# Analysis of Deformations of the Tunnel Excavation Face via Simplified Calculation Methods
## 3 Axisymmetric Numerical Modelling of the Excavation Face
The total number of quadrilateral elements used for each considered geometry is shown in Table 1.  
Each of the numerical models foresaw simulations of both excavation steps (each advancing step with a length of 1.2 m along the tunnel axis) and support installation, in order to be able to reproduce as faithfully as possible the real conditions during the tunnel construction: 17 advancement steps were implemented. The simulation began at the top border of the model and continued until it reached the centre. The excavation was simulated by simply removing elements in the tunnel area, while the support installation was considered by reactivating elements in the zone occupied by the support structure, attributing the mechanical properties of concrete. These elements were deemed to be in an initial state of zero stress when reactivated. For each studied case, the stresses and strains within the model were evaluated (in Figures 4 and 5, there is an example of the obtained results, with reference to the case of a tunnel with a radius of 2 m, lining thickness of 20 cm, in situ stress of 11 MPa, excavated in a Good Rock (type C). Another simplified numerical model was adopted in the study, maintaining the same geometries and rock types considered for the detailed model. The simplified model adopts a quick single-step simulation instead of a step-by-step approach. In the simplified model, the values of the rock elastic modulus E was varied (from the original value to 1/2, 1/3, 1/4, and 1/5 of it) so as to be able to verify whether an artificial modification of the rock stiffness is necessary in case one wishes to adopt simplified numerical modelling which does not reproduce the exact succession of the construction phases. After the analysis of the calculation results (Figures 6-8), it is clear that in the case of Good and Intermediate rock types, the face extrusion values obtained via the simplified model agree with those of the detailed model, maintaining the original value of the elastic modulus of the rock mass. However, in the context of poor mechanical properties (Type A of the rock mass), utilizing an elastic modulus of 1/5-E leads to face extrusion results of the simplified model better fitting those obtained from the calculation using the detailed model. Thanks to the comparison developed on the calculation results, it is therefore possible to state that the simplified axisymmetric model (one step approach) is able to analyze the mechanical and deformation behaviour of the excavation face in order to evaluate its stability conditions. From the conducted analyses, it was possible to detect how the simplified numerical modelling (one step approach) is able to correctly represent the deformation conditions of the excavation face in deep tunnels excavated in weak rocks, with the sole precaution of applying an artificial reduction in the elastic modulus of the rock mass at 1/5 of its real value, only when the mechanical characteristics are poor (i.e., type A of the rock mass). \begin{table}
\begin{tabular}{c c c c} \hline
**Radius** & **In the X Direction** & **In the Y Direction** & **Total Number of Quadrilateral Elements** \\ \hline
2.0 m & 65 & 100 & 6500 \\
3.5 m & 115 & 176 & 20,240 \\
5.0 m & 163 & 250 & 40,750 \\ \hline \end{tabular}
\end{table} 
Table 1: Geometrical characteristics of the numerical model, varying the tunnel radius.