# Failure in the Tension Zone around a Circular Tunnel Excavated in Saturated Porous Rock
## 4 Rock Failure in the Tension Zone during Tunnel Excavation: Results and Discussion
The radius of the tunnel is \(R\) = 2 m and the radial dimension \(r\) = 20 m. The simulations are set up with the data of Case A (Table 2). The excess pore pressure in the elastic (Equation (8)) and elastoplastic field generated by the excavation of the tunnel, is calculated with the initial pore pressure \(u_{ini}\) = 3 MPa (hydrostatic). The elastic constants, estimated from the available data [36], are \(E_{u}\) = 10 GPa  
Figure 7: (**a**) Variation of \(u_{lim}\) with \(\delta\) in all the Cases reported in Table 3; (**b**) Variation of \(u_{lim}\) with \(\delta\) in Case 2A and Case 3A at \(r\) = \(R\). Comparison with the initial pore pressure \(u_{ini}\) and the pore pressure in undrained condition \(\Delta u\) = 0 and \(\Delta u\
eq\) 0 (calculated with FLAC in Case 3A). \(v_{u}=0.33\). The strength parameters, used in the elastoplastic simulation, are \(m_{t}=11.6\)\(\sigma_{c}=36.6\) MPa \(\sigma_{t}=-3.1\) MPa (Case 3A). Figure 8 shows the results of the elastic and elastoplastic numerical models: in both cases, the average pore pressure calculated in the zones of the grid increases in the horizontal direction and decreases in the vertical direction, as expected. Figure 9a shows that the maximum pore pressure (with a positive excess pore pressure) at \(\vartheta=0^{\circ}\) is \(u\approx 4\) MPa in the elastic case (at \(r=R\)). Figure 9b shows the comparison between the trend of the pore pressures in the elastic and elastoplastic field at 30 cm from the wall. The occurrence of plasticity and the stress redistribution determine a pore pressure \(u\approx 3.2\) MPa at \(r=R\). The maximum pore pressure in the elastoplastic case is \(u\approx 3.9\) MPa and occurs at \(r=2.15\) m. The averaged pore pressure in 15 cm (\(2.00\leq r\leq 2.15\) m) is \(u=3.55\) MPa. The pore pressure at a radial distance \(r\geq 2.20\) m coincides with the elastic and elastoplastic solutions and the effect of the excavation on pore pressure disappears from a radial distance \(r\geq 12.5\) m. On the other hand, the pore pressure, at \(\vartheta=90^{\circ}\) and  
\(r=R\), is \(u=2.2\) MPa in both cases (negative excess pore pressure), because plasticity does not occur. A comparison between these pore pressures calculated with FLAC and \(u_{lim}\) is shown in Figure 7b. At \(\vartheta=0^{\circ}\) the total pore pressure is the averaged \(u=3.55\) MPa. The figure shows that in the azimuth range \(\vartheta=0^{\circ}\)-\(45^{\circ}\), there is the occurrence of the limit condition in Case 3A. In Case 2A, plasticity does not occur, and \(u_{lim}\) is always above \(u_{undrained}\). Figure 10 shows the hydro-mechanical response of the rock at \(r\geq R\) calculated with FLAC: the occurrence of the tension zone (red) around the tunnel and tensile failure (yellow symbols) when \(\Delta u=0\) and \(\Delta u\
eq 0\). Figure 10a refers to \(\Delta u=0\) and shows that tensile failure occurs from \(\vartheta=0^{\circ}\) to \(\vartheta=90^{\circ}\), in agreement with the analytical results (Figure 7b), and a thin plastic zone is generated. The tension zone is more extended in the vertical direction (\(\vartheta=90^{\circ}\)). Figure 10b refers to \(\Delta u\
eq 0\) and shows that the tension zone is more extended in the horizontal direction and the generation of a plastic zone from \(\vartheta=0^{\circ}\) to \(\vartheta=45^{\circ}\), with a thickness of about 15 cm at \(\vartheta=0^{\circ}\). Figure 11 shows the trend of the radial effective stresses with the radial distance \(r\geq R\) at different azimuths, when \(\Delta u=0\) and \(\Delta u\
eq 0\), calculated with FLAC. The numerical simulations indicate that, in both undrained conditions, the thickness of the tension zone varies from 0.80 m to 1 m. Figure 11a shows that the worst situation is at \(\vartheta=90^{\circ}\) when \(\Delta u=0\), because the total radial stress increases slowly at this azimuth.