# 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 figure shows the normalized strength envelopes and the effective state of stress of the correspondent 4 cases of Figure 5a. The two filled symbols lying on each envelope correspond to the two values of \(S\) at \(\theta=0^{\circ}\) (Case A, S = 16 MPa and Case B, S = 8 MPa). For any strength envelope, the symbols with the lower values of \(\sigma^{\prime}\)/\(\sigma_{c}\) correspond to lower values of \(X\). This indicates that low values of \(X\) can induce tensile failure close to the uniaxial tensile strength, in agreement with Equation (12). Furthermore, the assumption \(m_{c}=m_{t}\) overestimates the tensile strength of the rock in the tension zone \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(z\) & \(\sigma_{v}\) & \(\sigma_{B}\) & \(S\) at \(\theta=0^{\circ}\) & \(S\) at \(\theta=90^{\circ}\) \\  & **(m)** & **(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline \multirow{2}{*}{Case A} & \multirow{2}{*}{300} & \multirow{2}{*}{7.00} & \multirow{2}{*}{5.00} & 16 & \multirow{2}{*}{8} \\  & & & & & 65 \\ \hline Case B & 150 & 3.50 & 2.45 & 8 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stresses used in the analysis of \(u_{lim}\). The factor \(K=0.7\) in both cases. Figure 5: (**a**) Variation of \(u_{lim}\) with \(S\) (Table 2) calculated at \(r=R\) with different couples of \(m\) and \(\sigma_{c}\) found in the data fitting (Figure 2 and Table 1). Each \(u_{lim}\) is normalized by the correspondent values of \(\sigma_{c}\); (**b**) Variation of \(u_{lim}\) with \(m=5.6\)–\(11.6\) at fixed \(\sigma_{c}\) in Case A at \(\theta=0^{\circ}\) (\(S=16\) MPa). of \(u_{lim}\) between Case A and Case B decreases with increasing \(\theta\), because \(S\) reduces with \(\theta\). An analysis of the data reported in Table 4 demonstrates that at this depth (\(z\) = 150 m) failure in the tension zone is not expected to occur; because \(u_{lim}\) is very high compared with the hydrostatic pore pressure (1.5 MPa) and a very high excess pore pressure cannot be practically generated. This outcome indicates that, at shallow depths, failure in the tension zone should not occur with moderately weak rocks and geostatic stresses. A complete analysis of the Cases reported in Table 3 is shown in Figure 7a. The difference between these Cases can be interpreted as percentages of the \(u_{lim}\) calculated with the strength parameters of the tension zone: \(\sigma_{c}\) = 36.6 MPa and \(m_{t}\) = 11.6 (Case 3A). The highest increase is 82% and occurs in Case 2A, which refers to the strength parameters of the compression zone: \(\sigma_{c}\) = 42.2 MPa and \(m_{c}\) = 7.8. An increase of 49% occurs in Case 3A*. This outcome indicates the influence of \(m\) on the magnitude of \(u_{lim}\). Finally, an increase of 22% occurs in Case 2A*, indicating the influence of \(\sigma_{c}\) on the magnitude of \(u_{lim}\). The two fitted \(\sigma_{c}\) (36.6 MPa and 42.2 MPa) differ by 15%, which is a usual range of variability. A comparison between the hydrostatic pore pressure (Case A with z \(\approx\) 300 m) \(u_{ini}\) = 3 MPa and the limit pore pressures of Case 2A and Case 3A is shown in Figure 7b. If \(\Delta u\) = 0 the undrained pore pressure is equal to the initial one \(u_{unthin}\) = \(u_{ini}\) = 3 MPa. The figure shows that \(u_{lim}>u_{ini}\) in Case 2A (black line), indicating that failure does not occur. In Case 3A (green line) \(u_{lim}\leq u_{ini}\) = 3 MPa which indicates that at all azimuths failure in the tension zone can occur. The previous analyses refer to the boundary of the excavation \(r\) = \(R\) and \(\Delta u\) = 0. The generation of excess pore pressures \(\Delta u\
eq\) 0 and the hydro-mechanical response of the rock at \(r\) > \(R\) (with \(\Delta u\) = 0 and \(\Delta u\
eq\) 0) are calculated with FLAC. Given the symmetry of the tunnel with respect to its axis and assuming isotropic behavior of the rock, the numerical model represents one-quarter of the full section of a tunnel and the surrounding rock, with a radial grid with 60 \(\times\) 60 elements.