## 2 Modifications for Hoek-Brown Yield Criterion and the Pseudo-Dynamic Method on Seismicity Analysis
### External Force Work
#### 3.3.3 Anchoring Force Work  
Prestressed anchor cables are widely employed for slope reinforcement [47; 48]. When seismic events occur, the slope is in a state of critical instability. At this time, the anchor cables apply the anchoring force to the sliding block of the slope through axial tensile deformation. In the following section, the negative work done by the anchoring force on the slope, that is, keeping the slope in the steady state, is analyzed and derived in detail. As illustrated in Figure 6, the anchor cable numbers installed on the upper step slope are noted as \(T_{u1}\), \(T_{u2}\), \(\ldots\), \(T_{uii}\), \(\ldots\), \(T_{un}\), with a total of \(\mathbf{n}\) rows of anchor ropes; the anchor ropes on the lower step slope are similarly recorded as \(T_{l1}\), \(T_{l2}\), \(\ldots\), \(T_{lj}\), \(\ldots\), \(T_{lm}\), with \(\mathbf{m}\) rows of anchor cables overall. The above numbers also represent the anchoring force values of the corresponding anchor ropes. In addition, \(\delta_{ui}\) and \(h_{ui}\) are the anchorage angle and anchor head height of the i-th row of anchor cables on the upper slope of the step, respectively; likewise, \(\delta_{\bar{\eta}}\) and \(h_{lj}\) are the anchorage angle and anchor head height of the j-th row of the anchor ropes installed on the lower step slope. For the failure mechanism depicted in Figure 6, the external force work on the sliding block of the slope done by the i-th row of anchors on the upper step slope and the j-th row of anchors on the lower step slope can be expressed, respectively, as  
\[\dot{W}_{ui}=-(b+B_{ui}{}^{\prime})T_{ui}\omega r_{ui}\sin(\theta_{ui}-\delta_ {ui}) \tag{77}\]  
\[\dot{W}_{lj}=-(b+B_{lj}{}^{\prime})T_{lj}\omega r_{lj}\sin(\theta_{lj}-\delta_ {lj}) \tag{78}\]  
where \(B_{ui}{}^{\prime}\), \(r_{ui}\), and \(\theta_{ui}\) are the breadth for the 3D section of the two-stage slope at the altitude of the i-th row of anchor heads on the upper step slope as well as the polar diameter and polar angle of this row of anchor heads; \(B_{lj}{}^{\prime}\), \(r_{lj}\), and \(\theta_{lj}\) are the breadth for the 3D section of the two-stage slope at the altitude of the j-th row of anchor heads on the lower step slope, the polar diameter, and the polar angle of this row of anchor heads in the polar coordinate system, separately. The negative value of the anchoring force work indicates that the anchor cables do negative work on the sliding block of the slope and are the stabilizing factors of the slope. Moreover, \(\theta_{ui}\) in Equation (77) and \(\theta_{lj}\) in Equation (78) can be obtained by the geometric relationship shown in Figure 6.  
\[\theta_{ui}=\arctan\left[\frac{e^{(\theta_{h}-\theta_{0})\tan\varphi_{i}} \sin\theta_{h}-h_{ui}/r_{0}}{\cos\theta_{0}-\eta_{2}-(\eta_{1}-h_{ui}/r_{0}) \cot\beta_{1}}\right] \tag{79}\]  
Figure 5: Diagram of the horizontal stratification method: (**a**) full graph; (**b**) detail zoom. In combination with the dimensionless quantities \(f_{1}\), \(f_{2}\), \(f_{4}\) and \(f_{6}\) appearing and employed in the preceding chapters, the entire anchorage force work can be further written as  
\[ \begin{array}{l}W_{T}=\sum\limits_{i=1}^{n}\dot{W}_{ui}+\sum\limits_{j=1}^{m} \dot{W}_{lj}\\ \phantom{W_{T}=\sum\limits_{i=1}^{n}\dot{W}_{ui}+\sum\limits_{j=1}^{m}\dot{W} _{lj}}\\ \phantom{W_{T}=\sum\limits_{i=1}^{n}\dot{W}_{ij}+\sum\limits_{j=1}^{m}\dot{W} _{lj}}\\ \phantom{W_{T}=\sum\limits_{i=1}^{n}\dot{W}_{ij}+\sum\limits_{j=1}^{m}\dot{W} _{lj}}\end{array}