# Analysis of Deformations of the Tunnel Excavation Face via Simplified Calculation Methods
## 2 Stress and Strain Condition of the Tunnel Face via Hemispherical Approach  
Using a hemisphere, whose radius matches that of the tunnel and is positioned as shown in Figure 2, one can assess for a deep tunnel, with a certain degree of accuracy, the stress and strain behaviour of the excavation face and of the ground ahead of it. This study can lead to the evaluation of the type of behaviour of the soil or rock: elastic or elasto-plastic [7]. When a plastic zone is detected in a cortical portion of the hemisphere, this usually means that the ground on the excavation face and for a certain distance from it is unstable to some extent. The level of instability depends on how thick the plastic zone is and on the residual strength parameters of the soil or rock, i.e., the strength parameters shown by the ground within the plastic zone, the area where the ground behaves in the plastic field. In these circumstances, the excavation face may collapse or experience the breaking of pieces of material into different sizes. By approximating the excavation face as a hemisphere, it becomes feasible to evaluate the radial and circumferential stresses and radial displacements using the spherical symmetry as a powerful calculation tool. The mathematical treatment of the stress and strain behaviour of the ground around the spherical surface is reported in the paper [7]. In this work, steps are developed that allow the values of the stresses and strains in the material at the boundary of the spherical surface to be obtained, both in the elastic behaviour zone and in the plastic field. Several hypotheses have been proposed for the convergence-confinement method, which is utilized for analysing stresses in the ground [3; 24; 25; 26]. One such hypothesis assumes that the lithostatic stresses are homogenous and isotropic, i.e., the stresses in the medium before creating the hemispheric void are equal to \(P_{0}\) in all directions. Based on this assumption, the radial stresses \(\sigma_{r}\) (which act radially toward the sphere center), the circumferential stresses \(\sigma_{\theta}\) (oriented perpendicular to the radial direction), and the radial displacement ur in the ground around a sphere can be described under elastic behaviour conditions by simple equations [7]:  
\[\sigma_{\theta}=P_{0}+0.5\cdot(P_{0}-p)\cdot\frac{R^{3}}{r^{3}} \tag{1}\]  
\[\sigma_{r}=P_{0}-(P_{0}-p)\cdot\frac{R^{3}}{r^{3}} \tag{2}\]  
\[u_{r}=\frac{1+\
u}{2\cdot E}\cdot(P_{0}-p)\cdot\frac{R^{3}}{r^{2}} \tag{3}\]  
where the following are defined:  
\(P_{0}\): the lithostatic stress that exists at a tunnel depth;  
\(R\): the sphere radius;  
\(r\): the distance from the sphere centre (\(r\geq R\));  
\(p\): the radial pressure applied to the tunnel face (on the inner surface of the hemisphere);  
\(E\), \(n\): the elastic modulus and Poisson's ratio of the ground. On the surface of the hemisphere (\(r=R\)), the following value of the radial displacement \(u_{R}\) is obtained [7]:  
\[u_{R}=\frac{1+\
u}{2\cdot E}\cdot(P_{0}-p)\cdot R \tag{4}\]  
Figure 1: Ground deformation response during the full-face excavation on the tunnel face (extrusion) and the tunnel walls (radial displacements). If a plastic zone exists around the hemisphere, with an extension from \(R\) to the plastic radius \(R_{pl}\), the material within this zone is considered to exhibit a plastic behaviour.