## 2 Modifications for Hoek-Brown Yield Criterion and the Pseudo-Dynamic Method on Seismicity Analysis
### The 3D Horn-Shaped Rotating Failure Mechanism  
The limit analysis method (LA) requires that the assumed failure mechanism ought to meet the kinematically admissible conditions; specifically, the sliding surface of the mechanism must be continuous, i.e., without interruption points. In 2009, the 3D horn-shaped rotating failure mechanism was designed by [PERSON] and [PERSON] based on the 2D log-helix failure mechanism [18], in which the derived safety factor of 3D soil slopes is quite close to the engineering reality [45; 46]. Subsequently, the suitability of the above-mentioned mechanism for rock slopes was extensively discussed and demonstrated when the nonlinear strength criterion was converted to the linear, employing the generalized tangent technique. Therefore, the 3D rotating failure mechanism is adopted in the article for analyzing the safety of 3D rock slopes subjected to seismicity reinforced with prestressed anchor cables.  
As illustrated in Figure 2, the 3D rock slope disturbed by the seismicity is reinforced by adding a step at the appropriate vertical height of the slope and installing prestressed anchor cables on the inclined surface of the slope, in which the width CD of the added step is set to \(a\). Additionally, it can be seen that the 3D rotating failure mechanism is a horn-shaped cone with vertex angle \(2\varphi_{t}\), formed by radiating two log-spiral curves at the vertex O\({}^{\prime}\) and taking the ray segment from the origin O intercepted by these two log-helix curves as the diameter of the cone section, where the lower spiral curve passes precisely through the toe of the two-stage slope. The mathematical expressions for the upper and lower log-spiral curves can be given separately:  
\[r=r_{0}e^{(\theta-\theta_{0})\tan\varphi_{t}} \tag{17}\]  
and  
\[r^{\prime}=r^{\prime}{}_{0}e^{-(\theta-\theta_{0})\tan\varphi_{t}} \tag{18}\]  
where \(r\) and \(r^{\prime}\) are the polar diameters of the upper and lower log-spiral curves at any polar angle, respectively. As an illustration, \(r_{0}\) and \(r_{0}{}^{\prime}\) are the polar diameters (OA and OA\({}^{\prime}\)) of the upper and lower log-spiral curves at polar angle \(\theta_{0}\). According to Equations (15) and (16), the expression for the curve formed by the successive centers of the circular section of the horn-shaped cone at any polar angle can be deduced as  
\[r_{m}=\left(r+r^{\prime}\right)/2=r_{0}f_{1} \tag{19}\]  
The radius of the circular section for the horn-shaped cone at any polar angle can be similarly shown as  
\[R=\left(r-r^{\prime}\right)/2=r_{0}f_{2} \tag{20}\]  
where \(f1\), \(f2\) can be expressed as  
\[f_{1}=\frac{1}{2}\bigg{[}e^{(\theta-\theta_{0})\tan\varphi_{t}}+\frac{r^{ \prime}_{0}}{r_{0}}e^{-(\theta-\theta_{0})\tan\varphi_{t}}\bigg{]} \tag{21}\]  
\[f_{2}=\frac{1}{2}\bigg{[}e^{(\theta-\theta_{0})\tan\varphi_{t}}-\frac{r^{ \prime}_{0}}{r_{0}}e^{-(\theta-\theta_{0})\tan\varphi_{t}}\bigg{]} \tag{22}\]  
In addition, it can be seen that the overall height of the 3D two-stage rock slope is \(H\), and the heights of the upper and lower steps are \(\xi_{1}H\) and \(\xi_{2}H\), respectively, in which \(\xi_{1}\) and \(\xi_{2}\) are the height coefficients. Obviously,  
\[\xi_{1}+\xi_{2}=1 \tag{23}\]  
From the stereoscopic perspective, as shown in Figure 3, the 3D failure mechanism for the two-stage rock slope is inserted in a plane strain block with an arbitrary adjustable length, and with the ever-increasing length \(b\) of the block, the 3D slope will progressively change to the 2D case. Apparently, the following correlations exist between the width of the embedded plane strain block \(b\), the maximum width of the 3D portion \(B^{\prime}{}_{\text{max}}\), and the overall slope width \(B\):  
\[b=B-B^{\prime}{}_{\text{max}} \tag{24}\]  
The introduction of the plane strain block allows a comparison with the data results for the 2D slope condition in the subsequent sections, providing the design and maintenance basis for diverse slope engineering sites.