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Optimization Techniques in Slope Stability Analysis Methods

  • Koushik Pandit
  • Shantanu Sarkar
  • Mahesh Sharma
Chapter
Part of the Advances in Natural and Technological Hazards Research book series (NTHR, volume 50)

Abstract

The estimation of factor of safety (FoS) or design reliability of slopes is a pre-requisite for an efficient and safe application of landslide mitigation measures for ensuring long-term slope stability. The evaluation of stability of slopes, especially in a hilly region with wide variations in its geological formation is an upfront challenging task for geologists as well as for geotechnical engineers and till date, has been tackled using several optimization algorithms and slope stability analysis methods. The purpose of this book chapter is to present an up-to-date along with an overall review of the slope stability analysis methods which have used different optimization algorithms for deterministic and probabilistic or stochastic evaluation of FoS or reliability index, respectively, including some case studies from published literatures. This review shows that the FoS or reliability of slopes obtained by applying the commonly established analysis methods coupled with optimization algorithms, using both the deterministic and the probabilistic approaches, may vary in their values as well as in their computational effort and errors encountered.

Keywords

Slope stability Factor of safety Reliability index Optimization techniques Stochastic analysis Heuristic algorithms 

References

  1. 1.
    World Disasters Report (2016) Resilience: saving lives today, investing for tomorrow. Available from: http://media.ifrc.org/ifrc/publications/world-disasters-report-2016/. Last accessed on 28.03.2017
  2. 2.
    National Disaster Management Guidelines – Management of Landslides and Snow Avalanches (2009, June) A publication of the National Disaster Management Authority, Government of India, New DelhiGoogle Scholar
  3. 3.
    National Disaster Management Plan (2016, May. A publication of the National Disaster Management Authority, Government of India, New Delhi. Available from: http://www.ndmindia.nic.in. Last accessed on 28.03.2017
  4. 4.
    Disaster Management in India (2011) A publication of: Ministry of Home Affairs, Government of India, New Delhi. Available online from: www.undp.org/content/dam/india/docs/disaster_management_in_india.pdf. Last accessed on 28.03.2017
  5. 5.
    Taha MR, Khajehzadeh M (2010) Slope stability assessment using optimization techniques: an overview. Electron J Geotech Eng (EJGE) 15:1901–1915Google Scholar
  6. 6.
    Rao Singiresu S (2009) Engineering optimization: theory and practice, 4th edn. Wiley, HobokenGoogle Scholar
  7. 7.
    Ferguson TS (n.d.) LP – Linear programming: a concise introduction. UCLA Department of Mathematics. Available from: www.math.ucla.edu/~tom/LP.pdf. Last accessed on 29.03.2017
  8. 8.
    Dantzig GB (1963) Linear programming and extensions. Princeton University Press, PrincetonGoogle Scholar
  9. 9.
    Dantzig GB, Thapa MN (1997) Linear programming 1: introduction. Springer, New York. LLC. ISBN 978-0-387-22633-0Google Scholar
  10. 10.
    Morgan SS (1997) A comparison of simplex method algorithms. MSc. thesis. University of Florida. Available from: https://web.archive.org/web/20110807134509/http://www.cise.ufl.edu/research/sparse/Morgan/index.htm. Last accessed on 29.03.2017
  11. 11.
    Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Cambridge, MA. ISBN 1-886529-00-0Google Scholar
  12. 12.
    Kuhn HW, Tucker A (1951) Nonlinear programming. In: Proceedings of the 2nd Berkeley symposium on mathematical statistics and probability. University of California Press, BerkeleyGoogle Scholar
  13. 13.
    Fox RL (1971) Optimization methods for engineering design. Addison-Wesley, ReadingGoogle Scholar
  14. 14.
    Box MJ (1965) A new method of constrained optimization and a comparison with other methods. Comput J 8(1):42–52CrossRefGoogle Scholar
  15. 15.
    Cheney EW, Goldstein AA (1959) Newton’s method of convex programming and Tchebycheff approximation. Numer Math 1:253–268CrossRefGoogle Scholar
  16. 16.
    Kelly JE (1960) The cutting plane method for solving convex programs. J SIAM VIII(4):703–712Google Scholar
  17. 17.
