## 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  
#### 4.3.1 Solving the Stability Factor  
The diagram of the stability analysis for the second type of slip surface and the force diagram of soil strip \(j\) are shown in Figures 13 and 14.  
Figure 13: Diagram of stability analysis for the second type of slip surface.  
Soil strip on the left side of \(O\!F\).  
The force provided by the anchor-plates on the soil strip \(j\) is as follows:  
\[N_{uj}=[\sum_{f=1}^{t}\left(T_{uf}\sin\alpha_{f}+T_{uf}\cos\alpha_{f}F_{s}/\tan \varphi_{f}\right)]b_{j}\sec\alpha_{j}/L \tag{16}\]  
By Mohr-Coulomb criterion [30]:  
\[T_{j}d=N_{j}d\tan\varphi_{j}/F_{s}+c_{j}b_{j}d\sec\alpha_{j}/F_{s} \tag{17}\]  
where \(T_{j}d\) is the shear force at the bottom of the soil strip \(j\) at the left on point \(F\); \(N_{j}d\) is the normal force at the bottom of the soil strip \(j\) at the left on point \(F\); \(b_{j}\) is the width of the soil strip \(j\) at the left on point \(F\); \(c_{j}\) is the cohesion at the slip surface of the soil strip \(j\) at the left on point \(F\); \(\varphi_{j}\) is the angle of internal friction at the slip surface of the soil strip \(j\) at the left on point \(F\); \(\alpha_{j}\) is the angle between the tangent line and the horizontal plane at the slip plane of the soil strip \(j\) at the left on point \(F\).  
Consider the vertical force balance [30]:  
\[\left(N_{j}d-N_{uj}\right)\cos\alpha_{j}-T_{j}d\sin\alpha_{j}=W_{j}d \tag{18}\]  
where \(W_{j}d\) is the dead weight of the soil strip \(j\) at the left on point \(F\).  
Simultaneous Equations (17) and (18) can be obtained:  
\[N_{j}d=\left(W_{j}d+N_{uj}\cos\alpha_{j}+c_{j}b_{j}\tan\alpha_{j}/F_{s}\right) /m_{aj} \tag{19}\]  
\[T_{j}d=\left(W_{j}d\tan\varphi_{j}/F_{s}+N_{uj}\cos\alpha_{j}\tan\varphi_{j}/F_ {s}+c_{j}b_{j}d/F_{s}\right)/m_{aj} \tag{20}\]  
\[m_{aj}=\cos\alpha_{j}-\sin\alpha_{j}\tan\varphi_{j}/F_{s} \tag{21}\]  
Soil strip on the right side of \(O\!F\).  
The force analysis of the soil strip on the right side of \(O\!F\) is similar to that in the case of slip surface \(1\), and the final deduced results are Equations (4)-(6).  
Consider the moment balance to the center of the circle:  
\[\sum_{i=1}^{n}T_{i}dR+\sum_{j=1}^{m}T_{j}dR=\sum_{i=1}^{n}\left(W_{i}+q_{0}b_{ i}\right)d\sin\alpha_{i}R-\sum_{j=1}^{m}W_{j}d\sin\alpha_{j}R \tag{22}\]  
where \(n\) is the number of the strips of sliding body at the right on point \(F\); \(m\) is the number of the strips of sliding body at the left on point \(F\); \(R\) is the arc radius of the slip surface; \(T_{i}d\) is the shear force at the bottom of the soil strip \(i\) at the right on point \(F\); \(W_{i}d\) is the dead weight of the soil strip \(i\) at the right on point \(F\); \(\alpha_{i}\) is the angle between the tangent line and the horizontal plane at the slip surface of the soil strip \(i\) at the right on point \(F\).  
Figure 14: Force diagram of soil strip \(j\).  
Substitute Equations (5) and (20) into Equation (22) to obtain:  
\[F_{\rm s}=\frac{\sum\limits_{i=1}^{n}\left[(W_{i}+q_{0}b_{i})d\tan\varphi_{i}+c_{ i}b_{i}d+N_{ui}\cos\alpha_{i}\tan\varphi_{i}\right]R/m_{\alpha i}}{\sum\limits_{i=1}^{n} \left(W_{i}+q_{0}b_{i}\right)d\sin\alpha_{i}R-\sum\limits_{j=1}^{m}W_{j}d\sin \alpha_{j}R} \tag{23}\]  
where \(b_{i}=\) the width of the soil strip \(i\) at the right on point \(F\); \(c_{i}=\) the cohesion at the slip surface of the soil strip \(i\) at the right on point \(F\); \(\varphi_{i}=\) the angle of internal friction at the slip surface of the soil strip \(i\) at the right on point \(F\).  
Substitute Equations (1) and (16) into Equation (23) to obtain:  
\[F_{\rm s}=\frac{\sum\limits_{i=1}^{n}\left[(W_{i}+q_{0}b_{i})d\tan\varphi_{i}+ c_{i}b_{i}d+\sum\limits_{f=1}^{t}\left(T_{uf}\sin\alpha_{f}+T_{uf}\cos\alpha_{f}F_{s}/ \tan\varphi_{f}\right)b_{i}\tan\varphi_{i}/L\right]R/m_{\alpha i}}{\sum\limits _{i=1}^{n}\left(W_{i}+q_{0}b_{i}\right)d\sin\alpha_{i}R-\sum\limits_{j=1}^{m} W_{j}d\sin\alpha_{j}R} \tag{24}\]