## 3 Analytical Model for the Highway Subgrade Stability
### The Relationship between the Subgrade Stability Factors and the ELB Height  
Assuming that the strength of the ELB is great enough, ELB shear failure does not occur. The overturn failure will occur as shown in Figure 5. \(E_{a}\) is the active earth pressure; \(F_{n}\) is the landslide pushing force; and \(E^{\prime}{}_{p}\) is the passive earth pressure. \(l\) is the distance from the \(E^{\prime}{}_{p}\) application point to the ELB bottom. \(m\) is the distance from the \(F_{n}\) application point to the ELB bottom. When the ELB is in a critical state, the torque on the ELB equals zero. Then,  
\[M_{h}=0 \tag{7}\]  
The torque equilibrium equation is established as follows. \[E^{\prime}{}_{p}\cdot l=F_{n}\cdot(H-\frac{h_{1}}{2})+E_{a}\cdot m \tag{8}\]  
where \(l=\frac{H}{3}\frac{\gamma HK_{p}+6\zeta\sqrt{K_{p}}}{\gamma HK_{p}+4\zeta \sqrt{K_{p}}}\), \(m=\frac{\gamma HK_{q}-2\zeta\sqrt{K_{q}}}{3\gamma K_{q}}\). The passive earth pressure \(E^{\prime}{}_{p}\) can be calculated as follows. \[E^{\prime}{}_{p}=1/2\times\gamma H^{2}\tan^{2}(45^{{}^{\circ}}+\frac{\varphi}{ 2})+2 cH\tan(45^{{}^{\circ}}+\frac{\varphi}{2})K_{p}=\tan^{2}(45^{{}^{\circ}}+ \frac{\varphi}{2}) \tag{9}\]  
The active earth pressure \(E_{a}\) can be calculated as follows. \[E_{a}=1/2\times\gamma H^{2}\tan^{2}(45^{{}^{\circ}}-\frac{\varphi}{2})-2 cH\tan (45^{{}^{\circ}}-\frac{\varphi}{2})+\frac{2c^{2}}{\gamma}K_{a}=\tan^{2}(45^{{}^ {\circ}}-\frac{\varphi}{2}) \tag{10}\]  
Substituting Equations (2), (3), (9) and (10) into Equation (8), we obtain Equation (11). Since \(H\) is much larger than \(h_{1}\), \(H\)-\(h_{1}/2\) can be replaced by \(H\). \[F_{s}=PH^{2}+QH+R/H+U-F_{n-1}\psi_{n}/G_{n}\sin\beta_{n} \tag{11}\]  
Figure 5: Diagram of the ELB overturn failure. where  
\[P = \left(\frac{\gamma K_{p}}{6}\frac{\gamma HK_{p}+6c\sqrt{K_{p}}}{\gamma HK _{p}+4c\sqrt{K_{p}}}-\frac{\gamma K_{a}}{6}\right)/G_{n}\sin\beta_{n}\] \[Q = \left(\frac{2c\sqrt{K_{p}}}{3}\frac{\gamma HK_{p}+6c\sqrt{K_{p}}}{ \gamma HK_{p}+4c\sqrt{K_{p}}}+c\sqrt{K_{a}}\right)/G_{n}\sin\beta_{n}\] \[U = \left(G_{n}\cos\beta_{n}\tan\varphi_{n}+c_{n}l_{n}-\frac{2c^{2}}{ \gamma}\right)/G_{n}\sin\beta_{n}\] \[M_{h}=0 \tag{7}\]0  
In Equation (11), \(\gamma\) is soil unit weight; \(K_{a}\) is the active earth pressure coefficient; \(K_{p}\) is the passive earth pressure coefficient; \(F_{n}\) and \(F_{n-1}\) are the residual sliding force of the block \(n\) and block \(n-1\), respectively; \(\Psi_{n}\) is the transfer coefficient; \(F_{s}\) is the stability factor; \(G_{n}\) is the weight of the block \(n\); \(\varphi_{n}\) is the internal friction angle of block \(n\) along the sliding surface; \(c_{n}\) is the cohesion of block \(n\) along the sliding surface; \(\beta_{n}\) is the bottom slope angle of block \(n\); \(l_{n}\) is the length of block \(n\) along the sliding surface; \(h_{1}\) is the height of the ELB above the sliding surface. Equation (11) shows that the stability factor \(F_{s}\) is positively correlated with the square of ELB height \(H\). The stability factor \(F_{s}\) increases with the increase in the ELB height \(H\). In a word, the stability factor \(F_{s}\) increases when the ELB height \(H\) and width \(B\) increase. The above analytical model can be used to design the subgrade and evaluate the subgrade stability. There are some assumptions for the above derivation process. For example, the soil pressure on the ELB is simplified as a horizontal force. In addition, from Equations (6) and (11), it is known that there are other parameters influencing the subgrade stability, such as ELM density, cohesion, and internal friction. Therefore, it is necessary to further study the influences of these parameters on the stability factors since experimental tests have high requirements for equipment and costs [23] and it is difficult to explore the influences of these parameters on the stability factors using Equations (6) and (11). However, numerical simulation is an efficient method to further study the influences of these parameters on the ELB stability factors.