    Powell MJD (1978) A fast algorithm for nonlinearity constrained optimization calculations. In: Watson GA et al (eds) Lecture notes in mathematics. Springer-Verlag, BerlinGoogle Scholar
  18. 18.
    Zoutendijk G (1960) Methods of feasible directions. Elsevier, AmsterdamGoogle Scholar
  19. 19.
    Zoutendijk G (1966) Nonlinear programming: a numerical survey. SIAM J Control Theory Appl 4(1):194–210CrossRefGoogle Scholar
  20. 20.
    Rosen JB (1960) The gradient projection method of nonlinear programming, part I: linear constraints. SIAM J 8:181–217Google Scholar
  21. 21.
    Rosen JB (1961) The gradient projection method of nonlinear programming, part II: linear constraints. SIAM J 9:414–432Google Scholar
  22. 22.
    Gabriele GA, Ragsdell KM (1977) The generalized reduced gradient method: a reliable tool for optimal design. ASME J Eng Ind 99:384–400CrossRefGoogle Scholar
  23. 23.
    Box MJ (1966) A comparison of several current optimization methods and the use of transformations in constrained problems. Comput J 9:67–77CrossRefGoogle Scholar
  24. 24.
    Carroll CW (1961) The created response surface technique for optimizing nonlinear restrained systems. Oper Res 9:169–184CrossRefGoogle Scholar
  25. 25.
    Zangwill WI (1967) Nonlinear programming via penalty functions. Manag Sci 13(5):344–358CrossRefGoogle Scholar
  26. 26.
    Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4:303–320CrossRefGoogle Scholar
  27. 27.
    Rockafellar RT (1973) The multiplier method of Hestenes and Powell applied to convex programming. J Optim Theory Appl 12(6):555–562CrossRefGoogle Scholar
  28. 28.
    Chen J, Yin JH, Lee CF (2003) Upper bound limit analysis of slope stability using rigid finite elements and nonlinear programming. Can Geotech J 40:742–752.  https://doi.org/10.1139/T03-032CrossRefGoogle Scholar
  29. 29.
    Chinneck JW (2015) Practical optimization: a gentle introduction. Carleton University, Ottawa. Available online at: www.sce.carleton.ca/faculty/chinneck/po.html Last accessed on 30.03.2017Google Scholar
  30. 30.
    Bellman R (1957) Dynamic programming. Princeton University Press, PrincetonGoogle Scholar
  31. 31.
    Kremen A, Tsompanakis Y (2010) Application of dynamic programming to evaluate the slope stability of a vertical extension to a balefill. Waste Manag Res 28:373–382.  https://doi.org/10.1177/0734242X09354767CrossRefGoogle Scholar
  32. 32.
    Kremen A (2014) Improve your slope stability analyses by using dynamic programming. In: A blog post. Available online from: http://www.cornerstoneeg.com/2014/10/29/improve-slope-stability-analyses-dynamic-programming/. Last accessed on 30.03.2017
  33. 33.
    Dantzig GB, Thapa MN (2003) Linear programming 2: theory and extensions. Springer, New YorkGoogle Scholar
  34. 34.
    Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4:373–395CrossRefGoogle Scholar
  35. 35.
    Cook SA (1983) An overview of computational complexity. Commun ACM 26(6):401–408CrossRefGoogle Scholar
  36. 36.
    Rechenberg I (1965) Cybernetic solution path of an experimental problem. Library Translation 1122, Royal Aircraft Establishment, FarnboroughGoogle Scholar
  37. 37.
    Holland J (1975) Adaptation in natural and artificial systems. University of Michigan Press, OxfordGoogle Scholar
  38. 38.
    Solati S, Habibagahi G (2006) A genetic approach for determining the generalized interslice forces and the critical non-circular slip surface. Iran J Sci Technol Trans B Eng 30(B1):1Google Scholar
  39. 39.
    McCombie P, Wilkinson P (2002) The use of the simple genetic algorithm in finding the critical factor of safety in slope stability analysis. Comput Geotech 29:699–714CrossRefGoogle Scholar
  40. 40.
    Zolfaghari AR, Heath AC, McCombie PF (2005) Simple genetic algorithm search for critical non-circular failure surface in slope stability analysis. Comput Geotech 32(3):139–152CrossRefGoogle Scholar
  41. 41.
    Kirkpatrick S (1984) Optimization by simulated annealing—quantitative studies. J Stat Phys 34(5–6):975–986CrossRefGoogle Scholar
  42. 42.
    Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092CrossRefGoogle Scholar
  43. 43.
    Blum C, Li X (2008) Swarm intelligence in optimization. Swarm intelligence: introduction and applications, 1st edn. Springer-Verlag, Berlin, pp 43–85Google Scholar
  44. 44.
