## 1 Introduction  
Hydraulic tunnels play a crucial role in regional water transfer, hydropower generation, diversion and flood discharge, etc. Different water-flowing conditions are the key issues for structural design and safe operation [1; 2]. According to different stress states, hydraulic tunnels can be divided into non-pressure and pressure tunnels. The water flow in the non-pressure tunnel is open flow and the water level affects the internal force of the lining. The inside of the pressure tunnel is filled with water and the value of water conveyance pressure influences the internal force of the lining [3; 4]. Besides, the dynamic characteristics of water in the tunnel and the water seepage of the lining structure will also affect the internal force of the structure [5; 6]. At present, there is still a lack of general formula to solve the internal force of lining under different water conveyance conditions. The modified routine method improves the calculation theory of the routine method by introducing the bending stiffness to effectively consider the reduction effect of the bending stiffness of the joint. It has become a commonly used theoretical method for calculating lining internal force in tunnel structure design [7]. [PERSON] et al. [8] considered the situation that the groundwater level is located in the tunnel section and modified the formula of horizontal displacement of the structure in the modified routine method. This expands the applicable scene of the modified routine method. [PERSON] et al. [9] aim at the problem that the calculation theory of shield segments is insufficient to study the joints. They consider the allowable deformation of the segment joint. The elastic center method and superposition principle are adopted to deduce the calculation formula of lining internal force with the discontinuous joint. [PERSON] et al. [10] introduced a new analytical solution to calculate the internal force and deformation of the circular tunnel lining by using the curved Euler beam theory and the principle of minimum potential energy. This method can consider joints and loads with arbitrary distribution and has remarkable advantages in rigid body displacement treatment and soil reaction calculation. With the emergence of large-diameter and deep tunnels, the traditional design method will certainly impact the internal force calculation because it does not consider the pre-deformation and assembly errors in the segment assembly process. [PERSON] et al. [11] deduced an analytical solution to calculate the internal force of shield lining by assuming the assembly error of bolt holes and the pre-deformation of lining and using the principle of minimum potential energy. None of the above studies considered the internal force of the lining caused by the internal water pressure of the hydraulic tunnel. However, in actual use, whether in a tunnel with or without pressure, different internal water pressures influence the internal force of the lining [12; 13]. This paper refers to internal force positive and negative sign provisions in the modified routine method. The general formula for calculating the internal force of lining under different water levels and water conveyance pressures is derived by using the mechanical analysis method [10; 14]. The internal force generated by the lining structure of the tunnel under the external load can be calculated by the modified routine method or other methods [15]. This formula can be superimposed on the formula of internal force caused by external load given by a modified routine method, further improving the application scope of tunnel structure calculation theory. In this paper, the internal stress calculation model of the tunnel section is established according to structural mechanics theory. The calculation formula of lining internal force is deduced considering any water level height and different water conveyance pressures.