## 2 Mechanical Analysis of Subsea Tunnels Considering Ground Reinforcement
### Analytical Solutions for Various Configurations  
Assuming that stress components and seepage volume force do not vary with the angular orientation, then the equilibrium equation of the tunnel surrounding rock under seepage volume force reads:  
\[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}r}+\frac{\sigma_{r}-\sigma_{\theta}}{r}- \alpha\frac{\mathrm{d}p_{w}}{\mathrm{d}r}=0, \tag{2}\]  
where the subscripts \(r\) and \(\theta\) in order denote the radial and the tangential directions, \(\alpha\) is the effective area coefficient of the seepage body force. For the sake of safety, \(\alpha\) is assumed to be 1 in the study of the rock failure and stability [21]. Under the axisymmetric plane-strain condition, the relationship between the strain and the displacement can be expressed by:  
\[\varepsilon_{r}=\frac{\partial u_{r}}{\partial r},\;\varepsilon_{\theta}= \frac{u_{r}}{r}, \tag{3}\]  
The stress-strain relation can be obtained by applying the generalized Hooke's law to the elastic region under a plain-strain condition:  
\[\left\{ \begin{array}{l}\varepsilon_{r}=\frac{1-\mu^{2}}{E}(\sigma_{r}-\frac{ \mu}{1-\mu}\sigma_{\theta})\\ \varepsilon_{\theta}=\frac{1-\mu^{2}}{E}(\sigma_{\theta}-\frac{\mu}{1-\mu} \sigma_{r})\end{array} \right., \tag{4}\]  
where \(E\) and \(\mu\) in order are the elastic modulus and the Poisson's ratio, while \(\sigma_{r}\) and \(\sigma_{\theta}\) denote the radial and tangential stressors, respectively. Using the Mohr-Coulomb criterion, the ground behavior in the plastic zone is governed by:  
\[\frac{\sigma_{\theta}-\sigma_{r}}{2}=\frac{\sigma_{\theta}+\sigma_{r}}{2}\sin \varphi+c\cos\varphi, \tag{5}\]  
where \(c\) and \(\varphi\) are the cohesion and internal friction angle, respectively. The yield criterion can be applied to the plastic zones both in the reinforced ground and the natural ground. By taking the dilation effect into consideration, the flow rule of the plasticity can be stated as:  
\[\varepsilon_{r}^{p}+N_{\Psi}\varepsilon_{\theta}^{p}=0, \tag{6}\]  
where \(N_{\Psi}=(1+\sin\Psi)/(1-\sin\Psi)\), \(\Psi\) is the dilation angle. The compatibility relation is also given by:  
\[\frac{\mathrm{d}\varepsilon_{\theta}}{\mathrm{d}r}=\frac{\varepsilon_{r}- \varepsilon_{\theta}}{r}, \tag{7}\]  
In the context of small deformations, the total strains would be the sum of the elastic strain and the plastic strain components:  
\[\varepsilon_{\theta}=\varepsilon_{\theta}^{e}+\varepsilon_{\theta}^{p},\; \varepsilon_{r}=\varepsilon_{r}^{e}+\varepsilon_{r}^{p}, \tag{8}\]  
where the superscripts \(e\) and \(p\) represent elastic and plastic, respectively. According to [PERSON]'s law, the seepage quantity through the thickness of a circular section can be written as [22]:  
\[\mathrm{Q}=\left\{ \begin{array}{l}\frac{2\pi\mathrm{d}r^{\prime}}{\gamma_{w} }\frac{\mathrm{d}p_{w}}{\mathrm{d}r},\;r_{0}\leq r\leq r_{a}\\ \frac{2\pi\mathrm{d}r}{\gamma_{w}}\frac{\mathrm{d}p_{w}}{\mathrm{d}r},\;r_{a} \leq r\leq R_{0}\end{array} \right., \tag{9}\]  
where \(\gamma_{w}\) is the density of seawater, \(r_{a}\) is the radius of the reinforced zone, \(k\) and \(k^{\prime}\) in order are the permeability coefficient of the natural zone and that of the reinforced zone. Note that the seepage pressure at the fixed boundaries \(r=r_{0}\) and \(r=R_{0}\) are 0 and \(p_{a}\), respectively, and considering the seepage continuity assumption, the \(p_{w}\) in Equation (9) can be obtained as [22]:  
\[p_{w}=\left\{ \begin{array}{ll}p_{a}k\mathrm{Im}\frac{r}{r_{0}}/\left(k^{ \prime}\mathrm{Im}\frac{R_{0}}{r_{a}}+k\mathrm{Im}\frac{r_{a}}{r_{0}}\right),&r _{0}\leq r\leq r_{a}\\ p_{a}\left(k^{\prime}\mathrm{Im}\frac{r}{r_{a}}+k\mathrm{Im}\frac{r_{a}}{r_{0}} \right)/\left(k^{\prime}\mathrm{Im}\frac{R_{0}}{r_{a}}+k\mathrm{Im}\frac{r_{a }}{r_{0}}\right),&r_{a}\leq r\leq R_{0}\end{array} \right., \tag{10}\]  
According to the abovementioned basic equations as well as corresponding boundary conditions, the analytical solution for each configuration is developed in Appendix A.