## 1 Introduction  
When a shield tunnel is constructed, the choice of a suitable cross-section for the tunnel is very important, especially when the tunnel is built in a city center. Compared to traditional circular tunnels, the quasi-rectangular shield tunnel with improved cross-sectional shape, makes two-way subway transportation in one tube possible, as shown in Figure 1. This concept can reduce both the construction costs and the environmental impact. Moreover, when compared to a large-diameter circular shield tunnel, the underground space efficiency can be increased by 20%, and the buried depth can be decreased significantly [1; 2; 3]. Compared to other special-section shield tunnels, the following advantages of a quasi-rectangular tunnel are emphasized: easier control of the surface subsidence when compared to that of a double-O tube (DOT) in relation to improper backfilling grouting and the concave shape of the shield machine [4; 5], and the low cost of the segment linings due to the use of reinforced concrete compared to the composite steel-concrete segments in the rectangular cross-section tunnels in Rokujizo Station, the Ishida Station, and Hongqiao Linkong 11-3 Connection [6; 7]. Given the above, quasi-rectangular shield tunnels are expected to be used for a wide range of applications in the future. At present, the basic design principles for quasi-rectangular tunnels linings are identical to those for circular tunnels. Research related to calculation models for special-section tunnel linings is rarely published. To date, four types of calculation methods for tunnel linings can be used in practice: (a) the routine method, which fully ignores the effect of the segmental joints [8]; (b) the multi-hinge ring method, which treats the joints as free hinges [9]; (c) the modified routine method (MRM), which assumes an average uniform rigidity ring based on a rigidity reduction factor and bending moment transmission [10; 11]; (d) the beam-spring model (BSM), which treats the segments as beams and the joints as springs [12; 13]. Methods (a) and (b) cannot reflect the real structural behavior and are therefore not commonly used. Although the MRM method relies much on empirical parameters and the BSM method requires defining accurate spring parameters through experience, tests, analytical solution or numerical analysis, these two types are used widely in practice [12]. Furthermore, other calculation approaches have been developed in recent decades. Analytical solutions for circular and DOT tunnels considering the effects of joint stiffness are established by [PERSON] et al. [14] and [PERSON] et al. [15] respectively. However, only symmetric load distributions or anti-symmetric load distributions are possible for the analytical solutions. For cast-iron segmental tunnels, a 3D tunnel model in which the joint bolts are replaced by a set of springs is developed, and a good agreement between the 3D model and field measurements is achieved [16; 17]. A heterogeneous equivalent beam model is proposed, converting the local range of joints into a beam element according to the actual stress state on the joint section and then treating the non-continuous segment linings as a continuous heterogeneous structure [18]. It is obvious that all these models focus on how to deal with the joints, especially with the rotational behavior of the longitudinal joints. Figure 1: Comparison of two circular single track shield tunnels and a quasi-rectangular double track shield tunnel. Indeed, given the existence of the longitudinal and circumferential joints, segmental linings are a non-continuous structure. The effects of the joints on internal forces and lining displacements are significant and should be carefully taken into account in the design [14; 19; 20; 21]. In previous studies, the rotational stiffness of lining joints was considered as a constant parameter [14; 22; 23].