# Numerical Analysis of Friction Reduction of Grouting in Long-Distance Pipe Jacking
## 3 Case Study  
### Example Verification  
To ensure the reliability of the proposed numerical model, example verification was performed using a case provided in the article by [PERSON] et al. [16].  
The details of the case are provided in the article [16]. The method of processing the numerical simulation is similar to the subsequent simulation process. However, there are two main differences: the length of the pipe is 100 m in this example verification, which is significantly shorter than the following simulation; the bottom burial depth is 9.6 m, which is significantly shallower than the following simulation. The partial results are illustrated in Figures 3 and 4. According to the statistics in the article [16], the soil stratum through which the pipe traverses is silty clay. As shown in Table 1, the average frictional resistance ranges between 5 kN/m and 8 kN/m; these values are used to calculate the theoretical results, determining the minimum and maximum values. Similarly, according to Table 2[17], the interface friction angle ranges from 0.2 to 0.3; these two values are used to calculate the numerical results, determining their corresponding minimum and maximum values.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soil Type** & **Soft Clay** & **Silt** & **Silty Fine Soil** & **Medium-to-Coarse Sand** \\ \hline Reinforced Concrete Pipe & 3.0–5.0 & 5.0–8.0 & 8.0–11.0 & 11.0–16.0 \\ Steel Pipe & 2.0–4.0 & 4.0–7.0 & 7.0–10.0 & 10.0–13.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average frictional resistance between the pipe and soil for thixotropic grout (kN/m\({}^{2}\)).  
The interface reduction coefficient is set to 0.2 in Figure 3. The numerical result of the distribution is approximately circular and conforms to prevalent viewpoints [18] and Chinese regulations [13]. The distribution of axial force under different reduction coefficients is shown in Figure 4. It can be observed that as the distance from the launch shaft increases, the jacking force decreases approximately linearly, indicating that the distribution of shear stress along the pipe is relatively uniform. Additionally, the jacking force increases by approximately 1.5 times when the interface reduction coefficient is increased by the same factor, which aligns with common sense.  
From Table 3, the numerical results are compared with the theoretical results introduced in Section 2 and the field-measured results provided in the referenced article. It can be observed that the numerical results closely align with the theoretical results, and the field-measured results fall within both ranges.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Soil Type** & **Wet** & **Dry** \\ \hline Clay, Silty Clay & 0.2\(\sim\)0.3 & 0.4\(\sim\)0.5 \\ Sand, Silty Sand & 0.3\(\sim\)0.4 & 0.5\(\sim\)0.6 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Interface reduction coefficients between the pipe and the surrounding soil.  
Figure 4: The distribution of axial force under two different reduction coefficients.  
Figure 3: Shear stress distribution across a cross-section at 50 m.  
From the above example, the numerical results presented below were validated for reliability and trustworthiness. It should be noted that the main differences between the example verification and the subsequent simulation are the length and the depth of the pipe. If the length is significantly increased or the depth is considerably deeper, the theoretical results may lead to significant errors.