NURBS Surface-Altering Optimization for Identifying Critical Slip Surfaces in 3D Slopes
Publication: International Journal of Geomechanics
Volume 22, Issue 9
Abstract
The evaluation of slope stability of a three-dimensional slope requires identifying the critical slip surface with the minimum factor of safety, which is a complex optimization problem. Failure to identify the critical slip surface can lead to unconservative conclusions about the stability of a slope. This paper proposes a novel 3D surface-altering optimization method, which iteratively alters the geometry of a 3D slip surface to find the critical slip surface representing the minimum factor of safety in a slope. The geometry of the slip surface is defined via nonuniform rational basis spline (NURBS) curves formed over a plan grid of control points. The proposed method includes a series of five subroutines that apply various forms of transformations to the control points. These subroutines include minimization problems, which determine the optimal transformation parameters for minimizing the obtained factor of safety of the resulting slip surfaces. Given that any geometrically defined slip surface can be approximated using an equivalent series of NURBS control points, the proposed method can be used in efforts to further reduce the global factor of safety first obtained via conventional search methods, such as those involving spherical or ellipsoidal slip surfaces. To demonstrate its effectiveness, the proposed method was applied to further optimize the critical ellipsoidal slip surfaces reported in some numerical examples. Comparing the results with those limited to ellipsoidal slip surfaces, the proposed method was consistently able to identify slip surfaces with significantly lower factors of safety. The postaltered slip surfaces also matched closely with finite element shear strength reduction results. As such, the proposed method is effective in searching for critical slip surfaces and can be used as a final step in the critical surface searching routine.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request. The original software code is confidential; however, models prepared in for the paper can be provided upon request.
References
An, Y., Q. Wu, C. Shi, and Q. Liu. 2016. “Three-dimensional smoothed-particle hydrodynamics simulation of deformation characteristics in slope failure.” Géotechnique 66 (8): 670–680. https://doi.org/10.1680/jgeot.15.P.222.
Azzouz, A. S., and M. M. Baligh. 1955. “Loaded areas on cohesive slopes.” J. Geotech. Eng. 1983: 709–729.
Baligh, M. M., and A. S. Azzouz. 1975. “End effects on stability of cohesive slopes.” J. Geotech. Eng. 101 (11): 1105–1117.
Bharat, T. V., P. V. Sivapullaiah, and M. M. Allam. 2009. “Swarm intelligence-based solver for parameter estimation of laboratory through-diffusion transport of contaminants.” Comput. Geotech. 36 (6): 984–992. https://doi.org/10.1016/j.compgeo.2009.03.006.
Bishop, A. W. 1955. “The use of the slip circle in the stability analysis of slopes.” Géotechnique 5 (1): 7–17. https://doi.org/10.1680/geot.1955.5.1.7.
Bretas, T. C. 2020. “Retroanálise probabilística tridimensional por equilíbrio-limite de rupturas de talude em belo horizonte/mg.” M.S. thesis, School of Mines, Univ. Federal de Ouro Preto.
Bui, H. H., R. Fukagawa, K. Sako, and J. C. Wells. 2011. “Slope stability analysis and discontinuous slope failure simulation by elasto-plastic smoothed particle hydrodynamics (SPH).” Géotechnique 61 (7): 565–574. https://doi.org/10.1680/geot.9.P.046.
Cavounidis, S. 1987. “On the ratio of factors of safety in slope stability analyses.” Géotechnique 37 (2): 207–210. https://doi.org/10.1680/geot.1987.37.2.207.
Chen, J. 2004. “Slope stability analysis using rigid element.” Ph.D. thesis, Dept. of Civil and Structural Engineering, Hong Kong Polytechnic Univ.
Chen, B. R., and X. T. Feng. 2007. “CSV-PSO and its application in geotechnical engineering.” In Chap. 15 of Swarm intelligence: Focus on Ant and particle swarm optimization, edited by F. T. S. Chan and M. K. Tiwari, 263–288. Vienna, Austria: Itech Education and Publishing.
Chen, R. H., and J. L. Chameau. 1982. “Three-dimensional limit equilibrium analysis of slopes.” Géotechnique 32 (1): 31–40.
Chen, W., and P. Chen. 2011. “PSOslope: A stand-alone windows application for graphic analysis of slope stability.” In Vol. 6728 of Advances in swarm optimization, edited by Y. Tan, Y. Shi, Y. Chai, and G. Wang, 56–63. Berlin: Springer. Lecture Notes in Computer Science.
