# Numerical Analysis of Friction Reduction of Grouting in Long-Distance Pipe Jacking
## 3 Case Study
### Structure Parameter Settings  
The pipe is modeled using linear elastic plate elements. The outer diameter is 3600 mm, the inner diameter is 3000 mm, and the longitudinal length is 625 m. The bottom burial depth is 24.4 m. The structure model is shown in Figure 7.  
The unit weight of the pipe (\(\gamma\)) is 24.5 kN/m\({}^{3}\), and the thickness is 0.3 m. In order to consider the effect of the grouting, the interface between the pipe and the surrounding soil is set. Typically, the range of the interface reduction coefficient between the pipe and the surrounding soil is listed in Table 2[17].  
Different interface reduction coefficients are set on the interface to simulate different effects of grouting. The relationship between the interface friction angle and the internal friction angle can be calculated as  
\[tan\varphi_{inter}=tan\varphi\times R_{inter} \tag{4}\]  
In the formula,  
\(\varphi_{inter}\) is the interface friction angle;  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Soil Layer** & **Thickness/m** & \begin{tabular}{c} **Unit Weight** \\ **(Unsaturation)/** \\ **kN-m-3** \\ \end{tabular} & \begin{tabular}{c} **Unit Weight** \\ **(Saturation)/** \\ **kN-m-3** \\ \end{tabular} & \begin{tabular}{c} **Elastic** \\ **Modulus** \\ **/MPa** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Internal Friction** \\ **Angle/\({}^{c}\)** \\ **Angle/\({}^{c}\)** \\ \end{tabular} &
\begin{tabular}{c} **Cohesion/kPa** \\ \end{tabular} \\ \hline Fill Soil & 10.2 & 18.71 & 19.37 & 17.52 & 0.2 & 15.3 & 30.6 \\ Silty Clay & 1.9 & 18.57 & 19.11 & 14.13 & 0.3 & 14.5 & 26.9 \\ Silty Clay & 1.2 & 18.70 & 19.30 & 16.59 & 0.3 & 15.5 & 35.7 \\ clay & 2.3 & 18.77 & 19.20 & 21.33 & 0.35 & 12.1 & 43.7 \\ Silty Clay & 3.6 & 18.77 & 19.50 & 18.60 & 0.3 & 16.7 & 35.2 \\ Silty Clay & 3.4 & 18.96 & 19.60 & 22.14 & 0.3 & 16.2 & 37.5 \\ Silty Clay & 3.6 & 19.16 & 19.70 & 23.91 & 0.3 & 17.2 & 37.2 \\ clay & 8.2 & 19.12 & 19.60 & 27.96 & 0.35 & 14.2 & 49.7 \\ Silty Clay & 3.7 & 19.40 & 20.00 & 29.10 & 0.3 & 17.7 & 38.4 \\ clay & 5.5 & 19.48 & 19.80 & 33.51 & 0.35 & 14.7 & 54.4 \\ \hline \end{tabular}
\end{table}
Table 4: Basic parameters of the soil.  
Figure 6: Model of the soil.  
Figure 7: Structural model.  
\(\varphi\) is the internal friction angle;  
\(R_{inter}\) is interface reduction coefficient;  
In this case, the strata through which the pipe traverses consist of silty clay, and the groundwater level is located 11 m below the ground surface, resulting in wet soil conditions. According to Table 2, the Interface reduction coefficients between the pipe and the surrounding soil should be set within the range of 0.2 to 0.3. However, the friction-reduction effect of grouting may not be effective in long-distance pipe jacking. To address potential issues and ensure the safety of practical engineering applications, additional considerations may be required. Therefore, six interface reduction coefficients are set separately by 0.2, 0.24, 0.28, 0.32, 0.36, and 0.4.  
To simulate the pressure exerted by the soil on the front surface of the pipe, a force perpendicular to the surface is applied. The magnitude of this force increases along the vertical direction to reflect the rise in the lateral soil pressure with depth.