# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 5 Conclusions  
When the backfill width behind the wall is finite, this paper posits the formation of a trapezoidal slip wedge at the limit state. According to the static equilibrium condition of the slip wedge, a calculation method for the earth pressure considering the backfill width is proposed. The effects of the cohesion, internal friction angle and wall-soil friction angle on the earth pressure were systematically investigated. The principal findings can be summarized as follows:  
1. Under the active limit state, the earth pressure resultant force behind the wall exhibits a parabolic relationship with the backfill width. For cohesionless soil, the earth pressure continuously increases until it reaches the Coulomb active earth pressure, while clay experiences an initial increase followed by a decrease in earth pressure. 2. For cohesionless soils with a smooth wall back, the passive earth pressure resultant force obtained by the proposed method fully coincides with the Coulomb's solution within the critical width range, and the corresponding rupture angles equal to the Coulomb value. As the wall-soil friction angle increases, the influence of backfill width on the passive earth pressure becomes increasingly pronounced, while exhibiting distinct exponential decay characteristics with increasing backfill dimensions. Figure 22: (**a**) Effect of the wall-soil friction angle \(\delta\) on \(E_{p}\); (**b**) Effect of wall-soil friction angle \(\delta\) on the slip surface. Upon reaching the critical width, the approach presented in this paper can be simplified to the Coulomb theory for cohesionless soil. For clay, the results obtained in this paper are higher than those predicted by the Coulomb theory under an active state while being slightly lower under a passive state. 4. For a given backfill width, the cohesion \(c\) is inversely proportional to the resultant active limit earth pressure force \(E_{a}\) and directly proportional to the resultant passive limit earth pressure force \(E_{p}\). As the internal friction angle \(\varphi\) increases, \(E_{a}\) decreases nonlinearly, while \(E_{p}\) experiences a significant growth. Furthermore, variations in the wall-soil friction angle \(\delta\) have a relatively minor influence on \(E_{a}\), whereas \(E_{p}\) grows exponentially with increasing \(\delta\). Conceptualization, X.W. and F.D. ; writing--original draft preparation, X.W. and F.D. ; validation, X.W., L.Z. and J.G. ; formal analysis, X.C. and H.X. ; funding acquisition F.D., J.G. and H.X. All authors have read and agreed to the published version of the manuscript. This research was supported by the National Natural Science Foundation of China (No. 51979225, 52009107, 52309172), the Key Laboratory for Science and Technology Coordination and Innovation Projects of Shaanxi Province (No. 2014 SZS15-Z01) and the Natural Science Foundation of Shaanxi Province (No. 2020 JQ-584). Not applicable. The data used to support the findings of this study are available from the corresponding author upon reasonable request. The authors declare no conflict of interest. The resultant earth pressure on the retaining wall  
Reaction forces on the slip wedge by the soil below the slip surface  
The weight of the slip wedge  
The reaction force exerted on the slip wedge by the soil below the slip surface  
The resultant cohesive force on the slip surface  
The height of the retaining wall  
The width of backfill behind the wall  
The reaction coefficient  
The rupture angle  
The wall-soil friction angle  
The cohesive force  
The internal friction angle  
The unit weight of soil  
Intermediate variable (Tangent value of rupture angle \(\theta\))  
Intermediate variable, \(n=1\)-4. (Related to \(\gamma\), \(c\), \(\varphi\), \(\delta\), \(H\), \(L\), \(a_{s}\))  
Intermediate variable, \(n=1\)-2. (Related to \(A_{n}\)\(\sim\)\(G_{n}\))  
Intermediate variable, \(n=1\)-2. (Related to \(a_{n}\)\(\sim\)\(e_{n}\))  
The solution of the quartic equation of one variable, n = 1-2.