Generating Slip Surfaces Using the Logistic Function Integral
Publication: International Journal of Geomechanics
Volume 23, Issue 5
Abstract
How to generate slip surfaces efficiently is crucial for the stability analysis of slopes. Traditional approaches for generating slip surfaces rely directly on the geometric features of slopes, which makes it complicated to generate arbitrary slip surfaces. Moreover, the resulting surfaces are usually not smooth in many cases. This study proposes a method for generating an arbitrary slip surface utilizing the superposition of the integral of the logistic function. This technique can transform the traditional geometry problem into a parameter selection one, in which three groups of parameters in the function determine the shape of the slip surface. The higher-order continuity and differentiable property of this function ensure the smoothness of the slip surfaces. The paper investigates the impact of relevant parameters on the slip surface’s pattern and provides the upper and lower bounds of the control variables. This method further develops a procedure for producing arbitrary slip surfaces. The factor of safety and the critical slip surface are calculated using the Morgenstern–Price method and the biogeography-based optimization. After constructing the slip surfaces by this method, two cases in Association for Computer Aided Design show that this method is applicable and verify its calculation accuracy by comparing the factor of safety and the critical slip surface. The results indicate that this approach presents a new direction for the slip surface structure, which can be a general method extending from two to three dimensions.
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Acknowledgments
The research is funded by the National Natural Science Foundation of China (NSFC) (Grant Nos. 52108372 and 52090081).
Notation
The following symbols are used in this paper:
- a
- parameter of the logistic function;
- b
- parameter of the logistic function;
- C
- integral constant;
- c
- parameter of the logistic function;
- c0
- parameter of the additional linear function;
- d0
- parameter of the additional linear function;
- f(x)
- integration form of L(x);
- g(x)
- superposition of step functions;
- H(t)
- step function;
- L
- length of the slope;
- L(x)
- superposition of the logistic function;
- l(x)
- logistic function;
- n
- number of slices;
- s
- normalized abscissa;
- s(t)
- integral form of the logistic function;
- t
- normalized ordinate;
- xA
- abscissa of point A;
- xB
- abscissa of point B;
- yA
- ordinate of point A;
- yB
- ordinate of point B;
- y0
- integration constant;
- αi
- coefficient; and
- ξ
- coefficient.
References
Bai, T., T. Qiu, X. Huang, and C. Li. 2014. “Locating global critical slip surface using the Morgenstern–Price method and optimization technique.” Int. J. Geomech. 14 (2): 319–325. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000312.
Basudhar, P. K., Anubhav, and M. R. Lakshminarayana. 2017. “Three-dimensional limit-equilibrium stability analyses of slopes and effect of inclusion of soil nails.” Int. J. Geomech. 17 (9): 04017067. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000932.
Bıshop, A. W. 1955. “The use of the slip circle in the stability analysis of slopes.” Géotechnique 5: 7–17. https://doi.org/10.1680/geot.1955.5.1.7.
Bolton, H., G. Heymann, and A. Groenwold. 2003. “Global search for critical failure surface in slope stability analysis.” Eng. Optim. 35 (1): 51–65. https://doi.org/10.1080/0305215031000064749.
Chen, Z.-Y., and N. R. Morgenstern. 1983. “Extensions to the generalized method of slices for stability analysis.” Can. Geotech. J. 20 (1): 104–119. https://doi.org/10.1139/t83-010.
Cheng, Y. M., L. Li, and S. C. Chi. 2007a. “Performance studies on six heuristic global optimization methods in the location of critical slip surface.” Comput. Geotech. 34 (6): 462–484. https://doi.org/10.1016/j.compgeo.2007.01.004.
Cheng, Y. M., L. Li, S.-c. Chi, and W. B. Wei. 2007b. “Particle swarm optimization algorithm for the location of the critical non-circular failure surface in two-dimensional slope stability analysis.” Comput. Geotech. 34 (2): 92–103. https://doi.org/10.1016/j.compgeo.2006.10.012.
