# Prediction of Friction Resistance for Slurry Pipe Jacking
## 2 Overview of the Existing Prediction Models of Friction Resistance  
Numerous models that calculate the friction resistance of pipe jacking have been proposed by authors from all over the world. The superposition principle is usually used, which holds that the comprehensive outcome of two or more linear factors of a system is equal to the accumulation of the effect of each factor. In pipe jacking, the linear factors to generate friction resistance are due to the weight of pipe (\(f_{W}\)), soil pressure (\(f_{s}\)), slurry pressure (\(f_{m}\)), pipe-soil cohesion (\(f_{sc}\)), and pipe-slurry cohesion (\(f_{mc}\)). Their equations can be expressed as [5; 6; 9; 10; 14; 19; 21; 22; 25; 26]. \[f_{W} =\mu_{s}W \tag{1}\] \[f_{s} =\mu_{s}N\] (2) \[f_{m} =\mu_{m}N\] (3) \[f_{sc} =c_{s}B_{s}\] (4) \[f_{mc} =c_{m}B_{m} \tag{5}\]  
Figure 1: Schematic of slurry pipe jacking. where \(\mu_{s}\) and \(\mu_{m}\) are the kinematic friction coefficient of pipe-soil and pipe-slurry, respectively; \(W\) is the weight of pipe per unit length, kN/m; \(N\) is the total normal force acting on the pipe, kN/m; \(c_{s}\) and \(c_{m}\) are the pipe-soil cohesion resistance and pipe-slurry cohesion resistance, respectively, kPa; \(B_{s}\) and \(B_{m}\) are the pipe-soil contact width and pipe-slurry contact width, respectively, m.  
Some hypotheses have been made to establish the prediction models, by which one or some of the items listed above should be considered. Typical hypotheses are:  
**Hypothesis 1. **_The excavated tunnel is self-stable, the pipeline simply slides along the bottom of the tunnel due to its own weight (see Figure 2a) [9; 10; 14]._  
**Hypothesis 2. **_The angular space due to overcut is completely filled with lubricant slurry, and the excavated tunnel is stable under the slurry pressure (see Figure 2b) [22; 25]._  
**Hypothesis 3. **_The excavated tunnel is unstable, the surrounding soil collapses and is in full contact with the whole area of the jacking pipes (see Figure 2c) [5; 6; 21; 26]._  
**Hypothesis 4. **_The excavated tunnel is stable under the pressure of slurry, and part of the pipe comes in contact with the surrounding soil (see in Figure 2d) [3]._  
According to Hypothesis 1, there are two kinds of prediction models, in which the item of \(f_{\textit{sv}}\) has to be taken into consideration. The first one assumes that the friction resistance is only due to the weight of pipe [9; 21]. It is given as  
\[f=\mu_{s}W \tag{6}\]  
\[\mu_{s}=\tan(\varphi/2) \tag{7}\]  
The second one also takes the pipe-soil cohesion resistance (item of \(f_{\textit{sv}}\)) into account [10; 14; 21], which is given by  
\[f=\mu_{s}W+c_{s}B_{s} \tag{8}\]  
where the pipe-soil contact width \(B_{s}\) is calculated by Hertzian contact model, as [10; 14; 21]  
\[B_{s}=1.6(Pk_{d}C_{e})^{1/2} \tag{9}\]  
\[f_{s} =\mu_{s}N\]0  
where \(D_{c}\) and \(D_{p}\) are the internal diameter of cavity and external diameter of pipe, respectively, m; \(v_{p}\) and \(v_{s}\) are the [PERSON]'s ratio for pipe and soil material, respectively; \(E_{p}\) and \(E_{s}\) are the elasticity modulus of the pipe and soil, respectively, kPa; and \(P\) is the effective external force acting on the center of the pipe, kN/m, usually it is considered as equal to the weight of pipe per unit length \(W\). \[f_{s} =\mu_{s}N\]1 The models to calculate friction resistance according to different hypotheses: (**a**) Hypothesis 1; (**b**) Hypothesis 2; (**c**) Hypothesis 3; (**d**) Hypothesis 4. According to Hypothesis 2, the friction resistance is related only to the properties of lubricant slurry, and the only model found was one that takes both the items of \(f_{m}\) and \(f_{mc}\) into account [22; 25]. \[f_{s} =\mu_{s}N\]2  
\[f_{s} =\mu_{s}N\]3  
\[f_{s} =\mu_{s}N\]4  
where \(P_{m}\) is the mud slurry pressure. It is noted that most of the studies completed to date have focused exclusively on the prediction models established by Hypothesis 3.