## 1 Introduction  
With the rapid advancement of underground transportation, the utilization of shield tunnels has expanded considerably [1]. A shield tunnel comprises a tubular structure connected by bolts, and the existence of joints compromises the tunnel's overall rigidity [2]. When exposed to factors like seismic events [3, 4], fluctuations in water levels [5, 6], and the excavation of adjacent tunnels [7, 8], substantial longitudinal deformations can manifest within the tunnel, giving rise to potential concerns such as water seepage, structural deterioration, and localized damage. As an illustration, over the span of four years from 2005 to 2009, the maximum settlement value for Nanjing Metro Line 1 has reached 122 mm, coupled with noticeable structural cracking [9]. Likewise, in the construction of Shanghai Metro Line 4, the permafrost surrounding the shield tunnel was disturbed, causing the swift infiltration of soil and water into the tunnel, ultimately resulting in substantial damage to the lining (2004) [10]. Similar accidents were also observed in Foshan (2018) [11], Kaohsiung (2007) [10], Tianjin (2011) [12] and Nanjing (2007) [13], all with disastrous consequences. All the large-scale failures in these shield tunnels originated from small-scale local failures. In response to these phenomena, some shield tunnel projects [14; 15] adopted double-layered lining structures, using a secondary lining as a safety reserve or reinforcement measure to ensure the safety of the shield tunnel structure. Compared with single-layered shield tunnels, double-layered structures are applied in smaller areas and remain concentrated in large-diameter river-crossing or sea-crossing tunnels [16] and water transfer tunnels [17]. In the early double-layered shield tunnel projects, the segment lining was generally used as the load-bearing structure, while the secondary lining was used as an auxiliary structure to improve the durability and water resistance of the shield tunnel. For example, the Trans-Tokyo Bay Highway Tunnel used a secondary lining with a thickness of 35 mm as a structural safety buffer in the Tokyo Bay area with soft submarine strata and significant water pressure [18]. However, existing studies show [19] that the secondary lining can be used as one of the load-bearing structures in a shield tunnel to enhance the overall stiffness. Furthermore, with the development of urban rail transit, the shield tunnel is facing more complex engineering situations, and the design and usage requirements are constantly being improved. Compared with single-layered segment lining, a double-layered lining structure has more advantages in reinforcement, fire prevention, collision resistance, anti-settlement, anti-erosion, and water pressure resistance. Consequently, an increasing number of shield tunneling projects, such as the Rail-cum-Road Yangtze River Cross Tunnel in Wuhan [20], the Shiziyang Tunnel [21], and the Wusongkou Yangtze River-crossing tunnel in Shanghai [22], are opting for double-layered lining structures. Therefore, it is important to study the effect of the secondary lining on the mechanical properties of such shield tunnels. Previous studies have analyzed the mechanical properties of double-layered shield tunnels mainly through model experiments, theoretical analysis, and numerical simulation methods. Among them, the model experiment is an intuitive and reliable method for studying the mechanical properties of structure. [PERSON] et al. [23] investigated the differences in the mechanical properties of underwater shield tunnels between single-layered lining and double-layered lining structures through models and field tests. In terms of the theoretical method, [PERSON] et al. [24] proposed an analytical analysis method for double-layered linings that can consider both joint effects and the interactions between soil and lining, as well as lining and lining. [PERSON] et al.