# Research on Fracture Mechanism and Stability of Slope with Tensile Cracks
## 3 Analytical Method for Safety Factor of Slope with Crack  
### Analytical Method for Safety Factor of Toe Circle Slope  
#### 3.1.1 Calculated Model of Toe Circle Slope  
In view of fact that the slip surface of slope is composed of tension crack and shear surface, below the vertical crack area, the slip surface is simplified as a circle surface, and the slip body is treated as a rigid body, as shown in Figure 3. The calculated model of toe circle slope failure is established in the Cartesian coordinate system, and the center of the circle is at the toe of slope. The height of slope is \(H\), the angle of slope is \(\beta\), the unit weight of soil is \(\gamma\), and the vertical crack depth is \(\delta\). The parameters of the slip surface are as follows: the coordinates of the center of the circle is \(O\left(a,b\right)\), \(a\) and \(b\) are the coordinate values of \(X\) and \(Z\) in the direction of the coordinate system shown in Figure 3, the radius of the circle arc is \(r\), the length of the slip surface is \(l\), and the width of the top of the slip body is \(d\). The soil in the crack area is simplified as surcharge, and the surcharge function is defined as \(q(x)\).  
Figure 2: Slope failure mode.  
In the calculated model, based on the theory of the Swedish slice method, the slip surface is divided into n slices, and the inter slice forces are plotted in Figure 4 [18]. The weight of any soil slice is \(\mathrm{d}w\), the length of soil slice is \(\mathrm{d}x\), the height of soil slice is \(h(\mathrm{z})\), the length of the slip surface for soil slice is \(\mathrm{d}l\), the angle of the slip surface with the horizontal axis is \(\mathrm{\theta}\), and the normal and shear forces on the bottom of the soil slice are \(\mathrm{d}N\) and \(\mathrm{d}T\), respectively. The slip moment of soil slice is \(\mathrm{d}M\), and the surcharge is \(q(x)\).  
In the toe circle calculated model, slope face \(z_{1}\), slope upper surface \(z_{2}\), and slip surface \(z_{3}\) can be respectively expressed as follows:  
\[\left\{\begin{array}{l}z_{1}=x\tan\beta\\ z_{2}=H-\delta^{\prime}\\ z_{3}=b-\sqrt{r^{2}-\left(x-a\right)^{2}}\end{array}\right. \tag{7}\]  
where weight of the soil slice is expressed as  
\[\mathrm{d}w=\gamma h(z)\mathrm{d}x+q(x)\mathrm{d}x \tag{8}\]  
Figure 4: Forces acting on the slices subjected to surcharge when the slice forces are neglected.  
Figure 3: Toe circle slope calculated model.  
The normal and shear forces on the bottom of the soil slice are expressed as,  
\[\mathrm{d}T =\mathrm{d}w\sin\theta \tag{9}\] \[\mathrm{d}N =\mathrm{d}w\cos\theta \tag{10}\]  
According to the load distribution, the calculation region \(x_{1}\), \(x_{2}\), and \(x_{3}\) can be respectively written as follows:  
\[\left\{\begin{array}{l}x_{1}=(H-\delta^{\prime})\cot\beta\\ x_{2}=H\cot\beta\\ x_{3}=a+\sqrt{r^{2}-\left[(H-\delta^{\prime})-b\right]^{2}}\end{array}\right. \tag{11}\]  
For any soil slice, the relationship between the length of the slip surface \(\mathrm{d}l\) and the inclination of the slope \(\theta\) are expressed as,  
\[\left\{\begin{array}{l}\sin\theta=\frac{x-a}{r}\\ \cos\theta=\frac{\sqrt{r^{2}-(x-a)^{2}}}{r}\\ \mathrm{d}l=\frac{r}{\sqrt{r^{2}-(x-a)^{2}}}\mathrm{d}x\end{array}\right. \tag{12}\]  
The height of soil slice \(h(z)\) is written as  
\[h(z)=\left\{\begin{array}{l}z_{1}-z_{3}\ 0\leq x\leq x_{1}\\ z_{2}-z_{3}\ x_{1}\leq x\leq x_{3}\end{array}\right. \tag{13}\]  
The surcharge of crack region \(q(x)\) is written as  
\[q(x)=\begin{cases}0&0\leq x\leq x_{1}\\ \frac{\gamma}{\cot\beta}(x-x_{1})&x_{1}\leq x\leq x_{2}\\ \gamma\delta^{\prime}&x_{2}\leq x\leq x_{3}\end{cases} \tag{14}\]