Cheng, Y. M., L. Li, and S. S. Fang. 2011. “Improved harmony search methods to replace variational principle in geotechnical problems.” J. Mech. 27 (1): 107–119. https://doi.org/10.1017/jmech.2011.12.
Cheng, Y. M., L. Liang, S. C. Chi, and W. B. Wei. 2008. “Determination of the critical slip surface using artificial fish swarms algorithm.” J. Geotech. Geoenviron. Eng. 134 (2): 244–251. https://doi.org/10.1061/(ASCE)1090-0241(2008)134:2(244).
Deng, D. 2021. “Novel model for limit-equilibrium analysis of slope stability with a nonlinear strength criterion.” Int. J. Geomech. 21 (9): 06021021. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002137.
Donald, I. B., and P. S. Giam. 1992. “The ACADS slope stability programs review.” In Vol. 3 of Proc., 6th Int. Symp. Landslides, 1665–1670. Rotterdam, Netherlands: A.A. Balkema.
Fellenius, W. 1927. Erdstatische Berechnungen: Mit Reibung und Kohäsion Adhäsion und Unter Annahme Kreiszylindrisher Gleitflächen [Earth static calculations: With friction and cohesion adhesion and assuming circular cylindrisher sliding surfaces]. [In German.] Berlin: W. Ernst.
Gandomi, A. H., A. R. Kashani, M. Mousavi, and M. Jalalvandi. 2017. “Slope stability analysis using evolutionary optimization techniques.” Int. J. Numer. Anal. Methods Geomech. 41 (2): 251–264. https://doi.org/10.1002/nag.2554.
Gao, W. 2016. “Determination of the noncircular critical slip surface in slope stability analysis by meeting ant colony optimization.” J. Comput. Civil Eng. 30 (2): 06015001. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000475.
Gao, W. 2017. “Investigating the critical slip surface of soil slope based on an improved black hole algorithm.” Soils Found. 57 (6): 988–1001. https://doi.org/10.1016/j.sandf.2017.08.026.
Goh, A. T. C. 1999. “Genetic algorithm search for critical slip surface in multiple-wedge stability analysis.” Can. Geotech. J. 36 (2): 382–391. https://doi.org/10.1139/t98-110.
Hu, C., R. Jimenez, S.-c. Li, and L.-p. Li. 2015. “Determination of critical slip surfaces using mutative scale chaos optimization.” J. Comput. Civ. Eng. 29 (5): 04014067. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000373.
Janbu, N. 1973. Slope stability computations, 47–86. New York: Wiley. Embankment-Dam Engineering: Casagrande Volume.
Kahatadeniya, K. S., P. Nanakorn, and K. M. Neaupane. 2009. “Determination of the critical failure surface for slope stability analysis using ant colony optimization.” Eng. Geol. 108 (1): 133–141. https://doi.org/10.1016/j.enggeo.2009.06.010.
Kalatehjari, R., N. Ali, M. Kholghifard, and M. Hajihassani. 2014. “The effects of method of generating circular slip surfaces on determining the critical slip surface by particle swarm optimization.” Arabian J. Geosci. 7 (4): 1529–1539. https://doi.org/10.1007/s12517-013-0922-5.
Liu, X., D.-Q. Li, Z.-J. Cao, and Y. Wang. 2020. “Adaptive Monte Carlo simulation method for system reliability analysis of slope stability based on limit equilibrium methods.” Eng. Geol. 264: 105384. https://doi.org/10.1016/j.enggeo.2019.105384.
Malkawi, A. I. H., W. F. Hassan, and S. K. Sarma. 2001. “Global search method for locating general slip surface using Monte Carlo techniques.” J. Geotech. Geoenviron. Eng. 127 (8): 688–698. https://doi.org/10.1061/(ASCE)1090-0241(2001)127:8(688).
