## 4 Stability Analysis Model of Filled-Slope Reinforced by Frame with Prestressed Anchor-Plates
### The Ultimate Bearing Capacity of the Anchor-Plate  
For the calculation of the ultimate bearing capacity of the anchor-plate, on the basis of the method in [PERSON] (2007), the calculation method of the pullout resistance of the anchor-plate is improved. The force diagram of the anchor-plate is shown in Figure 16.  
According to [PERSON] (2007), the formula for calculating the pullout force of the anchor-plate is as follows:  
\[T_{uf}=\sum\min\{T_{s},T_{p}\} \tag{31}\]  
Figure 16: Force diagram of the anchor-plate.  
where \(T_{uf}\) is the ultimate bearing capacity of the \(f\)-layer anchor-plate; \(T_{s}\) is the ultimate tensile force of the steel rod; \(T_{p}\) is the limit value of the pullout force that the anchor-plate can provide, which is composed of the friction resistance of the interface between the anchor-plate and soil and the passive earth pressure of the anchor-plate.  
\[T_{s}=f_{y}A_{s} \tag{32}\]  
where \(f_{y}\) is the yield strength of the steel rod; \(A_{s}\) is the cross-sectional area of the steel rod.  
\[T_{p}=F_{f}+Q \tag{33}\]  
where \(F_{f}\) is the friction resistance on the anchor-plate; \(Q\) is passive earth pressure on the anchor-plate.  
Because [PERSON] (2016) did not consider the friction between the two sides of the anchor-plate and the soil when calculating the friction between the anchor-plate and the soil, this paper improves it, and the improved formula for calculating the friction between the anchor-plate and the soil is as follows:  
\[F_{f}=2(\tau_{1}A_{1}+\tau_{2}A_{2}) \tag{34}\]  
where \(\tau_{1}\) is the shear strength of the interface between the upper and lower interface of the anchor-plate and the soil; \(\tau_{2}\) is the shear strength of the interface between the front and rear interface of the anchor-plate and the soil; \(A_{1}\) is the upper surface area of anchor-plate; \(A_{2}\) is the side surface area of anchor-plate:  
\[\tau_{1}=\mu(\gamma h_{0}+q_{0}) \tag{35}\] \[\tau_{2}=k_{0}\mu(\gamma h_{0}+q_{0})\]  
where \(\gamma\) is the weight of the soil layer where the anchor-plate is located; \(h_{0}\) is the height from the top of the slope to the soil layer where the anchor-plate is located; \(q_{0}\) is the uniformly distributed load on the top of the slope; \(\mu\) is the coefficient of lateral earth pressure:  
\[Q=\sigma_{p}A_{3} \tag{36}\]  
where \(\sigma_{p}\) is the passive earth pressure of the soil layer where the anchor-plate is located; \(A_{3}\) is the front-end area of anchor-plate:  
\[\left.\begin{array}{l}A_{1}=lb\\ A_{2}=lh\\ A_{3}=bh\end{array}\right\} \tag{37}\]  
where \(l\) is the length of the anchor-plate; \(b\) = the width of the anchor-plate; \(h\) is the height of the anchor-plate.  
Because [PERSON] (2016) did not consider the cohesion of the fill soil when calculating the passive earth pressure on the front end of the anchor-plate, which obviously does not conform to the actual engineering situation, the improved formula is as follows:  
\[\sigma_{p}=k_{p}(\gamma h_{0}+q_{0})+2c\sqrt{k_{p}} \tag{38}\]  
where \(c\) is the soil cohesion of the soil layer where the anchor-plate is located; \(k_{p}\) is the Rankine passive earth pressure coefficient of the soil around the anchor-plate.