# Analysis of Deformations of the Tunnel Excavation Face via Simplified Calculation Methods
## 4 Reliability of the Hemispherical Approach  
After conducting 81 numerical simulations with the simplified model, significant and valuable results have been obtained on the deformation of the excavation face. These findings are illustrated in the accompanying figures, which provide visual representations of the outcomes. It was possible also to compare the obtained results with those from analytical methods (hemispherical approach). In this particular case, the extrusion at the centre of the face obtained from the numerical calculation is compared with the radial inward displacement of the sphere (Figures 9-11). Figure 3: (**a**) Scheme of the whole developed axisymmetric numerical model to simulate the tunnel excavation (the green zone is the area of the tunnel, the blue zone the rock mass surrounding the tunnel); (**b**) Detail of the mesh of the model close to the excavation face (the green zone refers to the 3 elements used to simulate the concrete lining). The comparison between the simulated extrusion values and the ones obtained from the spherical method facilitates a quantitative evaluation of the simulation's accuracy. If the results align closely, it provides confidence in the simulation model's ability to accurately represent the behaviour of the studied phenomenon. By incorporating this validation step, the researchers can ensure that the simulation results are trustworthy and can be used with confidence for further analysis and decision-making. It helps to establish the credibility and reliability of the simulation approach employed in the study. The comparison is performed for tunnels with radii of 2, 3.5, and 5 m with different values of lining thickness and in situ stress state, respectively. Figure 4: Vertical stresses in the rock (in front of the excavation face and at the contour of the excavated profile), for the case of a tunnel with a radius of 2 m, thickness of the lining 20 cm, \(P_{0}\) = 11 MPa, excavated in a Good rock mass (type C). Figure 5: Horizontal displacements (extrusion) in the rock ahead of the excavation face, for the case of a tunnel with a radius of 2 m, thickness of the lining 20 cm, \(P_{0}\) = 11 MPa, excavated in a Good rock mass (type C). Figure 8: Comparison of the tunnel face extrusion in Poor rock (type A): detailed numerical model vs. simplified numerical models (with the decreased value of the elastic modulus 1/5-E), varying the lining thickness S (10, 20, 30 cm). Figure 6: Comparison of the Tunnel Face Extrusion in Good rock (type C): Detailed Numerical Model vs. Simplified Numerical Models (with the original E value), varying the lining Thickness S (10, 20, 30 cm). Figure 7: Comparison of the tunnel face extrusion in Intermediate rock (type B): detailed numerical model vs. simplified numerical models (with the original E value), varying the lining thickness S (10, 20, 30 cm). The spherical method, which does not take into account the thickness of the lining, provides a single value for face extrusion. In contrast, the simulation results consider different lining thicknesses. It is worth noting that the face extrusion values obtained from the spherical method closely align with the average extrusion values observed for linings with thicknesses of 10, 20, and 30 cm. It is important to note that the results presented in this section pertain to a specific scenario where the face pressure is assumed to be zero. The purpose of this simulation was to replicate a mechanized excavation process, where the face pressure is not a significant factor. From the analysis of the figures, it is possible to note how the type of rock and the in situ stress have an important effect on the extent of extrusion of the excavation face. More specifically, for a \(P_{0}\) of 11 MPa (corresponding to a tunnel depth of 425-475 m) and poor characteristics of the rock mass (type A), a clear increase in extrusion is seen, denoting an intense plasticization of the of excavation face. This aspect is detectable for all three analysed tunnel radii.