## 3 Results
### Failure Criteria  
For a rock mass, the interface attitude greatly influences the strength. The relationship between the interface strength and the rock mass strength can be obtained based on the strength envelope of the Mohr-Coulomb strength criterion [17]. In this research, the formulas are derived based on the plane strain problem. Although the physical model is 3D, it has the same mechanic behavior along the z direction, which can be equal to the plane strain condition. According to the geometrical relationship in Figure 10, the following expressions can be obtained:  
\[\left\{ \begin{array}{l}\alpha_{1}=\frac{\varphi_{i}}{2}+\frac{1}{2}\sin^{-1 }\left[\frac{2\chi_{i}\cos\varphi_{i}+(\varphi_{1}+\varphi_{3})\sin\varphi_{i}} {\sigma_{1}-\sigma_{3}}\right]\\ \\ \alpha_{2}=\frac{\varphi_{i}}{2}-\frac{1}{2}\sin^{-1}\left[\frac{2\chi_{i}\cos \varphi_{i}+(\varphi_{1}+\sigma_{3})\sin\varphi_{i}}{\sigma_{1}-\sigma_{3}} \right]+\frac{\pi}{2}\end{array} \right. \tag{1}\]  
where \(\varphi_{i}\) is the internal friction angle of the interface, and because the interface was flat in this study, \(\varphi_{i}=36^{\circ}\) was determined by a tilt test; \(c_{i}\) is the cohesion of the interface, and \(c_{i}=780\) kPa was obtained by a direct shear test. When \(\sigma_{3}=0\) and \(\sigma_{1}=\sigma_{c}\), the UCS of the inact rock mass was 19.236 MPa. From this calculation, \(\alpha_{1}=38^{\circ}\) and \(\alpha_{2}=88^{\circ}\) can be obtained. When the dip angle \(\alpha_{1}<\alpha<\alpha_{2}\), the structural planes slip. When \(\alpha<\alpha_{1}\), and \(\alpha>\alpha_{2}\), the structural planes do not slip. In this test, the structural plane of the samples of \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), and \(90^{\circ}\) does not encounter slide deformation, while the structural plane of \(45^{\circ}\), \(60^{\circ}\), and \(75^{\circ}\) does encounter slide deformation. This study investigated the effects of the different orientations of the contact faces on the chamber, and two models are proposed: a complete rock mass model with an interface and a chamber arch model. These two models are simplified as Problem 1 and Problem 2, respectively (Figure 11). Problem 1:  
In this study, two different interface forms appeared: the through-through interface (\(0^{\circ}\)-\(45^{\circ}\)) and the non-penetrating interface (\(60^{\circ}\)-\(90^{\circ}\)) (Figure 12). Figure 11: Calculation model. Figure 12: Failure modes. (**a**) regards the sample with below \(45^{\circ}\) dip angle of structural plane, (**b**) regards the sample with above \(45^{\circ}\) dip angle of structural plane. Figure 10: Strength envelope. For failure mode (a), the strength is expressed as follows:  
\[\sigma_{1m}=\left\{ \begin{array}{ll}\sigma_{3}.\frac{1+\sin\varphi}{(1-\sin \varphi)}\cos\alpha+c\cdot\frac{2\cos\varphi}{(1-\sin\varphi)}\cos\alpha+c_{i} \frac{1}{\tan\varphi_{i}}&(\alpha<\alpha_{1})\\ \sigma_{3}.\frac{\sin(2\alpha-\varphi_{i})+\sin\varphi_{i}}{\sin(2\alpha- \varphi_{i})-\sin\varphi_{i}}+c_{i}\cdot\frac{2\cos\varphi_{i}}{\sin(2\alpha- \varphi_{i})-\sin\varphi_{i}}&(\alpha_{1}<\alpha<\alpha_{2})\\ \sigma_{3}.\frac{1-\mu}{\mu}+\sigma_{c}&(\alpha<\alpha_{2})\end{array} \right. \tag{2}\]  
where \(\varphi\) is the internal friction angle of the rock (\(38^{\circ}\)); \(c\) is the cohesion of rock (4 MPa); and \(\mu\) is Poisson's ratio of the rock (0.25). For failure mode (b), the strength is expressed as follows:  
\[\sigma_{1m}=\frac{2a}{2a+b}\sigma_{1}+\frac{b}{2a+b}\sigma_{1i} \tag{3}\]  
where \(\sigma_{1i}\) is the strength of the structural plane. Problem 2:  
The ability of the tunnel model with an interface to resist the compression exerted by the arch effect, and its UCS, were lower than those of the complete model, and the UCS was variable (Figure 13). Three assumptions were made: (1) the surrounding rock material is a homogeneous elastomer; (2) the stress distribution of the surrounding rock is uniform; and (3) the stress is uniformly distributed in the plastic zone.