflection profiles of 2D and 3D FEA are bent between the depths of 5.0 and 8.0 m. Below a depth of 8 m, the two profiles of pile deflections are approximately merged together. The change in the shape ratio for the footing and/or pile spacing for the pile-wall has a negligible impact on the pile deflection below a depth of 8.0 m. The lateral deflections of the pile toe (\(\delta_{\rm LT}\)) using both 2D and 3D FEA have an approximately identical deflection at 0.0015 m,\(d\) indicating a small influence from the adjacent footing pressure and an identical point of pile fixity.  
From the comparisons in Figure 4 for a pile-wall adjacent to a square footing, it is clear that 2D FEA in comparison with 3D FEA may underestimate the pile lateral deflection by more than 400% because 2D FEA cannot consider the footing shape ratio and pile spacing.