# Prediction of Friction Resistance for Slurry Pipe Jacking
## 3 Calculation of Friction Resistance for Slurry Pipe Jacking
For the determination of \(\delta\), most of the guidelines, such as PJA, JSA, JMTA, BS EN 1594-90, and GB 50332-2002 assume shear planes as perfectly rough and take an angle of friction in the shear planes \(\delta\) equal to the soil internal friction angle \(\varphi\). However, ATV-A 161 E-90 and ASTM F 1962-11 make a more cautious assumption and only takes into account half the internal friction angle \(\varphi/2\). For the lateral pressure coefficient \(K\) above the tunnel, [PERSON] assumes \(K\) coefficient is equal to 1, which corresponds to the range of values encountered in clayey soils. PJA, ASTM F 1962-11, and GB 50332-2002 suggest \(K=K_{a}\) (calculated by [PERSON]'s formula of active soil pressure coefficient), while BS EN 1594 and ATV-A 161 assume \(K=K_{0}\) (calculated by [PERSON]'s formula of soil pressure coefficient at rest). Moreover, according to ATV-A 161, this \(K\) coefficient is equal to 0.5, which corresponds to an internal soil friction angle of \(30^{\circ}\), a typical value for sandy soils. Parameters of \(b\), \(\delta\), and \(K\) chosen by the different authors have been summarized in Table 1.  
It is noted that none of the approaches use the same parameters. Consequently, the vertical soil stress calculated by these approaches would be quite different. Thus, it is not convincing to pick out an approach to use without checking the field data. This work will be carried out in the next section. Thus far, all the equations needed to calculate friction resistance have been determined. If the parameters needed for the prediction equations are quantified, by using Equations (18) and (22) the total normal force \(N\) can be determined, then together with Equations (23), (24), (27), (31), and (37), the contact angle \(2\varepsilon\), the effective friction coefficient \(\mu\), and the friction force \(f\) now can be uniquely identified. The flow chart is shown in Figure 6.  
Apparently, the effective friction coefficient here is not just related to the interfriction angle of soil \(\varphi\) but also the state of pipe-soil contact and the effect of lubrication. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \(\mathbf{b}\) & \(\delta\) & \(K\) & \(\mathbf{c}\) \\ \hline Terzaghi (Japan) & \(D_{p}(1+2\tan\,\alpha)\) & \(\varphi\) & 1 & \(c\) \\ JMTA (Japan) & (\(D_{p}+0.08\))tan \(\beta\) & \(\varphi\) & 1 & \(c\) \\ JSA (Japan) & (\(D_{p}+0.1\))tan \(\beta\) & \(\varphi\) & 1 & \(c\) \\ PJA (UK) & \(D_{p}\tan\,\beta\) & \(\varphi\) & \(K_{a}=\tan^{2}\,\alpha\) & \(c\) \\ BS EN 1594 (UK) & \(D_{p}(1+2\tan\,\alpha)\) & \(\varphi\) & \(K_{0}=1-\sin\,\varphi\) & \(c\) \\ AVT A-161 (Germany) & \(1.732D_{p}\) & \(\varphi/2\) & \(K_{0}=0.5\) & None \\ ASTM F 1962-11 (US) & \(1.5D_{p}\) & \(\varphi/2\) & \(K_{a}=\tan^{2}\,\alpha\) & None \\ GB 50332-2002 (China) & \(D_{p}(1+\tan\,\alpha)\) & \(\varphi\) & \(K_{a}=\tan^{2}\,\alpha\) & None \\ \hline \hline \end{tabular} Note: \(\alpha=\pi/4-\varphi/2\); \(\beta=\pi/4+\alpha/2\).  
\end{table} 
Table 1: Definition of the empirical parameters in [PERSON]’s silo model by different authors [37]. Figure 6: Flow chart of friction resistance prediction.