## 4 Establishment of Finite Element Model
### Boundary Conditions and Calculation Assumptions  
1. As shown in Figure 5, set the initial temperature of all soil layers to 20 \({}^{\circ}\)C, and the cooling plans for the circular freezing tubes in the concrete pipe and the profiled freezing tubes in the hollow pipe shall be taken based on their corresponding freezing system temperatures. Based on the engineering ambient temperature, the surface boundary (boundary 1) is set as a convective heat transfer boundary, while the borders on both sides (boundaries 2 and 3) and the bottom (boundary 4) are set as adiabatic, as shown in Figure 4a;
2. Without considering the surface load, the surface boundary (boundary 1) is regarded as a free boundary. Roller support is installed on both sides (boundaries 2 and 3), and only horizontal displacement is restricted. To restrict horizontal and vertical movement, fixed constraints are established at the bottom (boundary 4);
3. Assuming that the thermal resistance between different soil layers is zero, and ignoring the effect of soil moisture migration on the freezing process. The freezing point of all  
Figure 5: Cooling plan for two types of freezing tubes: (**a**) Circular freezing tubes; (**b**) Profiled freezing tubes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Soil Layer**} & \multicolumn{1}{c}{**Elastic Modulus**} & \multirow{2}{*}{**Poisson’s Ratio**} & \multirow{2}{*}{**Cohesion (kPa)**} & \multicolumn{1}{c}{**Internal Friction**} \\  & & **(MPa)** & & & **Angle (\({}^{\circ}\))** \\ \hline Artificial backfill & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 20 & 0.3 & 12 & 15 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 10.67 + 4.3 \(\mid\)T\(\mid\) & 0.26 & 1730 & 4.98 \\ \hline \multirow{2}{*}{Gravelly sand} & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 25 & 0.25 & 0 & 24 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 19.3 + 4.7 \(\mid\)T\(\mid\) & 0.29 & 1860 & 5.69 \\ \hline \multirow{2}{*}{Mucky soil} & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 4 & 0.35 & 16 & 14 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 10.67 + 7.8 \(\mid\)T\(\mid\) & 0.28 & 990 & 4.49 \\ \hline \multirow{2}{*}{Silty clay} & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 28 & 0.32 & 16 & 10 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 4.67 + 6.6 \(\mid\)T\(\mid\) & 0.31 & 1770 & 7.05 \\ \hline \multirow{2}{*}{Medium sand} & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 25 & 0.25 & 0 & 24 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 16 + 5.2 \(\mid\)T\(\mid\) & 0.3 & 1910 & 4.49 \\ \hline \multirow{2}{*}{Gravel clay} & \begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 30 & 0.25 & 12.5 & 10 \\  & \begin{tabular}{c} Frozen \\ \end{tabular} & 34 + 1.2 \(\mid\)T\(\mid\) & 0.3 & 1450 & 4.49 \\ \hline \multirow{2}{*}{Weathered granite} & \
\begin{tabular}{c} Unfrozen \\ Frozen \\ \end{tabular} & 29 & 0.26 & 80 & 40 \\  &
\begin{tabular}{c} Frozen \\ \end{tabular} & 29 + 1.2 \(\mid\)T\(\mid\) & 0.3 & 900 & 6.83 \\ \hline \multirow{2}{*}{Concrete} & \multicolumn{2}{c}{\(25\times 10^{3}\)} & 0.2 & — & — \\ \hline \multicolumn{1}{c}{Jacked pipe} & \multicolumn{2}{c}{\(210\times 10^{3}\)} & 0.27 & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of materials. soil layers is uniformly set at \(-1.5\)\({}^{\circ}\)C for the convenience of studying the size and shape of the frozen soil curtain. To begin, an initial geostress field equilibrium analysis is performed to eliminate the initial vertical displacement of soil caused by gravity stress. Second, before the frost heave analysis, the model is subjected to volume force (soil gravity) and corresponding displacement boundary conditions to analyze the internal stress field of the soil at this time, and the data are derived and applied to the subsequent frost heave analysis model as the initial stress field of the soil.