## 5 Numerical Analysis
### Simulation Results Validation  
As shown in Figure 12, the initial geostatic stress analysis observed a stress equilibrium, with a vertical displacement (the upward displacement is positive) less than 10-7 and the vertical stress (the downward stress is negative) exhibiting layered distribution, consistent with the initial stress conditions before the construction. The analysis focused on the stress distribution characteristics under different lubrication conditions to compare and verify the reliability of the segmented calculation model. (1) Earth pressure  
As shown in Figure 13, there was minimal difference in earth pressure under different contact states, overall presenting a pattern of bottom > top > left wall = right wall. The average normal pressures were extracted from Figure 13 and compared with calculated values (Table 11). Due to the small elastic modulus and thickness of the equivalent layers, the soil tended to contract towards the box, resembling active earth pressure (WL). However, the stress concentration occurred at both ends of the tunnel due to excavation, leading to a simulated earth pressure larger than the calculated values. Meanwhile, considering the effect of the lubricant, the simulated earth pressure at the box top fell between positive and negative soil arching pressures (FP). Overall, even though the numerical model could not simulate the actual injection and filtration of the lubricant, the simulation effect on the soil arching action is satisfactory. Figure 12: Initial geostatic stress. (2) Friction force  
As shown in Figure 14 and Table 12, there were significant differences in the friction force under different lubricant conditions. The magnitude of the friction force followed the order of full box-soil interaction > three-side box-soil interaction > two-side box-soil interaction > one-side box-soil interaction. As mentioned earlier, under different lubricant conditions, the earth pressure was generally consistent. The main reason for the differences in the friction force is the variation in contact area and friction coefficients. In particular, in Figure 14b-d, the friction force significantly decreased after the transition from box-soil friction to box-lubricant friction. As shown in Figure 15, in the FP stage, the simulated values for full, three, and two-side box-soil interactions exceeded the measured jacking force. Since there were changes in the box-soil contact state, friction coefficients, and friction types, the simulated values for one side box-soil interaction were the closest to the measured values. This once again proves that assuming full box-soil interaction and constant friction coefficients in the existing calculation models is inconsistent with actual construction. Due to the lubricant effects on the controlling factors, the outer surface of the box in a certain range does not interact with the soil, and the interface friction characteristics are also different, which corresponds with the assumption of the segmented calculation model in this paper. This demonstrates that the segmented calculation is more practical for guiding long-distance box jacking. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Position**} & \multicolumn{3}{c}{**Simulated Average Value (kPa)**} & \multicolumn{3}{c}{**Theoretical Calculated Value (kPa)**} \\ \cline{2-7}  & **a** & **b** & **c** & **d** & **WL** & **FP** \\ \hline Box top & 135 & 134 & 134 & 135 & 114 & 110/183 \\ Box bottom & 179 & 183 & 180 & 179 & 138 & / \\ Left wall & 72 & 69 & 67 & 67 & 67 & / \\ Right wall & 72 & 69 & 67 & 67 & 67 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison of calculated and simulated pressure. Figure 13: Normal pressure nephogram. Figure 14: Friction force nephogram.