## 2 Thrust, Torque, and Power Estimation Model for Modified EPB TBM in Mixed-Face Ground
### Thrust Model
#### 2.2.1 The Frictional Resistance between the TBM Body and the Surrounding Strata  
As shown in Figure 1, this part of the frictional resistance is primarily caused by the positive force of the surrounding strata acting on the shield surface, which is composed of two major components: the force generated by the gravity of the strata and the stratigraphic reaction force generated by the machine's self-weight. The force generated by the gravity of the strata acts on the TBM's upper part and lower part, which are equal in size and opposite in direction, as are the forces acting on the left and right sides. The frictional resistance caused by the gravity of the stratum can be calculated by decomposing the vertical earth pressure \(p_{1}\) and lateral earth pressure \(p_{2}\) at a location on the side of the body into positive pressures \(p_{1}^{\prime}\) and \(p_{2}^{\prime}\) perpendicular to the side of the body, as shown in Figure 2. For the RSI mixed-face strata, the frictional resistance generated by the gravity of both strata on the TBM shell body needs to be calculated separately [1]. Figure 1: The frictional force between shield and ground. In the soft strata, the vertical earth pressure \(p_{11}\) and lateral earth pressure \(p_{12}\) can be written as follows:  
\[p_{11}=\gamma_{1}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)} \tag{4}\]  
\[p_{12}=K\gamma_{1}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)} \tag{5}\]  
where \(\gamma_{1}\) is the volume weight of the soft ground, \(K\) is the coefficient of lateral earth for soft ground, \(H\) is the distance from the ground to cutterhead center, and \(D\) is the diameter of excavation surface. Decomposition of the above two forces into positive pressures \(p^{\prime}_{11}\) and \(p^{\prime}_{21}\) perpendicular to the side of the TBM body:  
\[p^{\prime}_{11}=\gamma_{1}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)}\sin^{2}\theta \tag{6}\]  
\[p^{\prime}_{21}=K\gamma_{1}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)}\cos^{2}\theta \tag{7}\]  
Therefore, the frictional resistance in the soft ground can be expressed as follows:  
\[F_{11}=\int_{\theta_{1}}^{\theta_{2}}\frac{D}{2}f_{1}p^{\prime}_{11}Ld\theta +\int_{\theta_{1}}^{\theta_{2}}\frac{D}{2}f_{1}p^{\prime}_{21}Ld\theta \tag{8}\]  
where \(f_{1}\) is the coefficient of friction between the soft ground and the shield, \(L\) is the length of the shield, and \(\theta_{1}\) is the angle of the intersection (interface between two layers and the cutterhead profile) with respect to the center of the cutterhead. In the hard strata, the vertical earth pressure \(p_{12}\) and lateral earth pressure \(p_{22}\) can be written as follows:  
\[p_{12}=\gamma_{2}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)} \tag{9}\]  
\[p_{22}=\lambda\gamma_{2}(H-\sin\theta) \tag{10}\]  
Decomposition of the above two forces into positive pressures \(p^{\prime}_{12}\) and \(p^{\prime}_{22}\) perpendicular to the side of the TBM body:  
\[p^{\prime}_{12}=\gamma_{2}\bigg{(}H-\frac{D}{2}\sin\theta\bigg{)}\sin^{2}\theta \tag{11}\]  
Figure 1:0  
Figure 1:1 Force diagram of TBM body. Therefore, the frictional resistance in the hard ground can be expressed as follows:  
Figure 1:2  
where \(f_{2}\) is the coefficient of friction between the hard rock and the shield. The ground reaction force created by the machine's self-weight is supposed to act on the lower half-circumference, and this component of the frictional resistance is equal to the product of the TBM's self-weight and the coefficient of friction of the ground and the shell [23]. The force includes two situations. As shown in Figure 2, when \(h\leq H\), the frictional resistance in this part is equal to the product of the self-weight of the TBM and the friction coefficient of the hard rock and the shield shell. It can be written as follows:  
Figure 1:3  
when \(h>H\), it is assumed that the frictional resistance in this part is related to the percentage of soft and hard rock in the lower half of the excavation face.