# Influence of Fracture Geometric Characteristics on Fractured Rock Slope Stability
## 4 Sensitive or Parametric Stability Analyses
### Influence of Fracture Dip  
In this section, we mainly investigate the effect of the fracture dip on the mean slope stability. Table 5 shows the statistics of the mean trace length for nine different cases and the calculated \(\mu_{Fs}\) values. Note that the mean trace lengths in Case 0 are chosen from Table 1.  
The \(\mu_{Fs}\) for the nine cases with different \(\mu_{D_{1}}\) and \(\mu_{D_{2}}\) is plotted in Figure 9. As is shown, the \(\mu_{Fs}\) slowly decreases when \(\mu_{D_{1}}\) increases from 5\({}^{\circ}\) to 15\({}^{\circ}\) (Figure 9a). In particular, when \(\mu_{D_{1}}\) increases from 5\({}^{\circ}\) to 10\({}^{\circ}\), the \(\mu_{Fs}\) changes less than 0.04. The result indicates that the slope stability remains unchanged when the dip of horizontal fractures varies in the range of 0-10\({}^{\circ}\). On the contrary, the \(\mu_{Fs}\) increases significantly (greater than 0.29) when \(\mu_{D_{2}}\) ranges from 40\({}^{\circ}\) to 60\({}^{\circ}\) (Figure 9b), meaning that the slope stability is more obviously affected by the change in inclined fracture dip. It is to be expected that the increase in inclined fracture dip results in an increase in the sliding force and decrease in the resistant sliding force, making the discrete blocks susceptible to slip instability along these inclined fractures.  
Moreover, we make the indicator \(\gamma\) (the ratio of \(\mu_{D_{2}}\) to \(\mu_{D_{1}}\)) to investigate the influence of two sets of fracture angles on the slope stability. The \(\gamma\) is constantly greater than 1; the closer \(\gamma\) is to 1, the smaller the angle between the two fracture sets. When \(\gamma\) = 1, the two sets of fractures have similar dips and should be classified as the same set. The \(\mu_{Fs}\) with nine different \(\gamma\) values is displayed in Figure 10a, showing that as \(\gamma\) increases from 1 to 12, the \(\mu_{Fs}\) shows an increasing trend. However, the \(Fs\) values of Cases 3, 6 and 9 (i.e., \(\mu_{D_{1}}\) equal to 5\({}^{\circ}\)) obviously deviate from the overall trend, as shown in Figure 10a. That is because when \(\mu_{D_{1}}\) ranges from 5\({}^{\circ}\) to 10\({}^{\circ}\), the \(\mu_{Fs}\) remains almost unchanged (Figure 9), while the \(\gamma\) changes. To accurately describe the relationship between \(\mu_{Fs}\) and \(\gamma\), Figure 10b removes the \(Fs\) values of Cases 3, 6 and 9. It is interesting that \(\mu_{Fs}\) and \(\gamma\) show a clear linear relationship and can be well-fitted by the regression equation \(\mu_{Fs}=0.171*\gamma+0.973\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mu_{D_{1}}\)/(\({}^{\circ}\)) & \(\mu_{D_{2}}\)/(\({}^{\circ}\)) & \(\gamma\) & \(\mu_{Fs}\) \\ \hline Case 0 & 3 & 50 & 16.7 & 1.78 \\ \hline Case 1 & 15 & 60 & 4 & 1.76 \\ Case 2 & 10 & 60 & 6 & 2 \\ Case 3 & 5 & 60 & 12 & 2.03 \\ \hline Case 4 & 15 & 50 & 3.3 & 1.63 \\ Case 5 & 10 & 50 & 5 & 1.83 \\ Case 6 & 5 & 50 & 10 & 1.87 \\ \hline Case 7 & 15 & 40 & 2.7 & 1.47 \\ Case 8 & 10 & 40 & 4 & 1.49 \\ Case 9 & 5 & 40 & 8 & 1.51 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistical table of the mean dip and mean stability.  
Meanwhile, these results also indicate that the stability of the rock slopes is closely related to the angle of two sets of fractures. When the angles between two sets of fractures are small, the blocks are mostly rhombus-shaped with a large aspect ratio, which leads to the rock slope being easily damaged by landslides along one of the fracture sets.