# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 5 Comparison with Existing Theories and Experimental Data  
To assess the validity and practical applicability of the passive earth pressure results derived in this study, a comparative analysis was conducted with existing theoretical solutions and available experimental data. Figure 11 compares the horizontal passive earth pressure distribution obtained from the present study with predictions from [PERSON] et al. [11], [PERSON] and [PERSON] [12], and classical approaches, such as the Coulomb, Rankine, and Caquot-Kerisel theories. Classical theories, such as those of [PERSON], Rankine, and Caquot-Kerisel, generally predict linear passive earth pressure distributions against the wall height, although the specific magnitudes vary based on their underlying assumptions regarding wall friction and failure surface geometry. As shown in Figure 11, for the depicted conditions, the results from the Coulomb and Caquot-Kerisel theories are relatively close to each other, and both predict higher pressures than Rankine theory. In contrast, the distributions derived from methods explicitly incorporating soil arching effects, including the present work, [PERSON] et al. [11], and [PERSON] and [PERSON] [12], are distinctly nonlinear. These curved profiles predict lower pressures than the classical linear theories in the upper part of the wall. However, they tend to predict significantly higher pressures near the wall base. This deviation from a linear profile reflects a redistribution of stresses attributed to the soil arching mechanism, which decreases pressure in the upper wall region while concentrating it near the base. Figure 12a presents a comparison of the calculated passive earth pressure coefficient \(K_{p}\) as a function of the internal friction angle \(\varphi\) against results from the aforementioned theories. All models predict an increase in \(K_{p}\) with increasing \(\varphi\), reflecting enhanced shear resistance with greater frictional strength. However, significant quantitative differences are observed between the theories. The predictions from [PERSON] et al. [11] yield the highest \(K_{p}\) values, while the results from the Coulomb theory and [PERSON] and [PERSON]'s method [12] are closely aligned. The Caquot-Kerisel theory predicts \(K_{p}\) values slightly lower than those  
of Coulomb. Among all theories compared, Rankine provides the lowest \(K_{p}\) predictions. Notably, the \(K_{p}\) values obtained using the present method lie between the predictions of the Caquot-Kerisel and Rankine theories for the range of \(\varphi\) considered. The divergence between the predictions from the various theories generally increases at higher internal friction angles. The influence of \(\varphi\) on the normalized height of the resultant force application point is compared in Figure 12b. Classical theories yield a constant \(h_{p}=H/3\), independent of \(\varphi\), consistent with a uniform linear distribution across the wall height. Conversely, \(h_{p}\), from this study and References [11; 12], decreases as \(\varphi\) increases, reflecting a shift in the pressure resultant due to curvilinear distributions. At \(\varphi=0\), all models converge at \(h_{p}=H/3\), indicative of a cohesionless, frictionless state with no arching. As \(\varphi\) rises, divergence emerges: the \(h_{p}\) from [PERSON] et al. [11] exhibits the most significant decline, followed by [PERSON] and [PERSON] [12] with a moderate reduction, while the present study shows the smallest decrease. Figure 13a illustrates the effect of the soil-wall friction angle \(\delta\) on the passive earth pressure coefficient \(K_{p}\) for a fixed internal friction angle \(\varphi=30^{\circ}\). As expected for a smooth wall condition, all theories converge to the same \(K_{p}\) value at \(\delta=0\), which corresponds to the Rankine passive earth pressure coefficient. As \(\delta\) increases from zero, different trends emerge. The result predicted by Rankine theory shows that \(K_{p}\) remains constant, as wall friction is neglected in the formulation.