# 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
#### 3.1.2 Determination of the Safety Factor  
In this study, the safety factor was determined by using the limit equilibrium method. The safety factor of slope can be defined as the ratio of resisting moment and sliding moment on slip surface. The soil slice element of slope toe circle is shown in Figure 4, so the safety factor is given by the following expression:  
\[F=\frac{M_{R}}{M_{S}}=\frac{\int r\mathrm{d}T^{\prime}}{\int r\mathrm{d}T}= \frac{\int\mathrm{d}T^{\prime}}{\int\mathrm{d}T}=\frac{T^{\prime}}{T} \tag{15}\]  
where \(F\) is safety factor, \(M_{R}\) is resisting moment, \(M_{S}\) is sliding moment, \(T\) is sliding force, and \(T^{\prime}\) is resisting force.  
(1) Calculation of the sliding force  
In the toe circle slope, the total sliding force induced by the self-weight, \(T_{toe}\), is the result from the shear forces on the bottom of the soil slice, \(\mathrm{d}T\) shown in Equation (9), acting alone the failure surface \(AN\), as  
\[\begin{array}{lcl}T_{toe}&=&\int_{AN}\mathrm{d}T\\ &=&\int_{0}^{A}\gamma(z_{1}-z_{3})\sin\theta\mathrm{d}x+\int_{x_{1}}^{x_{3}} \gamma(z_{2}-z_{3})\sin\theta\mathrm{d}x+\int_{x_{1}}^{x_{2}}\frac{\gamma}{ \cot\beta}(x-x_{1})\sin\theta\mathrm{d}+\int_{x_{2}}^{x_{3}}\gamma\delta^{ \prime}\sin\theta\mathrm{d}x\\ &=&\frac{\gamma(H-\delta^{\prime})^{2}}{r}\bigg{[}\frac{(a\cot\beta+b)}{\theta }-\frac{(H-\delta)\left(1+\cot^{\beta}\delta\right)}{\theta}\bigg{]}\\ &&\quad+\frac{\gamma\cot\beta}{r}\bigg{\{}\frac{\cot\beta\left[H^{3}-(H-\delta^ {\prime})^{3}\right]}{3}-\frac{\left[a+(H-\delta^{\prime})\cot\beta\right] \left[H^{2}-(H-\delta^{\prime})^{2}\right]}{2}+a\delta(H-\delta^{\prime}) \bigg{\}}\end{array} \tag{16}\]The resisting force can be calculated in two parts. One is induced by the cohesion of soil \(T^{\prime}_{toe-}\) and the other is induced by the friction angle of soil \(T^{\prime}_{toe-\varphi}\).  
The \(T^{\prime}_{toe-c}\) acting on the slip surface \(AN\) can be written as follows:  
\[\begin{array}{ll}T^{\prime}_{toe-c}&=\int_{AN}\mathrm{cd}l=\int_{0}^{x_{3}}c \frac{r}{\sqrt{r^{2}-(x-a)^{2}}}\mathrm{d}x\\ &=cr(\arcsin\frac{\sqrt{r^{2}-[(H-\delta^{\prime})-b]^{2}}}{r}+\arcsin\frac{a} {r})\end{array} \tag{17}\]  
Similarly, the \(T^{\prime}_{toe-\varphi}\) acting on the slip surface \(AN\) can be written as follows:  
\[\begin{array}{ll}T^{\prime}_{toe-\varphi}=\tan\varphi\!\int_{AN}dN\\ =\tan\varphi\!\int_{0}^{x_{1}}\gamma(z_{1}-z_{3})\cos\theta\mathrm{d}x+\tan \varphi\!\int_{x_{1}}^{x_{3}}\gamma(z_{2}-z_{3})\cos\theta\mathrm{d}x+\tan \varphi\!\int_{x_{1}}^{x_{2}}\frac{\gamma}{\mathrm{cot}\beta}(x-x_{1})\cos \theta\mathrm{d}x+\tan\varphi\!\int_{x_{2}}^{x_{3}}\gamma\delta^{\prime}\cos \theta\mathrm{d}x\\ =\frac{\gamma r\tan\varphi}{2}\!\left\{\begin{array}{c}[a\tan\beta-(H- \delta^{\prime})]\arcsin\frac{\left[(H-\delta^{\prime})\cot\beta-a\right]}{r} \\ -[b-(H-\delta^{\prime})]\arcsin\frac{\sqrt{r^{2}-[(H-\delta^{\prime})-b]^{2}}}{r }\\ -(b-a\tan\beta)\arcsin\frac{\delta}{r}\end{array}\right\}+\frac{\gamma\tan \varphi}{6r}\left|\begin{array}{c}\left\{4r^{2}-\left[b-(H-\delta^{\prime}) \right]^{2}\right\}\sqrt{r^{2}-\left[(H-\delta^{\prime})-b\right]^{2}}\\ +2b^{3}\tan\beta+3a^{2}b\tan\beta+4a^{3}+3 ab^{2}-\\ -(b-a\tan\beta)\arcsin\frac{\delta}{r}\end{array}\right.\\ +\frac{\gamma\delta^{\prime}\tan\varphi}{2r}\left\{\begin{array}{c}\sqrt{r^{ 2}-\left[(H-\delta^{\prime})-b\right]^{2}}[(H-\delta^{\prime})-b]\\ -(H\cot\beta-a)\sqrt{r^{2}-\left[(H-\delta^{\prime})\cot\beta-a\right]^{2}} \end{array}\right\}+\frac{\gamma\delta^{\prime}\tan\varphi}{2}\left\{\arcsin \frac{\sqrt{r^{2}-[(H-\delta^{\prime})-b]^{2}}}{r}-\arcsin\frac{H\cot\beta-a}{r }\right\} \tag{18}\]  
The resisting force \(T^{\prime}_{toe}\) can be expressed as  
\[T^{\prime}_{toe}=T^{\prime}_{toe-c}+T^{\prime}_{toe-\varphi} \tag{19}\]  
Therefore, the safety factor of toe circle slope is given by the following expression:  
\[F=\frac{T^{\prime}_{toe}}{T_{toe}}=\frac{T^{\prime}_{toe-c}+T^{\prime}_{toe- \varphi}}{T_{toe}} \tag{20}\]