## 3 Orthogonal Test Scheme and Numerical Model
### The Principle of RFPA3D  
A finite element software, RFPA3D (three-dimensional realistic failure process analysis), was applied for the numerical simulation. RFPA3D is an effective method based on the damage mechanics and the statistical theory to simulate the gradual failure process of rock mass. It is worth mentioning that the mechanical parameters (such as the strength and elastic modulus) of the model elements obey the Weibull distribution. Therefore, the distribution characteristics of a rock's mechanical parameters at the mesoscopic level can be considered in the RFPA3D. Additionally, the macroscopic failure is the accumulation process of mesoscopic element failure. For detailed information on the calculational principles and model validity of the RFPA3D software, please refer to papers [24; 30; 31]. The constitutive equations of the RFPA3D software are briefly summarized in this section. The elastic-brittle damage constitutive model was introduced into the RFPA3D to describe the stress-strain relationship. The constitutive relationship of the model is expressed as follows:  
\[\sigma=E\varepsilon=E_{0}(1-D)\varepsilon \tag{1}\]  
where \(\sigma\) and \(\varepsilon\) are the stress applied to the model and the corresponding strain, respectively; \(E\) represents the elastic modulus of the damaged element; and \(E_{0}\) and \(D\) denote the initial elastic modulus of the element and the damage variable, respectively. However, with increasing load, the elastic modulus of the element degrades gradually with the increase of the \(D\). When the model element is in a uniaxial tension state, the maximum tensile strain criterion and the Mohr Coulomb criterion were used in the RFPA3D. The corresponding \(D\) was defined as in Equation (2). \[D= \begin{cases}0&\varepsilon>\varepsilon_{to}\\ 1-\frac{\varepsilon_{to}}{\varepsilon E_{0}}&\varepsilon_{to}\geq \varepsilon\geq\varepsilon_{tu}\\ 1&\varepsilon<\varepsilon_{tu}\end{cases}  \tag{2}\]  
where \(\sigma_{\text{rt}}\) denotes the residual stress of the model element, and \(\sigma_{\text{rt}}=\lambda\sigma_{\text{t}}\). Among them, \(\sigma_{\text{t}}\) and \(\lambda\) are the tension stress applied on the model and the residual coefficient, respectively; \(\varepsilon_{to}\) is the failure strain threshold, which is the tensile strain corresponding to the elastic limit; \(\varepsilon_{tu}\) is the ultimate tensile strain coefficient, and \(\varepsilon_{tu}=\eta\varepsilon_{to}\); \(\eta\) denotes the ultimate strain coefficient. According to the elastic damage theory, the stress-strain relationship can be derived as follows:  
\[\sigma_{ij}= \begin{cases}2G\varepsilon_{ij}+\lambda\delta_{ij}\varepsilon_{kk }&\overline{\varepsilon}>\varepsilon_{to}\\ \frac{\varepsilon_{rt}}{E_{0}}(2G\varepsilon_{ij}+\lambda\delta_{ij} \varepsilon_{kk})&=\frac{\varepsilon_{rt}}{\overline{\varepsilon}}\Big{(} \frac{\varepsilon_{ij}}{1+\overline{\varepsilon}}+\frac{\delta_{ij} \varepsilon_{kk}}{(1+\overline{\varepsilon})(1-2\overline{\varepsilon})} \Big{)}\,\varepsilon_{to}\geq\overline{\varepsilon}\geq\varepsilon_{tu}\\ 0&\overline{\varepsilon}<\varepsilon_{tu}\end{cases}  \tag{3}\]  
where \(G\) and \(\
u\) are the shear modulus and the Poisson's ratio of the model element, respectively; when \(i=j,\delta_{ij}=1\); otherwise, \(\delta_{ij}=0\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & \multicolumn{4}{c}{**Influencing Factors**} \\ \cline{2-5}  & **Inclination/\({}^{c}\)** & **Dip Angle/\({}^{c}\)** & **Trace Length/m** & **Bulk Density/m\({}^{-3}\)** \\ \hline T-4 & Level 2 (94.49) & Level 1 & Level 2 & Level 3 \\ T-5 & Level 2 & Level 2 & Level 3 & Level 1 \\ T-6 & Level 2 & Level 3 & Level 1 & Level 2 \\ T-7 & Level 3 (141.73) & Level 1 & Level 3 & Level 2 \\ T-8 & Level 3 & Level 2 & Level 1 & Level 3 \\ T-9 & Level 3 & Level 3 & Level 2 & Level 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Moreover, the Mohr-Coulomb criterion is applied when the model element is in a uniaxial compression state.