## 2 MHB Failure Criterion  
The Mohr-Coulomb failure criterion is a linear criterion that requires shear strength parameters cohesion \(c\) and frictional angle \(\varphi\) to evaluate the stability problems. Due to its simplicity, it has been widely used to investigate the stability of various geotechnical engineering for many decades [14]. However, many previous studies show that the strength of rock mass is a non-linear stress function. Therefore, the Mohr-Coulomb failure criterion could not agree well with the rock mass failure envelope, especially for the stability problems of rock slope where the rock mass is in a state of low confining stresses, making the non-linear feature more prominent.  
An alternative failure criterion for the stability problem of rock mass is the MHB failure criterion involving the fitting of curves to the triaxial strength tests results of rock mass [15]. Since the MHB failure criterion was introduced, it has been widely adopted to analyze the stability of engineering structures, e.g., slope stability, strip footing stability, and tunnel stability [16; 17; 18]. The MHB failure criterion can be described as follows:  
\[\sigma_{1}=\sigma_{3}+\sigma_{ci}\bigg{(}m\frac{\sigma_{3}}{\sigma_{1}}+s \bigg{)}^{a}, \tag{1}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum effective principal stresses, respectively; \(\sigma_{ci}\) is the uniaxial compressive stress of the intact rock. Besides, the value of \(s\), \(a\), and \(m\) can be calculated by the following equations:  
\[m=m_{i}\exp\bigg{(}\frac{GSI-100}{28-14D_{i}}\bigg{)}, \tag{2}\]  
\[s=\exp\bigg{(}\frac{GSI-100}{9-3D_{i}}\bigg{)}, \tag{3}\]  
\[a=\frac{1}{2}+\frac{1}{6}\bigg{[}\exp\bigg{(}-\frac{GSI}{15}\bigg{)}-\exp \bigg{(}-\frac{20}{3}\bigg{)}\bigg{]}, \tag{4}\]  
where \(m_{i}\) denotes the Hoek-Brown constant; \(GSI\) denotes the geological strength index; \(D_{i}\) denotes the disturbance factor of the intact rock. Hence, for the MHB failure criterion, there is a total of four input parameters, namely, \(\sigma_{ci}\), \(GSI\), \(m_{i}\), and \(D_{i}\). These four material parameters for the MHB failure criterion could be obtained directly from [PERSON] [19].  
The generalized tangential technique conducted by [PERSON] et al. was adopted to transform the nonlinear criterion into a linear one to estimate the tunnel stability by the MHBfailure criterion in this study, shown in Figure 1[16]. It can be concluded that the tangential line technique could provide an upper-bound solution for the stability issues due to the tangential line exceeding the original non-linear envelope. The tangential line at point \(M\) could be expressed as the following classic equation:  
\[\tau=c_{t}+\sigma_{n}tan\varphi_{t}, \tag{5}\]  
where \(c_{t}\) and \(\varphi_{t}\) are the tangential cohesion and tangential friction angle, respectively; \(c_{t}\) can be calculated as follows:  
\[\frac{c_{t}}{\sigma_{c}}=\frac{\cos\varphi_{t}}{2}\left[\frac{ma(1-\sin\varphi _{t})}{2\sin\varphi_{t}}\right]^{\frac{\varphi_{t}}{1-a}}-\frac{\tan\varphi_{t }}{m}\left(1+\frac{\sin\varphi_{t}}{a}\right)\left[\frac{ma(1-\sin\varphi_{t}) }{2\sin\varphi_{t}}\right]^{\frac{\varphi_{t}}{1-a}}+\frac{s}{m}\tan\varphi_{t}, \tag{6}\]  
In Equation (6), the tangential cohesion \(c_{t}\) is the mutative parameter with the change of internal friction angle \(\varphi_{t}\).