# Prediction of Friction Resistance for Slurry Pipe Jacking
## 2 Overview of the Existing Prediction Models of Friction Resistance
This may be attributed to the assumption of full contact of the pipe and soil that leads to a large prediction value of friction resistance, or in other words, Hypothesis 3 is conservative. Because of that, this kind of model is widely accepted by authorities and standards from all over the world, such as Japan Sewage Association (JSA) [22], UK Pipe Jacking Association (PJA) [27], Chinese Trenchless Technology Association (CTTA) [28], and Germany Standard (AVT A-161) [29]. This kind of model can be summarized and divided into the following four categories. \[f_{s} =\mu_{s}N\]5  
\[f_{s} =\mu_{s}N\]6  
\[f_{s} =\mu_{s}N\]7  
\[f_{s} =\mu_{s}N\]8  
The fourth kind of model (Equation (17)) introduces an empirical constant \(\beta\) (smaller than 1) on the basis of the third model (Equation (16)) to reflect the effect of lubrication. For the models summarized above, the item of \(f_{s}\) (=\(\mu_{s}N\)) has to be taken into consideration. Thus, the key problem for this kind of model is exactly focused on the calculation of soil pressure \(N\). CTTA suggests \(N\) to be calculated by [PERSON]'s formula, which gives that [28]  
\[f_{s} =\mu_{s}N\]9  
\[f_{m} =\mu_{m}N\]0  
\[f_{m} =\mu_{m}N\]1  
where \(\sigma_{v}\) is the vertical soil stress; \(K\) is the coefficient of soil pressure above the pipe; \(\gamma\) is the unit weight of soil; \(h\) is the overburden depth of the pipeline; and \(\phi\) is the internal friction angle of soil. However, JSA suggests using [PERSON]'s silo model, the expression of \(N\) is then expressed as [22]  
\[f_{m} =\mu_{m}N\]2  
\[f_{m} =\mu_{m}N\]3  
where \(c_{s}\) is the cohesion of soil; \(\delta\) is the friction angle between the pipe and soil; \(b\) is the influencing silo width of soil above the pipe, and the other symbols have the same meanings as before. Although a lot of prediction equations of friction resistance have been proposed and even some of them have been applied to engineering practice, it is obvious that their hypotheses are quite different, and even for the same hypothesis, models and parameters can be different too. Thus, it is bound to make the prediction friction resistances vary greatly. Furthermore, apart from Equations (11) and (17), the other models completely ignore the effect of lubrication, which is very important to slurry pipe jacking. In the design philosophy of slurry pipe jacking, the angular space due to overcut is expected to be completely filled with lubricant slurry, to reduce the friction resistance with maximum efficiency, creating a 'filter cake' layer around the cavity and is then pressurized to the support pressure required for the soil (see Figure 2b) [3; 10]. In this case, the friction resistance should be only related to the slurry pressure and the friction coefficient between slurry and the pipe. From this point of view, the expression Equation (11) seems a convincing explanation here. However, the more general case is that the excavated tunnel is stable under pressure of slurry and part of the pipe inevitably comes in contact with the soil (see Figure 2d) [3]. The reasons for the occurrence of pipe-soil contact can be complex, such as insufficient design and control of grouting amount of slurry, the pipeline deviates from the intended line and level, irregular deformation of the surrounding soil, and interpenetration between the soil and slurry. Thereby, the state of contact can change from 'pipe-slurry' into 'pipe-soil-slurry' (Figure 2b,d). In such a case, a simple form of Equation (17) seems more suitable to reflect the effect of lubricant slurry. However, it is logical that the greater the contact width between the pipe and the soil, the smaller the effect of lubrication will be, and, therefore, the greater the friction resistance will be. Thus, the value of \(\beta\) should be highly affected by the pipe-soil contact width. For different soils, grouting amount of slurry and design parameters (such as buried depth and overcut) is bound to lead to completely different contact widths of pipe-soil.