## 1 Introduction
[17] investigated the influence of the internal structure of the shield tunnel on its longitudinal stiffness by model test and pointed out that the longitudinal stiffness of the tunnel structure is significantly improved when the internal structure is considered. [PERSON] et al. [18] introduced the concrete parameters into numerical simulation and investigated the influence of the nonlinear damage characteristics of concrete materials on the longitudinal deformation characteristics of shield tunnels. Model tests or numerical simulations can intuitively observe the longitudinal deformation characteristics of shield tunnels, but they involve higher operating costs and have certain limitations in engineering analysis. Building upon the results of numerical simulations and model tests, many researchers have conducted theoretical studies to establish a longitudinal analytical model for shield tunnels. [PERSON] et al. [19] proposed a tunnel-soil interaction model based on the Timoshenko beam simplified model for a tunnel on an elastic foundation, which could consider the shearing dislocation in circumferential joints. [PERSON] et al. [20] present a simplified method to evaluate the longitudinal behavior of shield tunnels adjacent to braced excavations. An appropriate model was adopted to study both shear and bending deformation of segmental rings, based on the Timoshenko beam theory. [PERSON] et al. [21] proposed a numerical model using homogeneous solid segmental ring-spring elements to understand the longitudinal structural deformation mechanism of shield tunnel linings, which includes a novel treatment for circumferential joints validated with field data. [PERSON] et al. [22] introduced an analytical solution for the longitudinal response of shield tunnel linings, explicitly considering ring-to-ring interactions via two-directional joint springs and employing the state space method. [PERSON] et al. [23] demonstrated the inherent nonlinearity of segmental joints through analytical derivation and determined the stiffness of segmental joints' 3D surfaces under various loadings. They subsequently proposed an iterative algorithm to compute the stiffness of segmental joints based on their internal forces. [PERSON] et al. [24] introduced a longitudinal nonlinear stiffness model for shield tunnels that accounts for both longitudinal axial forces and bending moments. Their findings highlighted the significant impact of the elastic-plastic state of longitudinal bolts on structural deformation. To account for the circumferential joint's actual operational range and the coupling effect of the shield tunnel's transverse and longitudinal performance, [PERSON] et al. [25] introduced the impact factor (\(\lambda\)) for circumferential joints and the impact factor (\(\psi\)) for the transverse bending stiffness of segmental linings. [PERSON] et al. [26] proposed an approach to evaluate the longitudinal deformation of shield tunnels due to potential disturbances post-construction. The aforementioned studies expand the understanding of LECM in shield tunnels but omit a comprehensive discussion on the longitudinal deformation mode under combined bending moment and axial force. In summary, when shield tunnels are subjected to adverse conditions such as earthquakes or uneven settlement during construction or operation, axial forces and bending moments exist simultaneously along the longitudinal direction of the tunnel. Using LECM to calculate the longitudinal forces in shield tunnels can reduce the accuracy of the results. Employing numerical simulation or model testing methods can obtain more reasonable longitudinal forces in shield tunnels. However, to achieve more accurate results, discrete segment rings (segments) and longitudinal bolts must be used during numerical modeling or model preparation. Extensive contact increases the modeling and computational costs sharply and significantly raises the difficulty of preparing experimental models.