Chen, Z., X. Wang, C. Haberfield, J.-H. Yin, and Y. Wang. 2001a. “A three-dimensional slope stability analysis method using the upper bound theorem, Part 1: Theory and methods.” Int. J. Rock Mech. Min. Sci. 38: 369–378. https://doi.org/10.1016/S1365-1609(01)00012-0.
Chen, Z., J. Wang, Y. Wang, J.-H. Yin, and C. Haberfield. 2001b. “A three-dimensional slope stability analysis method using the upper bound theorem, Part 2: Numerical approaches, applications and extensions.” Int. J. Rock Mech. Min. Sci. 38: 379–397. https://doi.org/10.1016/S1365-1609(01)00013-2.
Cheng, Y. M., T. Lansivaara, and W. B. Wei. 2007a. “Two-dimensional slope stability analysis by limit equilibrium and strength reduction methods.” Comput. Geotech. 34 (3): 137–150. https://doi.org/10.1016/j.compgeo.2006.10.011.
Cheng, Y. M., L. Li, and S. C. Chi. 2007b. “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, Y. J. Sun, and S. K. Au. 2012. “A coupled particle swarm and harmony search optimization algorithm for difficult geotechnical problems.” Struct. Multidiscip. Optim. 45 (4): 489–501. https://doi.org/10.1007/s00158-011-0694-z.
Cheng, Y. M., H. T. Liu, W. B. Wei, and S. K. Au. 2005. “Location of critical three-dimensional non-spherical failure surface by NURBS functions and ellipsoid with applications to highway slopes.” Comput. Geotech. 32 (6): 387–399. https://doi.org/10.1016/j.compgeo.2005.07.004.
Cheng, Y. M., and C. Yip. 2007. “Three-dimensional asymmetrical slope stability analysis extension of bishop’s, Janbu’s, and Morgenstern—Price’s techniques.” J. Geotech. Geoenviron. Eng. 133 (12): 1544–1555. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:12(1544).
Dawson, E. M., W. H. Roth, and A. Drescher. 1999. “Slope stability analysis by strength reduction.” Géotechnique 49 (6): 835–840. https://doi.org/10.1680/geot.1999.49.6.835.
Diederichs, M. S., M. Lato, P. Quinn, and R. Hammah. 2007. “Shear strength reduction (SSR) approach for slope stability analyses.” In Proc., 1st Canada–US Rock Mechanics Symp., edited by E. Eberhardt, D. Stead and T. Morrison, 319–327. London: Taylor & Francis.
Donald, I. B., and S. K. Giam. 1988. “Application of the nodal displacement method to slope stability analysis.” In Proc., 5th Australia–New Zealand Conf. on Geomechanics, 456–460. Australia: Institution of Engineers.
Farzaneh, O., and F. Askari. 2003. “Three-dimensional analysis of nonhomogeneous slopes.” J. Geotech. Geoenviron. Eng. 129 (2): 137–145. https://doi.org/10.1061/(ASCE)1090-0241(2003)129:2(137).
Fredlund, D. G., J. Krahn, and D. E. Pufahl. 1981. “The relationship between limit equilibrium slope stability methods.” In Vol. 3 of Proc., 10th Int. Conf. on Soil Mechanics and Foundation Engineering, 409–416. Stockholm, Sweden: Publications Committee of X. ICSMFE.
Fu, W., and Y. Liao. 2010. “Non-linear shear strength reduction technique in slope stability calculation.” Comput. Geotech. 37 (3): 288–298. https://doi.org/10.1016/j.compgeo.2009.11.002.
Gandomi, A. H., A. R. Kashani, M. Mousavi, and M. Jalalvandi. 2015. “Slope stability analyzing using recent swarm intelligence techniques.” Int. J. Numer. Anal. Methods Geomech. 39 (3): 295–309. https://doi.org/10.1002/nag.2308.
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.
Giger, M. W., and R. J. Krizek. 1975. “Stability analysis of vertical Cut With variable corner angle.” Soils Found. 15 (2): 63–71. https://doi.org/10.3208/sandf1972.15.2_63.
Goodale, E. I. T. J., and B. Fisher. 2021. “Global stability assessment of open pit slopes using LEM and FEM: A comparison between the factor of safety and strength reduction factor.” In The Evolution of Geotech, 1st ed., edited by R. E. Hammah, T. E. Yacoub, A. McQuillan, and J. Curran, 436. London: CRC Press.
Greco, V. R. 1996. “Efficient Monte Carlo technique for locating critical slip surface.” J. Geotech. Eng. 122 (7): 517–525. https://doi.org/10.1061/(ASCE)0733-9410(1996)122:7(517).
