## Appendix A The Convergence-Confinement Method  
The stress redistribution and three-dimensional deformation induced by tunnel excavation and lining installation during construction is a very complex three-dimensional problem. To simulate tunnel construction, the convergence-confinement method in plane strain conditions is widely used. The method relies on the ground reaction curve, which describes the response of the ground around the tunnel [26; 27], and the support reaction curve, which describes the lining response. The two-dimensional tunneling problem is illustrated in Figure 11. The tunneling process is represented by increasing the \(\lambda\) value from zero to one. Figure 11 shows the main characteristics of the convergence-confinement method. The convergence curve corresponds to the internal pressure versus the tunnel radial displacement. The radial displacement of the tunnel increases as the internal pressure decreases. The tunnel can be self-stabilized without a liner (Curve [a]) in Figure 11) or the surrounding ground can fail which leads to an increase in the ground load acting on the tunnel lining (Curve [b] in Figure 11). In the latter case, a liner must be installed to keep the stability of the tunnel. Based on the convergence-confinement method, [PERSON] and [PERSON] [11] proposed a set of analytical equations valid for deep circular tunnels in rock. These equations are used to ensure the technical viability of the shotcrete solutions proposed in a simplified manner, bypassing complex numerical simulations necessary for shallow tunnels and non-circular shapes, but without loss of generalization capacity of the conclusions. The problem is studied as a two-dimensional plane strain case, in which the pressure, \(p_{i}\), and the radial displacement, \(u_{r}\), are constant in the tunnel walls. The ground reaction curve is obtained from the elastoplastic solution of a circular opening subjected to a hydrostatic stress field, \(\sigma_{0}\). Equations (11) and (12) present the elastic and plastic portions of the curve, respectively. \[u_{r}^{cl}=\left(\frac{S_{o}-p_{i}}{2G_{rm}}\right)R \tag{13}\]where \(u_{r}^{el}\) is the elastic radial displacement of the massif, \(S_{o}\) the uniform tension around the tunnel, \(p_{i}\) the pressure of the walls at the tunnel, and \(G_{rm}\) the shear modulus of the rock massif. \[ \begin{array}{c}u_{r}^{pl}=u_{r}^{el}\left\{\frac{K_{\psi}-1}{K_{\psi}+1}+\frac {2}{K_{\psi}+1}\!\!\left(\frac{R_{pl}}{R}\right)^{K_{\psi}+1}+\frac{1-2\
u}{4 \left(S_{o}-P_{i}^{\alpha\alpha}\right)}\left[ln\!\left(\frac{R_{pl}}{R}\right) \right]^{2}-\left[\frac{1-2\
u}{K_{\psi}+1}\frac{\sqrt{P_{i}^{\alpha\alpha}}}{ S_{o}-P_{i}^{\alpha\alpha}}+\right.\\ \left.\frac{1-\
u}{2}\frac{K_{\psi}-1}{\left(K_{\psi}+1\right)^{2}}\frac{1}{S_ {o}-P_{i}^{\alpha}}\right]\left[\left(K_{\psi}+1\right)ln\!\left(\frac{R_{pl}}{ R}\right)-\left(\frac{R_{pl}}{R}\right)^{K_{\psi}+1}+1\right]\right\}\end{array} . \tag{10}\]  
where \(K_{\psi}\) is the dilatancy coefficient, \(P_{i}^{\alpha\alpha}\) the critical pressure, related with the transition of an elastic to plastic behavior of the tunnel walls, and \(R_{pl}\) the radius of the failed region, developed when \(p_{i}<P_{i}^{\alpha\alpha}\). The shotcrete support is modeled with perfect elastoplastic behavior, with an elastic radial stiffness \(k_{s}\), and maximum pressure, \(p_{s,max}\). Equations (10)-(11) describe the support behavior [28]. \[p_{s}=K_{s}u_{r} \tag{11}\]  
\[p_{s}^{m\alpha x}=\frac{f_{cd}}{2}\!\!\left[1-\frac{\left(R-t_{c}\right)^{2}}{ R^{2}}\right] \tag{12}\]  
\[K_{s}=\frac{E_{c}}{(1+v_{c})R}\ \frac{R^{2}-\left(R-t_{c}\right)^{2}}{(1-2v_{c})R ^{2}+\left(R-t_{c}\right)^{2}} \tag{13}\]  
where \(p_{s}\) is the pressure in the support, \(f_{cd}\) the design compressive strength of the shotcrete, \(u_{r}\) the radial displacement of the support, \(v_{c}\) the Poisson's coefficient of the shotcrete, \(E_{c}\) the modulus of elasticity of the shotcrete, \(R\) the radius of the shotcrete ring and \(t_{c}\) representing its thickness.