## 2 Finite Element Modeling
### Model Parameters
#### 2.2.2 Parameters of Pipe Piles and Pile Caps  
This study utilizes a two-dimensional calculation model for the finite element calculation. The elastic-plastic model was selected for both the pile cap and the pile body. To simulate the PHC pipe pile, a plate unit representation is employed, converting the actual three-dimensional pile into a two-dimensional equivalent wall based on the principle of equivalent stiffness. The vertical stiffness of the slab-soil composite roadbed remains the same before and after the equivalence. The two-dimensional equivalent slab body is modeled using a six-node triangular unit based on [PERSON]'s slab theory [26]. Tables 2 and 3 present the calculated parameters for the PHC pipe pile and pile cap, respectively, when the diameter of the pile diameter is 0.3 m.  
The parameters in the table are calculated as follows.  
1. Width of the equivalent plate body. According to the principle that the area of soil treated by a single pipe pile is equal, the area replacement rate is calculated via Equation (1).  
\[m=\frac{d^{2}}{d_{\mathrm{e}}{}^{2}} \tag{1}\]  
According to the principle that the area replacement rates of the pile and plate to soil are the same, the width of the two-dimensional equivalent plate body is calculated by Equation (2).  
\[B=m\times S \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Soil Name** & \
\begin{tabular}{c} _7 unsat_ \\ **(kN/m\({}^{3}\))** \\ \end{tabular} & \(E\) (kN/m\({}^{2}\))** &
\begin{tabular}{c} \(c\) \\ **(kN/m\({}^{2}\))** \\ \end{tabular} & \(\varphi\) (\({}^{\circ}\)) & \(\mu\) & **Model** \\ \hline Embankment fill & 20.0 & \(2.5\times 10^{4}\) & 25.0 & 28.0 & 0.30 & Mohr–Coulomb \\ Bedding & 25.0 & \(5.0\times 10^{4}\) & 1.0 & 36.0 & 0.25 & Mohr–Coulomb \\ Farming soil & 17.8 & 4760 & 13.0 & 11.5 & 0.32 & Mohr–Coulomb \\ Soft soil & 14.6 & 1370 & 10.0 & 8.1 & 0.33 & Mohr–Coulomb \\ Mucky silty clay & 17.0 & 6200 & 3.7 & 6.9 & 0.33 & Mohr–Coulomb \\ Sandy soil & 19.5 & \(3.5\times 10^{4}\) & 0 & 27.0 & 0.30 & Mohr–Coulomb \\ Silty clay & 19.0 & 9500 & 12.8 & 12.1 & 0.31 & Mohr–Coulomb \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soil parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pile Spacing (m)** & \(E\)**A (kN)** & \(E\)**I (kN-m\({}^{2}\))** & \(B\) (m)** & \(W\) (kPa)** & \(\mu\) \\ \hline
2.0 & \(2.68\times 10^{6}\) & 277.20 & 0.035 & 0.31 & 0.15 \\
2.4 & \(2.68\times 10^{6}\) & 192.50 & 0.029 & 0.23 & 0.15 \\
2.6 & \(2.68\times 10^{6}\) & 164.03 & 0.027 & 0.24 & 0.15 \\
3.0 & \(2.68\times 10^{6}\) &Axial stiffness and flexural stiffness of the equivalent plate body.  
Since the area and elastic modulus of sheet and pile are the same in the composite subgrade per unit area, the vertical compressive stiffness of the composite subgrade before and after equivalence should also be the same. The axial stiffness and flexural stiffness of the plate are defined to obtain the internal force of the pile. The axial stiffness and flexural stiffness of the two-dimensional equivalent plate are calculated by Equations (3) and (4).  
\[EA=E\times B\times S \tag{3}\]  
\[EI=E\times S\times\frac{B^{3}}{12} \tag{4}\]  
3. Equivalent plate body bulk weight.  
In the computational model, the plates are stacked on a continuum, so the plates overlap the soil. In order to accurately calculate the bulk density of the soil and structure in the finite element model, the bulk density of the plate material should be subtracted from the bulk density of the soil, so the bulk weight of the equivalent plate is calculated by Equation (5).  
\[w=(\gamma_{concrete}-\gamma_{soil})\times B \tag{5}\]