Study on the limit equilibrium slice method considering characteristics of inter-slice normal forces distribution: the improved Spencer method

  • Shunchuan Wu
  • Longqiang Han
  • Ziqiao Cheng
  • Xiaoqiang Zhang
  • Haiyong ChengEmail author
Original Article


The limit equilibrium slice method is a universal method to determine the safety factor of a slope. However, the forms of the inter-slice normal forces distribution are not considered in the traditional limit equilibrium slice method, and the influence of different positions of thrust lines on the safety factor is not considered. Thus, an improved Spencer method is developed here, wherein the characteristics of the inter-slice normal forces distribution are considered. The safety factor of a deep concave sliding surface is calculated, and a difference of approximately 20% is found between the safety factors that are obtained using the improved method and those obtained using the traditional method. The influence rule of different forms of the inter-slice normal forces distribution on the safety factor has been studied, and the influences of various factors on the inter-slice forces contribution, the stress state of the slice and the stress state of the slip surface have been analysed. Finally, two case studies are combined to illustrate the impact of the improved method on engineering safety and economic benefits.


Limit equilibrium slice method Inter-slice normal forces Thrust line Safety factor Improved Spencer method 



The work is supported by the National Key Research and Development Funding projects of China (Nos. 2017YFC08053003), the National Natural Science Foundation of China (51934003), the Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province, and the Research Start-up Fund for Introduced Talent of Kunming University of Science and Technology (KKSY201821024).


  1. Bishop AW (1955) The use of the slip circle in the stability analysis of earth slopes. Geotechnique 5:7–17CrossRefGoogle Scholar
  2. Bishop AW, Morgenstern NR (1960) Stability coefficients for earth slopes. Geotechnique 10(4):129–150CrossRefGoogle Scholar
  3. Chen Z-Y (1983) The general method of soil slope stability analysis and its improvement. J Geotech Eng 5(4):11–27 (in Chinese) Google Scholar
  4. Chen Z-Y, Mi H-L, Wang X-G (2001) A three-dimensional limit equilibrium method for slope stability analysis. J Geotech Eng 23(5):525–529 (in Chinese) Google Scholar
  5. Duncan JM (1996) State of the art: limit equilibrium and finite-element analysis of slopes. J Geotech Eng 122(7):577–596CrossRefGoogle Scholar
  6. Fan K, Fredlund DG, Wilson GW (1986) An interslice force function for limit equilibrium slope stability analysis. Can Geotech J 23(3):287–296CrossRefGoogle Scholar
  7. Fredlund DG, Krahn J (1977) Comparison of slope stability methods of analysis. Can Geotech J 14(3):429–439CrossRefGoogle Scholar
  8. Fredlund DG, Krahn J, Pufahl DE (1981) The relationship between limit equilibrium slope stability methods. Proceedings of the International Conference on Soil Mechanics and Foundation Engineering. 3:409–416Google Scholar
  9. Fredlund DG, Zhang ZM, Lam L (1992) Effect of the axis of moment equilibrium in slope stability analysis. Can Geotech J 29(3):456–465CrossRefGoogle Scholar
  10. Janbu N (1954) Application of composite slip surface for stability analysis. Proceedings of European Conference on Stability of Earth Slopes, Sweden, 3:43–49Google Scholar
  11. Lam L, Fredlund DG (1993) A general limit equilibrium model for three-dimensional slope stability analysis. Can Geotech J 30(6):905–919CrossRefGoogle Scholar
  12. Liu X-J (2016) The improvement of Spencer method on slope stability analysis. Geotech Eng Tech 2(4):186–188 (in Chinese) Google Scholar
  13. Morgenstern NR, Price VE (1965) The analysis of the stability of general slip surfaces. Geotechnique 15(1):79–93CrossRefGoogle Scholar
  14. Rahardjo H, Fredlund DG (1984) General limit equilibrium method for lateral earth force. Can Geotech J 21(1):166–175CrossRefGoogle Scholar
  15. Sarma SK (1973) Stability analysis of embankments and slopes. Geotechnique 23(3):423–433CrossRefGoogle Scholar
  16. She Q-R (2018) Optimization design for deep foundation supporting structure in Jianyang. Fujian Archit Constr 1:46–49 (in Chinese) Google Scholar
  17. Spencer E (1967) A method of analysis of the stability of embankments assuming parallel interslice forces. Geotechnique 17(1):11–26CrossRefGoogle Scholar
  18. Spencer E (1973) Thrust line criterion in embankment stability analysis. Geotechnique 23(1):85–100CrossRefGoogle Scholar
  19. Stianson JR, Fredlund DG, Chan D (2011) Three-dimensional slope stability based on stresses from a stress-deformation analysis. Can Geotech J 48(6):891–904CrossRefGoogle Scholar
  20. Stianson JR, Chan D, Fredlund DG (2015) Role of admissibility criteria in limit equilibrium slope stability methods based on finite element stresses. Comput Geotech 66:113–125CrossRefGoogle Scholar
  21. Wang Z, Ye X-M, Liu Y-X (2018) Improved Janbu slices method considering progressive destruction in landslide. Rock Soil Mech 39(2):675–682 (in Chinese) Google Scholar
  22. Wu S-C, Jin A-B, Liu Y (2017) Slope engineering. Metallurgical industry press, Beijing (in Chinese) Google Scholar
  23. Zhang Q, Wang Y, Cao X (2011) Backfill strength prediction based on the grey-neural network. Ind Miner Process 40(12):26–28Google Scholar
  24. Zheng Y-R, Shi W-M, Yang M-C (2004) Discussion on imbalance thrust force method and Srarm’s method. J Rock Mech Eng 23(17):3030–3036 (in Chinese) Google Scholar
  25. Zhou Y, Guo N, Yang X-H (2015) Deformation analyses and reinforcement treatment of an Overbreak Deep Foundation pit supported with pile-anchor retaining structure. J Undergr Sp Eng S1:211–216 (in Chinese) Google Scholar
  26. Zhu D-Y, Lee C-F, Qian Q-H et al (2001) A new procedure for computing the factor of safety using the Morgenstern-Price method. Can Geotech J 38(4):882–888CrossRefGoogle Scholar
  27. Zhu D-Y, Lee C-F, Jiang H-D (2003) Generalised framework of limit equilibrium methods and numerical procedure for slope stability analysis. Geotechnique 53(4):377–395CrossRefGoogle Scholar
  28. Zhu D-Y, Lee C-F, Huang M-S, Qian Q-H (2005) Modifications to three well-known methods of slope stability analysis. J Rock Mech Eng 24(02):183 (in Chinese) Google Scholar
  29. Zou G-D, Jiang W-Y (2003) Rebuilt slice method for slope stability analysis. J Rock Mech Eng 22(12):1953–1959 (in Chinese) Google Scholar

Copyright information

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Authors and Affiliations

  1. 1.Faculty of Land Resources EngineeringKunming University of Science and TechnologyKunmingPeople’s Republic of China
  2. 2.Key Laboratory of Ministry for Efficient Mining and Safety of Metal MinesUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  3. 3.Power China Road Bridge Group Co., Ltd.BeijingPeople’s Republic of China

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