## 1 Introduction
[25] presented a new longitudinal structural model considering the shear-dislocation between rings, and proposed an equivalent shear stiffness to consider the influence of joints. As a more comprehensive, cost-effective, and efficient calculation method, numerical simulation has become increasingly important in studying the mechanical properties of double-layered shield tunnels. Based on double-layered lining joint load tests, [PERSON] et al. [26; 27] pioneered three contact surface connection models of double-layered lining according to different simplifications of the contact surface of the double-layered lining. [PERSON] et al. [28; 29] proposed a contact surface combined model of a double-layered lining with a compression bar and shear spring on the basis of the shear compression resistance model after comparing and analyzing the differences in mechanical properties of the above three models. Most of the subsequent studies of double-layered lining are based on the combined model with a compression bar and shear spring. For example, [PERSON] et al. [30] established a three-dimensional double-layered lining model of shield tunnels with segments and joints, and simulated the contact surface with a shell, spring, and a combination of a compression bar and shear spring. [PERSON] et al. [31] constructed a calculation model of double-layered shield tunnels that can reflect the non-linearity of the flexural stiffness of the joint and the failure mechanism of compression-shear spring of contact surface between segment and secondary lining. However, the above studies focused on the analysis of lateral mechanical properties of double-layered shield tunnels and ignored the influence of the longitudinal mechanical properties. Previous studies [32] found that the presence of segments staggered assembly and bolt joints would reduce the overall longitudinal stiffness of shield tunnels and renders the structure more flexible. When the geological conditions change, the surrounding soil is excavated, or the ground is stacked unevenly [33; 34], the shield tunnels are susceptible to uneven longitudinal settlements, local damage being induced to segments, and water leakage in joints. Therefore, the longitudinal mechanical properties of double-layered shield tunnels deserve attention. In recent years, some scholars have studied the longitudinal equivalent bending stiffness and deformation distribution of double-layered shield tunnels using theoretical methods or numerical simulation, but the studies have mostly focused on the effects of load, variation of material property and thickness of the secondary lining [35; 36; 37]. The application of the secondary lining at a reasonable time can better improve the overall internal force distribution and give full play to the structural load-bearing capacity of doubled-layered shield tunnels [38]. However, previous studies [39; 40] on the construction time of the secondary lining mainly take a single segment ring or several representative segment rings as the research object to investigate the lateral mechanical properties of segment lining structure; however, they did not consider the longitudinal mechanical properties, and ignore the adverse effects of the longitudinal deformation of shield tunnels on their overall performance, thus possessing certain limitations. This study aims to investigate the influence of the construction time of the secondary lining on the longitudinal mechanical properties of the double-layered lining of shield tunnels, as well as to determine the optimal construction time for the secondary lining. To achieve these objectives, numerical simulation methods are employed to analyze the distribution of internal forces and deformations within shield tunnels under different construction time scenarios for the secondary lining. By considering the load-bearing characteristics of various sections in the double-layered shield tunnel, the study identifies the most suitable construction time for the secondary lining.