## 2 Internal Force Calculation of Lining  
In the process of tunnel water conveyance, the lining structure is under the pressure of water and the internal force calculation formulas at different water levels are different. When water is delivered without pressure, there is only the water's gravity inside the lining. When water is delivered under pressure, the lining bears the water's gravity and water conveyance pressure. The external rock and soil of the tunnel give the vertical reaction force to the lower part of the lining to counteract the water's gravity. As the two forces are a pair of interaction forces acting on the lower lining, the influence of water gravity on the lining's internal force in the vertical direction can be ignored [16; 17]. Therefore, it is not necessary to consider the external reaction force of the lining caused by internal water gravity in the vertical direction when calculating with the modified routine method [18]. Taking the pressurized water conveyance tunnel as an example, the calculation model of the internal force of the tunnel section is established, as shown in Figure 1.  
In the figure, \(r\) is the inner radius of the lining; \(H_{w}\) is the water level height; \(P\) is the water conveyance pressure; \(q\) is the gravity of the water at the bottom of the tunnel, \(q=\gamma_{w}\cdot H_{w}\); \(\gamma_{w}\) is the weight of water. The load and structure on both sides of the tunnel are symmetrical and there is no vertical pressure. The basic system is shown in Figure 2[8]. The lining is a secondary statically indeterminate structure and the typical equation of the force method is as follows:  
\[ \begin{cases}\delta_{11}X_{1}+\delta_{12}X_{2}+\Delta_{1P}=0\\ \delta_{21}X_{1}+\delta_{22}X_{2}+\Delta_{2P}=0\end{cases}  \tag{1}\]  
where \(\delta_{11}\), \(\delta_{12}\), \(\delta_{21}\) and \(\delta_{22}\) are the coefficient of the typical equation; \(\Delta_{1P}\) and \(\Delta_{2P}\) are the free terms of the typical equation; \(X_{1}\) and \(X_{2}\) are the redundant unknown force. According to the modified routine method, it is stipulated that the positive and negative signs of internal forces are as follows. The bending moment is positive when the inner side is tension; the axial force is positive when this is compression; the shear force is positive with the counterclockwise direction around the isolator [10,14]. The unit bending moment diagrams \(\overline{M}_{1}\) and \(\overline{M}_{2}\) are shown in Figure 3.  
Figure 1: Diagram of internal load in the section of pressurized water conveyance tunnel. Figure 2: Basic system for internal force calculation of pressurized water conveyance tunnel. The distribution forms of internal water pressure in pressurized water conveyance and pressureless water conveyance are different, but the basic structure and unit bending moment diagram of internal force calculation in the two cases are the same. Therefore, the formulas for calculating the coefficients and free terms of typical equations in the two water conveyance forms as follows:  
\[\delta_{11}=\int_{0}^{\pi}\frac{\overline{M}_{1}^{2}}{EI}rd\theta=\frac{\pi r}{ EI} \tag{2}\]  
\[\delta_{12}=\delta_{21}=0 \tag{3}\]  
\[\delta_{22}=\int_{0}^{\pi}\frac{\overline{M}_{2}^{2}}{EI}rd\theta=\frac{\pi r^ {3}}{2 EI} \tag{4}\]  
\[\Delta_{1P}=\int_{0}^{\pi}\frac{\overline{M}_{1}M_{P}}{EI}rd\theta=\frac{r}{ EI}\int_{0}^{\pi}M_{P}d\theta \tag{5}\]  
\[\Delta_{2P}=\int_{0}^{\pi}\frac{\overline{M}_{2}M_{P}}{EI}rd\theta=-\frac{r^{2 }}{EI}\int_{0}^{\pi}M_{P}\cos\theta d\theta \tag{6}\]  
where \(EI\) is the bending stiffness, which will be eliminated later; \(M_{P}\) is the bending moment produced by water pressure in the basic structure; \(\theta\) is the angle between the radius at the calculated position and the \(x\)-axis. The coefficients and free terms are substituted into the typical equation and the formulas for calculating the redundant unknown forces \(X_{1}\) and \(X_{2}\) are obtained. \[\left\{