## 2 Basic Principles and Methods
### The Effects of Internal and External Water Pressure  
The two analytical functions of annular lining structure under the action of internal and external water pressure can be expressed according to the Laurent series as follows [4]:  
\[\left\{\begin{array}{l}\varphi_{1}(\mathrm{z})=\sum\limits_{k=1}^{\infty}a_ {k}z^{-k}+\sum\limits_{k=1}^{\infty}b_{k}z^{k}\\ \psi_{1}(\mathrm{z})=\sum\limits_{k=1}^{\infty}c_{k}z^{-k}+\sum\limits_{k=1}^{ \infty}d_{k}z^{k}\end{array}\right. \tag{1}\]  
where \(a_{k}\), \(b_{k}\), \(c_{k}\), and \(d_{k}\) are undetermined coefficients and \(z=re^{i\theta}\) represents the position of any point in the study domain.  
Assuming that the external and internal water pressures are \(p_{1}\) and \(p_{2}\), respectively, then, according to the external and internal water pressure conditions on the lining-surrounding rock contact surface \(L_{1}\) and the lining-inner wall boundary \(L_{2}\), we can obtain [4]:  
\[\left\{\begin{array}{l}2 Re[\varphi^{\prime}{}_{1}(\mathrm{z})]-e^{2i\theta}[ \overline{z}\varphi_{1^{\prime\prime}}(\mathrm{z})+\psi^{\prime}{}_{1}( \mathrm{z})]=-p_{1},z=R_{1}e^{i\theta}\\ 2 Re[\varphi^{\prime}{}_{1}(\mathrm{z})]-e^{2i\theta}[\overline{z}\varphi_{1^{ \prime\prime}}(\mathrm{z})+\psi^{\prime}{}_{1}(\mathrm{z})]=-p_{2},z=R_{2}e^{ i\theta}\end{array}\right. \tag{2}\]  
where \(R_{1}\) and \(R_{2}\) represent the outer and inner radii of the lining, respectively. In this paper, tensile stress is positive, compressive stress is negative, and \(p_{1}\) and \(p_{2}\) are positive, which respectively represent the internal and external hydraulic pressure. SubstitutingEquation (1) into Equation (2), it can be seen that if \(b_{1}\) and \(c_{1}\) are not zero, the remaining undetermined coefficients are equal to zero, and \(b_{1}\) and \(c_{1}\) are equal to:  
\[\left\{\begin{array}{l}b_{1}=\frac{p_{0}R_{1}^{-2}-p_{1}R_{0}^{-2}}{2(R_{0}^{- 2}-R_{1}^{-2})}\\ c_{1}=\frac{p_{1}-p_{0}}{R_{0}^{-2}-R_{1}^{-2}}\end{array}\right. \tag{3}\]  
That is to say, the analytical function corresponding to only the annular lining under the action of internal and external water pressure is:  
\[\left\{\begin{array}{l}\varphi_{1}(z)=\frac{p_{0}R_{1}^{-2}-p_{1}R_{0}^{-2}} {2(R_{0}^{-2}-R_{1}^{-2})}z\\ \psi_{1}(z)=\frac{p_{1}-p_{0}}{R_{0}^{-2}-R_{1}^{-2}}z^{-1}\end{array}\right. \tag{4}\]  
Substituting \(\varphi_{1}(z)\), \(\psi_{1}(z)\) into \(\varphi_{j}(z)\), \(\psi_{j}(z)\) of Equation (5) [3] can obtain the stress components at any position (\(\sigma_{1}^{r}\), \(\sigma_{1}^{\theta}\), \(\tau_{1}^{\theta}\)) in the lining at the time, and substituting \(\varphi_{1}(z)\), \(\psi_{1}(z)\) into Equation (6) [3] can obtain the radial displacement components \(u_{1}^{\prime}\) and tangential displacement components \(u_{1}^{\theta}\) at any position in the lining:  
\[\left\{\begin{array}{l}\sigma_{j}^{r}+\sigma_{j}^{\theta}=4 Re\left[\varphi_{ j}^{\prime}(z)\right]\\ \\ \sigma_{j}^{\theta}-\sigma_{j}^{r}+2i\tau_{j}^{r\theta}=2e^{2i\theta}\left[ \overline{z}\varphi^{\prime\prime}{}_{j}(z)+\psi_{j}^{\prime}(z)\right]\end{array}\right. \tag{5}\]  
\[u_{j}^{r}+iu_{j}^{\theta}=\frac{1}{2G_{n}}e^{-i\theta}\left[\kappa_{n}\varphi_ {j}(z)-z\overline{\varphi_{j}^{\prime}(z)}-\overline{\psi_{j}(\overline{ \zeta})}\right] \tag{6}\]  
where \(G_{n}\) is shear modulus, \(G_{n}=E_{n}/[2(1+\mu_{n})]\), \(\kappa_{n}=3\)-\(4\mu_{n}\), \(E_{n}\) is [PERSON]'s modulus, \(\mu_{n}\) is [PERSON]'s ratio; when \(n\) is equal to 1, it corresponds to the surrounding rock; when \(n\) is equal to 2, it corresponds to the lining.  
It is verified that the results obtained are consistent with those obtained for a thick-walled cylinder, indicating that the potential function corresponding to the action of internal and external water pressure on the lining has been derived correctly.