# Analysis of Deformations of the Tunnel Excavation Face via Simplified Calculation Methods
## 3 Axisymmetric Numerical Modelling of the Excavation Face  
A detailed numerical model was developed using the two-dimensional explicit finite difference code FLAC2D 8.1 in the axisymmetric configuration. The model was used to analyse the stress and strain conditions in the ground during the construction phases of a circular tunnel supported by a concrete lining. An homogeneous and hydrostatic stress state was assumed, where the initial state of stress was kept constant in the model regardless of the direction. The boundary conditions of the model involve the application of an in situ stress equal to \(P_{0}\) on all borders, except on the axis of axial symmetry of the tunnel (y axis in Figure 3). The following further assumptions were made:  
The chosen criterion for the ground failure is the Mohr-Coulomb one, which is commonly used for weaker rocks that exhibit less curvature in their strength criterium. The post-failure behaviour of the rock was assumed to be ideal elastic-plastic, with residual strength parameters equal to the peak ones. The elastic modulus was assumed to be constant in both the elastic and plastic phases. The dilatancy angle, denoted as \(\psi\), was assumed to be zero (the minimum value that can be reached). This means that the strain behaviour in the plastic zone is characterized by constant volume plastic strains, following the approach described by [27; 28; 29] for deep rocks with weak mechanical properties. The concrete lining was assumed to have a constant elastic modulus of 30,000 MPa and Poisson's ratio of 0.15, which are typical values for concrete materials. A parametric analysis was developed considering three types of weak rocks, where problems regarding the stability of the excavation face can be encountered: rock with poor mechanical properties (Type A), medium mechanical properties (Type B), and good mechanical properties (Type C). Type A had a cohesion (\(c_{p}=c_{r}\)) of 0.3 MPa, friction angle (\(\Phi_{p}=\Phi_{r}\)) of 25\({}^{\circ}\), and an elastic modulus E of 4000 MPa. Type B with a cohesion (\(c_{p}=c_{r}\)) of 0.9 MPa, friction angle (\(\Phi_{p}=\Phi_{r}\)) of 30\({}^{\circ}\), and an elastic modulus E of 8000 MPa. Type C with a cohesion (\(c_{p}=c_{r}\)) of 1.5 MPa, friction angle (\(\Phi_{p}=\Phi_{r}\)) of 35\({}^{\circ}\), and an elastic modulus E of 12,000 MPa. The Poisson's ratio was considered 0.3 for all three studied rock types. A total of 27 numerical models were developed for each rock type, considering different combinations obtained by changing the following geometrical parameters:  
Tunnel radius, \(R\) (2, 3.5, and 5 m);  
Thickness of the concrete lining (0.1, 0.2, and 0.3 m);  
In situ vertical stress (\(P_{0}\)) (1.1, 5.5, and 11 MPa). The purpose of this analysis was to analyse as many instances as possible of those that may occur during the construction of medium-to-large diameter horizontal tunnels in weak rock formations at different depths from the surface. Figure 3 illustrates the scheme of the developed axisymmetric numerical model: the bottom and top of the model are constrained in the Y direction and the in situ pressure is applied on the right side. The tunnel axis coincides with the Y axis, and also, only one half of the tunnel and of the surrounding ground is simulated in the model thanks to the symmetry of the problem. The simulation of the tunnel advancement and the lining construction starts on the top of the model towards its centre, where the stress and strain state is analysed. The size of the model varies with the radius of the simulated tunnel: in the radial direction, it goes from 65 m (case of tunnel with radius 2m) to 163 m (tunnel with radius 5 m); in the longitudinal one from 100 m (radius 2 m) to 250 m (radius 5 m). The number of quadrilateral elements used to simulate the excavated zone is 10. Quadrilateral elements are also adopted to represent both the concrete lining and the rock surrounding the tunnel.