## 5 Calibration and Application
### Stability of a Rock Slope with Three Joints and a Rock Bridge  
Example 2 is an artificial jointed rock slope with three joints, as shown Figure 12, in which \(J2\) and \(J3\) are fully persistent joints, and \(J1\) is a non-persistent joint. A rock bridge is in the intersection region between joint \(J1\) and the slope-free surface. This example is used to examine the influence of the joint persistence \(k\) of joint \(J1\) and the shear strength parameters of the rock bridge on the strength reserve coefficient of the slope. The physical and mechanical parameters of the slope material are listed in Table 3. This slope is discretized using mixed numerical discrete methods, a rigid block element is employed to simulate the existing rock blocks, and a triangular finite element grid is used in the rock bridge with medium grid density. Figure 13 shows the divisions of the mixed discretization with different values of \(k\) of \(J1\). Figure 11: Plot of ultimate shear load vs. joint persistence \(k\). Figure 10: Plot of the ultimate shear load vs. grid density of finite elements for \(p=8\) and \(k=60\%\). The statically admissible stress fields are established using the proposed method. The strength reserve coefficients of the slope are calculated by an iterative algorithm for various values of \(k\) and different cohesions of rock bridge. The results are plotted in Figures 14 and 15, in which the results using the _Sama_ and _Janbu_ methods are plotted for comparison. The _Sama_ method and the _Janbu_ method assume that the shear failure will occur along the slip surface \(ABC\) in the slope, in which \(AB\) is the non-persistent joint \(J1\), and \(BC\) is a fictitious fracture. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Name** & **Unit Weight (\(k\)N/m\({}^{3}\))** & **Cohesion (\&Pa)** & **Internal Friction Angle (\(\gamma\))** \\ \hline \hline Interact rock mass & 25.50 & 1000.00 & 36.00 \\ Rock bridge & 25.50 & 1000.00 & 36.00 \\ Jointed plane & / & 0.05 & 30.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Physical and mechanical parameters of example 2. Figure 12: Sketch of jointed rock slope of example 2 (unit: m). Figure 14 shows that the strength reserve coefficients computed by the proposed method are slightly smaller than those computed by the _Sama_ method and _Janbu_ method for \(c=1000\) kPa. It is seen that the strength reserve coefficient decreases roughly linearly with the increase in \(k\). The lower bound method in this study provides a rigorous lower bound solution in contrast to the _Sama_ method. This means that the lower bound solution is a safer method for practical design. Figure 15 shows the variation in the strength reserve coefficient with connectivity rate \(k\) for various cohesions of the rock bridge \(c\) (\(c=1000\) kPa, 800 kPa, 600 kPa, 400 kPa). The results computed by the _Sama_ method are plotted for comparison. It is seen that the strength reserve coefficient varies roughly linearly with \(k\) for various cohesions of the rock bridge. Figure 15 also shows that the strength reserve coefficient decreases with \(c\). The strength reserve coefficient calculated using the lower bound method proposed in this study is smaller than that obtained from the _Sama_ method. Figure 16 is a sketch of the yield zone of the slope for different values of joint persistence \(k\) and \(c=1000\) kPa, which is obtained from the lower bound method. The results show that: (1) The slope failure mode is the occurrence of shear failure in three joints and the rock bridge. Joint /1 completely yields, whereas joints /2 and /3 partially yield. (2) The failure zone of the rock bridge formed by the non-persistent joint /1 is not always in the direction of line \(BC\), being mainly manifested as a random and irregular region according to the results of the lower bound method. (3) Traditional methods such as the _Sama_ and  
Figure 14: Strength reserve coefficient \(\lambda_{m}\) vs. for joint persistence \(k\) for different calculation methods.