    Soliman MM, Hassanien AE, Onsi HM (2014) Bio-inspiring techniques in watermarking medical images: a review. In: Hassanien AE, Kim TH, Kacprzyk J, Awad AI (eds) Bio-inspiring cyber security and cloud services: trends and innovations, vol 70, 1st edn. Springer, Berlin, pp 93–114.  https://doi.org/10.1007/978-3-662-43616-5CrossRefGoogle Scholar
  45. 45.
    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceeding of the IEEE international conference on neural networks, Perth, Australia. pp 1942–1948Google Scholar
  46. 46.
    Pant M, Thangaraj R (2007) Particle swarm optimization: performance tuning and empirical analysis. Stud Comput Intell 203:101–128Google Scholar
  47. 47.
    Kennedy J, Eberhart R, Shi Y (2001) Swarm intelligence. Morgan Kaufmann Academic Press, San Francisco, pp 1931–1938Google Scholar
  48. 48.
    Dorigo M, Maniezzo V, Colorni A (1996) The ant system optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern B 26(1):29–41CrossRefGoogle Scholar
  49. 49.
    Dorigoa M, Stutzle T (2004) Ant colony optimization. MIT Press, CambridgeGoogle Scholar
  50. 50.
    Dorigoa M, Blum C (2005) Ant colony optimization theory: a survey. Theor Comput Sci 344(2005):243–278CrossRefGoogle Scholar
  51. 51.
    Blum C (2005) Ant colony optimization: introduction and recent trends. Phys Life Rev 2(4):353–373CrossRefGoogle Scholar
  52. 52.
    Geem ZW, Kim JH, Loganathan GV (2001) A new heuristic optimization algorithm: harmony search. Simulation 76:60–68CrossRefGoogle Scholar
  53. 53.
    Yang XS (2009) Harmony search as a metaheuristic algorithm. In: Geem ZW (ed) Music-inspired harmony search algorithm. Springer-Verlag, Berlin., SCI 191, pp 1–14Google Scholar
  54. 54.
    Omran M, Mahdavi (2008) Global-best harmony search. Appl Math Comput 198:643–656Google Scholar
  55. 55.
    Fellenius W (1936) Calculation of the stability of earth dams. In: Proceedings of the 2nd Congress on large dams, International Commission on large dams of the World Power conference, vol 4, pp 445–462Google Scholar
  56. 56.
    Taylor DW (1948) Fundamentals of soil mechanics. Wiley, New YorkGoogle Scholar
  57. 57.
    Bishop AW (1955) The use of the slip circle in the stability analysis of earth slopes. Geotechnique 5(1):7–17CrossRefGoogle Scholar
  58. 58.
    Morgenstern NR, Price VE (1965) The analysis of the stability of general slip surfaces. Geotechnique 15(1):79–93CrossRefGoogle Scholar
  59. 59.
    Spencer E (1967) A method of analysis of the stability of embankments assuming parallel inter-slice forces. Geotechnique 17(1):11–26CrossRefGoogle Scholar
  60. 60.
    Janbu N (1973) Slope stability computations. In: Embankment dam engineering. Wiley, New YorkGoogle Scholar
  61. 61.
    Sarma SK (1973) Stability analysis of embankments and slopes. Geotechnique 23(3):423–433CrossRefGoogle Scholar
  62. 62.
    Chen Z, Ugai K (2008) Limit equilibrium and finite element analysis – a perspective of recent advances. Landslides and engineered slopes – Chen et al. Taylor & Francis Group, London, 25–38CrossRefGoogle Scholar
  63. 63.
    Spall JC (2003) Introduction to stochastic search and optimization. Wiley, Hoboken. ISBN 0-471-33052-3CrossRefGoogle Scholar
  64. 64.
    Haldar A, Mahadevan S (1999) Probability, reliability, and statistical methods in engineering design. Wiley, New York. ISBN: 978-0-471-33119-3Google Scholar
  65. 65.
    Coates DF (1981) Rock mechanics principles. Monograph 874. Canada Centre for Mineral and Energy Technology (CANMEeT, formerly Mines Branch, Energy, Mines and Resources Canada), OttawaGoogle Scholar
  66. 66.
    Hasofer AM, Lind NC (1974) An exact and invariant first-order reliability format. J Eng Mech ASCE 100:111–121Google Scholar
  67. 67.
    Breitung K, Hohenbichler M (1989) Asymptotic approximations for multivariate integrals with an application to multinormal probabilities. J Multivar Anal 30:80–97.  https://doi.org/10.1016/0047-259X(89)90089-4CrossRefGoogle Scholar
  68. 68.
    Rubinstein RY, Kroese DP (2008) Simulation and the Monte Carlo method, 2nd edn. WileyGoogle Scholar
  69. 69.
    Slide Version 6.0 (2010) User’s guide. Rocscience, Toronto. Available from: https://www.rocscience.com/help/slide/webhelp/slide_interpret/probability/Probabilistic_Analysis_Overview.htm. Last accessed on 06.04.2017Google Scholar
  70. 70.
    Yang XL, Yin JH (2004) Slope stability analysis with nonlinear failure criterion. J Eng Mech 130(3):267–273.  https://doi.org/10.1061/(ASCE)0733-9399(2004)130:3(267)CrossRefGoogle Scholar
  71. 71.
