# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 4 Parametric Analysis
### Effect of Various Parameters on the Passive Limit State  
The influence of cohesion (\(c\)) on the passive earth pressure resultant force (\(E_{p}\)) is depicted in Figure 19a. \(E_{p}\) increases with the rise in cohesion, while the critical width (\(L_{p}\)) decreases slightly as cohesion increases. When \(L/H>4/3\), the effect of backfill width \(L\) on \(E_{p}\) decreases significantly. For cohesionless soil (\(c=0\) kPa), the results of this study align with Coulomb theory when \(L=L_{p}\). For clay, the results in this study are marginally less than the Coulomb results when \(L=L_{p}\). This discrepancy occurs because the slip surface determined in this study is more critical than that predicted by the Coulomb theory. The Coulomb solution represents a special case in this study rather than the minimum value of the function \(E_{x2}(\theta)\). The intersection points of the \(E_{x2}(\theta)\) functions derived from both assumptions correspond to the Coulomb solution, as shown in Figure 19b. As the backfill width behind the wall approaches the critical width (\(L_{p}\)), the discrepancy between the results obtained by the proposed method and those derived from Coulomb theory under passive state is significantly smaller than the discrepancy observed under active conditions. This discrepancy arises from the fact that the cohesion resultant force (\(C\)) under the passive state has a significantly smaller magnitude compared to the self-weight (\(W\)) and the counterforce (\(R\)), rendering the variation in \(C\) less influential on \(E_{p}\). For the specific case studied here (\(c=15\) kPa, \(L=11.74\) m), a comparison of resultant  
Figure 19: (**a**) Effect of cohesion \(c\) on \(E_{p}\); (**b**) Comparison of \(E_{x1}(\theta)\) formed by triangular and trapezoidal wedges. Figure 18: (**a**) Effect of wall-soil friction angle \(\delta\) on \(E_{a}\); (**b**) Effect of wall-soil friction angle \(\delta\) on the slip surface. force under passive conditions is illustrated in Figure 20a. Cohesion (\(c\)) also affects the distribution of the slip surface, as shown in Figure 20b. For a constant backfill width (\(L\)), both the rupture angle (\(\theta\)) and the height of the slip surface inflection point (\(L\)tan\(\theta\)) exhibit an inverse relationship with cohesion \(c\). The effect of \(c\) on these parameters will be intensified as \(L\) increases. The rupture angle \(\theta\) decreases linearly with increasing \(L\). At the critical width, the difference between the rupture angle predicted by this study and [PERSON] theory grows with increasing cohesion. The effect of the friction angle (\(\varphi\)) on the passive earth pressure resultant force (\(E_{p}\)) is shown in Figure 21a. For a constant backfill width (\(L\)), \(E_{p}\) increases nonlinearly with increasing \(\varphi\). The smaller the backfill width (\(L\)), the more pronounced the effect of \(\varphi\) on \(E_{p}\). The critical width (\(L_{p}\)) is directly proportional to \(\varphi\). As backfill width (\(L\)) decreases, \(E_{p}\) grows exponentially. As the backfill consists of cohesionless soil, the results of this study at the critical width are consistent with [PERSON]'s solution. The rupture angle (\(\theta\)) decreases linearly with increasing \(\varphi\), and this trend remains unaffected by variations in backfill width. A linear relationship also exists between backfill width and rupture angle. As the backfill width increases, the passive rupture angle obtained in this paper gradually decreases until it reaches the Coulomb rupture angle, as illustrated in Figure 21a. Figure 21: (**a**) Effect of the friction angle \(\varphi\) on \(E_{p}\); (**b**) Effect of the friction angle \(\varphi\) on the slip surface. Figure 20: (**a**) Relationship between the resultant forces and the failure angle; (**b**) Effect of cohesion \(c\) on the slip surface. The impact of the wall-soil friction angle (\(\delta\)) on the passive earth pressure resultant force (\(E_{p}\)) is depicted in Figure 22a.