Griffiths, D. V., and P. A. Lane. 1999. “Slope stability analysis by finite elements.” Géotechnique 49 (3): 387–403. https://doi.org/10.1680/geot.1999.49.3.387.
Hovland, H. J. 1977. “Three-dimensional slope stability analysis method.” J. Geotech. Eng. Div. 103: 971–986. https://doi.org/10.1061/AJGEB6.0000493.
Huang, C.-C., and C.-C. Tsai. 2000. “New method for 3D and asymmetrical slope stability analysis.” J. Geotech. Geoenviron. Eng. 126 (10): 917–927. https://doi.org/10.1061/(ASCE)1090-0241(2000)126:10(917).
Huang, C.-C., and C.-C. Tsai. 2002. “General method for three-dimensional slope stability analysis.” J. Geotech. Geoenviron. Eng. 128 (10): 836–848. https://doi.org/10.1061/(ASCE)1090-0241(2002)128:10(836).
Hungr, O. 1987. “An extension of Bishop’s simplified method of slope stability analysis to three dimensions.” Géotechnique 37 (1): 113–117. https://doi.org/10.1680/geot.1987.37.1.113.
Janbu, N. 1973. “Slope stability computations.” In Embankment Dam engineering: Casagrande memorial volume, edited by R. C. Hirschfeld and S. J. Poulos, 47–86. New York: Wiley.
Javankhoshdel, S., B. Cami, T. Yacoub, T. Ma, and Y. Abolfazlzadeh. 2021. “Multi modal failure mechanism in open pit mines using LEM and FEM approaches.” In Proc., 55th US Rock Mechanics/Geomechanics Symp.
Jellali, B., and W. Frikha. 2017. “Constrained particle swarm optimization algorithm applied to slope stability.” Int. J. Geomech. 17 (12): 06017022. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001019.
Jiang, J. C., and T. Yamagami. 1999. “Determination of the sliding direction in three-dimensional slope stability analysis.” In Proc., 44th Symp. on Geotechnical Engineering, 193–200. Reno, Nevada: University of Nevada.
Jiang, J. C., T. Yamagami, and R. Baker. 2003. “Three-dimensional slope stability analysis based on nonlinear failure envelope.” Chin. J. Rock Mech. Eng. 22 (6): 1017–1023.
Jiang, Q., and C. Zhou. 2018. “A rigorous method for three-dimensional asymmetrical slope stability analysis.” Can. Geotech. J. 55: 495–513. https://doi.org/10.1139/cgj-2017-0317.
Kalatehjari, R., N. Ali, M. Hajihassani, and M. K. Fard. 2012. “The application of particle swarm optimization in slope stability analysis of homogeneous soil slopes.” Int. Rev. Modell. Simul. 5 (1): 458–465.
Kalatehjari, R., A. S. A. Rashid, N. Ali, and M. Hajihassani. 2014. “The contribution of particle swarm optimization to three-dimensional slope stability analysis.” Sci. World J. 2014: 973093. https://doi.org/10.1155/2014/973093.
Kennedy, J., and R. Eberhart. 1995. “Particle swarm optimization.” In Proceedings of ICNN'95 – International Conference on Neural Networks, 1942–1948. Perth, WA: IEEE. https://doi.org/10.1109/ICNN.1995.488968.
Kumar, U., and S. S. Jayadeva. 2016. “Benchmarking NLopt and state-of-art algorithms for continuous global optimization via hybrid IACOR.” Swarm Evol. Comput. 27: 116–131. https://doi.org/10.1016/j.swevo.2015.10.005.
Li, L., Y. Wang, L. Zhang, C. Choi, and C. W. W. Ng. 2019. “Evaluation of critical slip surface in limit equilibrium analysis of slope stability by smoothed particle hydrodynamics.” Int. J. Geomech. 19 (5): 04019032. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001391.
Li, L., G. Yu, X. Chu, and S. Lu. 2009. “The harmony search algorithm in combination with particle swarm optimization and its application in the slope stability analysis.” In Vol. 2 of Proc., of the Int. Conf. on Computational Intelligence and Security, 133–136. Beijing: IEEE.
Li, S., B. Cami, S. Javankhoshdel, B. Corkum, and T. Yacoub. 2020. “Considering multiple failure modes in limit equilibrium slope stability analysis: Two methods.” In GeoVirtual 2020, the Canadian Geotechnical Society. Surrey, BC: Canadian Geotechnical Society.
Lim, K., A. V. Lyamin, M. J. Cassidy, and A. J. Li. 2016. “Three-dimensional slope stability charts for frictional fill materials placed on purely cohesive clay.” Int. J. Geomech. 16 (2): 04015042. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000526.
