A Mathematical Extremum Method to Determine the Critical Slip Surface Using the Swedish Slice Method
- 1School of Architecture and Civil Engineering, Langfang Normal University, Langfang 065000, China
- 2School of Engineering and Technology, China University of Geosciences, Beijing 100083, China
- 3Department of Geological Hazard Survey, China Institute of Geo-Environment Monitoring (Guide Center of Prevention Technology for Geo-Hazards, MNR), Beijing 100081, China
Abstract
To obtain the critical slip surface (CSS) and the factor of safety (FOS) using the Swedish slice method (SSM), a mathematical extremum method (MEM) is proposed. In the MEM, the complete mathematical expressions of essential equations determining the CSS and the FOS are derived. It is discovered that a set of approximate explicit solutions of the equations exists. The equations are written into an EXCEL spreadsheet and solved using the solver tool. During the solution process, the discovered approximate explicit solutions are used as iterative initial values. Results are obtained with just one iteration. Compared with the empirical method and the computer search method, it is verified that the MEM has good applicability. Compared with the graphical trial procedure, the commercial software, and the stability charts, it is further demonstrated that the MEM has acceptable generality. Additionally, assessments are made on some of the existing research based on the MEM. It is noted that some existing findings are either inaccurate or only applicable under specific conditions. These findings provide valuable references for engineers and researchers.
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BibTeX
@article{hua2026mathematical,
title = {{A Mathematical Extremum Method to Determine the Critical Slip Surface Using the Swedish Slice Method}},
author = {Chengya Hua and Yunxia Zhang and Chenguang Song and Huanhuan Yin and Qian Wang},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213882},
url = {https://www.mdpi.com/2227-7390/14/21/3882},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Mathematical Extremum Method to Determine the Critical Slip Surface Using the Swedish Slice Method
AU - Chengya Hua
AU - Yunxia Zhang
AU - Chenguang Song
AU - Huanhuan Yin
AU - Qian Wang
PY - 2026
DA - 2026/10/24
JO - Mathematics
DO - 10.3390/math14213882
UR - https://www.mdpi.com/2227-7390/14/21/3882
AB - To obtain the critical slip surface (CSS) and the factor of safety (FOS) using the Swedish slice method (SSM), a mathematical extremum method (MEM) is proposed. In the MEM, the complete mathematical expressions of essential equations determining the CSS and the FOS are derived. It is discovered that a set of approximate explicit solutions of the equations exists. The equations are written into an EXCEL spreadsheet and solved using the solver tool. During the solution process, the discovered approximate explicit solutions are used as iterative initial values. Results are obtained with just one iteration. Compared with the empirical method and the computer search method, it is verified that the MEM has good applicability. Compared with the graphical trial procedure, the commercial software, and the stability charts, it is further demonstrated that the MEM has acceptable generality. Additionally, assessments are made on some of the existing research based on the MEM. It is noted that some existing findings are either inaccurate or only applicable under specific conditions. These findings provide valuable references for engineers and researchers.
SN - 2227-7390
ER -