# Calculation Method of Earth Pressure Considering Wall Displacement and Axial Stress Variations
## 3 Example Verification
### Discrete Element Experiment  
The discrete element software PFC2d was used in this study, with the Adhesive Rolling Resistance Linear Model (ARRL) as the contact model. The contact model was fitted to the clay of the Doumen Reservoir, and its macroscopic mechanical parameters were determined using the GDS stress-path triaxial instrument. These parameters are listed in Table 1.  
The contact model parameters were calibrated using numerical biaxial tests, as shown in Figure 7. The stress-strain curve resulting from the calibrated parameters is illustrated in Figure 8, which exhibits good agreement with the results of the indoor experiments. Figure 8: Soil stress-strain curve. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specific Gravity of Soil Particle/\(Gs\)** & **Density/g cm\({}^{-3}\)** & **Void Ratio/\(e\)** & \(c\)/kPa** & \(\varphi^{O}\) \\ \hline
2.7 & 1.78 & 0.808 & 14 & 24.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Macroscopic mechanical parameters of clay. Figure 7: Sample model after servo completion. The calibrated parameters are given in Table 2. The cohesion and friction angle were obtained as 13.8 kPa and 24.32\({}^{\circ}\), respectively, using the stress circles formed by the numerical experiments under three different confining pressures (Figure 9), which are in good agreement with the results from the indoor tests. The model shown in Figure 7 was also used to simulate the change in earth pressure behind the retaining wall at a given point during wall displacement. Prior to loading, the model was subjected to \(K_{0}\) consolidation with a consolidation factor of \(\mu/(1-\mu)\) to simulate soils with different burial depths by adjusting the confining pressure. (1) Active earth pressure  
In the active pressure zone, the soil can be considered to undergo a process from the static state to the active limit state, where the vertical stress \(\sigma_{z}\) remains constant and the lateral stress \(\sigma_{x}\) decreases continuously until failure. In this process, \(\sigma_{z}\) is set to 100 kPa, 150 kPa, and 200 kPa and held constant, while \(\sigma_{x}\) decreases continuously until failure. The Duncan model is used with a failure ratio \(R_{f}\) of 0.75, a modulus parameter \(K\) of 145, a dimensionless index \(n\) of 0.75, and a Poisson's ratio \(\mu\) of 0.35. The numerical results were compared with the results of this study, [PERSON] [49] and Rankine theory, as shown in Figure 10.  
As depicted in Figure 10, it is evident that \(\sigma_{x}\) exhibits a decreasing trend with increasing strain. Once the strain exceeds 1.5%, \(\sigma_{x}\) shows fluctuations within a specific range. This fluctuation is attributed to the unevenness of the failure surface, which causes the soil on both sides of the surface to interact under vertical pressure, leading to \(\sigma_{x}\) fluctuations. The upper limit of this fluctuation corresponds to the result of [PERSON] [49], while the lower limit corresponds to the method in this paper, which exhibits good consistency with the numerical results. Within this range, the principal stress direction remains constant, and \(\sigma_{y}\) is the intermediate principal stress. Therefore, the difference between the method in this paper and [PERSON] [49] in the active earth pressure region is not significant, differing by \(\mu^{2}E_{t}\varepsilon_{x}/(1-\mu^{2})\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Particle** & **Particle** & **Particle** & **Effective** & **Stiffness** & **Friction** & **Rotational** & **Maximum** & **Domain of** \\
**Density** & **Radius** & **Radius** & **Modulus** & **Ratio**\(k^{*}\) & **Coefficient** & **Friction** & **Attraction** & **Mutation** \\
**kg/m\({}^{3}\)** & **\(R_{max}\)/mm** & **\(R_{min}\)/mm** & **\(E^{*}\)/MPa** & **Ratio**\(k^{*}\) & **\(\mu\)** & **Coefficient** & **N** & **Attraction** \\ \hline
2650 & 0.714 & 0.1785 & 33 & 3.6 & 0.21 & 0.8 & 10 & 0.5 \(R_{max}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Discrete element model parameters.