## 1 Introduction  
With the increasing demand for infrastructure construction in China, the construction of more and more tunnels has been put on the agenda, such as the Qichongcun Tunnel in Guizhou, Hanjialing Tunnel, Yangzong Tunnel, and Gongbei Tunnel of the Hong Kong-Zhuhai-Macao Bridge Project [1; 2; 3; 4]. The safety of such tunnels during and after construction is lower than that of conventional tunnels. In order to ensure construction safety, the primary support of tunnels not only consists of bolts and shotcrete but steel arch frames or steel grids as well [1] (see Figure 1). The displacement of tunnel walls and the bearing capacity of the primary support are important indicators to evaluate the safety of tunnel structures and guide the tunnel support scheme. Generally, engineers will use numerical simulation software to perform calculations [5; 6; 7; 8]. However, due to the computational speed and complexity of numerical simulation software, there is an urgent need to propose a simple and efficient calculation method. Analytical methods are simple and convenient, and they are widely used in tunnel mechanics. Many scholars have studied the analytical methods of tunnel stress and displacement. [PERSON] (1898) solved the stress, strain, and displacement distribution of deep-buried circular tunnels using a plane strain model based on the elastic mechanics theory [9]. [PERSON] (1938) proposed the Fenner formula considering the plastic deformation of surrounding rock based on the Mohr-Coulomb constitutive model [10]. The function of a complex variable proposed by [PERSON] (1963) has been widely used by many researchers in tunnel mechanics [11]. [PERSON] et al. (2006) and [PERSON] et al. (2001) analyzed the stress distribution of unlined tunnels using this method [12]. Compared with previous studies using the theory of complex variables, it is more convenient to calculate the stress distribution of deep-buried tunnels in a non-hydrostatic stress field [13]. The analytical method of tunnel mechanics proposed above gives the solution of the stress and displacement of unlined tunnels but does not consider the effect of the lining on the surrounding rock. [PERSON] (2001) improved the Fenner formula by providing an elastic-plastic solution considering the lining based on the operational characteristics of hydraulic tunnels and compared it with the original Fenner formula; the results show that the improved Fenner formula can better calculate the plastic zone and stress distribution of hydraulic tunnels during operation [14]. [PERSON] et al. (2022) considered the variability of surrounding rock and lining performance and provided an analytical solution for pressure tunnels under long-term operation, which is of great significance for the study of the long-term durability of tunnel structures [15]. [PERSON] et al. (2020) provided analytical solutions for the stress and deformation of noncircular tunnel lining and surrounding rock based on the theory of complex variables, which can better analyze the interaction between the surrounding rock and the support structure [16]. [PERSON] et al. (2013) deduced the stress and displacement of the tunnel surrounding the rock in elastic semi-infinite space based on the analytical continuation method in the function of a complex variable, which solved the mechanical problems of shallow buried tunnels [17]. [PERSON] et al. (2021) deduced the stress field and displacement field of multiple tunnels in deep strata by using the function of a complex variable and compared the contact mode between the surrounding rock and the lining during the solution [18]. [PERSON] et al. (2013) considered the stress release effect when deducing the stress and deformation of deep-buried tunnels [19]. The convergence constraint method, first proposed in 1978, has been a guiding method for tunnel support design until now.