## 2 Modifications for Hoek-Brown Yield Criterion and the Pseudo-Dynamic Method on Seismicity Analysis
### External Force Work
#### 3.3.2 Seismic Force Work  
Seismicity is the other critical factor of slope instability. The seismic force can be decomposed into a horizontal seismic force and a vertical seismic force, where the vertical seismic force has much less impact on slope stability than the horizontal seismic force at \(k_{\mathrm{c}}\leq 0.5k_{h}\), according to the results of [PERSON] et al. [44], which can be neglected to simplify the computational model. In the later paper, the horizontal stratification method is employed to analyze the effect of the horizontal seismic force on slope stability, since the seismic acceleration stays invariant in the arbitrary horizontal plane. As depicted in Figure 5a, a horizontal plane, denoted as PP\({}^{\prime}\), is taken at random height Z of the slope of height H, and the polar angles corresponding to the two intersection points P and P\({}^{\prime}\) of this horizontal plane and the slope profile are denoted as \(\alpha\) and \(\alpha^{\prime}\), separately. Therefore, an arbitrary height Z within the range of slope height H in the Cartesian coordinate system \(x\)-\(E\)-\(z\), as originated by the foot of the slope, can be stated as a function of the polar angle \(\alpha\). \[z(\alpha)=r_{0}e^{(\theta_{h}-\theta_{0})\tan\varphi_{t}}\sin\theta_{h}-r_{p}\sin\alpha \tag{61}\]  
where \(r_{p}\) is the polar diameter corresponding to point P, which could be shown as  
\[r_{p}=r_{0}e^{(\alpha-\theta_{0})\tan\varphi_{t}} \tag{62}\]  
By putting Equation (61) into Equation (14), we can obtain seismic accelerated speed \(a_{h}\) on the level plane of any height Z within the height H of the slope. And furthermore, as shown in Figure 5b, the work exerted by a level seismic force within volume micro-element d\(V\) of height d\(z\) can be expressed as  
\[\mathrm{d}W_{s}=\frac{a_{h}}{g}\gamma v\sin\theta\mathrm{d}V \tag{63}\]  
Of these, the geometric equation for the volume micro-element in the 3D portion of the slope, as shown in Figure 5b, can be derived as \[\mathrm{d}V_{3\mathrm{D}}=l(\theta)\frac{\sin\alpha}{\sin^{2}\theta}r_{p}\mathrm{d} \theta\mathrm{d}z \tag{65}\]  
Likewise, as illustrated in Figure 5b, the volume micro-element in the embedded plane strain block of the slope can be similarly shown as  
\[\mathrm{d}V_{\mathrm{embed}}=b\frac{\sin\alpha}{\sin^{2}\theta}r_{p}\mathrm{d} \theta\mathrm{d}z \tag{66}\]  
In Equation (65), \(l(\theta)\) is the breadth for the 3D section of the two-stage slope at an arbitrary polar angle \(\theta\) (\(\theta_{0}\leqslant\theta\leqslant\theta_{h}\)), which is shown as  
\[l(\theta)=2\sqrt{R^{2}-d^{2}} \tag{67}\]  
\[d=r_{p}\frac{\sin\alpha}{\sin\theta}-r_{m}=r_{0}f_{12} \tag{68}\]  
where \(f_{12}\) is a non-dimensional equation, listed as  
\[f_{12}=e^{(\alpha-\theta_{0})\tan\varphi_{t}}\frac{\sin\alpha}{\sin\theta}-f_ {1} \tag{69}\]  
Also, the micro-element \(\mathrm{d}z\) in Equations (65) and (66) can be stated as  
\[\mathrm{d}z=\frac{\cos(\alpha-\varphi_{t})}{\cos\varphi_{t}}r_{p}\mathrm{d}\alpha \tag{70}\]  
Thereby, with the combination of Equations (62)-(70), we can obtain the seismic force work within the 3D portion for the slope and embedded plane strain block, individually. \[W_{s-3D}=\omega\gamma\int_{\theta_{0}}^{\theta_{h}}\int_{\alpha}^{\alpha^{ \prime}}l(\theta)r_{p}^{3}\frac{\sin^{2}\alpha}{\sin^{2}\theta}\frac{\cos( \alpha-\varphi_{t})}{\cos\varphi_{t}}\frac{a_{h}}{g}\mathrm{d}\alpha\mathrm{d}\theta \tag{71}\]  
\[r_{p}=r_{0}e^{(\alpha-\theta_{0})\tan\varphi_{t}} \tag{62}\]0  
The integration upper limit \(\alpha^{\prime}\) of the dual integrals in Equations (71) and (72) can be derived from \(\alpha\), whose segmental function expression is represented as  
\[\alpha^{\prime}=\left\{