## 4 Theoretical Model for Friction Force
### Comparison of Friction Force Solutions  
The calculated values of the existing models and the segmented calculation model were contronged with the measurements. The calculation parameters are provided in Table 8. According to the results of the shear tests, the friction coefficients for Cases A and B were 0.603 during the WL stage and 0.207 during the GP stage. In the FP stage, an additional high-viscosity bentonite slurry was injected at the box top in Case A, resulting in a clay-concrete contact interface at the top of the box, and a friction coefficient of 0.062 was used. For Case B, the average of the friction coefficients for clay and sand was used, resulting in a value of 0.1345. Additionally, to compare the applicability of the existing models, TSP's friction coefficient was assumed to be the same as that of the FP stage, BMDN's friction coefficient was calculated using the specified formula, and the unit area friction force for TCB/TSR was selected to be within the recommended range of 8 to 11 kN/m\({}^{2}\). As shown in Figure 10, the segmented calculation model proposed in this paper demonstrated good predictive accuracy in the three segments of Case A and Case B. In contrast, the existing models failed to predict the friction force for different lubrication conditions during long-distance jacking, especially in the WL and FP stages. The friction force was significantly underestimated or overestimated, posing challenges to cost savings and construction safety. In particular, during the 0-17 m in Case A and 0-11 m in Case B without lubrication, both the existing model and the segmented calculation model adopted the assumption of full box-soil interaction. However, the segmented model considered the actual contact interface of this condition, resulting in calculated values closer to the measured values, while the existing model's assumption about lubrication condition friction coefficients was clearly inconsistent with the actual situation. In the GP stage, the friction force gradually decreased with increasing jacking length. Due to the unstable changes in the contact state and friction coefficients, the overall results showed fluctuations around the calculated values of the three-side and two-side box-soil interaction models. The FP stages of 80-233.6 m in Case A and 31-163 m in Case B, where slurry lubrication played a predominant role, accounted for the largest proportion of the entire jacking length. The proposed model considered the effect of earth pressure on the friction force during the lubricant injection-filtration process. The predicted curve of the maximum and minimum values for these stages essentially covered the measured values, with the actual friction forces fluctuating between the curves. The errors between the calculated values and measured values for the maximum friction force are presented in Table 9. It can be observed that, despite considering the lubrication-reducing effect, the existing models still overestimate the maximum friction \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Calculation Models**} & \multirow{2}{*}{**Interaction State**} & \multicolumn{2}{c}{**Friction Coefficient \(\mu\)**} \\ \cline{3-5}  & & & **Case A** & **Case B** \\ \hline \multirow{3}{*}{Existing models} & TSP & & 0.062 & 0.1345 \\ \cline{2-5}  & BMDN & Full box-soil interaction & 0.2905 & 0.2726 \\ \cline{2-5}  & TCB/TSR & & \(f_{u}=8\)–11 kN/m\({}^{2}\) \\ \hline \multirow{3}{*}{Models in this paper} & WL & Full box-soil interaction & 0.603 \\ \cline{2-5}  & GP & Three/two side box-soil interaction & 0.207 \\ \cline{1-1} \cline{2-5}  & FP & One side box-soil interaction & 0.062 & 0.1345 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The calculation parameters for friction force. force. In Case A, the errors of the minimum and maximum friction force calculated by the proposed model were \(-37.1\%\) and \(4.6\%\), respectively. However, although TSP adopts the same friction coefficient as the proposed model, the error in the maximum jacking force still reached \(198.1\%\).