## 2 Engineering Overview and Numerical Modeling
### Model Assumptions, Element Mesh, and Boundary Conditions  
The following assumptions are made for efficient model operation.  
1. In the process of simulation excavation, the site is flat after each excavation, and each over-excavation is 0.5 m. Anchor construction and the application of prestressing are completed instantly, and there is no loss of prestress.
2. Since the time for excavation is relatively short, the consolidation analysis is ignored ([PERSON] et al., 2016).
3. Slope spray mixing is considered by the shell element, the anchors are considered by cable elements, and piles are considered by equivalent solid elements, pile elements, and liner elements, respectively, based on different methods of pile equivalent calculation. Crown beams are considered by solid elements and beam elements, and waist beams are divided into two cases considering waist beams (which are considered by  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Symbol** & **Description** & **Element** & **Soil 1** & **Soil 2** & **Soil 3** & **Soil 4** \\  & & & **(0–1 m)** & **(1–21.5 m)** & **(21.5–26.3 m)** & **(26.3–50 m)** \\ \hline \(E_{50,\textit{ref}}\) & Primary loading & & & & & \\  & stiffness (reference) & \(\mathrm{kN/m^{2}}\) & 6500 & 2210 & 72,000 & 100,000 \\  & Oedometric & & & & & \\  & stiffness (reference) & \(\mathrm{kN/m^{2}}\) & 6500 & 2210 & 72,000 & 100,000 \\  & Un/reloading & & & & & \\  & stiffness (reference) & \(\mathrm{kN/m^{2}}\) & 19,500 & 6630 & 216,000 & 300,000 \\  & Poisson’s ratio un/reloading & & & & & \\  & Cohesion & \(\mathrm{kN/m^{2}}\) & 8 & 25 & 30 & 30 \\  & Friction angle & \(\circ\) & 12 & 22 & 25 & 35 \\  & Dilatancy angle & \(\circ\) & 0 & 0 & 0 & 5 \\  & Stress dependency index & & & & & \\  & Soil weight & \(\mathrm{kN/m^{3}}\) & 19 & 19.5 & 19.5 & 20.5 \\  & Earth pressure coefficient & & & & & \\  & at rest & – & 1-sin\(\varphi\) & 1-sin\(\varphi\) & 1-sin\(\varphi\) & 1-sin\(\varphi\) \\  & Reference pressure & \(\mathrm{kN/m^{2}}\) & 100 & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input soil parameters in PH model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Support** & **Element** & **Young’s** \\  & **Structure** & **Weight (\(\mathrm{kN/m^{3}}\))** & **Modulus (Mpa)** & **Poisson’s Ratio** \\ \hline
1 & Crown beam & 24 & 28,000 \\
2 & Waist beam & 24 & 28,000 \\
3 & Spray concrete & 24 & 25,500 \\
4 & Anchor cable & 25 & 195,000 \\
5 & Retaining pile & 24 & 28,000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Supporting structure parameters of foundation pit.  
beam elements) and without waist beams. For the pile model, more details are shown in Section 2.5.  
Based on the typical profile and bolt spacing, the positions of the first two anchor points along the outer extension direction of the vertical profile are indicated by black triangles (Figure 3b), and the third layer of anchors is a black rectangular box. Considering the geometric spacing of the pile anchors and the grid quality, the model is 7.8 m long in the direction of extension outside the vertical profile. It is divided into 12 equally spaced grids, with spacing of 0.65 m each. This division allows for the spatial location in the anchor head to be coupled to the mesh nodes, in order to characterize the stiffening effect during the actual construction (Figure 3a). The soil mesh near the pile body is locally refined to ensure that it will not affect the accuracy of the calculation results, while reducing the calculation time, and this operation is generally recognized by related scholars [28; 29].  
Horizontal motion is constrained at the lateral boundary, while both horizontal and vertical motions are constrained at the bottom boundary (Figure 3b). In addition, the distance between the pile and the outer boundary of the mesh is ensured to be larger than three times the final excavation depth, to minimize the boundary effect [6].