## 1 Introduction  
Shield tunnels are assembled structures consisting of concrete segments and connecting bolts that exhibit discontinuities in both the transverse (ring) and longitudinal directions [1, 2]. Generally speaking, for the study of shield tunnel transverse direction, the problem can be simplified to plane strain when the joints in each segmental ring are limited and the stress characteristics of the shield tunnel are relatively clear [3, 4, 5]. In contrast, the large longitudinal scale of the shield tunnel, the complex working conditions of the traversing strata, and the large number of circumferential joints between the rings make the longitudinal stress and deformation of the shield tunnel very complicated. For the longitudinal deformation analysis of shield tunnels, the most applied model is the longitudinal equivalent continuous model (LECM) proposed by [PERSON] [6]. The model considers the longitudinal tension, bending, and compression of the shield tunnel independently and gives the solution method for its deformation stiffness, respectively. [PERSON] et al. [7] investigated the longitudinal earthquake response characteristics of the shield tunnels subjected to single-degree-of-freedom seismic excitation and pointed out that when the seismic load is excited horizontally, horizontal bending mainly occurs in shield tunnels, and longitudinal tensile and compressive compression mainly occur in shield tunnels when the seismic load is excited longitudinally. Similarly, based on the LECM, [PERSON] et al. [8] carried out a series of model tests to investigate the seismic response characteristics of shield tunnels across soft and hard, uneven strata. [PERSON] et al. [9] conducted centrifuge shaking-table tests by considering the shield tunnel as a continuous structure in the longitudinal direction. The above results indicate that LECM has certain engineering applicability, and the calculations are relatively simple. However, when shield tunnels deform under complex external loads, the accuracy of tunnel internal forces calculated using LECM may decrease. In order to better clarify the longitudinal deformation characteristics of shield tunnels, many experts conduct research using numerical simulation or model tests. The longitudinal deformation characteristics of shield tunnels could be visualized and clarified by discretizing the segments and bolts using the above methods. [PERSON] et al. [10] prepared a discrete model to investigate the longitudinal bending stiffness of shield tunnels under varying axial forces. [PERSON] et al. [11] investigated the longitudinal seismic performance of shield tunnels using incremental dynamic analysis and the spring-beam model. [PERSON] et al. [12] used the Shantouwan shield tunnel as the background to establish a 3D fine finite element model to research the damping behavior of shape memory alloy (SMA) flexible joints under seismic loading. [PERSON] et al. [13] established a 3D finite element model of segment-bolt joints and investigated the seismic deformation characteristics of different types of bolt joints, and the results showed that the stress in the oblique bolt is the maximum, whereas the stress in the curved bolt is the minimum. [PERSON] et al. [14] established a refined numerical model of the anchor joint and investigated the tension and stress distribution characteristics of the joint under different directions of earthquake. [PERSON] et al. [15] used cement mortar to prepare the joints, simulated bolts with steel wire, and conducted shaking table tests under soft soil conditions to investigate the seismic response differences of shield tunnels to near-field and far-field ground motion. [PERSON] et al. [16] investigated the earthquake response characteristics of tunnel-shaft junctions under longitudinal earthquake excitation by using discrete segmental splicing. [PERSON] et al.