## 3 Numerical Analysis Modelling  
### Finite Element Model  
In this study, finite element analysis was conducted using the Midas Gts Nx program. A three-dimensional model is established to study the influence of inclined and straight combination support piles on the bending moment and displacement of adjacent pile foundations during the construction of foundation pits. Considering the boundary effect on the accuracy of the numerical results, the influence of excavation of the foundation pit on adjacent pile foundations is chosen to establish a 3D finite element model. Hence, the accuracy of the analysis was significantly impacted by the mesh density inside the excavation, while the mesh density outside the excavation seemed to have less of an impact [20]. Therefore, a fine mesh was used near the pile and around the excavation zone.  
Figure 3: Layout of Foundation pit section. (unit: mm).  
Figure 2: Sketch of the double-row pile.  
Meanwhile, a coarser mesh was used outside these zones. In this study, the numerical model is shown in Figure 4. The dimensions of the foundation pit are 45 \(\times\) 45 m and the mesh dimensions are 130 m (about three times the width of the pit) width in a transverse direction, 130 m (about 3 times the width of the pit) in a longitudinal direction, and 70 m (equal to seven times the excavation depth) depth in the vertical direction. Therefore, these dimensions are suitably large to reduce boundary effects in the numerical analysis because the massive increase in mesh size does not affect the analysis results. The boundary conditions are set as follows: the bottom surfaces are constrained in all directions while the top surface is free. In addition, the four sides of the model are restrained in the normal direction only without considering the influence of groundwater.  
The constitutive model type is very important, and it affects the accuracy of the numerical analysis results. Many authors, such as [21; 22; 23; 24; 25] show that the Modified Mohr-Coulomb (MMC) is suitable to simulate all soil types (both stiff and soft soils). In addition, in the case of excavation of the foundation pit, the loading and unloading behavior of the soil must be simulated [26; 27]. Moreover, the MMC model is considered more accurate because it depends on the three soil stiffness parameters, which correspond to three loading conditions: triaxial loading stiffness, oedometer loading stiffness, and unloading-reloading modulus. Therefore, the soil strata adopt the Modified Mohr-Coulomb model. The required parameters are shown in Table 1. On the other hand, the linear elastic model is based on Hooke's law. Therefore, it is very suitable for simulating stiff structures in the soil. Hence, the pile body and the crown of the connecting beam have C50 high-strength concrete. So, the linear elastic constitutive model was chosen.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** Soil Layer & **Mixed Fill** & **Clay** & **Silty Clay** & **Powdery Clay** \\ \hline Thickness/m & 4.4 & 4.52 & 12.3 & 7 \\ Poisson ratio & 0.2 & 0.2 & 0.2 & 0.2 \\ \(\gamma/\left(\text{kN}/\text{m}^{-3}\right)\) & 18.5 & 19.4 & 18.0 & 20.3 \\ \(\text{E}_{50}^{\text{ref}}/\text{Mpa}\) & 5 & 8.6 & 3 & 10.6 \\ \(\text{E}_{\text{cur}}^{\text{ref}}/\text{Mpa}\) & 6 & 10.32 & 3.6 & 12.72 \\ \(\text{E}_{\text{cur}}^{\text{ref}}/\text{Mpa}\) & 45 & 77.4 & 27 & 95.4 \\ \(\text{S}_{0}^{\text{ref}}/\text{Mpa}\) & 121.5 & 209 & 72.9 & 257.6 \\ c/(kpa) & 5 & 15 & 1 & 1 \\ \(\varphi/(^{\circ})\) & 24 & 31 & 10 & 26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structural properties adopted in the numerical analysis.  
Figure 4: 3D finite element model.