# Numerical Analysis of Friction Reduction of Grouting in Long-Distance Pipe Jacking
## 3 Case Study
### Results Analysis  
Figure 8 shows the distribution of sectional frictional resistance at the same position (100 m from the launch shaft) under different interface reduction coefficients. Figure 9 illustrates the variation in the standard deviation and average value with respect to different interface reduction coefficients. It can be observed that as the coefficient increases in the range of 0.2 to 0.4, the average frictional resistance around the pipe increases approximately linearly, and the value of the standard deviation of the stress points around the pipe also increases linearly. This reveals that as the interface reduction coefficient increases, the distribution of the stress around the pipe becomes more uneven. Thus, if the interface friction angle is sufficiently large, the variation in shear stress across the same section should be considered.  
Figure 10 shows the axial jacking force along the length of the pipe under different interface reduction coefficients. It can be observed that as the distance from the launch shaft increases, the reduction in the jacking force slows down, indicating a slight decrease in the frictional resistance around the pipe. The reason may be the difference in displacement between the pipe and the surrounding soil. There should be sufficient relative distance to develop frictional resistance. Since the pipe is not modeled as rigid, the displacement near the launch shaft is greater than that at the pipe's end, reducing the frictional resistance at the end.  
The theoretical and numerical results of the jacking force are summarized in Figure 11. Regarding the numerical results, as the interface friction angle increases, the total jacking force of the pipe increases approximately linearly. It can be inferred that in long-distance pipe jacking projects, the proportion of front resistance is very small, and the main resistance originates from the frictional resistance between the pipe and the surrounding soil.  
Figure 8: Sectional frictional resistance at the same position under different reduction coefficients.  
Figure 11: Contrast between the theoretical and numerical results.  
Figure 10: The distribution of axial force under different reduction coefficients.  
Figure 9: Variation in standard deviation and average value.  
The theoretical method is presented in Section 2, indicating that the frictional resistance is 10 \(\mathrm{kN}/\mathrm{m}^{2}\), as shown in Table 1. The jacking force remains constant because the frictional resistance is defined as a fixed range. The different values can be applied within this range, but the value is too imprecise to select, so the fixed value is applied here. The simulated jacking force is much larger than the theoretical value because the theoretical value does not consider the increasing trend of frictional resistance with the depth of the pipe. In this simulation, the pipe's bottom end is buried at a depth of 24.4 m, where the normal stress on the interface between the soil and the pipe is quite large, as shown in Figure 12 (100 m from the launch shaft). This results in significantly larger friction resistance around the pipe, much greater than the average friction resistance between the pipe and soil selected in the theoretical method as shown in Table 1.