## 5 Calibration and Application
### Ultimate Shear Load of Direct Shear Specim with Coplanar Intermittent Joints  
The first numerical example is a direct shear test specimen of the rock mass reported in [PERSON] et al. (2015), as shown in Figure 7. The calculation parameters are listed in Tables 1 and 2. The rock specimen contains the two coplanar intermittent joints. The rock bridge between adjacent joints plays an important role in the bearing capacity of the rock specimen. The ultimate shear load of the specimen with different combinations of joint persistence and normal stress is calculated using the proposed method. The calculated results are compared with analytical solutions.  
The mixed numerical discrete method is used to discretize the rock specimen, as schematically shown in Figure 8. As shown in Figure 8, to evaluate the sensitivity of the computational accuracy to the finite element grid density, the specimen is divided into grids with four rigid block elements (\(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\)) and 8, 16, 26, 50, and 116 finite elements, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Name** & **Unit Weight (\(\mathbf{k}\)N/m\({}^{\mathbf{3}}\))** & **Cohesion (\(\mathbf{k}\)Pa)** & **Internal Friction Angle (\({}^{\star}\))** \\ \hline Rock mass & 15.00 & 4230.00 & 26.55 \\ Jointed plane & / & 0.00 & 35.20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical and mechanical parameters of example 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Joint Persistence \(\mathbf{k}\) (\%)** & **Normal Stress \(\sigma_{n}\) (MPa)** \\ \hline
90 & 2.0 \\
80 & 2.0 \\
70 & 2.0 \\
60 & 0.0 \\
60 & 1.0 \\
60 & 2.0 \\
60 & 3.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation parameters of the direct shear test specimen.  
Figure 7: Sketch of direct shear test specimen with two coplanar intermittent joints.  
The ultimate shear loads for various conditions are calculated. The results are listed in Figures 9 and 10. The analytical solutions, which are obtained based on the assumption of pure shear failure in the jointed plane direction, are listed for comparison. In general, the numerical simulation error decreases with the increase in \(p\) and grid density. Obviously, the computational accuracy will also significantly increase with the increase in \(p\) and grid density. In this study, for efficient computation with sufficient accuracy, the value of \(p\) is taken as 8 to 16, and the grid density is taken as medium density. The proposed method is an approximate method for solving the limit state of the non-persistent jointed rock. Numerical error has three sources: (1) linearization of the Mohr-Coulomb yield criteria; (2) the grid density of triangular finite elements; (3) the linear programming solver.  
Figure 8: Mixed numerical discretization with different finite element grid densities (\(k=60\%\), \(n_{b}=4\)): (a) \(n_{e}=8\), (b) \(n_{e}=16\), (c) \(n_{e}=26\), (d) \(n_{e}=50\), (e) \(n_{e}=116\).  
Figure 9: Plot of ultimate shear load vs. edge quantity \(p\) of polygon for \(k=0.6\).  
Figure 11 shows the variation in the ultimate shear load with joint persistence \(k\) for a normal stress \(\sigma_{n}=2.0\) MPa. It is seen that the ultimate shear load decreases with the increase in \(k\), indicating that the mechanical effect of the rock bridge is reduced when \(k\) increases. The numerical results agree well with the analytical solutions.