# Rankine Active Earth Pressure of Unsaturated Filling under Earthquake
## 2 Principals and Methods
### Pseudo-static Analysis of Seismic Active Earth Pressure  
#### 2.2.1 Stress Analysis of The Rhombic Element  
The basic idea of the pseudo-static method is to simplify the seismic action into an inertial force acting on the research object. As shown in Figure 1, \(H\) is the wall height, \(\beta\) is the dip angle of the backfill, \(\delta\) represents the interface friction angle between the retaining wall and the backfill, \(\alpha\) is the inclination of the wall back, \(k_{\rm h}\) and \(k_{\rm v}\) are the horizontal and vertical seismic coefficients, \(E_{\rm a}\) is the resultant force of the active earth pressure, \((\sigma_{v}-u_{a})\) and \((\sigma_{h}-u_{a})\) are the net normal stresses, \(d_{\rm 0a}\) denotes the gravitational distance from the resultant action position to the bottom, and \(z_{\rm 0a}\) is the crack depth. Assuming the direction of the vertical seismic force to be upward and the direction of the horizontal seismic force to be outward, the seismic angle \(\eta\) could be obtained by  
\[\tan\eta=\frac{k_{\rm h}}{1-k_{\rm v}} \tag{6}\]  
Taking a rhombus element (\(abcd\)) as the study object, \(ad\) and \(bc\) are parallel to the wall back, while \(ab\) and \(cd\) are parallel to the surface of the backfill. During an earthquake, the research element turns into \(abc^{\prime}d^{\prime}\), where \(ad\) and \(bc\) are rotated by seismic angle \(\eta\) and the net normal stress \((\sigma_{\rm v}\)-\(u_{\rm a})\) turns into \((\sigma_{\rm v}\)-\(u_{\rm a})^{\prime}\) with an intersection angle of \(\eta\). If the pore air in the soil communicates with external atmosphere, \(u_{\rm a}=0\). Thus, we get  
\[(\sigma_{\rm v}-u_{\rm a})^{\prime}=\frac{\gamma z(1+k_{\rm v})\cos\beta}{\cos\eta} \tag{7}\]  
in which \(\gamma\) represents the actual volumetric weight of the backfill, and \(z\) is the depth. The stress characteristics of the active limit equilibrium state on the rhombus unit could be presented by the Mohr circle in Figure 2. The process could be stated as follows [20]. (1) Draw a line \(OB\) with an intersection angle of \(\eta+\beta\) from the coordinate origin, and mark \(B\) by a length of \((\sigma_{\rm v}-u_{\rm a})^{\prime}\), which is the net stress acting on \(ab\). (2) Determine a Mohr circle \(O_{1}\) with a radius of \(R\) through point \(B\), tangent to shear strength envelope \(O_{2}E\). (3) The circle intersects the line \(OB\) at the point \(A\), which represents the net stress \((\sigma_{\rm h}-u_{\rm a})^{\prime}\) acting on \(ad^{\prime}\). Figure 1: Calculation model. Figure 2: The Mohr stress circle of rhombic unit. As shown in Figure 2, we have  
\[\left\{ \begin{array}{ll}AA^{\prime}=R\sin[\lambda-(\beta+\eta)]\\ BB^{\prime}=R\sin[\lambda+(\beta+\eta)]\end{array} \right. \tag{8}\]  
\[OA=OB\cdot\frac{AA^{\prime}}{BB^{\prime}}=OB\frac{\sin[\lambda-(\beta+\eta)]}{ \sin[\lambda+(\beta+\eta)]}=\gamma z(1+k_{\rm v})\frac{\cos\beta}{\cos\eta} \frac{\cos(\beta+\eta)\tan\lambda-\sin(\beta+\eta)}{\cos(\beta+\eta)\tan \lambda+\sin(\beta+\eta)} \tag{9}\]  
In the triangle \(O_{1}EO_{2}\),  
\[\sin\varphi^{\prime}_{t}=\frac{O_{1}E}{O_{1}O_{2}}=\frac{R}{OA\cos(\beta+\eta )+R\cos[\lambda-(\beta+\eta)]+\left[c^{\prime}_{t}+(u_{a}-u_{w})\tan\varphi^{ b}_{t}\right]\cot\varphi^{\prime}_{t}} \tag{10}\]  
In triangles \(O_{1}BB^{\prime}\) and \(OB^{\prime}\), following equation could be obtained:  
\[\left\{ \begin{array}{ll}O_{1}B^{\prime}&=BB^{\prime}\cot[\lambda+(\beta+ \eta)]\\ &=OB\sin(\beta+\eta)\cot[\lambda+(\beta+\eta)]\\ &=\gamma z(1+k_{\rm v})\frac{\cos\beta}{\cos\eta}\sin(\beta+\eta)\cot[ \lambda+(\beta+\eta)]\\ O_{1}B^{\prime}&=R\cos[\lambda+(\beta+\eta)]\end{array} \right.