## 2 Thrust, Torque, and Power Estimation Model for Modified EPB TBM in Mixed-Face Ground
### Thrust Model
#### 2.2.2 Cutting Resistance by Cutters  
In the RSI mixed strata, the resistance of the cutterhead in the hard rock is mainly caused by the normal force for disc cutters to penetrate the rock, and the resistance of the cutterhead in the soft strata is primarily caused by the normal force of the drag cutters cutting the soil. In the following, the method for calculation normal force is explained: Figure 3 shows the normal and rolling forces of the disc cutter during the cutting process. The normal force is perpendicular to the direction of the excavation face, and the direction of the rolling force is parallel to the excavation face. [PERSON] et al. [6] calculated the combined force of the disc cutter rolling in hard rock. It can be expressed as Equation (17):  
\[F_{ti}=\frac{\phi TR}{1+\psi}p^{0} \tag{17}\]  
where \(F_{ti}\) is the total force of a disc cutter, \(\psi\) is a constant for the pressure distribution function, \(\phi\) is the angle of the contact between the rock and disc cutter \(\phi=\arccos(\frac{R-p}{R})\), \(p\) is the penetration rate per revolution, \(T\) is the width of the cutter tip, \(R\) is the radius of the disc cutter, \(P^{0}\) is the pressure of the broken zone \(P^{0}=C\sqrt{\frac{s\sigma_{c}^{2}\sigma_{c}}{\phi\sqrt{RT}}}\), \(C\) is a dimensionless parameter and approximately equals to \(2.12\), \(s\) is the cutter spacing, \(\sigma_{c}\) is the uniaxial compressive strength (USC) of the rock, and \(\sigma_{t}\) is the tensile strength. The normal force (\(F_{ni}\)) of the disc cutter can be expressed as Equation (18):  
\[F_{ni}=F_{ti}\cos\frac{\phi}{2} \tag{18}\]Then the normal force of the disc cutter can be expressed as follows:  
\[F_{ni}=C\frac{\phi TR}{1+\psi}\sqrt[3]{\frac{\sqrt{gc_{c}^{2}\sigma_{t}^{2}}}{ \phi\sqrt{RT}}}\cos\frac{\phi}{2} \tag{19}\]  
When the EPB TBM advances, the drag cutter penetrates the soft ground and then has a squeezing and sliding effect on the soil as the cutterhead rotates. The drag cutter calculation models of [PERSON] et al. [7] and [PERSON] et al. [1] neglected the effect of soil friction on the cutter. The cutter calculation model of [PERSON] et al. [24] analyzed the friction between soil and cutter but ignored the impact of chamber pressure on the drag cutter and the soil. In this paper, a new model for determining the drag cutter force is developed, as shown in Figure 4. The soil being stripped is approximate as a triangular prism with \(\Delta\)MI] as the base and the width of the cutter as the height. The height of the bottom triangle represents the drag cutter's penetration rate, \(d\). Therefore, the gravity of the spalled soil can be expressed as Equation (20):  
\[G=\rho gwS_{\Delta MIJ}=\frac{1}{2}\rho gwd^{2}(\tan\alpha+\cot\alpha) \tag{20}\]  
where \(G\) is the gravity of the stripped soil, \(w\) is the width of the drag cutter, \(\alpha\) is the front angle of drag cutter, and \(d\) is the cutting depth per revolution of one drag cutter. According to [PERSON] et al. [1], the cutting depth of the two types drag cutters are, respectively, \(d_{1}\) and \(d_{2}\); \(d_{1}=\frac{\delta}{1+\frac{\beta^{\prime}}{b}}\), \(d_{2}=\frac{\delta}{1+\frac{\beta}{b^{\prime}}}\), \(\delta\) is the cutting depth when the cutterhead rotates at \(\beta\) angle, and \(\beta^{\prime}\) is the angle between the adjacent drag cutters. Figure 4: Force analysis diagram of drag cutter and scraped soil: (**a**) force analysis diagram of drag cutter; (**b**) force analysis diagram of scraped soil. Figure 3: Force diagram of the disc cutter during the cutting process. We conducted a static force study of the drag cutter and the sliced soil using [PERSON]'s earth pressure theory. The forces on the drag cutter are shown in Figure 4a. \(F_{ni}\) is the drag cutter's normal force, which is in the same direction as the EPB TBM's forward motion, and \(F_{ij}\) is the drag cutter's cutting force, which is parallel to the excavation surface. \(P\) is the force of soil acting on the front surface of the cutter.