## 4 Sensitivity Analyses
### Mechanical Responses of the Rock Bolts
#### 4.1.2 Displacement Ratio  
To further discuss the influence of bolt displacement changes on the surrounding rock support, the ratio of rock bolt end displacement \(\Delta_{1}\) to tail displacement \(\Delta_{2}\) under different stresses is analyzed. The ratio  
\[\alpha=\frac{\Delta_{1}}{\Delta_{2}} \tag{36}\]  
is defined as the rock bolt that constrains the front end of the rock bolt hole to penetrate the bedrock as seen in Figure 8. This ratio is used to characterize the anchoring capacity of the bolt. The smaller the ratio, the better the bolt anchoring capacity.  
It can be seen from Figure 9 that when there is Class A large extrusion deformation, the displacement ratio of the rock bolt monotonously decreases as time increases and before finally becoming stable. The capacity gradually increases with time, and finally, the surrounding rock and the bolt support form a bearing circle. When there is Class B large extrusion deformation, with the continuous increase in time and after the bolt is excavated and supported in the tunnel for a specific time, the ratio first reaches its peak value and then gradually decreases to become stable. The curve in Figure 9c,d shows the non-monotonous development of the displacement ratio \(\alpha\), under high in situ stress conditions. This indicates that as time increases, the anchoring end (\(\Delta_{1}\)) becomes loose. As a consequence, the anchoring capacity of the bolt gradually decreases.  
The bolt can effectively restrain the deformation and displacement of the surrounding rock of the tunnel after excavation and can improve the stability of the surrounding rock. However, the selection of the length of the bolt is also critical. The relationship between the displacement ratio of the end and the bolt length was analyzed, as shown in Figure 9. It can be seen from the figure that under the same in situ stress conditions, as the length of the bolt increases, the value of \(\alpha\) also tends to be stable when it reaches a specific value, and in different periods, the \(\alpha\) of the bolt first decreases to a specific value and stabilizes afterward. With the increase in the in situ stress, the \(\alpha\) of the same bolt gradually decreases  
Figure 8: Schematic diagram of the anchor ratio (\(\Delta_{1}/\Delta_{2}\)).  
at the same excavation time. When there is Class C or Class D large extrusion deformation (\(P_{i}\) = 10 MPa and T = 1000 h), \(\alpha\) does not change much when the bolt length is longer than 9 m. This shows that the length of the bolt affects its ability to support the surrounding rock of the tunnel. Generally speaking, the longer the length of the bolt \(L\) is, the more favorable it will be for the stability of large deformation development in the surrounding rock after the tunnel is excavated. However, the longer the bolt length \(L\), the greater the corresponding engineering investment costs and the inconvenience of the drilling and anchoring work. The construction difficulty will increase sharply, as will the construction complexity. Considering the economy as well as safety, the length of the bolt represents a critical expense. In addition, when the in situ stress is different, the change of the \(\alpha\) of the bolt of the same length will also be different, indicating that the difference in in situ stress will affect the deformation of the end of the bolt. Then, the yield of tunnel excavation will also change. It is worth noting that no matter how the ground stress changes, after the bolt length reaches 9 m, the \(\alpha\) value is \(\leq\)30%. The restriction on the deformation of the surrounding rock is no longer significant, indicating that it is feasible to constrain the deformation of the surrounding rock of the tunnel wall.