# Key Technology of TBM Excavation in Soft and Broken Surrounding Rock
## 4 Countermeasures for TBM Construction Risk in the Weak and Broken Stratum
### TBM Driving Parameter Control in Soft and Broken Stratum
#### 4.3.2 Quantitative Analysis of the Relationship between TBM Key Parameters  
The thrust and torque of the TBM are composed of several parts, but the thrust and torque used for the rock breaking of cutters are the effective thrust and torque of the TBM. At this time, the quantitative relationship between the TBM effective thrust, effective torque, TBM penetration, and cutter head speed is mainly discussed. **Derivation of the relationship between TBM effective thrust and penetration**  
TBM effective thrust is generated by the vertical force of multiple cutters to break the rock. TBM effective thrust \(F_{1}\):  
\[F_{1}=n1\cdot F\mathrm{v}1\]  
In the above equation, \(n_{1}\) is the number of cutters; \(F_{V1}\) is the vertical force of the hob to break the rock. The vertical force \(F_{V1}\) of the rock-breaking force derivation model of the cutter is:  
\[F_{V1}=0.8\cdot T^{\frac{5}{6}}S^{\frac{1}{3}}\cdot D_{0}^{\frac{1}{2}}\cdot \sigma_{c}\cdot h^{\frac{1}{3}}\]  
So, TBM effective thrust \(F_{1}\) is:  
\[F_{1}=0.8\cdot T^{\frac{5}{6}}S^{\frac{1}{3}}\cdot D_{0}^{\frac{1}{2}}\cdot n 1\cdot\sigma_{c}\cdot h^{\frac{1}{3}}\]  
From the above formula, it can be obtained that the effective thrust of the TBM has a power function relationship with the penetration \(h\), a direct proportion relationship with the formation compressive strength, and a power function relationship with the cutter diameter. 2. **Derivation of the relationship between TBM effective torque and penetration** The effective torque of the TBM is generated by the rolling force of multiple cutters, and the effective thrust \(T_{1}\) of the TBM is: \[T_{1}=n_{1}F_{R}R_{i}\] In the above equation, \(n_{1}\) is the number of cutters; \(F_{R}\) is the rolling force of the cutter to break the rock; \(R_{i}\) is the average installation radius of the cutter. The rolling force \(F_{R}\) of the rock-breaking force derivation model of the cutter is: \[F_{R}=0.8\cdot T^{\frac{5}{6}}S^{\frac{1}{3}}\cdot\sigma_{c}\cdot h^{\frac{5 }{6}}\] So, the effective torque \(T_{1}\) of the cutter is: \[T_{1}=0.8\cdot T^{\frac{5}{6}}S^{\frac{1}{3}}\cdot n_{1}\cdot R_{i}\cdot \sigma_{c}\cdot h^{\frac{5}{6}}\] From the above formula, it can be obtained that the effective torque of the cutter head has a power function relationship with the penetration \(h\), is proportional to the compressive strength of the formation, and has nothing to do with the diameter of the cutter. 3. **Relationship between TBM torque and cutter head speed** A TBM generally adopts a variable-frequency electric drive, and the relationship between the driving power, torque, and cutter head speed is shown in the following formula: \[P=\frac{T\cdot n}{9550}\] In the above equation, \(P\) is the driving power, and the unit is kW; \(T\) is torque, and the unit is kW: m; \(n\) is the speed of the cutter head, and the unit is r/min. Before the power consumed by the cutter head rotation reaches the rated power of the motor, there is no limit between the torque and the cutter head speed. When the power consumed by the cutter head rotation reaches the rated power of the motor, the torque and the cutter head speed are mutually limited. If sufficient torque is required, the cutter head speed must be controlled. 4. **Experimental verification of the quantitative relationship between TBM key parameters** A. **Experimental equipment and rock samples** The rock-breaking experiment of the cutter is carried out by using the rock-breaking and wear experimental device of the contraction rule cutter (as shown in Figure 11), and the matching relationship among the penetration speed, blade spacing, and vertical force of the cutter is verified. Figure 11: Contraction rule cutter and rock-breaking experimental device. The cutter used in the experiment is a contraction rule cutter. The size of the cutter ring is designed according to the scale of 1:10. The contraction rule and prototype cutters are made of the same materials and from the same production process.