## 2 Convergence-Confinement Method and Displacement Release Coefficient  
### Principles of Convergence-Confinement Method  
In actual engineering, tunnel excavation is a complex three-dimensional change process, especially in the vicinity of the excavation surface, where the unexcavated rock mass in front has a \"semi dome \"spatial constraint effect on the surrounding rock at a certain distance behind. It produces a radial constraint on the surrounding rock nearby, which is equivalent to applying a virtual support force, so that the stress of the surrounding rock after excavation cannot be released at one time in a very short time, but will continue to be released as the tunnel face gradually goes away. In view of the spatial effect near the excavation surface, the stress mechanism of surrounding rock is simplified as a cantilever beam under the uniformly distributed load \(q\), the excavation surface O is simplified as a rigid constrained end [14], and the excavation surface C, which is not supported far away from the excavation surface, is simplified as a free end. From this, we can clearly understand the deformation process of surrounding rock under the action of spatial effect: when a certain point B of the tunnel is close to the excavation face O, it will receive the support force F provided by the excavation face and the virtual support force will make the surrounding rock in a state of stress balance. With the continuous advance of the excavation surface, the smaller the support force received at point B, the greater the deformation. Figure 1 shows the schematic diagram of the spatial constraint effect. Therefore, whether or not it is possible to effectively grasp the early deformation of the surrounding rock and the interaction between the rock mass and the supporting structure after excavation, or to effectively estimate the various stresses and displacements of the rock mass and the supporting system, based on the on-site monitoring and measurement data, design finding the best support opportunity and support system is the main purpose of the theoretical analysis of the CCM. The basic principle of the CCM is shown in Figure 2, including the Ground Reaction Curve (GRC), the Longitudinal Deformation Profile (LDP) and the Support Characteristic Curve (SCC); \(u\) is the radial displacement of the surrounding rock; \(x\) is the distance to the tunnel face; \(P_{i}\) is the support resistance; \(u_{0}\) is the early deformation of the monitoring section; point E is the boundary point of the elastic-plastic deformation of the surrounding rock; and point D is the support of the balance point between the characteristic curve and the surrounding rock characteristic curve. Figure 1: Schematic diagram of spatial effect of excavation face: (**a**) Semi dome effect; (**b**) Cantilever beam assumption of excavation face. Figure 2: Schematic diagram of CCM. GRC: It expounds the relationship between radial displacement and radial pressure of tunnel surrounding rock. When the radial displacement is 0, the surrounding rock pressure is equal to the initial in situ stress \(P_{0}\). When the surrounding rock pressure is 0, the tunnel radial is at its maximum \(u_{\rm max}\), which is the case without any support. The characteristic curve of surrounding rock can be roughly divided into three stages: the straight-line segment is the elastic deformation stage, the curve segment is the plastic deformation stage and the imaginary segment is the instability failure stage. LDP: It depicts \(x\) (generally 3-5 times the tunnel diameter) within a certain distance behind the front of the tunnel. The excavated surrounding rock has spatial constraints on the excavated surrounding rock and \(u_{0}\) is the early deformation of the surrounding rock of the monitoring section. SCC: It refers to the action process of support structures with different stiffnesses.