## 5 Conclusions  
1. Through a large number of numerical simulation tests, it is concluded that the geometric characteristics of the slope cannot be taken into account to a certain extent when calculating the permanent displacement of sliding soil of multi-stage loess slope under the action of earthquake; in addition, it is feasible to use the comprehensive slope to calculate the permanent displacement of the sliding body of multi-stage loess slope under the action of earthquake.
2. On the premise of using the comprehensive slope rate to calculate the permanent displacement of multi-stage loess slope under earthquake, based on the energy method, combined with the Newmark slide displacement method, considering soil deformation and damping, the formulas for calculating positive and negative critical acceleration and permanent displacement of sliding soil are reasonable.
3. The critical acceleration of the sliding soil is mainly affected by the inclination angle of the sliding surface, but not by the soil parameters. The earthquake is only an inducing factor for the permanent displacement of the sliding soil, and the decrease in gravitational potential energy plays a key role in the permanent displacement.
4. The method derived in this paper comprehensively considers the influence of negative critical acceleration on the solution of permanent displacement when the inclination angle of slip surface is small and the seismic acceleration is large, and based on the principle of energy balance in the process of calculation and derivation, the calculation results are thus more accurate.  
Conceptualization: S.Y; data curation: [PERSON]. ([PERSON]); formal analysis: [PERSON]. ([PERSON]), [PERSON] ([PERSON]) and [PERSON]; investigation: [PERSON]. ([PERSON]), [PERSON] ([PERSON]) and [PERSON]; methodology: [PERSON]. ([PERSON]) and S.Y; software: [PERSON]. ([PERSON]); writing--original draft: [PERSON]. ([PERSON]) and [PERSON] writing--review and editing: [PERSON]. ([PERSON]), [PERSON] ([PERSON]) and [PERSON] All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (No. 52168050) and the National Natural Science Foundation of China (No. 51768040).  
**Institutional Review Board Statement:** Not applicable.  
Figure 14: The variation curve on the ratio of seismic input energy to gravitational potential energy and the permanent displacement with inclination angle of potential slip surface.  
**Informed Consent Statement:** The authors listed in this paper all agree to publish this paper.  
**Data Availability Statement:** The data of simulation and theoretical calculation in the article are not freely available due to legal concerns and commercial confidentiality. Nevertheless, all the concepts and procedures are explained in the presented research, and parts of the research may be available upon request.  
**Acknowledgments:** The authors of the present work feel grateful and would like to thank Northwest Research Institute Co., Ltd. of C.R.E.C, Shandong Jianzhu University, National Natural Science Foundation of China, Lanzhou University of Technology, for providing financial and technical support in research.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest.