# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 3 Analysis of Soil Arching Under Passive Conditions
### Derivation of Soil Arch Shape  
Stress analysis within the wedge-shaped backfill reveals that the major principal stress varies in magnitude and direction at a given height. This observation stands in contrast to the assumptions commonly adopted in horizontal layer methods incorporating soil arching effects, which typically assume that the magnitude of the major principal stress remains constant along each horizontal layer, with only its orientation changing.  
The stress state of an arbitrary soil element along the soil arch is illustrated in Figure 5. The slope of the tangent to the soil arch curve is determined to be the tangent of the angle between the major principal stress direction and the horizontal axis, expressed as \(-\tan\xi_{\tau}t\). The negative sign indicates that the arch exhibits a downward concave profile. The relationship depicted in Figure 5 is given by the following:  
\[\tan 2\xi_{\tau}^{\prime}=\frac{2\tau_{xx}}{\sigma_{x}-\sigma_{z}}, \tag{24}\]  
where \(\xi_{\tau}^{\prime}\) denotes the angle between the major principal stress direction and the horizontal.  
From this, we can obtain the following:  
\[\tan\xi_{\tau}t=\left\{\begin{array}{ll}\frac{-1+\sqrt{\tan^{2}2\xi_{\tau}^{ \prime}t+1}}{\tan 2\xi_{\tau}^{\prime}t}&\tan 2\xi_{\tau}t^{\prime}\geq 0\\ \frac{-1-\sqrt{\tan^{2}2\xi_{\tau}^{\prime}t+1}}{\tan 2\xi_{\tau}^{\prime}t}& \tan 2\xi_{\tau}t^{\prime}<0\end{array}\right.. \tag{25}\]  
Substitution of Equation (24) into Equation (25) yields the following:  
\[\tan\xi_{\tau}t(\sigma)=\left\{\begin{array}{ll}\frac{-1+\sqrt{\frac{\tan 2\tau_{xx}}{ \sigma_{x}-\sigma_{z}}^{2}+1}}{\frac{\sigma_{x}-\sigma_{z}}{\sigma_{x}-\sigma_ {z}}}&\sigma_{x}\geq\sigma_{z}\\ \frac{-1-\sqrt{\frac{\tan 2\tau_{xx}}{\sigma_{x}-\sigma_{z}}^{2}+1}}{\frac{\sigma_{x}- \sigma_{z}}{\sigma_{x}-\sigma_{z}}}&\sigma_{x}<\sigma_{z}\end{array}\right., \tag{26}\]  
Figure 5: Stress state of soil element along the soil arch.  
The expression of the soil arch curve, denoted as \(f_{\text{arch}}(x)\), is obtained by integrating the tangent slope \(-\tan\xi_{\tau}t\) with respect to \(x\), given by the following:  
\[f_{\text{arch}}(x,z)=-\int_{0}^{x}\tan\xi_{\tau}t(t,z)dt+z. \tag{27}\]  
Equation (27) provides the expression for the soil arch curve at a given height \(z\). By substituting the stress solutions (Equations (14)-(16)) into Equation (27) and performing the integration, an explicit expression for the soil arch curve is obtained. For the passive sliding wedge considered in this study, the range of \(x\) is defined as (0, \(z\cot\phi\)).  
The distribution of the soil arch within the backfill is shown in Figure 6 for \(\varphi=30^{\circ}\)\(q=0.5\gamma H\), under varying soil-wall friction angles (\(\delta=0.3\varphi\) in Figure 6a, \(\delta=0.5\varphi\) in Figure 6b). The major and minor principal stress directions are indicated by dark green and red lines, respectively. The results show that soil arch curves at different depths exhibit similar shapes. The slopes of the arches at different heights are nearly constant at the wall face (\(x=0\)) and gradually decrease with increasing horizontal distance, eventually approaching zero at the sliding surface. This trend is consistent with the initial assumption of no principal stress rotation along the sliding boundary. Additionally, increasing the soil-wall friction angle leads to steeper arch profiles, reflecting a more pronounced soil arching effect within the backfill.