# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 3 Analysis of Soil Arching Under Passive Conditions
### Comparison with Circular and Parabolic Arches  
Expressions for circular (arc) and parabolic arches under the same sliding surface are derived for comparative analysis. (1) Circular Arch  
In horizontal layer methods that incorporate soil arching effects, the sliding backfill is assumed to be in a limit equilibrium state at every point. Based on the stress conditions at the wall (see Figure 2), the angle \(\theta_{\text{w}}\) between the major principal stress and the horizontal direction can be determined as follows:  
\[\theta_{\text{w}}=(\sin^{-1}(\sin\delta/\sin\varphi)+\delta)/2. \tag{28}\]  
According to the Mohr-Coulomb failure criterion, at limit equilibrium, the angle between the major principal stress direction and the failure plane is \(\pi/4-\varphi/2\). Consequently,  
Figure 6: Distribution of principal stress directions and soil arch curves in the sliding wedge: (**a**) \(\delta=0.3\varphi\); (**b**) \(\delta=0.5\varphi\). at the sliding surface itself, the major principal stress experiences no rotation, as shown in Figure 7. The circular arch geometry in Figure 7 satisfies the following:  
\[R\sin\theta_{\rm w}=z/\tan\phi,\ R\cos\theta_{w}=\overline{CO}. \tag{29}\]  
Thus, the vertical distance \(\overline{CO}\) is given by the following:  
\[\overline{CO}=z/(\tan\phi\cdot\tan\theta_{\rm w}). \tag{30}\]  
The center of the circular arch at height \(z\) is thus located at \((z/\tan\phi,z(1+1/\tan\phi\tan\theta_{\rm w}))\), and the function of the circular arch is given as follows:  
\[y=\sqrt{\left(z/\left(\tan\phi\sin\theta_{\rm w}\right)\right)^{2}-\left(x-z /\tan\phi\right)^{2}}\ +z(1+1/(\tan\phi\tan\theta_{\rm w})). \tag{31}\]  
where \(y\) is used to denote the vertical coordinate along the arch curve (to distinguish it from the depth coordinate \(z\)). (2) Parabolic Arch  
The parabolic arch is assumed to share the same boundary conditions for the major principal stress orientation as the circular arch: at \(x=0\): \(y=z\), \(y^{\prime}=-\tan\theta_{\rm w}\); at \(x=z/\tan\phi\): \(y^{\prime}=0\). Applying these boundary conditions allows the determination of the parabolic arch equation at height \(z\):  
\[y=(\frac{1}{2}\tan\theta_{\rm w}\tan\phi/z)x^{2}-x\tan\theta_{\rm w}+z. \tag{32}\]  
The soil arch curve derived in this study is compared with the conventional assumed parabolic and circular arches, as shown in Figure 8. For ease of comparison, the horizontal \(x\) and vertical \(y\) coordinates are normalized by \(L\), defined as the length of the horizontal layer at \(z=0.4H\). The coordinate origin in the figure corresponds to the point \((0,0.4H)\) within the wedge. Figure 8a,b shows that the derived soil arch closely aligns with both the circular and parabolic arches when the soil-wall friction angle is small (\(\delta<0.1\varphi\)). This suggests that, under low soil-wall friction conditions, the major principal stress trajectory can be reasonably approximated by a circular or parabolic shape. As the soil-wall friction angle increases, the derived soil arch becomes steeper, and the deviation from the assumed shapes  
Figure 7: Schematic diagram of circular arch shape under passive state conditions. becomes more pronounced. This clearly indicates that higher soil-wall friction enhances the soil arching effect within the backfill. Throughout the range examined, the difference between the parabolic and circular arches remains relatively minor, with the parabolic arch generally lying slightly below the circular one. The derived arch curve, however, exhibits the most significant sensitivity to changes in soil-wall friction and consistently lies below the parabolic arch. This greater curvature indicates that the proposed analytical method captures a stronger soil arching effect. A comparison between Figure 8a,b reveals that variations in \(q\) have no effect on the shapes of the circular and parabolic arches. This is because these assumed shapes are solely determined by the stress conditions at the wall back and the sliding surface, both of which are idealized as being in limit equilibrium.