## 4 Modified Solution of Cutterhead Torque
### Modified Solution Derivation  
Table 7 shows the statistics of several shield projects on the torque composition of the cutterhead [39,40]. It shows that CT is mainly composed of overcoming formation resistance and friction. \(T_{1}\) and \(T_{2}\) account for about 80-90% of the total torque. Based on  
Figure 9: Cutterhead torque of the Krause empirical model. this statistical data, this paper simplified the calculation model due to the unavailability of some shield mechanical parameters, as shown in Equation (4). \[T=1.2(T_{1}+T_{2}) \tag{4}\]  
\(T_{1}\) and \(T_{2}\) can be obtained by the following method. During the advancing process of the shield, the resistance torque generated by the cutter cutting the soil is:  
\[T_{1}=\int_{0}^{R_{0}}q_{u}h_{\max}dr=0.5q_{u}h_{\max}rR_{0}^{2} \tag{5}\]  
where \(q_{u}\) is unconfined compressive strength of soil; \(h_{\max}\) is maximum cutting depth per revolution of the cutterhead; \(R_{0}\) is the radius of the outermost tool. When the cutterhead is cutting, the front and side surfaces of the cutterhead rub against the soil, resulting in frictional resistance torque. The specific calculation formula can be expressed as follows:  
\[T_{2}=T_{21}+T_{22} \tag{6}\]  
\[T_{21}=\int\limits_{0}^{2\pi}\int\limits_{0}^{\frac{Q_{2}}{\gamma}}(1-\xi)f \cdot\sigma_{nf}\cdot r^{2}drd\theta \tag{7}\]  
\[T_{22}=\int_{0}^{2\pi}\frac{D^{2}}{4}fWp_{1}\sin^{2}\theta d\theta+\int_{0}^{2 \pi}\frac{D^{2}}{4}fWK_{a}p_{2}\cos^{2}\theta d\theta \tag{8}\]  
where \(T_{21}\) is the friction torque between the front of the cutterhead and the soil; \(T_{22}\) is the friction torque between the side of the cutterhead and the soil. \(\sigma_{nf}\) is the normal earth pressure on the front of the cutterhead; \(f\) is the friction coefficient of the soil; \(\xi\) is the opening ratio of the cutterhead; \(r\) is the turning radius of the cutterhead; \(\theta\) is the angle between the calculation unit on the cutterhead and the vertical axis (see Figure 10); \(W\) is the weight of the shield; \(p_{1}\) is the vertical pressure acting on the side of the cutterhead, \(p_{2}\) is the horizontal pressure acting on the side of the cutterhead; \(K_{a}\) is the active earth pressure coefficient. The data of rolling angle and cutterhead torque between Ring No. 480 and Ring No. 530 are shown in Figure 10. The points where the rolling angle is greater than 0 and \begin{table}
\begin{tabular}{c c c c} \hline
**Project** & \(T_{1}\) + \(T_{2}\) (kN-m) & **Total Torque (kN-m)** & **Percentage (\%)** \\ \hline Izmir Metro, Turkey & 4297 & 5289 & 81.0 \\ London Heathrow Airport Tunnel & 13,806 & 18,842 & 73.0 \\ Beijing Metro Line 9, China & - & - & 91.2 \\ Beijing Metro Line 10, China & - & - & 93.8 \\ \hline \end{tabular}
\end{table} 
Table 7: Related projects. Figure 10: Cutter torque calculation diagram. the points less than or equal to 0 are respectively fitted, and two fitted straight lines are obtained. The fitting parameters are shown in Table 8. As shown in Figure 11, as the rolling angle increases, CT also increases, and the relationship between the two is approximately linear. The theoretical solution of the torque in this area lies between the intercept values of the two straight lines. This is because the theoretical calculation model of CT established does not consider the correlation between rolling angle and torque, and the default is that the rolling angle is 0 mm/m. When the rolling angle reaches +5 mm/m (the set limit value), CT deviation is nearly 20% more than the value obtained from the theoretical model. It has been proved that the rolling angle is a factor that cannot be ignored.