# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 2 Derivation of Equilibrium Differential Equations and Solution for Passive Earth Pressure
### Boundary Conditions  
(1) Upper Surface (\(z=H\)):  
The backfill surface is horizontal and subjected to a uniform load \(q\). Thus, at \(z=H\), \(\sigma_{z}=q\), \(\tau_{zx}=0\).  
(2) Soil-Wall Interface (\(x=0\)):  
At the soil-wall interface (\(x=0\)), the soil element reaches a state of limit equilibrium, with the soil-wall friction angle (external friction angle) \(\delta\) fully mobilized. The stress state, represented by a Mohr's circle in Figure 2, yields the following relationships:  
\[\tau_{w}(z)=\mu_{w}\sigma_{wx}(z); \tag{3}\]  
\[\sigma_{wx}(z)=K_{\mathrm{wp}}\sigma_{wx}(z). \tag{4}\]  
where \(\tau_{w}\), \(\sigma_{wx}\), and \(\sigma_{wz}\) represent the shear stress, horizontal normal stress, and vertical normal stress, respectively, acting on the soil element adjacent to the wall face; \(\mu_{w}\) is the external friction coefficient, defined as \(\mu_{w}=\tan\delta\); and \(K_{\mathrm{wp}}\) (distinct from the passive earth pressure coefficient \(K_{p}\)) is the lateral stress ratio at the wall. From Figure 2, the geometric relationships are as follows:  
\[\theta_{1}=\theta_{2}+\delta,\,\sin\theta_{2}=\sin\delta/\sin\varphi. \tag{5}\]  
Thus,  
\[\theta_{1}=\delta+\sin^{-1}(\sin\delta/\sin\varphi), \tag{6}\]  
and we can obtain the following:  
\[K_{\mathrm{wp}}=\frac{1+\cos(\sin^{-1}(\sin\delta/\sin\varphi)+\delta)\sin \varphi}{1-\cos(\sin^{-1}(\sin\delta/\sin\varphi)+\delta)\sin\varphi}. \tag{7}\]  
Substituting Equation (4) into Equation (3), the stress boundary condition at \(x=0\) becomes the following:  
\[\tau_{w}(z)=A_{\mathrm{p}}\sigma_{wx}(z), \tag{8}\]  
Figure 2: Mohr circle representation of the stress state for a soil element adjacent to the wall (\(x=0\)) under the passive limit equilibrium state.  
where \(A_{\rm p}=\mu_{\rm w}K_{\rm wp}\).  
(3) Potential Sliding Surface (\(x=z\cot\phi\)):  
The classical limit equilibrium method developed by [PERSON] assumes a planar sliding surface for passive earth pressure behind retaining walls, with its inclination angle \(\phi\) determined by the internal friction angle \(\varphi\) and the wall friction angle \(\delta\). However, this analysis neglected the soil arching effect. Subsequent studies attempted to incorporate soil arching by considering the equilibrium of horizontal soil layers [11; 12; 13]. While advancing the analysis, these methods often relied on the simplifying assumption of zero shear stress acting on horizontal layers, which may not accurately represent actual soil behavior in the field.  
On the other hand, research has shown that, under passive conditions, the inclination of the sliding surface is affected to some extent by soil-wall friction angle but is mainly governed by the internal friction angle. Furthermore, it has been suggested that variations in the specific shape and inclination angle of the sliding surface tend to have a relatively minor impact on the magnitude of the calculated earth pressure [21]. Based on these findings, several researchers, including [PERSON] and [PERSON] [22] and [PERSON] and [PERSON] [23], assumed that the inclination of the sliding surface depends only on the internal friction angle. By combining this assumption with the equilibrium of horizontal soil layers and incorporating soil arching effects, their analyses yielded passive earth pressure values that were reported to align well with experimental data.  
For the sake of analytical simplicity and ease of calculation, this study adopts the assumption proposed in References [22; 23], where the inclination of the potential sliding surface with respect to the horizontal is considered a function of the internal friction angle only (i.e., \(\phi=\pi/4-\varphi/2\)). Under the Mohr-Coulomb criterion, this assumption implies that the principal stress directions remain unchanged along the sliding plane.