# Prediction of Friction Resistance for Slurry Pipe Jacking
## 3 Calculation of Friction Resistance for Slurry Pipe Jacking  
The general contact state of pipe-soil-slurry due to the interaction of the pipe, surrounding soil, and lubricant slurry is shown in Figure 3. In the picture, the position of pipe-soil contact is arbitrary, with a contact width of \(B_{s}\) and the corresponding contact angle of \(2\varepsilon\). From Figure 3, it is obvious that to calculate friction resistance \(f\), both of the items of \(f_{s}\) and \(f_{m}\) have to be taken into account, which can be expressed as  
\[f=\mu N=f_{s}+f_{m}=\mu_{s}N_{s}+\mu_{m}N_{m} \tag{23}\]  
Figure 3: The general contact state of pipe-soil-slurry. where \(\mu\) is an effective friction coefficient introduced to reflect the effect of lubrication and the influence of pipe-soil contact width. It is generally accepted that \(\mu_{s}=\tan(\varphi/2)\) for the coefficient of kinematic friction between soil and the pipe [21; 22]; \(\mu_{m}\) for the coefficient of kinematic friction between mud slurry and the pipe can be taken as 0.01, according to the test result reported by [PERSON] [30]. \(N_{s}\) and \(N_{m}\) are the total normal force of pipe-soil and pipe-slurry in contact, respectively. To calculate \(N_{s}\) and \(N_{m}\) precisely, the location of pipe-soil contact and the magnitude of contact angle (or contact width) and the contact force have to be determined. For various reasons leading to the occurrence of pipe-soil contact, it seems impossible to calculate these quantities in a target section of the pipeline. However, if taking the whole pipeline into consideration, and assuming that the pipe-soil contact can occur at any position of a section of the pipeline with a same probability and the contact force is approximately equal to the soil pressure in the contact area, this problem can be greatly simplified. In this case, we have the following equations:  
\[P=N_{s}=\frac{B_{s}}{C}N=\frac{\varepsilon}{\pi}N \tag{24}\]  
\[N_{m}=\frac{B_{m}}{C}N \tag{25}\]  
where \(C\) (=\(\pi D_{p}\)) is the external circumference of pipe and \(\varepsilon\) is the semi-angle of contact (Figure 3). By substituting Equations (24) and (25) in Equation (23), after some algebra, giving that  
\[\mu=\mu_{s}\lambda_{s}+\mu_{m}\lambda_{m} \tag{26}\] \[\lambda_{s}=\frac{B_{s}}{C}=\frac{\varepsilon}{\pi}\ \,\ \ \lambda_{m}= \frac{B_{m}}{C} \tag{27}\]  
\[B_{m}=C-\frac{B_{s}}{1+e} \tag{28}\]  
where \(e\) is the void ratio of soil. By substituting Equation (27) in Equation (26), the expression of \(\mu\) can be further rewritten as  
\[\mu=\mu_{s}\frac{\varepsilon}{\pi}+\mu_{m}\bigg{(}1-\frac{\varepsilon}{\pi(1+ e)}\bigg{)} \tag{29}\]  
According to Equation (29), the calculation of \(\varepsilon\) is essential to calculate \(\mu\). Hertzian model provides a simple way for the calculation of the width of contact (or contact angle) as we have mentioned before (see Equations (8) and (10)); however, the Hertzian contact problem is approached only when the applied force is small, or the large radial clearance is large, and the limited angle of contact is smaller than about \(30^{\circ}\)[31]. Due to the technical limitations, most of the pipe jacking projects encounter clay or sandy soils and with a small overcut, it is therefore important that the applicability of the Hertzian contact model should be extremely limited here. Actually, the Hertzian contact model is just a special case of the [PERSON]'s contact model with a small contact width (or angle) [31]. If a large contact angle (larger than \(30^{\circ}\)) occurs, the more general contact model proposed by [PERSON] should be taken as the first choice. The following singular integro-differential governing equation of contact angle is derived by [PERSON], as [31; 32]  
\[B=4(1-\beta)-2(1-\alpha)\int\limits_{-\xi}^{+\xi}q(t)\frac{dt}{1+t^{2}}-\frac {\pi}{2}(1+\alpha)\frac{E_{p}\Delta R}{(1-v_{p}^{2})P} \tag{30}\]  
or  
\[f=\mu N=f_{s}+f_{m}=\mu_{s}N_{s}+\mu_{m}N_{m} \tag{23}\]0The involved auxiliary variables are defined as [31]  
\[