# Prediction of Friction Resistance for Slurry Pipe Jacking
## 3 Calculation of Friction Resistance for Slurry Pipe Jacking
\begin{array}{l}\Delta R=\frac{D_{z}-D_{p}}{2}\ \,\ \ \ \xi=\tan\left(\frac{z}{2} \right)\,\\ \alpha=\frac{1-\eta}{1+\eta}\ \,\ \ \beta=\frac{\lambda}{2(1+\eta)}\ \,\\ \eta=\frac{E_{p}}{E_{s}}\frac{1-v_{s}^{2}}{1-v_{p}^{2}}\ \,\ \ \lambda=\ \frac{1-2v_{p}}{1-v_{p}}-\eta\frac{1-2v_{s}}{1-v_{s}}\end{array}  \tag{31}\]  
After some approximate treatments, the key terms of Equation (30) have been solved by [PERSON] [32], as follows:  
\[\int\limits_{-\xi}^{+\xi}q(t)\frac{dt}{1+t^{2}}=\frac{1}{2\pi}\frac{I_{b}}{\xi ^{2}(\xi^{2}+1)}+\frac{\xi^{2}}{\xi^{2}+1} \tag{32}\]  
\[I_{b}=\pi\log(\xi^{2}+1) \tag{33}\]  
\[B=\frac{2\xi^{4}+2\xi^{2}-1}{\xi^{2}(\xi^{2}+1)} \tag{34}\]  
By substituting these equations into Equation (29), [PERSON] obtained an approximate form of general contact angle relation. \[\frac{\pi(\alpha+1)E_{p}\Delta R}{(1-v_{p}^{2})P}=\frac{(\alpha-1)[\ln(\xi^{2 }+1)+2\xi^{4}]+2}{(\xi^{2}+1)\xi^{2}}-4\beta \tag{35}\]  
As compared with Equation (9), Equation (35) is a far more complex nonlinear equation. It can be further simplified with respect to that the elastic modulus of soil \(E_{s}\) is generally much smaller than that of pipe \(E_{p}\) (the difference between the two can be three orders of magnitude). Thus, from Equation (30), the magnitude of auxiliary variable \(\eta\) should be very large, and, therefore, the following approximate relations can be obtained:  
\[ \begin{array}{l}\frac{\pi(\alpha+1)E_{p}}{(1-v_{p}^{2})}\approx\frac{2\pi E_ {s}}{(1-v_{s}^{2})},\\ \alpha\approx-1,\\ \beta\approx\frac{1-2v_{s}}{2(1-v_{s})}\end{array}  \tag{36}\]  
Using Equation (36), Equation (35) can be then simplified as  
\[\frac{\pi E_{s}\Delta R}{(1-v_{s}^{2})P}+\frac{1-2v_{s}}{1-v_{s}}=\frac{1-[ \ln(\xi^{2}+1)+2\xi^{4}]}{(\xi^{2}+1)\xi^{2}} \tag{37}\]  
From Equation (37), it is essential to calculate \(P\), which requires one to calculate the total normal force \(N\). It can be gained by integrating the normal stress \(\sigma_{n}\) on an element of the pipe surface and is determined on the basis of vertical and horizontal soil stresses. \[\sigma_{n}=\sigma_{v}\sin\theta+\sigma_{a}\cos\theta \tag{38}\]  
\[N=4\!\!\int_{0}^{\pi/2}\sigma_{n}\frac{D_{p}}{2}d\theta \tag{39}\]  
where \(\theta\) is defined as the angle between the corresponding radius line and the horizontal line at each point of the pipe (Figure 4). By substituting Equation (38) in Equation (39), it is easy to obtain the equation of \(N\), which has the same form as Equation (18). To calculate Equation (18), the vertical soil stress \(\sigma_{v}\) has to be first determined. It is noted that at the present time, by far the most commonly used model for soil pressure calculation is [PERSON]'s silo model (Equation (22)) [5; 6; 9; 19; 21; 26]. According to Equation (22), the calculation of the vertical soil stress requires some physical parameters that may be determined with some accuracy, such as the height of cover \(h\), the cohesion \(c_{s}\), and the unit weight of soil \(\gamma\), but also some empirical parameters, such as \(b\), \(\delta\), and \(K\). The definition of these empirical parameters varies from one author to another. Here, typical approaches of [PERSON], Germany Standard ATV-A 161 E-90 [29], Chinese Standard GB 50332-2002 [33], UK Standard BS EN 1594-09 [34], US Standard ASTM F 1962-11 [35], UK PJA [27], Japan JMTA [36], and Japan JSA [22] would be discussed and compared. For the calculation of silo width \(b\), three kinds of boundary planes of wedge failure assumed by different authors and the corresponding equations are clearly illustrated in Figure 5. The width of the boundary plane is related to the 'vault' effect of soil. Generally, a smaller \(b\) means a lower 'vault' effect of soil, leading to a larger vertical soil stress. Figure 4: The earth pressure and the normal stress acting on the pipe. Figure 5: Boundary planes of wedge failure assumed by different authors. JSA: Japan Sewage Association; PJA: UK Pipe Jacking Association.