## 1 Introduction
However, many studies and experiments have shown that there are a number of influential factors on the rotational stiffness in longitudinal joints, such as their inability to transfer tensile stress, the different section sizes, the number of bolts, the bolt pretightening forces and the structural geometries [24; 25; 26]. The rotational stiffness is also related to the level of internal forces, including both axial forces and bending moments [26; 27], which we define in this paper as moment-axial force interaction behavior. A bilinear or multi-linear constitutive relation to describe moment-axial force interaction behavior of the various joints is introduced based on experiments or numerical approaches [28; 29; 30]. Although the moment-axial force interaction behavior of longitudinal joints is recognized [20; 31], research about the calculation method on how to fully consider this joint characteristic in the full ring calculations and the related effect is rarely found, especially on special-section tunnels where this is more relevant. The related studies [32; 33] have observed the moment-axial force characteristic of the joints in quasi-rectangular tunnels. However, [PERSON] et al. [32] presented full-scale ring experiments and focused on the bearing mechanism and failure process of the overall lining structure, whereas [PERSON] et al. [33] validated a 3D joint model by comparing results from model to experiments. How to take account of the joints' moment-axial force characteristic into the general structural model and its effect on the structural behavior were not further studied. Moreover, also the shear stiffness of the joint in quasi-rectangular tunnels varies with the joint's axial forces, which we define in this paper as shear-axial force interaction behavior. Since relatively larger bending moments and shear forces occur in special-shaped shield tunnels compared to those in circular tunnels [34], the joints' moment-axial force and shear-axial force interaction behavior should be given appropriate attention. On the other hand, the sizes of the segments in quasi-rectangular tunnels are restricted by the narrow assembly space in the tunnel boring machine, so the joints are arranged much closer. It means that the influence of the joints on the overall structural performance may be larger than in other shield tunnels. All these factors make the effect of the joints' moment-axial force and shear-axial force interaction behavior non-negligible. In this paper, an iterative incremental beam-spring model is used to consider the stiffness variation caused by moment-axial force and shear-axial force interaction behavior for each joint. The precise model parameters are derived from a nonlinear 3D finite element analysis of the local joint behavior and from the results of joint experiments. The iterative model is verified by an indoor full-scale ring experiment, and the significance of the joints' nonlinear interaction behavior is emphasized through a comparison between the results from the proposed iterative model and models with constant joint stiffness. Finally, for this new type of tunnel, the presented model reveals some unique aspects of quasi-rectangular linings. These aspects can be of relevance in further quasi-rectangular tunnel applications and are a possible reference for other special-section tunnels.