## 4 Stability Analysis Model of Filled-Slope Reinforced by Frame with Prestressed Anchor-Plates
### Stability Analysis in the Case of the First Type of Slip Surface  
#### 4.2.1 Solving the Stability Factor  
The diagram of the stability analysis for the first type of slip surface is shown in Figure 9.  
When analyzing the stability of the slope shown in Figure 9 based on the simplified Bishop method, due to the supporting effect of the frame with prestressed anchor-plates on the slope, the force of the soil strip is different from that of the soil strip without reinforcement. Thus, the resistant shear force provided by the anchor-plates and the resistant shear force of the soil itself cannot be simply superimposed. As shown in Figure 10, \(T_{u}{}^{\prime}\) and \(T_{u}{}^{\prime\prime}\) can be obtained by decomposing the ultimate bearing capacity provided by anchor-plates. \(T_{u}{}^{\prime}\) and soil have the same direction of resistant shear force of the soil, which directly provides resistant shear force for the sliding body. \(T_{u}{}^{\prime\prime}\) is perpendicular to \(T_{u}{}^{\prime}\), and it provides resistant shear force for the sliding body through friction with the soil outside the sliding surface. In order to derive the formula for calculating the slope stability under the action of the supporting structure, \(T_{u}{}^{\prime}\) is equivalent to the force in the same direction  
Figure 9: Diagram of stability analysis for the first type of slip surface. as \(T_{u}\)\({}^{\prime\prime}\) according to Equation (2). The force provided by the anchor-plates is uniformly distributed on the slip surface as the force perpendicular to the tangent direction of the slip surface, so the force provided by the anchor-plates on the soil strip \(i\) is as follows:  
\[N_{ui}=[\sum_{f=1}^{t}\,(T_{uf}\sin\alpha_{f}+T_{uf}\cos\alpha_{f}F_{s}/\tan \varphi_{f})]b_{i}\sec\alpha_{i}/L \tag{1}\]  
where \(t\) is the total number of layers of the prestressed anchor-plate; \(T_{uf}\) is the ultimate bearing capacity of the \(f\)-layer anchor-plate, which can be determined by Equation (31); \(\alpha_{f}\) is the angle between the tangent line and the horizontal direction at the intersection of the \(f\)-layer anchor-plate and the slip surface; \(\varphi_{f}\) is the internal friction angle of the soil at the intersection point of the \(f\)-layer anchor-plate and the slip surface; \(L\) is the total arc length of the slip surface. The force diagram of the soil strip considering the action of the supporting structure is shown in Figure 11, according to the Mohr-Coulomb criterion [30]:  
\[T_{i}d=N_{i}d\tan\varphi_{i}/F_{s}+c_{i}b_{i}d\sec\alpha_{i}/F_{s} \tag{2}\]  
where \(F_{s}\) is the stability factor of slope; \(T_{i}d\) is the shear force at the bottom of the soil strip \(i\); \(N_{i}d\) is the normal force at the bottom of the soil strip \(i\); \(\alpha_{i}\) is the angle between the tangent line and the horizontal direction at the slip surface of the soil strip \(i\); \(c_{i}\) is the cohesion at the slip surface of the soil strip \(i\); \(\varphi_{i}\) is the angle of internal friction at the slip surface of the soil strip \(i\); \(b_{i}\) is the width of the soil strip \(i\); \(d\) is the thickness of sliding element. Considering the vertical force balance [30]:  
\[(N_{i}d-N_{ui})\cos\alpha_{i}+T_{i}d\sin\alpha_{i}=(W_{i}+q_{0}b_{i})d \tag{3}\]  
where \(W_{i}d\) is the dead weight of the soil strip \(i\); \(q_{0}\) is the uniformly distributed load on the top of the slope. Simultaneous Equations (2) and (3) can be obtained:  
\[N_{i}d=[(W_{i}+q_{0}b_{i})d+N_{ui}\cos\alpha_{i}-c_{i}b_{i}d\tan\alpha_{i}/F_{ s}]/m_{ai} \tag{4}\]  
\[T_{i}d=[(W_{i}+q_{0}b_{i})d\tan\varphi_{i}/F_{s}+N_{ui}\cos\alpha_{i}\tan \varphi_{i}/F_{s}+c_{i}b_{i}d/F_{s}]/m_{ai} \tag{5}\]  
\[m_{ai}=\cos\alpha_{i}+\sin\alpha_{i}\tan\varphi_{i}/F_{s} \tag{6}\]  
Figure 11: Force diagram of soil strip \(i\). Figure 10: Resolution diagram of the ultimate bearing capacity of the anchor-plate.