## 5 Conclusions  
The effects of the secondary lining's construction time on the longitudinal mechanical properties of double-layered shield tunnels were investigated by using numerical simulation methods, and the optimal construction time of the secondary lining was determined by considering structural reliability and construction economics. The following conclusions were obtained:  
(1) The construction of the secondary lining effectively reduced the vertical deformation and dislocation deformation of the segments, enhancing the integrity of the double-layered lining tunnel. However, this effect gradually diminished as the construction time of the secondary lining was delayed;  
(2) The vertical deformation of the tunnel was predominantly concentrated at the position of maximum load, specifically near the middle of the tunnel. Conversely, the dislocation deformation primarily occurred in areas with uneven loading, namely along both sides of the tunnel. The alteration curve of segment dislocation essentially mirrored the first derivative of the vertical deformation curve of the segment;  
(3) Following the construction of the secondary lining, the longitudinal shear force in the segments notably increased and the longitudinal bending moments significantly decreased. However, as the construction time of the secondary lining was delayed, the longitudinal internal forces within the segments gradually reverted to their levels prior to the secondary lining construction. The longitudinal internal forces within the secondary lining diminished progressively from their maximum value to zero, in tandem with the postponed construction of the secondary lining;  
(4) Regarding the longitudinal deformation behavior, the early installation of the secondary lining resulted in the secondary lining assuming a higher load-bearing role. This subsequently led to diminished deformation within the segment lining. Nevertheless, this early construction also brought about an underutilization of the segment lining's load-bearing potential, ultimately resulting in the inefficient utilization of the structure's performance. Therefore, combined with the internal force regime on the double-layered lining structure, the reasonable degree of internal force of the secondary lining was introduced as the judgment criterion, and the reasonable construction time was determined to be 20% to 40% by considering the structural reliability and construction economy, meaning that the secondary lining was constructed when the deformation of the segmental lining was between 20% to 40% of the final deformation of the single-layered shield tunnel.  
The research findings of this paper can be applied in the field of shield tunneling, serving as a reference for the design and construction of shield tunnels. Due to the limited availability of relevant experiments and engineering data for this topic, a significant portion of the research has relied on numerical simulations. In subsequent studies, endeavors will be made to undertake model experiments, thereby delving more deeply into this topic.  
Conceptualization, S.C. and B.H.; funding acquisition, S.C., J.C. and X.F.; investigation, Y.Y., J.H. and Y.Z.; methodology, S.C., B.H. and H.W.; resources, J.C. and X.F.; supervision, Y.Z.; validation, X.F.; writing--original draft, B.H., H.W. and J.H.; writing--review & editing, Y.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 520/9135), the Natural Science Basic Research Plan in Shaanxi Province (No. 2020 JM-234), and the Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education and the Fundamental Research Funds for the Central Universities, CHD (No. 300102282201).  
**Data Availability Statement:** The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.  
**Conflicts of Interest:** The authors declare no conflict of interest.