## 2 Friction Resistance Analysis and Calculation  
### Resistance Analysis during Pipe Jacking Construction Based on a Literature Review  
Through literature research, the frictional resistance of slurry pipe jacking was analyzed and summarized. Six main factors affecting the frictional resistance of pipe jacking were identified: lubrication, pipe-soil contact, soil type, design parameters, soil pressure, and the interaction between the pipe and slurry. For clarity, they are displayed as follows:  
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The black arrows, in Figure 1, indicate the rough range and direction of the pressure of the surrounding soil on the pipe. And in Figure 1b, the black cross signs indicate the points of contact between the pipe and the soil. As shown in Figure 1b, in the case of grouting, some predictive friction resistance models are proposed. The Persson model, established by [PERSON] and [PERSON], is suitable [18; 31; 33], as illustrated in Figure 2.  
The dashed lines and the black arrows marked in Figure 2 indicate the axis of the two non-concentric circles. The blue arrows indicate the range and direction of the pressure of the soil on the pipe when the pipe is in contact with the soil. The central force, denoted as P, is primarily focused at the arc of contact between the pipe and the adjacent soil [18; 30]. Equations (1) and (2) delineate the contact angle and pressure:  
\[\frac{\pi(1+\alpha)E_{p}\Delta R}{\left(1-v_{p}^{2}\right)P}=\frac{(\alpha-1 )\left[\ln(b^{2}+1)+2b^{4}\right]+2}{(b^{2}+1)b^{2}}-4\beta \tag{1}\]  
\[\frac{R_{s}p(\theta)}{P}=\frac{1}{2\pi(b^{2}+1)b^{2}}\ln(\frac{\sqrt{b^{2}+1}+ \sqrt{b^{2}-e^{2}}}{\sqrt{b^{2}+1}-\sqrt{b^{2}-e^{2}}})+\frac{2\sqrt{b^{2}-e^{ 2}}}{\pi\sqrt{b^{2}+1}(1+e^{2})} \tag{2}\]  
where \(v_{p}\) is the Poisson's ratio of the pipe, \(E_{p}\) is the elastic modulus of the pipe (kPa), and \(R_{s}\) is the radius of the excavation tunnel. \(\alpha\) and \(\beta\) are related to the material parameters  
Figure 1: (a) Full contact schematic diagram and (b) pipe–soil partial contact model in grouting. Figure 2: Persson contact plane strain model. of the two contact bodies, respectively, and parameter b is related to the pipe-soil contact angle [18]. The computation of the associated parameters is delineated in Equation (3):  
\[\left\{ \begin{array}{l}\alpha=(1-\eta)/(1+\eta)\\ \beta=\lambda/(2(1+\eta))\\ \eta=E_{p}\big{(}1-v_{s}^{2}\big{)}/\big{(}E_{s}\big{(}1-v_{p}^{2}\big{)}\big{)} \\ \lambda=\big{(}1-v_{p}/\big{(}1-v_{p}\big{)}\big{)}-\eta(1-v_{s}/(1-v_{s}))\\ \Delta R=R_{s}-R_{p}\\ e=\tan(\theta/2)\\ b=\tan(\varepsilon/2)\end{array} \right. \tag{3}\]  
where \(v_{s}\) denotes the Poisson's ratio pertinent to the soil, \(E_{s}\) is the elastic modulus of the pipe (kPa), and \(R_{p}\) is the radius of the pipe. \(p(\theta)\) represents the contact pressure at location \(\theta\). Equation (1) can be determined by applying numerical analysis techniques [18, 30]. P, the contact pressure, can be obtained by taking it into Equation (2). In Ref. [32], it was assumed that the angle of contact (\(2\varepsilon\)) between the pipe and the soil can be approximated as a fixed value of 180 degrees, and the aggregate contact pressure is approximately 1.35 times the intrinsic weight of the pipeline. It can be seen from Equations (1) and (2) that the pipe-soil contact characteristics mainly depend on the elastic modulus and Poisson's ratio of the pipe and soil.