## 3 Analytical Model for the Highway Subgrade Stability  
### The Relationship between the Subgrade Stability Factors and the ELB Width  
The ELB and subgrade were made from compacted soil. Their strength is much greater than that of the soft soil. When the ELB strength is not enough, the subgrade is prone to shear failure as shown in Figure 3. To quantitatively analyze the subgrade stability factor, the following assumptions are made. The position of the sliding surface does not change with the increase in subgrade width and height. That is, \(h_{1}\) remains unchanged.  
When the ELB cannot provide adequate resistance, subgrade shear failure will occur and the ELB breaks along the sliding surface. The ELB can be divided into two parts by the sliding surface, as shown in Figure 4. Part I is above the sliding surface while part II  
Figure 3: Diagram of the subgrade shear failure mode.  
is below the sliding surface. The force on part I is demonstrated in Figure 4. The passive earth pressure on part I is \(E_{p}\). The landslide pushing force on part I from the sliding body is \(F_{n}\). The force on part I from part II is \(f\). When part I is in a critical state, then  
\[F_{n}=E_{p}+f \tag{1}\]  
The landslide body can be divided into many separate blocks. The landslide pushing force \(F_{n}\) can be calculated by the limiting equilibrium theory and the transfer coefficient method. The detailed calculation process is performed using the following equation.  
\[F_{n}=F_{n-1}\psi_{n}+F_{s}G_{n}\sin\beta_{n}-G_{n}\cos\beta_{n}\tan\varphi_{n }-c_{n}l_{n} \tag{2}\]  
\[\psi_{n}=\cos(\beta_{n-1}-\beta_{n})-\sin(\beta_{n-1}-\beta_{n})\tan\varphi_{n} \tag{3}\]  
where \(F_{n}\), \(F_{n-1}\) are the residual sliding force of block \(n\) and block \(n-1\); \(\Psi_{n}\) 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 the block \(n\) along the sliding surface; \(\beta_{n}\), \(\beta_{n-1}\) are the slope angle of the block \(n\), block \(n-1\) at the bottom, respectively; \(l_{n}\) is the length of the block \(n\) along the sliding surface.  
The force \(f\) is composed of two parts. One is the cohesion on the sliding surface, and the other one is the friction on the sliding surface. Then,  
\[f=B\times c+Bh_{1}\rho\tan\varphi \tag{4}\]  
In the equation, \(\rho\) is the density of the ELB body; \(h_{1}\) is the height of part I; \(c\) is the cohesion of the ELB body, and \(\varphi\) is the internal friction angle of the ELB body; \(B\) is the width of the body.  
The passive earth pressure \(E_{p}\) can be calculated according to the following equation.  
\[E_{p}=1/2\times\gamma h_{1}^{2}\tan^{2}(45^{\circ}+\frac{\varphi}{2})+2 ch_{1} \tan(45^{\circ}+\frac{\varphi}{2}) \tag{5}\]  
By substituting Equations (2)-(5) into Formula (1), we obtain Equation (6).  
\[F_{s}=B(c+h_{1}\rho\tan\varphi)/(G_{n}\sin\beta_{n})+W \tag{6}\]  
where  
\[W=\tan\beta_{n}\tan\varphi_{n}+\left(c_{n}l_{n}+\frac{1}{2}\times\gamma h_{For a given subgrade and the site, the physical and mechanical parameters are determined. Then, \(W\) is a constant value. In Equation (6), only \(F_{s}\) and \(B\) are variables. From Equation (6), it can be seen that the stability factor \(F_{s}\) is positively correlated with the ELB width \(B\). With the increase in the ELB width, the subgrade stability factor \(F_{s}\) increases accordingly.