# Research on an Evaluation Method for the Adaptability of TBM Tunnelling
## 3 Selection of Evaluation Index and Establishment of Fuzzy Function
### Establishment of the Fuzzy Function  
#### 3.2.1 Membership Function of the Tunnelling Parameters  
(1) Membership function of the RPM  
The main working parameters of the TBM cutting mechanism are the RPM (rotations per minute), torque and thrust. [PERSON] et al. [9] carried out a statistical study on the RPM and TBM diameter based on the HC surrounding rock classification method for hydraulic and hydropower projects in China [10], and obtained Equation (15):  
\[\text{RPM}=\frac{v_{\text{cutter}}}{\pi D}\times\alpha \tag{15}\]  
In the equation, \(v_{\text{cutter}}\) is the limit linear speed of the cutter. For the cutter of this project, its range is 120-165 m/min, \(D\) is the diameter of the TBM cutter head and \(\alpha\) is the reduction coefficient, which can be determined by the following Equation (16):  
\[\alpha=0.1417\times T^{0.4463} \tag{16}\]  
In the above equation, \(T\) is the comprehensive index in the HC surrounding rock classification method, which is composed of the uniaxial compressive strength, rock mass integrity coefficient, groundwater conditions, structural plane conditions, and structural plane orientation. If \(T\geq 80\), \(\alpha\) takes 1. The RPM has different values in hard rock, soft rock, and the hard-soft rock transition section. It is about two r/min for soft rock and about six r/min for hard rock. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **11** & **12** & **13** & **14** & **15** & **16** & **17** \\ \hline
**11** & 1 & 2 & 1/2 & 4 & 8 & 9 & 8 \\
**12** & 1/2 & 1 & 1/4 & 2 & 4 & 4.5 & 4 \\
**13** & 2 & 4 & 1 & 1/8 & 9 & 9 & 9 \\
**14** & 1/4 & 1/2 & 8 & 1 & 2 & 9/4 & 2 \\
**15** & 1/8 & 1/4 & 1/9 & 1/2 & 1 & 9/8 & 1 \\
**16** & 1/9 & 2/9 & 1/9 & 4/9 & 8/9 & 1 & 8/9 \\
**17** & 1/8 & 1/4 & 1/9 & 1/2 & 1 & 9/8 & 1 \\ \hline \end{tabular}  
* 11. Fault fracture zone, 12. large deformations of the surrounding rock, 13. Water inrush, 14. rock burst, 15. high geotemperature, 16. harmful gases, and 17. weak and uneven stratum.  
\end{table} 
Table 6: Scoring of adverse geology. Figure 3: Evaluation index system of the TBM tunnelling adaptability of Xinjiang hydraulic engineering. Since most of the surrounding rocks exposed in the project are hard rocks, the membership function is as follows in Equation (17):  
\[U_{1}(x)=\left\{ \begin{array}{l}\frac{x}{6}(0<x\leq 6)\\ 1(x>6)\end{array} \right. \tag{17}\]  
(2) Membership function of the torque  
The design of the torque mainly depends on the type of TBM, the cutter head form and the type and quantity of cutters. Under normal conditions, the torque should be controlled within a reasonable range to reflect the adaptability of TBM construction and formation. Improper torque control of the cutter head will not only affect the tunnelling efficiency of the TBM construction, but also cause engineering accidents in serious cases [11,12]. [PERSON] et al. studied the design parameters of 265 TBMs after 1895. Based on [PERSON]'s torque equation [13], they proposed Equation (18) [14]:  
\[T_{open}=F_{R}\cdot N_{c}\cdot\frac{D}{4}F_{l} \tag{18}\]  
where \(T_{open}\) is the minimum torque, \(F_{R}\) is the average rolling force of the cutter, \(N_{c}\) is the number of props, \(D\) is the radius of the TBM cutter head and \(F_{l}\) is the friction loss constant; \(F_{l}\) is 1.2 for the cutter. [PERSON] summarized the value of the TBM torque and considered that the torque of the TBM is composed of \(T_{open}\), the rock slag mixing resistance moment, \(T_{2}\), and the cutter head self-weight resistance moment, \(T_{3}\), as shown in Equation (19):  
\[\left. \begin{array}{l}T_{2}=Q\pi R^{3}P\eta\\ T_{3}=W\eta R\\ T_{total}=T_{open}+T_{2}+T_{3}\end{array} \right\} \tag{19}\]  
In general, the actual torque of the TBM is between \(T_{open}\) and \(T_{total}\). Therefore, the membership function of the torque is constructed as follows in Equation (20):  
\[U_{2}(x)=\left\{ \begin{array}{l}0(x<T_{open})\\ \frac{x-T_{open}}{T_{total}-T_{open}}(T_{open}<x\leq T_{total})\\ 1(x\geq T_{total})\end{array} \right.