# Rankine Active Earth Pressure of Unsaturated Filling under Earthquake
## 2 Principals and Methods  
### Unified Shear Strength of Unsaturated Soil  
The shear strength of unsaturated soils is one of the basic problems in unsaturated soil mechanics. Much research has been carried out and various calculation formulas have been suggested [18; 19]. However, based on Mohr-Coulomb strength theory, these formulas cannot reflect the effect of intermediate principal stress. Consequently, they cannot fully estimate the strength potential and self-supporting capacity of unsaturated backfills.  
The unified strength theorem and the calculation formula of unsaturated soil could be denoted as follows [11]:  
\[\left\{\begin{array}{l}F=(\sigma_{1}-u_{a})-\frac{\sigma_{c}}{\sigma_{t}(1+ b)}[b(\sigma_{2}-u_{a})+(\sigma_{3}-u_{a})]=\sigma_{c},\,\left(\sigma_{2}<\frac{ \sigma_{1}\sigma_{t}+\sigma_{3}\sigma_{c}}{\sigma_{c}+\sigma_{t}}\right)\\ F^{\prime}=\frac{1}{1+b}[(\sigma_{1}-u_{a})+b(\sigma_{2}-u_{a})]-\frac{\sigma_ {c}}{\sigma_{t}}(\sigma_{3}-u_{a})=\sigma_{c},\,\left(\sigma_{2}\geq\frac{ \sigma_{1}\sigma_{t}+\sigma_{3}\sigma_{c}}{\sigma_{c}+\sigma_{t}}\right)\end{array}\right. \tag{1}\]  
\[\tau_{t}=c^{\prime}_{t}+(\sigma-u_{a})\tan\varphi^{\prime}_{t}+(u_{a}-u_{w}) \tan\varphi^{b}_{t} \tag{2}\]where  
\[\left\{\begin{array}{l}c^{\prime}t=\frac{2(1+b)c^{\prime}\cos\phi^{\prime}}{2+b(1 +\sin\phi^{\prime})}\frac{1}{\cos\phi^{\prime}t},\\ \sin\phi^{\prime}t=\frac{2(1+b)\sin\phi^{\prime}}{2+b(1+\sin\phi^{\prime})}\\ \sin\phi^{\rm b}_{\rm t}=\frac{2(1+b)\sin\phi^{\rm b}}{2+b\left(1+\sin\phi^{ \rm b}\right)}\end{array}\right. \tag{3}\]  
In the above equations, \(F\) and \(F^{\prime}\) are yield functions under different stress conditions, \(\sigma\) denotes the total normal stress, \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) represent the principal stresses and \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\); \(u_{a}\), \(u_{w}\) are the pore air pressure and pore water pressure. Variables \(\sigma_{t}\) and \(\sigma_{c}\) are the tensile strength and compressive strength of the filling; \(\tau_{f}\) is the shear strength of unsaturated soil; \(c^{\prime}\) and \(\phi^{\prime}\) are the effective cohesion and effective friction angle of the saturated soil, respectively; \((\sigma-u_{a})\) is the net normal stress acting on the sliding surface; \((u_{a}-u_{w})\) is the matric suction on the sliding surface; \(\phi^{\rm b}\) is the friction angle attributed by the matric suction; \(c^{\prime}t\) and \(\phi^{\prime}t\) are the unified effective cohesion and unified effective friction angle, respectively; \(\phi^{\rm b}t\) represents the unified friction angle attributed by the matric suction; and \(b\) is the unified strength factor (\(0\leq b\leq 1\)). The lower bound \(b=0\) refers to the Mohr-Coulomb strength criterion. On the contrary, the upper bound \(b=1\) refers to the twin shear strength theory. It can be obtained by fitting Equation (2) with the true triaxial test results.  
Rewriting Equation (2), we get  
\[\tau_{\rm f}=c_{\rm tt}+(\sigma-u_{\rm a})\tan\phi^{\prime}t \tag{4}\]  
in which the apparent cohesion \(c_{\rm tt}\) is  
\[c_{\rm tt}=c^{\prime}t+(u_{\rm a}-u_{\rm w})\tan\phi^{\rm b}_{\rm t} \tag{5}\]