## 2 Modifications for Hoek-Brown Yield Criterion and the Pseudo-Dynamic Method on Seismicity Analysis
### Modified Pseudo-Dynamic Method  
Seismicity could dramatically enhance induced force for slope instability and decrease the shearing intensity for rock mass, which is an essential factor leading to the failure of rocky slopes [38; 39]. Within the classical pseudo-static model, the seismic effect is normally simplified to a uniform inertial force, and this approximate treatment ignores the dynamic characteristics of seismicity [40; 41]. In the conventional pseudo-dynamic approach, the seismic behavior is typically denoted in the form of a seismic wave motion [42]. Nevertheless, it postulates that the slope medium is a linear elastomer, which causes the seismic acceleration to be infinitely amplified, deviating from the actual reality [43]. To overcome these drawbacks, a pseudo-dynamic approach based on viscoelastic standing waves is proposed [32; 33], in which the geotechnical medium is presumed to be the Kelvin-Voigt (K-V) constitutive model; thus, the damping effects of the geotechnical material are considered, such that the seismic acceleration is not infinitely amplified. In accordance with the boundary conditions: (1) the free surface shear stress at the top of the slope is zero; (2) the displacement at the foot of the slope (\(z=0\)) (\(u_{ht}=u_{h0}\cos(\omega t)\)) and the explicit equation of the horizontal displacement of transverse wave within the slope mechanism can be obtained based on the improved pseudo-dynamic method. \[u_{\rm h}(z,t)=\frac{u_{\rm h0}}{C_{\rm S}^{2}+S_{\rm S}^{2}}[(C_{\rm S}C_{ \rm SZ}+S_{\rm S}S_{\rm SZ})\cos(\omega t)+(S_{\rm S}C_{\rm SZ}+C_{\rm S}S_{ \rm SZ})\sin(\omega t)] \tag{7}\]  
in which  
\[C_{sz}=\cos\!\left(y_{\rm S1}\frac{H-z}{H}\right)\cos\!\left(y_{\rm S2}\frac{ H-z}{H}\right) \tag{8}\]  
\[S_{sz}=-\sin\!\left(y_{\rm S1}\frac{H-z}{H}\right)\sin\!\left(y_{\rm S2}\frac{ H-z}{H}\right) \tag{9}\]  
Figure 1: The envelope curve of the nonlinear H-B yield criterion and its outer tangent. \[C_{S}=\cos(y_{S1})\cos\hbar(y_{S2}) \tag{10}\] \[S_{S}=-\sin(y_{S1})\sin\hbar(y_{S2})\] (11) \[y_{s1}=\frac{\omega H}{V_{s}}\sqrt{\frac{\sqrt{1+4D^{2}}+1}{2(1+4D ^{2})}}\] (12) \[y_{s2}=-\frac{\omega H}{V_{s}}\sqrt{\frac{\sqrt{1+4D^{2}}-1}{2(1+4D ^{2})}} \tag{13}\]  
By second-order differentiation of Equation (7), the horizontal seismic acceleration can be deduced as  
\[a_{\rm h}(z,t)=\frac{k_{\rm h}g}{C_{\rm S}^{2}+S_{\rm S}^{2}}[(C_{\rm S}C_{\rm SZ }+S_{\rm S}S_{\rm SZ})\cos(\omega t)+(S_{\rm S}C_{\rm SZ}+C_{\rm S}S_{\rm SZ}) \sin(\omega t)] \tag{14}\]  
where \(k_{h}g=-\omega^{2}u_{h0}\); \(k_{h}\) and \(g\) are the level seismicity factor and gravity accelerated speed, respectively. \(y_{s1}\) and \(y_{s2}\) are defined as the correlation functions of \(\omega H/V_{s}\) and \(D_{r}\), where \(H\) is the slope's height and \(V_{s}\) is the speed of the transverse wave propagation. Accordingly, the displacement and acceleration of the longitudinal wave propagation per unit rock layer at a certain moment can be given similarly. However, relevant research has shown that the vertical seismic acceleration during longitudinal wave propagation is almost negligible, with \(k_{v}\leq 0.5k_{h}\)[44]. Hence, the horizontal seismic effect, which has the most impact on slope stability, is taken into consideration in this paper with enough comparisons of the effects of lateral and longitudinal seismic accelerations. Model for 3D Two-Stage Rock Slopes Reinforced with Prestressed Anchor Cables Effected by Seismicity and Its Analytical Solution  
The upper limit approach of plastic limit analysis is the application of a set of theorems and assumptions [9], for ideally plastic materials, which permits the mechanism to strictly yield to the minimum upper limit of energy dissipation when failure occurs, and it is consistent with the deformation of the rock mass governed by the associated flow law. As a result, the principal strain rates are required to fulfill such a relationship.