# Calculation of Displacement-Dependent Active Earth Pressure for Deep Excavations in Soft Soil
## 3 Expression for Displacement-Dependent Active Earth Pressure
### Validation of Earth Pressure Prediction  
To validate the applicability of the model, an excavation case with homogenous soil was constructed. The final excavation depth was 9.5 m, and the depth of the diaphragm wall was 20 m. The properties of the homogeneous soil were the weighted averages of the  
Figure 11: Simplified soil strain distribution. values in Table 1: the unit weight was 18.79 kN/m\({}^{3}\), the cohesion was 17.07 kPa, and the friction angle was 15.61\({}^{\circ}\). For excavation to the bottom of the pit, the relationship curves between earth pressure and displacement at 1 m, 5 m, 10 m, 15 m, and 20 m below the surface are shown in Figure 12.  
As shown in Figure 12, the soil pressure decreased with increasing displacement at different depths. For the same deformation amount, the deeper the position, the more the soil pressure changed. The earth pressure variations at 1 m and 10 m below the surface were 6.3 kPa and 27.89 kPa, respectively, for the deformation from the resting condition to 20 mm. As the deformation increased, the reduction rate of the earth pressure decreased. For example, 10 m below the surface, the earth pressure decreased by 15.49% and 2.63% for deformation from the resting condition to 10 mm and from 40 mm to 50 mm, respectively. From the curve of the earth pressure at 1 m below the surface, the proposed model can develop a tensile stress area in the cohesion soil; that is, the earth pressure is negative. After 50 mm deformation, the earth pressure at different depths tends to be constant, and approaches the limiting active state. However, the deformation required for Rankine active earth pressure is variable at different depths. Through inversion of Equation (11), the active deformation corresponding to the limiting state at different depths is shown in Figure 13.  
Figure 12: The curves of \(\epsilon_{\delta}\)-\(\delta\). Figure 13: The movements required to active earth pressure. The deeper the soil layer, the smaller the deformation that corresponds to the limiting active earth pressure. In this case, the deformation required for fully active earth pressure near the excavation surface was 100 mm, while the required deformations were 150 mm and 46 mm at 5 m and 20 m below the surface, respectively. In practical deep foundation pit engineering, the allowable deformations controlled in the design and monitoring processes are the maximum lateral deformation of the supports, so it is impossible to reach a fully active earth pressure at the same time; the earth pressure of the support above the excavation surface should be greater than the active limiting value. This is based on the assumption that the support deflection mode is the quadratic parabola and that maximum deformation occurs at the excavation surface. If the active earth pressure behind the wall at the bottom of the pit reached the limiting value, then the earth pressure at 5 m below the surface would be 16.46% more than the Rankine active earth pressure, while the earth pressure at 15 m below the surface would have already reached the limiting value. In order to verify the proposed model, two cases with similar soil characteristics were selected from the published literature. A centrifuge model test was reported by [PERSON] et al. (2018) for a 38 m deep, multi-strutted, soft ground foundation pit. The test included constant centrifugal acceleration of 120\(\times\)\(g\), the height of the model wall was 542 mm, and the excavation depth in stage 9 was 316 mm. The unit weight, the cohesion and the friction angle of the soft clay were 17.3 kN/m\({}^{3}\), 19 kPa, and 20.1\({}^{\circ}\) respectively. The unit weight of the sand was 19.5 kN/m\({}^{3}\) and the friction angle was 26.9\({}^{\circ}\). The measured earth pressures were compared with the predictions under different test displacements, as shown in Figure 14. The measured earth pressures showed that the soil was under compression within 10-30 m depth, while the lateral wall moved towards the pit according to the measured displacements.