# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 3 Experimental Verification
### DEM Numerical Experiment
#### 3.2.2 Active Limit Earth Pressure  
To investigate the effect of the backfill width (\(L\)) on the active limit earth pressure, the soil height (\(H\)) behind the model wall is taken to be 6 m. According to Equation (10), the maximum width (\(L_{a}\)) satisfying the finite soil conditions is 3.92 m. Therefore, the backfill width is varied as follows: 0.5 m, 1 m, 1.5 m, 2 m, 2.5 m, 3 m, and 3.5 m for this study. Particles are generated using the stratified rainfall method. The particle radius needs to be increased to simplify computations due to the relatively large model size compared to the particle radius. Previous studies [46; 47] have demonstrated that the influence of particle size becomes negligible when the ratio of the shortest side of the model to the maximum particle radius (\(R_{max}\)) exceeds 100. Consequently, in this study, the particle size increased by a factor of 5 to 15, depending on the backfill width behind the wall. The convergence of the calculation results is evaluated based on the ratio of the average unbalanced force to the average contact force (referred to as the ratio-average). The models generated for each width are illustrated in Figure 7.  
The left wall of the model was displaced at a rate of 1 mm/s while the right wall remained stationary. The numerically calculated time steps ranged between \(10^{-5}\) and \(10^{-6}\) s. The earth pressure behind the wall was determined using measurement circles with a radius of 0.1 m and center spacing of 0.2 m. The measured results were then compared with the results from the Jaky formula [48], as shown in Figure 8a. The distribution of the static earth pressure under various backfill widths shows good agreement with the Jaky formula, confirming the rationality of the initial state simulation. Data were recorded at 0.5 mm intervals until the wall displacement reached 10 mm, after which recording intervals \begin{table}
\begin{tabular}{c c c c c c c c} \hline \begin{tabular}{c} **Particle** \\ **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} **Grain Size** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Effective** \\ **Modulus** \\ **\(E^{*}\)(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Stiffness** \\ **Ratio** \\ \(k^{*}\) \\ \end{tabular} & \begin{tabular}{c} **Friction** \\ **Coefficient** \\ **\(\mu\)** \\ \end{tabular} & \begin{tabular}{c} **Rolling** \\ **Friction** \\ **Coefficient** \\ **\(\mu\)** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum** \\ **Attractive** \\ **Force (N)** \\ \end{tabular} &
\begin{tabular}{c} **Attraction** \\ **Range** \\ **(mm)** \\ \end{tabular} \\ \hline
2650 & 0.71–0.18 & 35 & 3.6 & 0.21 & 0.8 & 10 & 0.35 \\ \hline \end{tabular}
\end{table} 
Table 2: Microscopic parameters. Figure 7: Numerical model under the active state. were extended to 1 mm. The active limit earth pressure in this paper was determined as the average of five data points, two preceding the minimum value and two subsequent to it. The relationship between the earth pressure resultant force and the wall displacement for each backfill width is shown in Figure 8b. As shown in Figure 8b, as the wall displacement increases, the smaller the backfill width (\(L\)) behind the wall, and the faster the earth pressure on the retaining wall decreases. When \(L>2\) m, the rate of decrease in earth pressure diminishes rapidly. Furthermore, the wall displacement required to reach the limit state varies with backfill width, exhibiting a proportional relationship between these two parameters. The influence of backfill width on earth pressure becomes significantly reduced when the width exceeds one-third of the wall height. To validate the method proposed in this paper, the DEM results are compared with the results of this paper, as illustrated in Figure 9a. The Pearson correlation coefficient between the two sets of results was 0.993, showing a consistent trend. The error between the two datasets decreases non-linearly as backfill width increases, ranging from a maximum of 36.2% to a minimum of 6.8%.