## 5 Conclusions  
In this paper, a geometric model of a ridge slope was constructed, based on the unified strength theory, and the stability of a ridge slope under different working conditions was studied by using strength reduction method. The following conclusions were reached.  
1. No displacement of the ridge slope occurred under different \(\alpha\) values when the reduction factor was less than six. The smaller the \(\alpha\) of the ridge slope, the worse its stability. When \(\alpha=20^{\circ}\), the stability coefficient of the ridge slope was 8, and when \(\alpha\) was greater than \(20^{\circ}\), the stability coefficient of the ridge slope increased to 10.
2. When the \(\beta\) was small, the plastic area was distributed in a flank slope position, and the plastic area gradually concentrated to the ridge as \(\beta\) increased. When the \(\beta\) was 70 \({}^{\circ}\), the stabilization coefficient of the ridge slope was 7. When the \(\beta\) decreased to 40\({}^{\circ}\) and 30\({}^{\circ}\), the stability coefficient of the ridge slope reached 15. Under the same reduction factor, the larger the \(\beta\) of the ridge slope, the larger the displacement, and the ridge slope was more prone to instability.
3. The higher the slope height was, the larger the displacement at the top of the slope was. The reduction coefficient of the ridge slope curve increased gently, and the instability phenomenon of the slope hardly occurred. When the slope height increased to 30 m, the displacement at the top of the slope was 14.7 m, and the safety and stability coefficient of the slope was 4.  
Figure 19: The displacement cloud map and plastic zone cloud map of ridge slope after excavation. (a) Displacement cloud map; (b) plastic zone cloud map.  
Figure 18: Model diagram of the ridge slope after slope cutting (\(H=30\) m, \(\alpha=30^{\circ}\), \(\beta=40^{\circ}\)).  
When the excavation slope rate was large, the bedrock exhibited a large displacement; the plastic zone and a large displacement began to appear on the excavation slope surface because of the close proximity of the excavation surface position to the back edge of the ridge slope. Therefore, the ridge slope is more prone to instability.  
Conceptualization, J.W.; Methodology, J.W., P.S. and F.W.; Software, J.W. and P.S.; Validation, J.W., P.L., P.S., H.L., F.W., Y.S. ([PERSON]), Y.L., A.C., Y.S. ([PERSON]) and Q.Z.; Formal analysis, [PERSON] and P.S.; Investigation, J.W., P.S., H.L., F.W., Y.S. ([PERSON]), Y.L., A.C., Y.S. ([PERSON]) and Q.Z.; Data curation, J.W., P.S., H.L., F.W., Y.S. ([PERSON]), Y.L., A.C., Y.S. ([PERSON]) and Q.Z.; Writing--original draft, [PERSON], P.L. and P.S.; Writing--review & editing, J.W. and P.L.; Supervision, J.W.; Funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Shanghai Municipal Science and Technology Project (18 DZ1201300;19 DZ1200900); the Xiamen Road and Bridge Group (XM2017-TZ0151; XM2017-TZ0117); the project of the Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education (CJ202101); the Shanghai Municipal Science and Technology Major Project (2021 SHZDZX0100) and the Fundamental Research Funds for the Central Universities; the Key Laboratory of Land Subsidence Monitoring and Prevention, Ministry of Natural Resources of China (No. KLLSMP202101; KLLSMP202201); Suzhou Rail Transit Line 1 Co. Ltd., and China Railway 15 Bureau Group Co. Ltd.  
Not applicable.  
The authors declare no conflict of interest.