## 2 Modifications for Hoek-Brown Yield Criterion and the Pseudo-Dynamic Method on Seismicity Analysis
### Internal Energy Dissipation  
When the 3D two-stage rock slope is in the critical failure state, the deformation of the rock body \(D_{V}\) and the sliding along the failure surface \(S_{t}\) between the slope body and the rock base \(D_{t}\) are the primary contributors of the energy dissipation inside the slope to retain its original condition. According to the theory of plastic limit analysis, \(D_{V}\) and \(D_{t}\) can be deduced, individually, as  
\[D_{V}=\iiint_{V}c_{t}\cos\varphi_{t}(\dot{e}_{1}-\dot{e}_{2}-\dot{e}_{3})\,\mathrm{ d}V \tag{25}\]  
and  
\[D_{t}=\iint_{S_{t}}c_{t}v\cos\varphi_{t}\mathrm{d}S_{t} \tag{26}\]  
\(v\) and \(V\) are velocity at the arbitrary micro-element and the deformation volume generated inside the 3D stepped slope failure mechanism, respectively, in which \(v\) can be calculated as  
\[v=(r_{m}+y)\omega \tag{27}\]  
In the above equation, \(y\) represents the value that corresponds to the coordinates of the \(Y\)-axis in the regional Cartesian system, which is established on any circular slice of the horn-shaped cone mechanism and originates at the center of the cone section, as depicted  
Figure 3: Stereoscopic diagram for 3D failure mechanism with the plane strain block and anchor group reinforcement. Figure 2: Three-dimensional horn-shaped rotating model for stepped rock slope reinforced with anchor groups. in Figure 4. The total internal energy dissipation could be stated as follows, based on the divergence theorem of [PERSON]:  
\[ \begin{array}{l}D=D_{t}+D_{V}\\ =c_{t}\cot\varphi_{t}\big{[}\beta_{S_{r}}\mathbf{v}\cdot\mathbf{nd}S_{r}\end{array}  \tag{28}\]  
in which \(S_{r}\) represents the residual surface section for the kinematically admissible model after reducing the 3D sliding surface. \(\mathbf{v}\) denotes the speed vector on \(S_{r}\), \(\mathbf{n}\) represents the unit normal vector extrversely, and \(\mathbf{v}\cdot\mathbf{n}\) is their dot product. According to Figure 3, the internal energy dissipation of the two-stage rock slope is separated into two parts, namely the 3D section and the 2D insert section. Likewise, the internal energy dissipation along the outer contour for the two-stage rock slope could be separated into four parts, namely \(D_{\text{AB}}\), \(D_{\text{BC}}\), \(D_{\text{CD}}\), and \(D_{\text{DE}}\). In accordance with Equation (28), the 3D section can be sequentially derived as  
\[D_{\text{AB-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{0}}^{\theta_{\text{ B}}}\int_{0}^{x_{1}^{*}}r_{0}^{2}\sin^{2}\theta_{0}\frac{\cos\theta}{\sin^{3} \theta}\text{d}x\text{d}\theta \tag{29}\]  
\[D_{\text{BC-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{\text{B}}}^{\theta _{\text{C}}}\int_{0}^{x_{2}^{*}}r_{0}^{2}\sin^{2}(\theta_{\text{B}}+\beta_{1} )\frac{\sin^{2}\theta_{0}}{\sin^{2}\theta_{\text{B}}}\frac{\cos(\theta+\beta_ {1})}{\sin^{3}(\theta+\beta_{1})}\text{d}x\text{d}\theta \tag{30}\]  
\[D_{\text{CD-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{\text{C}}}^{\theta _{\text{D}}}\int_{0}^{x_{3}^{*}}(r_{0}\sin\theta_{0}+\xi_{1}H)^{2}\frac{\cos \theta}{\sin^{3}\theta}\text{d}x\text{d}\theta \tag{31}\]  
\[D_{\text{DE-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{\text{D}}}^{\theta _{\text{B}}}\int_{0}^{x_{4}^{*}}r_{0}^{2}e^{2(\theta_{\text{h}}-\theta_{0}) \tan\varphi_{t}}\sin^{2}(\theta_{\text{h}}+\beta_{2})\frac{\cos(\theta+\beta_ {2})}{\sin^{3}(\theta+\beta_{2})}\text{d}x\text{d}\theta \tag{32}\]  
The second integral's upper bound for integration \(x_{i}^{*}(i=1,2,3,4)\) in the dual integration formulas above can be expressed as \(x_{i}^{*}=\sqrt{R^{2}-d_{i}^{2}}\).