## 1 Introduction  
Many segment ring models, such as the uniform rigidity ring model [1], modified uniform rigidity ring model [2], multi-hinge ring model [3, 4], beam-spring model [5, 6, 7], beam-discontinuous joint model [8], and shell-spring model [9], are used for the transverse analysis of the internal force and deformation of shield-driven tunnels. The main difference between all these models is the way that the flexural behavior of the longitudinal joints is considered. Indeed, the way to address the segment longitudinal joint is critical for the analysis of the shield tunnel structure [10, 11, 12, 13].  
At present, the more popular methods for the analysis of the tunnel structure are the beam-spring model and modified uniform rigidity ring model. The beam-spring model, which involves the longitudinal joint bending stiffness as the key parameter, behaves close to the actual state of the installed segment ring [14, 15]. The existing research regarding the beam-spring model mainly concentrates on the factors affecting the longitudinal joint bending stiffness and how the longitudinal joint bending stiffness affects the internal force and deformation [3, 16, 17, 18, 19]. Because of the limitations of all these methods, deviations among these studies are expected.  
Moreover, it is difficult to establish the beam-spring model in the numerical simulation, and the application of the beam-spring model is relatively complex. For the beam-spring model, longitudinal joints are treated as three-dimensional springs with flexural stiffness, axial stiffness, and shearing stiffness. In contrast, for the modified uniform rigidity ring model, the longitudinal joints of the installed segment ring are neglected. However, the effect of the longitudinal joints is still considered in the modified uniform rigidity ring model by reducing the transverse bending rigidity of a uniform rigidity ring. Thus, the bending rigidity of the modified uniform rigidity ring model can be obtained by multiplying the bending rigidity of the transverse section of the segment by the effective ratio of the transverse bending rigidity \(\eta\). Compared to the generally accepted but more comprehensive beam-spring model, the modified uniform rigidity ring model is simpler and easier to use. However, as the key parameter of the modified uniform rigidity ring model, the value of \(\eta\) is difficult to determine [2; 20; 21].  
Many studies have focused on the methods of how to obtain the value of \(\eta\)[20; 22; 23; 24; 25], and the factors influencing \(\eta\)[4; 24; 25; 26] have been reported. The existing methods to determine \(\eta\) are often too complex or lack a theoretical basis, and empirical equations based on model tests [27] are unable to correctly characterize the correlation between \(\eta\) and its influencing factors, such as the longitudinal joint bending stiffness, the diameter of the segment ring, the thickness of the segment, and the number of joints [28].  
In this article, the correlation between the longitudinal joint bending stiffness of an installed segment ring and the effective ratio of the transverse bending rigidity of a modified uniform rigidity ring is investigated. Then, formulas that express the correlation between the effective ratio of the transverse bending rigidity and its influencing factors are derived according to the longitudinal joint bending stiffness. With the knowledge of this effective ratio, the bending rigidity of a modified uniform rigidity ring is fully defined. Finally, the validity of the established algorithm for calculating the effective ratio of the transverse bending rigidity is verified by comparing the responses of the tunnel rings using the modified uniform rigidity ring models, which are established according to the effective ratio obtained by different methods, with those obtained with the generally accepted but more complex beam-spring model.