# Analysis of Deformations of the Tunnel Excavation Face via Simplified Calculation Methods
## 4 Reliability of the Hemispherical Approach
As regards the influence of the thickness of the concrete lining, it is appreciable only in the case of a very deep tunnel (\(P_{0}=11\) MPa) and rock mass with poor mechanical characteristics: in this case, a variability of approximately 10% of the extrusion can be detected as a function of the thickness of the lining (the variability increases to 15% for a tunnel with a small radius (\(R=2\) m). In all other cases, the influence of the thickness (and, therefore, of the stiffness) of the lining on the extrusion of the excavation face is to be considered negligible. Furthermore, as can be seen from the same Figures 9-11, the estimate of the extrusion obtained from the simplified hemispherical method is in line with the results obtained from the numerical modelling, for each analysed case; for the case of a deep tunnel (\(P_{0}=11\) MPa) and Poor rock (type A) the result of the simplified hemispherical method is intermediate to the variability range of the extrusion due to the different thickness of the concrete lining. [PERSON] et al. [30] analysed the stability of the excavation faces of deep tunnels through three-dimensional numerical modelling. After an extensive parametric analysis, by varying the main geometric parameters of the tunnel (two tunnel sizes, three tunnel depths) and geomechanical parameters of the rock mass (two horizontal thrust coefficients \(K_{0}\) and  
Figure 9: Face extrusion at the centre of the tunnel face: simplified numerical modelling and hemispherical approach (\(R=2\) m). Key: S: the concrete lining thickness; Good: rock mass type C; Intermediate: type B; Poor: type A; dashed lines: maximum admissible extrusion values for type C (green), type B (yellow), and type A (red), according to [30]. four sets of material parameters), they were able to obtain the dimensionless trend of the face extrusion parameter \(\Omega_{f}\) as the dimensionless face stability parameter \(\Lambda_{f}\) varies [30]:  
\[\Omega_{f}=1.4\cdot\Lambda_{f}^{-1.2} \tag{10}\]  
where the following are defined:  
\[\Omega_{f}=\left(\frac{u_{h}}{2\cdot R}\right)\cdot\left(\frac{E}{P_{0}}\right)\]  
\(u_{h}\): face extrusion (horizontal displacement);  
\[\Lambda_{f}=3.8\cdot\left(\frac{\sigma_{cm}}{P_{0}\cdot\sqrt{1+\frac{2}{3} \cdot K_{0}}}\right)\cdot\left(\frac{P_{0}}{2\cdot\gamma\cdot R}\right)^{0.35}\]  
\(\sigma_{cm}\): rock mass uniaxial strength:  
Figure 4:0  
\(\gamma\): rock mass specific weight. According to the authors [30], when \(\Lambda_{f}\) is equal to 1, the rate of increase in the face extrusion parameter \(\Omega_{f}\) increases considerably, indicating the approach to a limit condition of stability. \(\Lambda_{f}=1\), therefore representing a limit condition for stability. Setting \(\Lambda_{f}=1\) in Equation (10), the maximum admissible extrusion to guarantee the stability conditions of the excavation face becomes [30]:  
Figure 4:1  
Figure 4:2 Face extrusion at the centre of the tunnel face: simplified numerical modelling and hemispherical approach (\(R=3.5\) m). Key: S: the concrete lining thickness; Good: rock mass type C: Intermediate: type B; Poor: type A; dashed lines: maximum admissible extrusion values for type C (green), type B (yellow), and type A (red), according to [30]. In Figures 9-11, these limit extrusion values are represented with dashed lines, with the colours referring to the considered different types of rock (types A, B, and C). Extrusions that are higher than the limit values denote potential conditions of instability of the excavation face. For the analysed cases, these conditions are detected only for rocks of type A, for in situ stresses higher than 5.8-6.2 MPa, depending on the tunnel size. The limit extrusions, however, are not reached for type B and C rocks, not even for the maximum in situ stress values adopted in the study (\(P_{0}=11\) MPa).