## 2 Methodology
### Establishment of the Mechanical Model  
Segments are assembled into rings within the shield shell by the segment erector. In the assembly process, the sealing gaskets and rubber gaskets in the circumferential joints are compressed under shield thrust. The segment rings are then pushed out one by one as the shield drives forward. The compression force between rings is always equal to the shield thrust F\({}_{t}\), and the opening of circumferential joints is a fixed value x\({}_{0}\), as shown in Figure 1.  
In the process of shield underground docking, the disassembly operation will cause shield thrust unloading. The sealing gaskets, rubber gaskets and longitudinal bolts between segment rings will be stretched longitudinally under the action of segment weight and friction force f, and the opening of the circumferential joint will change to x\({}_{i}\). The load characteristics and longitudinal compression forces of the segment ring change, eventually reaching a new equilibrium, as shown in Figure 2.  
Figure 1: Schematic diagram of segment rings and circumferential joints under shield thrust. According to the above analysis, the force between segment rings undergoes two processes of compression and unloading during shield docking and disassembly. In the compression process, due to the extremely strict requirements on the formation condition and accuracy control during underground docking, the shield thrust always remains stable, and the initial opening of circumferential joints under thrust is fixed x\({}_{0}\). In the thrust unloading process, the compression force is released, which is resisted by lining-stratum friction and the opening of circumferential joints. The derivation process is complicated and difficult to understand if the opening of circumferential joints is calculated according to the directly released compression force. In this paper, the above processes are considered with a new idea and the mechanical model is simplified. As shown in Figure 3, the thrust unloading process is considered to be a tension process connected by multiple blocks. To simplify the mechanical model, thrust unloading is assumed to apply an equal and opposite tensile force, and the circumferential joint is considered to be an incompressible spring. The release of the compression force between the ring circumferential joint is equivalent to the same tensile force, and the compression modulus of the ring's circumferential joint is equivalent to the tensile stiffness of the spring. Each segment ring is assumed to be a separate block. The initial opening of circumferential joint x\({}_{0}\) is the initial distance between the blocks which are connected by a variable stiffness spring k.  
The calculation mechanism in Figure 3 is simple and clear, and can be used to accurately calculate the theoretical value of the opening after thrust unloading without any reinforcement measures. In practical engineering, thrust unloading leads to the excessive opening of circumferential joints, which leads to waterproofing failure and even the instability of the lining structure. Therefore, longitudinal reinforcement, such as adding a steel channel, is usually carried out on the segment structure before thrust unloading. Steel channel reinforcement, connecting the first segment ring and the back segment ring, can greatly reduce the openings of circumferential joints. According to the reinforcement principle of steel channels, the steel channel is equivalent to the reinforcement spring K, and the segment ring load model under the reinforcement condition of steel channels is established, as shown in Figure 4.  
Figure 3: Simplified schematic diagram of load model of segment lining under thrust unloading. Figure 2: Schematic diagram of segment rings and circumferential joints after thrust unloading.