## 4 Stability Analysis Model of Filled-Slope Reinforced by Frame with Prestressed Anchor-Plates
### Stability Analysis in the Case of the Second Type of Slip Surface
#### 4.3.2 Search for Models of Slip Surfaces  
(1) The center coordinates and radius of the slip surface  
The search model of the second kind of slip surface is shown in Figure 15. Since the slip surface always changes in the _xoy_ plane, it is simplified to two-dimensional coordinates. In the Cartesian coordinate system, \(P(x_{c},y_{c})\), \(C(e,H)\), \(A\)(-\(e_{0}\), \(0\)), \(O(0\), \(0)\), the angle between the tangent of point \(A\) and the horizontal is \(\theta\), and the slope of the tangent is \(k^{\prime},k^{\prime}=-tan\theta\).  
Then the slip surface equation is:  
\[\left(x-x_{c}\right)^{2}+\left(y-y_{c}\right)^{2}=R^{2} \tag{25}\]  
And satisfies:  
\[\left(-e_{0}-x_{c}\right)^{2}+\left(H-y_{c}\right)^{2}=R^{2} \tag{26}\] \[\left.x_{c}=-\left(k^{\prime}y_{c}+e_{0}\right)\right.\]  
Figure 15: Search model of the second kind of slip surface.  
Combining Equations (25) and (26) yields:  
\[\begin{array}{l}x_{c}=-e_{0}-\frac{k^{\prime}[(e+e_{0})^{2}+H^{2}]}{2H-2k^{ \prime}(e+e_{0})}\\ y_{c}=\frac{(e+e_{0})^{2}+H^{2}}{2H-2k^{\prime}(e+e_{0})}\\ R=\frac{\sqrt{1+k^{2}}[(e+e_{0})^{2}+H^{2}]}{2H-2k^{\prime}(e+e_{0})}\end{array} \tag{27}\]  
Therefore, the slip surface is controlled by \(\theta\), \(e_{0}\) and \(e\). The value range of \(\theta\), \(e_{0}\) and \(e\) is determined, and the appropriate step size is set respectively, and the search of slip surface can be realized by constantly changing.  
(2) The angle between the tangent of any point on the arc and the horizontal plane  
At any point \(M^{\prime}(x_{k},y_{k})\) on the arc, the angle between the tangent and the horizontal plane is \(\alpha_{k}\), which can be obtained from the geometric relationship:  
\[\sin\alpha_{k}=\frac{|x_{c}-x_{k}|}{R} \tag{28}\]  
\[\cos\alpha_{k}=\frac{y_{c}-y_{k}}{R} \tag{29}\]  
(3) The self-weight of the soil block \(i\) and \(j\)  
It can be seen from Figure 12 that the calculation of the self-weight of the soil strip under the second type of slip surface is more complicated. The coordinates of the soil strip \(i\) at the slip surface are \((x_{i},y_{i})\). When the upper end of the soil strip \(i\) is on the \(OB\), its upper coordinate is \((x_{i1},y_{i1})\). The coordinates of the soil strip \(j\) at the slip plane are \((x_{j},y_{j})\). When the upper end of the soil strip \(j\) is on \(OB\), its upper coordinate is \((x_{j1},y_{j1})\). When the abscissa of the zero-boundary point of the slip surface crossing from the filling layer to the foundation soil layer is \(x_{0}\), the calculation formulas of the self-weight of the soil strip \(i\) and \(j\) are as follows:  
\[W_{ij}=\left\{\begin{array}{l}\gamma_{1}b_{i}(H-y_{i})\;\;x_{i}\geq\frac{H}{ \tan\beta}\\ \gamma_{1}b_{i}(y_{i1}-y_{i})\;\;x_{0}\leq x_{i}<\frac{H}{\tan\beta}\\ \gamma_{1}b_{i}y_{i1}-\gamma_{2}b_{i}y_{i}\;0\leq x_{i}<x_{0}\\ \\ -\gamma_{2}b_{i}y_{i}\;\;x_{c}\leq x_{i}<0\\ -\gamma_{2}b_{i}y_{j}\;\;-e_{0}\leq x_{j}<x_{c}\end{array}\right. \tag{30}\]  
where \(\gamma_{1}\) is the weight of fill soil; \(\gamma_{2}\) is the weight of foundation soil.