Mafi, R., S. Javankhoshdel, B. Cami, R. J. Chenari, and A. H. Gandomi. 2021. “Surface altering optimization in slope stability analysis with non-circular failure for random limit equilibrium method.” Georisk: Assess. Manage. Risk Eng. Syst. Geohazards 15 (4): 260–286. https://doi.org/10.1080/17499518.2020.1771739.
Matsui, T., and K.-C. San. 1992. “Finite element slope stability analysis by shear strength reduction technique.” Soils Found. 32 (1): 59–70. https://doi.org/10.3208/sandf1972.32.59.
Michalowski, R. L. 1989. “Three-dimensional analysis of locally loaded slopes.” Géotechnique 39 (1): 27–38. https://doi.org/10.1680/geot.1989.39.1.27.
Mishra, M., V. R. Gunturi, and T. F. D. S. Miranda. 2019. “Slope stability analysis using recent metaheuristic techniques: A comprehensive survey.” SN Appl. Sci. 1 (12): 1–17. https://doi.org/10.1007/s42452-019-1707-6.
Mowen, X. J. E. 2004. “A simple Monte Carlo method for locating the three-dimensional critical slip surface of a slope.” Acta Geol. Sin. 78 (6): 1258–1266. https://doi.org/10.1111/j.1755-6724.2004.tb00783.x.
Naylor, D. J. 1981. Finite elements and slope stability. Lisbon, Portugal: Proceedings of the NATO Advanced Study Institute.
Nocedal, J., and S. Wright. 2006. Numerical optimization. Dordrecht, Netherlands: Springer.
Piegl, L., and W. Tiller. 1997. The NURBS book. 2nd ed. Berlin: Springer.
Powell, M. J. 2009. The BOBYQA algorithm for bound constrained optimization without derivatives. Rep. No. NA2009/06. Cambridge: University of Cambridge.
Qian, Z. G., A. J. Li, R. S. Merifield, and A. V. Lyamin. 2015. “Slope stability charts for two-layered purely cohesive soils based on finite-element limit analysis methods.” Int. J. Geomech. 15 (3): 06014022. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000438.
Rogers, D. F. 2001. An introduction to NURBS: With historical perspective. San Francisco: Morgan Kaufmann.
Shen, J., and M. Karakus. 2014. “Three-dimensional numerical analysis for rock slope stability using shear strength reduction method.” Can. Geotech. J. 51 (2): 164–172. https://doi.org/10.1139/cgj-2013-0191.
Shukha, R., and R. Baker. 2003. “Mesh geometry effects on slope stability calculation by FLAC strength reduction method—Linear and non-linear failure criteria.” In Proc., 3rd Int. Conf. on FLAC and Numerical Modeling in Geomechanics, 109–116. Sudbury, ON: CRC Press.
Song, E. 1997. “Finite element analysis of safety factor for soil structures.” Chin. J. Geotech. Eng. 19 (2): 1–7.
Spencer, E. 1967. “A method of analysis of the stability of embankments assuming parallel inter-slice forces.” Géotechnique 17: 11–26. https://doi.org/10.1680/geot.1967.17.1.11.
Su, Z., and L. Shao. 2021. “A three-dimensional slope stability analysis method based on finite element method stress analysis.” Eng. Geol. 280: 105910. https://doi.org/10.1016/j.enggeo.2020.105910.
Sun, C., J. Chai, T. Luo, Z. Xu, X. Chen, Y. Qin, and B. Ma. 2021. “Nonlinear shear-strength reduction technique for stability analysis of uniform cohesive slopes with a general nonlinear failure criterion.” Int. J. Geomech. 21 (1): 06020033. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001885.
Sun, C., J. Chai, Z. Xu, and Y. Qin. 2017. “3D stability charts for convex and concave slopes in plan view with homogeneous soil based on the strength-reduction method.” Int. J. Geomech. 17 (5): 06016034. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000809.
Taha, M. R., M. Khajehzadeh, and A. El-Shafie. 2010. “Slope stability assessment using optimization techniques: An overview.” Electron. J. Geotech. Eng. 15 (Q): 1901–1915.
Tian, D., L. Xu, and S. Wang. 2009. “The application of particle swarm optimization on the search of critical slip surface.” In Proc., of the Int. Conf. on Information Engineering and Computer Science, 1–4. Wuhan: IEEE.
Toufigh, M., A. Ahangarasr, and A. Ouria. 2006. “Using non-linear programming techniques in determination of the most probable slip surface in 3D slopes.” World Acad. Sci. Eng. Technol. 17: 30–35.
