# Rankine Active Earth Pressure of Unsaturated Filling under Earthquake
## 4 Conclusions  
1. By combining the shear strength model of unsaturated soil and the unified strength theory, the calculation model of seismic active earth pressure for unsaturated inclined backfill was deduced. The proposed method could be simplified to those in the previous literature under certain conditions. It extends the application range of the classic Rankine theory by taking earthquake, unsated soil, inclined rough wall back, non-horizontal backfill surface, and the intermediate principal stress into consideration.
2. As the seismic coefficients increase, the intensity of the Rankine active earth pressure will become larger. The earth pressure at 4 m depth with \(b=0.5\) and \(k_{v}=0\) increased by 35% when \(k_{h}\) increased from 0 to 0.1. With the decrease of the seismic coefficients or increase of the unified strength factor, the acting scope of Rankine active earth pressure moves downward slightly and the intensity reduces. The earth pressure at 4 m depth with \(k_{h}=0.1\) and \(k_{v}=0.1\) decreased by 44% when \(b\) increased from 0 to 1.
3. The horizontal seismic coefficient had a more significant effect on the resultant point location. For instance, in the case of \(k_{h}=0.1\) and \(b=1\), the resultant point of earth pressure varied from 2.20 m to 2.31 m when \(k_{v}\) increased from 0 to 0.2. Comparatively, for \(k_{v}=0.1\) and \(b=1\), the resultant point moved from 2.08 m to 2.38 m when \(k_{h}\) increased from 0 to 0.2. The intermediate principal stress caused a reduction of resultant acting height as well. In the case of \((u_{a}-u_{w})=30\) kPa and \(\varphi^{b}=14^{\circ}\), the resultant point moved downwardly from 2.43 m to 2.02 m when the intermediate principal stress was taken fully into consideration.  
Figure 7: Comparative research of seismic cases.  
As verification shows, the proposed model and algorithm are fit for seismic or unsaturated conditions. Owing to the scarce examples reported in the research subdivision, the validity of the proposed solution needs to be further examined by more examples. In any case, due to the consideration of the intermediate principal stress in the soil, the calculation result would be closer to the real solution theoretically. However, since it is difficult to determine an objective and accurate value for the unified strength factor (\(b\)), the solution is still a certain distance from engineering application, and more comparative research should be performed.  
Conceptualization, W.F.; methodology, W.L.; software, H.S., W.L.; validation, H.S., W.L. and Y.W.; formal analysis, W.L. and H.S.; writing--original draft preparation, H.S., W.L.; writing--review and editing, W.F., H.S. and Y.W.; visualization, H.S. and Y.W.; supervision, W.F.; project administration, W.F.; funding acquisition, W.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation, China, grant number: 52178413, and the Natural Science Foundation of Hunan Province, China, grant number: 2022 JJ30593.  
A part of the raw data cannot be shared at this time, as the data also forms part of an ongoing study.  
The authors declare no conflict of interest.