# Prediction of Friction Resistance for Slurry Pipe Jacking
## 5 Influence of Design Factors on Lubrication Efficiency and Friction Resistance
### Influence of Design Factors on Lubrication Efficiency  
Except for friction resistance, engineers are also concerned about the lubrication efficiency [9; 42; 43]. According to Equation (27), the magnitude of \(\mu\) is between the pipe-slurry friction coefficient \(\mu_{m}\) and the pipe-soil friction coefficient \(\mu_{s}\). If there is no contact between the pipe and the soil, the angular space due to overcut is completely filled with lubricant slurry, the effective friction coefficient is equal to \(\mu_{m}\); and if the soil is in full contact with the external surface of the pipe, the effective friction coefficient mainly depends on the pipe-soil nature and approximately equals to \(\mu_{s}\). Thus, the lubrication efficiency can be defined as  
\[\chi=\left(1-\frac{\mu-\mu_{m}}{\mu_{s}}\right)\times 100\% \tag{40}\]  
By substituting Equation (28) in Equation (31), and considering that \(\mu_{m}\) is far smaller than \(\mu_{s}\), the \(\chi\) can approximately be expressed by Equation (41).  
\[\chi=\left(1-\frac{\varepsilon}{\pi}\right)\times 100\% \tag{41}\]  
If pipe-soil contact angle \(\varepsilon=0\), \(\chi=100\%\) for maximum lubrication efficiency, and if \(\varepsilon=\pi\), \(\chi=0\%\) for minimum lubrication efficiency. It is noted that \(\chi=0\%\) is not going to happen. According to the Passon model (Equation (37)), the result calculated by the left terms of Equation (37) should be not less than zero, while the right term of Equation (37) is a monotonically decreasing function of pipe-soil contact angle. In other words, when the right term of Equation (37) is equal to zero, the pipe-soil contact angle reaches its maximum value, which is solved by \(\varepsilon=\varepsilon_{max}=72^{\circ}=0.4\pi\), corresponding to the minimum lubrication efficiency of \(60\%\) (i.e., \(\chi\) is theoretically between \(60\%\) and \(100\%\) on the basis of Passon contact model). Although it is not theoretically correct as compared to that counted by [PERSON] and [PERSON] as between \(45\%\) and \(90\%\)[9] and tested by [PERSON] as between \(47.8\%\) and \(78.6\%\)[5], it seems practical to estimate the efficiency of lubrication by the approach of this paper.  
The pipe-soil contact angle and the corresponding lubrication efficiency calculated by Equation (41) has been shown in Figure 9(a)-c, Figure 9(d)-f, respectively. It is found that, as compared to the low effect of overburden depth, special attention should be paid to the effect of pipe diameter and the overcut.  
Figure 9(f) shows that the lubrication efficiency strongly increases from \(64\%\) to \(91\%\), while the overcut increases only from \(0\) to \(15\) mm, and after that, the effect of overcut is significantly reduced. This observation confirms the importance of overcut, which has to be sufficiently wide so that the decrease of tunnel diameter induced by the elastic ground unloading does not lead to the closure of the annular space. Moreover, Figure 9(e) shows that the increasing of pipe diameter from \(1\) to \(4\) m causes obvious efficiency losses of lubrication from \(99\%\) to \(82\%\). Thus, one can conclude that the larger the pipe diameter, the lager the overcut is needed.  
The buried depth \(h\) is determined by the intended line and level of the tunnel, which is often limited by geological conditions and distributions of the existing buildings and structures, while the pipe diameter \(D_{p}\) highly depends on the practical use or traffic requirements. It seems that there are not many options for the buried depth \(h\) and pipe diameter \(D_{p}\) in the design. From this point of view, both for lubrication efficiency and friction reaction, more attention should be paid to the design of overcut.  
Parts of the conclusions analytically discussed above have been verified by authors from field observations [9; 20], which in turn again confirm the feasibility of the approach of this paper.