Ugai, K. 1989. “A method of calculation of total safety factor of slope by elasto-plastic FEM.” Soils Found. 29 (2): 190–195. https://doi.org/10.3208/sandf1972.29.2_190.
Ullah, S., M. U. Khan, and G. Rehman. 2020. “A brief review of the slope stability analysis methods.” Geol. Behav. 4 (2): 73–77. https://doi.org/10.26480/gbr.02.2020.73.77.
Wei, W. B., Y. M. Cheng, and L. Li. 2009. “Three-dimensional slope failure analysis by the strength reduction and limit equilibrium methods.” Comput. Geotech. 36 (2009): 70–80. https://doi.org/10.1016/j.compgeo.2008.03.003.
Xie, M., Z. Wang, X. Liu, and B. Xu. 2011. “Three-dimensional critical slip surface locating and slope stability assessment for lava lobe of Unzen Volcano.” J. Rock Mech. Geotech. Eng. 3 (1): 82–89. https://doi.org/10.3724/SP.J.1235.2011.00082.
Yamagami, T., and J.-C. Jiang. 1997. “A search for the critical slip surface in three-dimensional slope stability analysis.” Soils Found. 37 (3): 1–16. https://doi.org/10.3208/sandf.37.3_1.
Yang, X. S., and S. Deb. 2009. “Cuckoo search via lévy flights.” In 2009 World Congress on Nature & Biologically Inspired Computing, 210–214. New York: IEEE.
Yazdi, J. S., F. Kalantary, and H. S. Yazdi. 2012. “Calibration of soil model parameters using particle swarm optimization.” Int. J. Geomech. 12 (3): 229–238. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000142.
You, G., M. A. Mandalawi, A. Soliman, K. Dowling, and P. Dahlhaus. 2017. “Finite element analysis of rock slope stability using shear strength reduction method.” In International Congress and Exhibition “Sustainable Civil Infrastructures: Innovative Infrastructure Geotechnology,” 227–235. Cham, Switzerland: Springer.
Zhao, H. B., Z. S. Zou, and Z. L. Ru. 2008. “Chaotic particle swarm optimization for non-circular critical slip surface identification in slope stability analysis.” In Proc., of the Int. Young Scholars’ Symp. on Rock Mechanics: Boundaries of Rock Mechanics Recent Advances and Challenges for the 21st Century, edited by M. Cai and J. Wang, 585–588. London: CRC Press.
Zheng, Y. R., S. Y. Zhao, W. X. Kong, and C. J. Deng. 2005. “Geotechnical engineering limit analysis using finite element method.” Rock Soil Mech. 26 (1): 163–168.
Zhou, X. P., and H. Cheng. 2013. “Analysis of stability of three-dimensional slopes using the rigorous limit equilibrium method.” Eng. Geol. 160: 21–33. https://doi.org/10.1016/j.enggeo.2013.03.027.
Zienkiewicz, O. C., C. Humpheson, and R. W. Lewis. 1975. “Associated and non-associated visco-plasticity and plasticity in soil mechanics.” Géotechnique 25 (4): 671–689. https://doi.org/10.1680/geot.1975.25.4.671.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 22 • Issue 9 • September 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Oct 5, 2021
Accepted: Apr 24, 2022
Published online: Jul 11, 2022
Published in print: Sep 1, 2022
Discussion open until: Dec 11, 2022
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Ivan Chen, Terence Ma, Brigid Cami, Sina Javankhoshdel, Brent Corkum, A Heuristic Search Method for Critical Slip Surfaces with Weak Layers, Geo-Congress 2024, 10.1061/9780784485330.014, (126-134), (2024).
- Terence Ma, Brigid Cami, Sina Javankhoshdel, Brent Corkum, Thamer Yacoub, Optimization of Spline Slip Surfaces Using Metaheuristic Search in LEM, Proceedings of the TMIC 2022 Slope Stability Conference (TMIC 2022), 10.2991/978-94-6463-104-3_13, (130-139), (2023).
- Terence Ma, Brigid Cami, Sina Javankhoshdel, Brent Corkum, Nicolas Chan, Amir H. Gandomi, Spline Search for Slip Surfaces in 3D Slopes, International Journal of Geomechanics, 10.1061/IJGNAI.GMENG-8459, 23, 8, (2023).
- Scott D. Cylwik, Sina Javankhoshdel, Brigid Cami, Terence Ma, Characteristic Strength of a Slope with Spatial Variability and Cross-Correlation, Geo-Risk 2023, 10.1061/9780784484999.012, (114-123), (2023).