# Research on Fracture Mechanism and Stability of Slope with Tensile Cracks
## 4 Results Analysis
### The Effects of Parameters on the Stability of Slope
#### 4.2.6 The Effects of Optimal Crack Parameters and Slip Body Morphology on the Safety Factor  
If the optimal crack depth \(\delta^{\prime}\) and the horizontal distance of the crack from the slope crest, \(d\) (see Figures 3, 5 and 6), is known, the slip body morphology is introduced. When the optimal crack depth is determined, the horizontal location of optimal crack depth can be expressed as  
\[ \begin{array}{ll}d&=x_{3}-x_{2}\\ &=a+\sqrt{r^{2}-\left[(H-\delta^{\prime})-b\right]^{2}}-H\cot\beta\end{array}  \tag{29}\]  
Hence, a composite slip surface consists of the tensile crack, and a shear surface is obtained. The slip body morphology of case 2 under different \(\delta^{\prime}/H\) can be seen in Figure 14. For the sake of explanation, the results of slip surface obtained for the values of \(\delta^{\prime}/H\) will be illustrated. It can be seen from Figure 14 that the slip surface moves towards slope face with the increase of values of \(\delta^{\prime}/H\). The \(\delta^{\prime}/H\) ranges from 0 to 0.45, and it shows that the larger its value is, the larger the crack depth is. As is to be expected, for \(\delta^{\prime}/H=0\), the slip surface is completely shear plane and the safety factor is the maximum. Higher values of \(\delta^{\prime}/H\) can lead to lower safety factors, and the safety factor decreases by approximately 6.23% when the value of \(\delta^{\prime}/H\) varies from 0 to 0.25 (from 1.624 to 1.522). However, the minimum value of safety factor appears when the \(\delta^{\prime}/H=0.25\), and the crack does not extents to the half height of the slope. It shows that the crack depth is theoretically not unlimited extension, and an optimal value should exist. As shown in the above example, it can be assumed that \(\delta^{\prime}/H=1\), the slip body morphology coincides with that studied by Utili, the crack coming progressively near to the slope face and at the limit (\(d{\rightarrow}0\)) coinciding with it. The \(\delta^{\prime}/H{\rightarrow}1\), which does not exist in the slope itself. Additionally, it is noted that the crack occurs only at the upper surface of slope, which is an important prerequisite for stability analysis in this study. An Engineering Example Application: Determination of the Crack Depth and Safety Factor for a Highway Landslide  
Jinchangling-Liuku highway is an important transportation hub in southwest China, and runs across the Yunnan and Tibet Province. A lot of landslides have occurred on both sides of the road, taking a typical one as an example. The landslide parameters are as follows: the height difference between the toe and the top is 103 m; the length of horizontal  
Figure 14: Slip body morphology and location of cracks for various values of \(\delta^{\prime}/H\). projection of the landslide is 120 m; the landslide angle is 50\({}^{\circ}\); the volume of the landslide is estimated to be 5.6 \(\times\) 10\({}^{4}\) m\({}^{3}\); the inclination of slip surface is about 40\({}^{\circ}\). The morphology of slope after collapse is shown in Figure 15. The yellow dotted line represents the slide boundary of slip body, and the red dotted line represents the slip surface of landslide. Based upon a great amount of geological investigation data and indoor soil mechanic results, the medium weathered sandstone is the essential component in this area, so this landslide can be treated as a soil-like one. The physical parameters of landslide are as follows: \(c\) = 22.4 kPa, \(\varphi\) = 28\({}^{\circ}\), \(\gamma\) = 19.5 kN/m\({}^{3}\). According to the engineering geological exploration for the landslide, the morphology of slip body and actual failure surface are shown in Figure 16. Based on the above parameters, the minimum safety factor of the landslide is estimated at 0.98 by using the specification method, and the investigation shows that the crack depth at the tailing edge of landslide is about 8.5m. Since the toe of slope is the outlet of shear slip surface, the toe circle failure model is selected first in this method.