# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 4 Parametric Analysis
### Effect of Various Parameters on the Active Limit State  
The impact of cohesion (\(c\)) on the active limit earth pressure resultant force (\(E_{a}\)) behind the wall is illustrated in Figure 15a. When the backfill consists of cohesionless soil, \(E_{a}\) increases nonlinearly with growing backfill width \(L\), but the rate of increase gradually decreases. \(E_{a}\) reaches its maximum value at \(L\) = 4.19 m, where the corresponding width represents the critical width \(L_{a}\). At the critical width, both \(E_{a}\) and \(\theta\) obtained by the proposed method in this paper show complete consistency with [PERSON]'s results. For clay, the relationship between backfill width (\(L\)) and resultant active earth pressure force (\(E_{a}\)) transitions from being initially proportional to subsequently inversely proportional. As cohesion (\(c\)) increases, the critical width (\(L_{a}\)) progressively decreases. At \(L\) = \(L_{a}\), the results presented in this paper slightly exceed [PERSON]'s theoretical predictions, with the discrepancy gradually increasing with higher cohesion values. This phenomenon arises because [PERSON]'s triangular wedge assumption constitutes a special case of the trapezoidal wedge assumption introduced in this study. Specifically, the slip surface in [PERSON]'s theory does not correspond to the maximum value of \(E_{x1}(\theta)\). The intersection point of the \(E_{x1}(\theta)\) functions resulting from these two assumptions is the maximum value of the triangular wedge assumption ([PERSON] theory), as depicted in Figure 15b. The impact on the slip surface is shown in Figure 16. As the backfill width (\(L\)) increases, the rupture angle (\(\theta\)) derived from the method presented in this study continuously decreases and gradually approaches the Coulomb rupture angle. Simultaneously, the slip surface inflection point gradually approaches the ground surface. For the same backfill width, the influence of increasing cohesion on the slip surface gradually decreases. As shown in Figure 16, cohesionless soil and clay exhibit marked differences in the variation range of rupture angles with respect to backfill width, which is attributed to the inconsistency in the values of the reaction coefficient \(a_{s}\). The influence of the internal friction angle (\(\varphi\)) on the active earth pressure resultant force (\(E_{a}\)) is depicted in Figure 17a. When the backfill width is constant, \(E_{a}\) decreases inversely with increasing \(\varphi\), and the effect of \(\varphi\) on \(E_{a}\) gradually decreases. The leftward shift of the peak in the \(E_{a}\)-\(L\) curve indicates an inverse proportionality between the critical width \(L_{a}\) and \(\varphi\). Since \(c\) = 0 kPa, the results in this paper at the critical width are consistent with the Coulomb earth pressure. As \(\varphi\) increases, the rupture angle \(\theta\) exhibits a linear increase, while variations in backfill width do not alter the magnitude of the change in the  
Figure 15: (**a**) Effect of cohesion \(c\) on \(E_{a}\); (**b**) Comparison of \(E_{x1}(\theta)\) formed by triangular and trapezoidal wedges. rupture angle. Conversely, the rupture angle \(\theta\) decreases linearly with increasing backfill width, asymptotically approaching the Coulomb rupture angle. The height at the slip surface inflection point demonstrates a nonlinear increase with higher \(\varphi\). Notably, when \(L=L_{a}\), the rupture angle in this study coincides with the Coulomb rupture angle, and the slip surface inflection point coincides with the ground surface. Comparative analyses under varying backfill widths are depicted in Figure 17b. The impact of the wall-soil friction angle (\(\delta\)) on the active earth pressure resultant force (\(E_{a}\)) and the slip surface is illustrated in Figure 18a,b. \(E_{a}\) is inversely proportional to \(\delta\), and its influence diminishes rapidly as \(\delta\) increases. The critical width \(L_{a}\) increases linearly with \(\delta\).