## 3 Analysis of the Construction Time of the Secondary Lining
### Criteria for Determining the Construction Time of the Secondary Lining  
In practical engineering, the deformation forms of the segment lining vault included vertical deformation (settlement) and dislocation. Therefore, it was necessary to select an appropriate deformation type to determine the construction time of the secondary lining. The deformation form needed to meet the following characteristics:  
1. The deformation range was large enough to ensure sufficient resolution corresponding to different secondary lining construction times. 2. Deformation needed to be easy to measure, meaning it was necessary to consider the construction characteristics of the actual shield tunnel and the challenges associated with deformation measurement. In practical engineering, the vertical deformation of the vault of a single-layered shield tunnel can be obtained by conventional monitoring means, while the final vertical deformation can be predicted using the refined numerical simulation method. The vertical deformation cloud map of the single-layered shield tunnel in the stratum was displayed in Figure 9. The segment rings were numbered from 1 to 100 (left to right), and the vault vertical deformation at the midpoint of the width of each ring was used to create the tunnel longitudinal deformation curve, as shown in Figure 10. Due to the symmetry of the model, the vertical deformation of the shield tunnel was also symmetrical. When subjected to a non-uniform load, the vertical deformation of the shield tunnel became notably non-uniform. Varied degrees of vertical deformation occurred in distinct sections of the tunnel. At both ends of the shield tunnel, the vertical deformation of the tunnel was maintained at a low level, measuring only about 2.5 mm. This was due to the constraints imposed by the boundary conditions and the minimal soil and water load resulting from the shallow burial depth of the tunnel. Subsequently, the soil and water load uniformly increased along the tunnel's direction, leading to a significant surge in the vertical deformation of the tunnel at this stage. In the central segment of the tunnel, the soil and water load reached its maximum and remained constant, resulting in a gradual rise in the vertical deformation of the tunnel along its longitudinal axis. The highest value was attained at the 40 th ring of the shield tunnel, stabilizing at around 30 mm. When the shield tunnel was subjected to uneven loads, the segment lining experienced uneven settlement and the joints of adjacent segments would become misaligned. This kind of differential displacement between segments was referred to as dislocation. The dislocation of segment rings was often adopted to reflect the local deformation, and its variation along the longitudinal direction was also an important indicator of the longitudinal deformation of a shield tunnel. As shown in Figure 11, the dislocation of segment rings was the difference between the vertical deformation of adjacent segment rings, which could be considered the first order derivative of the vertical deformation distribution curve of the tunnel. It could also be interpreted as the rate of change of the vertical deformation, which was often used to reflect the degree of uneven deformation of the shield tunnel. Figure 12 shows the longitudinal distribution of segment dislocation. It was observed that the dislocations of the segments exhibited clear discretization along the longitudinal direction. The distribution curve, as a whole, presented a cubic function distribution, and the specific fitting formula is: \(y=0.0000227x^{3}-0.00341x^{2}+0.110x+0.198\). Within the 0-30 and 70-100 rings of the segment lining, the segment dislocation reached a maximum value of 1.81 mm. This was due to the gradual increase in load on both sides of the tunnel, leading to the pronounced uneven deformation of the segment lining and resulting in significant dislocation deformation.