## 1 Introduction  
Tunnel boring machine (TBM) technology has been widely used for the construction of the urban subway tunnel. The longitudinal performance research of the TBM tunnel based on the equivalent continuous model and elastic foundation theory is a hotspot. The longitudinal affects the safe operation of the subway, which has also been widely studied through field monitoring, numerical analysis and analytical solution [1; 2; 3; 4; 5]. The research on the tunnel longitudinal performance focuses on the longitudinal settlement and horizontal axis deviation, which are caused by the coupling effects of multi-load [6; 7; 8; 9]. [PERSON] et al. [10] and [PERSON] et al. [11] were the pioneers developing the analytical solution for calculating the longitudinal settlement based on longitudinal bending stiffness. Elastic foundation models such as the Winkler model [12], the Pasternak model [13], and the Kerr model [14] are the most widely used theoretical calculation models in the longitudinal deformation research of tunnels. After that, the generalized longitudinal equivalent continuous model was proposed by [PERSON] et al. [15]. [PERSON] et al. [16] proposed the model considering the mechanical behavior of blots. [PERSON] et al. [17] and [PERSON] and [PERSON] [18] used the continuous elastic beam to simulate the longitudinal structure model of the shield tunnel, which can analyze the longitudinal property of the shield tunnel. [PERSON] et al. [19] derived an analytical solution for the longitudinal bending stiffness of a segmental liner, which was mainly utilized in the shield tunnels. This method is verified by the simulation results from the finite element program. The longitudinal settlement mechanism of the tunnel can also be analyzed by the two-dimensional and three-dimensional numerical simulation methods [20; 21; 22]. [PERSON] et al. [23] developed a finite element method for analyzing the longitudinal performance of shield tunnels considering the longitudinal variation of geotechnical parameters. [PERSON] et al. [9] analyzed the deformation of the shield tunnel and the surrounding soil by building a three-dimensional model of shield tunnel. A numerical model was established to simulate the earth pressure balance (EPB) in excavation processes--the simulation results were compared with those obtained by the field measurement [24; 25]. Also, field monitoring is an important method to analyze the longitudinal settlement of the shield tunnel. [PERSON] [26] studied the relationship between shield parameters and the shield surface settlement through field surface settlement measurements. [PERSON] et al. [27] collected the measurements of tunneling-induced settlements during the construction of the new Milan underground line 5, which were back analyzed with the method of the classical Gaussian empirical expression. Then, the detailed description of the EPB tunneling performance in the transverse and longitudinal directions could be achieved. The research on the longitudinal performance of shield tunnels is mainly focused on the longitudinal settlement of tunnels. Limited research has been conducted on the problem of horizontal axis deviation of the small radius TBM interval construction. In the design of urban subway lines, the sharp curve of the shield tunnel with a small curvature radius is usually adopted to avoid the adverse geological conditions and surrounding existing buildings [28; 29]. When the curvature radius is less than 40 times of the TBM's diameter, the construction difficulty and quality risk will be increased significantly. The damage caused by the horizontal axis deviation of the constructed tunnel is particularly prominent, which can easily affect the assembly quality of the segments. The horizontal axis deviation can result in tunnel mud leakage and local structural damage, which pose a severe threat to the safety of metro operation. Therefore, it is of great significance to study the effect of the construction load on the horizontal axis deviation of a small radius TBM interval tunnel.