## 1 Introduction
Therefore, this article is also based on research conducted on circular tunnels. Elastoplastic analysis of lined tunnels is also an important content of engineering safety evaluation. [PERSON] et al. [19] pointed out that the first principal stress used in Mohr-Coulomb yield conditions for lined circular tunnels should be adjusted according to load and in situ stresses. [PERSON] et al. [20] put forward an elastic-plastic analytic solution for rock-liner interaction under general loading, in which the constitutive models of the surrounding rock and the liner were the Mohr-Coulomb and bilinear model respectively, determining the displacement, sensitivity of the displacement, and plastic zone of surrounding rock varying with general loading. In practice, the interaction between surrounding and lining changes over time. The main factors affecting the observed time dependency in tunnel construction are the sequence of excavation [21; 22; 23], the number of linings [24; 25; 26], their time of installation [4; 5; 9; 10; 11; 12; 27; 28], and the rheological properties of the host rock [24; 29; 30]. This paper uses the parameter of stress release coefficient to characterize the timing of support, which depends on the distance between the tunnel face and the lining section. Other research has focused on the influence of the anisotropy of rock mass on the interaction between the surrounding rock and lining. [PERSON] and [PERSON] [31; 32] obtained closed-form solutions for a deep circular tunnel excavated in transversely anisotropic rock, including the displacements and stresses of both the lining and the surrounding rock, and [PERSON] et al. [33] then expanded these solutions, which were suitable for any hole shapes. On the basis of [PERSON]'s works, [PERSON] et al. [34] proposed analytical solutions for an arbitrary-shaped tunnel with lining in anisotropic rock mass in conjunction with the consideration of different contact modes. In addition, there are many other factors that affect the interaction mechanism between surrounding rock and lining, such as the seepage pressure [35; 36; 37; 38; 39], multiple tunnel excavation [40], coupling with temperature field [41; 42], and void phenomena [43]. There are some meaningful examples within actual engineering practice. Based on the kinematic upper bound theorem and reliability theory, [PERSON] et al. studied the stability of a deep buried tunnel roof in nonlinear Hoek-Brown medium [44]. [PERSON] et al. introduced a method of using drilling and blasting techniques to distinguish and quantify technical and geological over-excavation in hard rock tunnel construction and demonstrated its application on an 8.1 km section of a 12 km pressure tunnel [45]. In summary, numerous factors influence the interaction between surrounding rock and lining support, necessitating specialized analysis and demonstration based on the primary contradictions between the engineering environment and construction needs. The focal point of this paper is the comprehensive analysis of the bearing capacity of long-distance water supply pipe tunnel support structures. It delves into the combined effects of internal and external water pressure, along with the surrounding rock pressure resulting from tunnel excavation. This aspect has been notably absent in previous research on lining support and this study is intended to address this gap. In recent years, numerous water conveyance tunnel projects have been confronted with high internal water pressure conditions. For instance, the Yellow River Tunnel of the South-to-North Water Transfer Project in China has a designed internal water pressure of 0.517 MPa, while the Guangzhou Xijiang Water Diversion Project faces a maximum internal water pressure of 0.9 MPa. Furthermore, water resources allocation projects in the Pearl River Delta may encounter internal water pressures reaching 1.05 MPa, and even 1.5 MPa under exceptional circumstances [25].