# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 6 Conclusions  
This study introduces an analytical method for passive earth pressure using 2D differential element equilibrium, critically eliminating a priori soil arch assumptions. The approach yields the internal stress field and pressure distribution and allows the passive arch shape to be derived analytically from principal stress trajectories. Parametric studies and comparisons with established theories and experimental data validated the method's effectiveness in capturing essential arching effects and passive earth pressure behavior. The principal conclusions are as follows:  
1. Owing to the passive arching effect, the limit equilibrium state is achieved only locally near the wall back and lower sliding surface, which contrasts with methods that assume the entire wedge to be in a limit equilibrium state. The derived major principal stress varies in magnitude and direction at constant height. 2. The derived passive soil arch shape lies below conventional parabolic/circular assumptions, becoming significantly steeper with increasing soil-wall friction angle. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \begin{tabular}{c} **Proposed** \\ **Method** \\ \end{tabular} & **[PERSON]** & **[PERSON]** & \begin{tabular}{c} **[PERSON] and** \\ **[PERSON] [1]** \\ \end{tabular} & \
\begin{tabular}{c} **[PERSON] et al.** \\ **[11]** \\ \end{tabular} &
\begin{tabular}{c} **[PERSON] and** \\ **[PERSON] [12]** \\ \end{tabular} \\ \hline SSE for earth pressure & 3.9272 & 21.9925 & 4.2317 & 14.7391 & 6.6133 & 5.149 \\ \(R^{2}\) for earth pressure & 0.8108 & \(-\)0.0595 & 0.7961 & 0.2899 & 0.6814 & 0.7519 \\ Application height error (\%) & \(-\)10.12 & 13.07\% & 13.07 & 13.07 & \(-\)45.33 & \(-\)24.31 \\ \hline \end{tabular} Note: A negative value for application height error indicates that the predicted application height is lower than the experimental result.  
\end{table} 
Table 1: Comparison of passive earth pressure prediction accuracy among different methods. Figure 14: Comparison between theoretical predictions [1,11,12] and experimental results [27]. Unlike fixed-geometry assumptions, it is also sensitive to surcharge, reflecting its dependence on the overall stress field. 3. The passive earth pressure coefficient increases with the internal friction angle and surcharge, consistent with established theories. However, it shows a non-monotonic relationship with the soil-wall friction angle, peaking at an intermediate value before decreasing. 4. Comparisons indicate the proposed method reasonably predicts pressure distributions observed in experiments, notably showing closer agreement in the lower wall regions than other theories. It should be noted, however, that, due to space constraints, the reasons behind some results remain insufficiently explored, particularly those exhibiting marked deviations from existing theories, such as the non-monotonic trend of \(K_{p}\) with increasing \(\delta\). Subsequent work is required to integrate extensive experimental data and numerical simulations to provide a more detailed interpretation and analysis of these theoretical outcomes. Additionally, the assumptions adopted in this study, while facilitating analytical tractability, inevitably introduce certain limitations. The assumptions of plane strain conditions, homogeneous cohesionless soils, planar sliding surfaces, and full mobilization of wall-soil friction simplify the analysis but may not fully reflect complex field conditions such as soil heterogeneity, cohesion effects, irregular failure surfaces, or incomplete interface mobilization. These idealizations restrict the direct applicability, particularly for cohesive or highly heterogeneous backfills, and do not account for 3D effects or complex geometries. Therefore, for practical engineering applications, especially under conditions significantly different from the idealizations, a comprehensive assessment can be achieved by integrating the results of the proposed method with site-specific data or numerical analyses. Writing--original draft preparation, P.Y.