# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 3 Experimental Verification
### DEM Numerical Experiment
#### 3.2.3 Passive Limit Earth Pressure  
In the passive state, the numerical model maintains a wall height of 6 m. As determined by Equation (19), the maximum width (\(L_{p}\)) of the finite soil is 11.76 m. Consequently, the backfill widths (\(L\)) were configured as 3 m, 3.75 m, 4.5 m, 6 m, 7.5 m, 9 m, and 10.5 m. The numerical models generated for each width are presented in Figure 11.  
The static earth pressure distributions under different widths are shown in Figure 12a, which demonstrates good agreement with the Jack formula, confirming the validity of the initial simulation. The variation pattern of earth pressure resultant force with wall displacement is illustrated in Figure 12b. The passive earth pressure resultant force (\(E_{p}\)) was calculated as the average of the maximum values and its four adjacent data points.  
Figure 10: Distribution of new contacts under different backfill widths.  
Figure 9: (**a**) Relationship between the width of the soil behind the wall and \(E_{a}\); (**b**) Distribution of earth pressure at various widths.  
In Figure 12b, it is evident that in the passive state, the rate of increase in earth pressure behind the wall is inversely proportional to the backfill width (\(L\)). The resultant force of the passive limit earth pressure gradually decreases with increasing backfill width. Furthermore, the wall displacement required to reach the limit state exhibits a direct proportion to the backfill width. As \(L\) increases from 3.5 m to 10.5 m, the wall displacement needed for the soil behind the wall to reach the limit state progressively increases from 5% to 8.5% of the wall height.  
The passive limit earth pressure resultant force (\(E_{p}\)) under various backfill widths are shown in Figure 13a. As the backfill width (\(L\)) behind the wall decreases, \(E_{p}\) demonstrates an exponential growth trend. When \(L/H>1\), the effect of backfill width on the earth pressure decreases sharply. At this point, the results from this study show close agreement with the Coulomb results while being marginally higher than the experimental measurements. The Pearson correlation coefficient between the two sets of results is 0.986, indicating a broadly consistent trend. The errors are positively correlated with the backfill width, ranging from a maximum of 18.4% to a minimum of 1.3%. The distribution of the earth pressure is shown in Figure 13b, and the variation trend of the results in this paper is in good agreement with the experimental results. Likewise, owing to the nonlinear increase in the soil modulus with burial depth, the increment of the earth pressure is proportional to the burial depth.  
Figure 11: Numerical model under the passive state.  
Figure 12: (**a**) Static earth pressure with different backfill widths; (**b**) Relationship between the passive earth pressure resultant force and displacement.  
The slip surface under the passive state for various widths is depicted in Figure 14a. As the backfill width increases, the angle between the slip surface and the horizontal plane decreases linearly, and the slip surface inflection point gradually approaches the ground surface. A comparison of the earth pressure resultant forces on both sides of the slip wedge above the slip surface inflection point is shown in Figure 14b. The magnitudes of both forces are essentially identical, which validates the rationality of the assumptions proposed in this paper.