## 4 Finite Element Calculation Model
### Establishment and Verification of a Model  
A three-dimensional numerical model was constructed with ABAQUS finite element software, maintaining identical dimensions to those employed in the indoor model test. The schematic representation of the three-dimensional numerical model and its mesh division are depicted in Figure 10. The soil was characterized using the Mohr-Coulomb constitutive model, while the composite foundation adopted the elastic constitutive model. The pile-soil contact surface incorporated a friction model [16]. In the friction model, the normal behavior adopted \"hard\" contact and the tangential behavior followed the penalty friction formula. The parameter settings aligned with those of the model test. The detailed parameter settings are shown in Table 3. The lateral displacement was constrained throughout the numerical model and both lateral and vertical displacements were constrained at the bottom. The top featured a free boundary condition. The numerical simulation underwent an initial ground stress balance before applying a displacement load of 2 mm/min to the pile top. Figure 10: Numerical model and grid distribution. Figure 9: Soil pressure increment distribution of a loose sand pile-bucket diameter ratio of 0.357. (**a**) Front side of the foundation; (**b**) Back side of the foundation. To affirm the reliability and validity of the numerical model, the numerical simulation results were compared with the load-displacement curves of the model test, as shown in Figure 11. The load-displacement curve generated by the numerical simulation exhibited a consistent pattern with the model test, demonstrating a noteworthy alignment between the two curves. The ultimate bearing capacities of the composite foundation with pile-bucket diameter ratios of 0.5, 0.417, and 0.357 obtained from the model tests under dense sand conditions were 6.24 N, 6.89 N, and 7.32 N, respectively. The relative errors compared to the model test results were 1.1%, 1%, and 2.1%, respectively. Therefore, it is verified that the above numerical model is reliable. Analysis of the Influence of the Pile-Bucket Diameter Ratio on the Bearing Capacity of Composite Foundation  
Figure 12 illustrates the correlation between horizontal load and displacement for various pile-bucket diameter ratios under dense sand conditions, comparing both the model test and numerical simulation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Setting Object** & **Angle of Friction/\({}^{\circ}\)** & **Elastic Modulus/MPa** \\ \hline loose sand & 29 & 10 \\ medium dense sand & 33 & 15.5 \\ dense sand & 38 & 20 \\ \hline \hline  & density/(g/cm\({}^{3}\)) & elastic modulus/GPa & poisson ratio \\ \hline Pile-bucket model & 7.85 & 200 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical simulation parameters. Figure 11: Comparison of numerical simulation and model test load–displacement results. Figure 12: Load–displacement relationship under different pile-bucket diameter ratios. Inspection of Figure 12 reveals that under different pile-bucket diameter ratios, the initial horizontal displacement of the composite foundation increases linearly with increasing load. Once a critical load is attained, the displacement increases significantly. For a given load, the larger pile-bucket diameter ratios exhibit a higher rate of displacement increment compared to their smaller counterparts. Therefore, the increase in bucket diameter has a significant impact on the horizontal bearing characteristics of the pile-bucket composite foundation. The pile's ultimate bearing capacity, determined at a pile head rotation of 0.5\({}^{\circ}\), was obtained and the resulting variation curve of ultimate bearing capacity with the pile-bucket diameter ratio is depicted in Figure 13.  
In Figure 13, it is evident that as the pile-bucket diameter ratio gradually decreases within the range of 1 to 0.25; the ultimate bearing capacity of the composite foundation progressively increases.