Mishra, M., V. R. Gunturi, and T. F. Da Miranda. 2019. “Slope stability analysis using recent metaheuristic techniques: A comprehensive survey.” SN Appl. Sci. 1 (12): 1674. https://doi.org/10.1007/s42452-019-1707-6.
Moayedi, H., A. Osouli, D. T. Bui, and L. K. Foong. 2019. “Spatial landslide susceptibility assessment based on novel neural-metaheuristic geographic information system based ensembles.” Sensors 19 (21): 4698. https://doi.org/10.3390/s19214698.
Morgenstern, N. R., and V. E. Price. 1965. “The analysis of the stability of general slip surfaces.” Géotechnique 15 (1): 79–93. https://doi.org/10.1680/geot.1965.15.1.79.
Petterson, K. E. 1955. “The early history of circular sliding surfaces.” Géotechnique 5 (4): 275–296. https://doi.org/10.1680/geot.1955.5.4.275.
Rocscience. 2021. “Slide2 user guide.” Accessed December 13, 2022. https://www.rocscience.com/help/slide2/documentation.
Sarma, S. K. 1973. “Stability analysis of embankments and slopes.” Géotechnique 23 (3): 423–433. https://doi.org/10.1680/geot.1973.23.3.423.
Shinoda, M., and Y. Miyata. 2019. “PSO-based stability analysis of unreinforced and reinforced soil slopes using non-circular slip surface.” Acta Geotech. 14 (3): 907–919. https://doi.org/10.1007/s11440-018-0678-x.
Singh, J., H. Banka, and A. K. Verma. 2019. “Locating critical failure surface using meta-heuristic approaches: A comparative assessment.” Arabian J. Geosci. 12 (9): 1–20. https://doi.org/10.1007/s12517-019-4435-8.
Soranzo, E., C. Guardiani, A. Saif, and W. Wu. 2022. “A Reinforcement Learning approach to the location of the non-circular critical slip surface of slopes.” Comput. Geosci. 166: 105182. https://doi.org/10.1016/j.cageo.2022.105182.
Spencer, E. 1967. “A method of analysis of the stability of embankments assuming parallel inter-slice forces.” Géotechnique 17 (1): 11–26. https://doi.org/10.1680/geot.1967.17.1.11.
Zhou, X. P., X. C. Huang, and X. F. Zhao. 2020. “Optimization of the critical slip surface of three-dimensional slope by using an improved genetic algorithm.” Int. J. Geomech. 20 (8): 04020120. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001747.
Zhu, D.-Y. 2001. “A method for locating critical slip surfaces in slope stability analysis.” Can. Geotech. J. 38 (2): 328–337. https://doi.org/10.1139/t00-118.
Zhu, J.-f., and C.-f. Chen. 2014. “Search for circular and noncircular critical slip surfaces in slope stability analysis by hybrid genetic algorithm.” J. Cent. South Univ. 21 (1): 387–397. https://doi.org/10.1007/s11771-014-1952-1.
Zolfaghari, A. R., A. C. Heath, and P. F. McCombie. 2005. “Simple genetic algorithm search for critical non-circular failure surface in slope stability analysis.” Comput. Geotech. 32 (3): 139–152. https://doi.org/10.1016/j.compgeo.2005.02.001.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 23 • Issue 5 • May 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: May 3, 2022
Accepted: Oct 31, 2022
Published online: Feb 22, 2023
Published in print: May 1, 2023
Discussion open until: Jul 22, 2023
ASCE Technical Topics:
- Arbitration
- Business management
- Design (by type)
- Dispute resolution
- Engineering fundamentals
- Freight transportation
- Geomechanics
- Geometrics
- Geotechnical engineering
- Highway and road design
- Hydraulic engineering
- Hydraulic properties
- Infrastructure
- Integrals
- Legal affairs
- Logistics
- Mathematics
- Parameters (statistics)
- Practice and Profession
- Public administration
- Public health and safety
- Safety
- Slope stability
- Slopes
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- Transportation engineering
- Water and water resources
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