    Baker R (1980) Determination of the critical slip surface in slope stability computations. Int J Numer Anal Methods Geomech 4:333–359CrossRefGoogle Scholar
  72. 72.
    Chen ZY, Shao CM (1988) Evaluation of minimum factor of safety in slope stability analysis. Can Geotech J 25:735–748CrossRefGoogle Scholar
  73. 73.
    Chen ZY, Morgenstern NR (1983) Extensions to the generalized method of slices for stability analysis. Can Geotech J 20:104–119CrossRefGoogle Scholar
  74. 74.
    Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313.  https://doi.org/10.1093/comjnl/7.4.308CrossRefGoogle Scholar
  75. 75.
    Arfken G (1985) The method of steepest descents. §7.4 in mathematical methods for physicists, 3rd edn. Academic, Orlando, pp 428–436Google Scholar
  76. 76.
    Davidon WC (1959) Variable metric method for minimization. A.E.C. Res. and Develop. Report ANL-5990 (Rev. TID-4500, 14th ed.)Google Scholar
  77. 77.
    Fletcher R, Powell MJD (1963) A rapidly convergent descent method for minimization. Comput J 6(2):163–168.  https://doi.org/10.1093/comjnl/6.2.163CrossRefGoogle Scholar
  78. 78.
    Bardet JP, Kapuskar MM (1989) A simplex analysis of slope stability. Comput Geotech 8:329–348CrossRefGoogle Scholar
  79. 79.
    Goh A (1999) Genetic algorithm search for critical slip surface in multiple-wedge stability analysis. Can Geotech J 36(2):382–391CrossRefGoogle Scholar
  80. 80.
    Donald IB, Giam PSK (1989) Improved comprehensive limit equilibrium stability analysis. Department of Civil Engineering Report No. 1/1989, Monash University, Melbourne, AustraliaGoogle Scholar
  81. 81.
    Cheng YM (2003) Location of critical failure surface and some further studies on slope stability analysis. Comput Geotech 30:255–267CrossRefGoogle Scholar
  82. 82.
    Karaulov AM (2005) Statement and solution of the stability problem for slopes and embankments as a linear-programming problem. Soil Mech Found Eng 42(3)CrossRefGoogle Scholar
  83. 83.
    Cheng YM, Li L, Chi SC, Wei WB (2007) Particle swarm optimization algorithm for the location of the critical non-circular failure surface in two-dimensional slope stability analysis. Comput Geotech 34:92–103CrossRefGoogle Scholar
  84. 84.
    Sun J, Li J, Liu Q (2008) Search for critical slip surface in slope stability analysis by spline-based GA method. J Geotech Geoenviron 134(2):252–256CrossRefGoogle Scholar
  85. 85.
    Cheng YM, Li L, Lansivaara T, Chi S, Sun Y (2008) An improved harmony search minimization algorithm using different slip surface generation methods for slope stability analysis. Eng Optim 40(2):95–115CrossRefGoogle Scholar
  86. 86.
    Li L, Yu G, Chu X, Lu S (2009) The harmony search algorithm in combination with particle swarm optimization and its application in the slope stability analysis. In: International conference on computational intelligence and security. IEEE, pp 133–136Google Scholar
  87. 87.
    Kahatadeniya KS, Nanakorn P, Neaupane KM (2009) Determination of the critical failure surface for slope stability analysis using ant colony optimization. Eng Geol 108:133–141CrossRefGoogle Scholar
  88. 88.
    Sabhahit N, Rao A (2011) Genetic algorithms in stability analysis of non-homogeneous slopes. Int J Geotech Eng 5(1):33–44CrossRefGoogle Scholar
  89. 89.
    Gao W (2015) Determination of the noncircular critical slip surface in slope stability analysis by meeting ant colony optimization. J Comput Civ Eng 30(2):1–10Google Scholar
  90. 90.
    Kalatehjari R, Arefnia A, Rashid ASA, Ali N, Hajihassani M (2015) Determination of three-dimensional shape of failure in soil slopes. Can Geotech J 52(9):1283–1301.  https://doi.org/10.1139/cgj-2014-0326CrossRefGoogle Scholar
  91. 91.
    Yamagami T, Jiang JC (1997) A search for the critical slip surface in three dimensional slope stability analysis. Soils Found 37(3):1–16CrossRefGoogle Scholar
  92. 92.
    Huang CC, Tsai CC (2000) New method for 3D and asymmetrical slope stability analysis. J Geotech Geoenviron 126(10):917–927CrossRefGoogle Scholar
  93. 93.
    Regmi RK, Jung K (2016) Application of dynamic programming to locate the critical failure surface in a rainfall induced slope failure problem. KSCE J Civ Eng 20(1):452CrossRefGoogle Scholar
  94. 94.
    Donald I, Chen ZY (1997) Slope stability analysis by the upper bound approach: fundamentals and methods. Can Geotech J 34(6):853–862.  https://doi.org/10.1139/t97-061CrossRefGoogle Scholar
  95. 95.
    Kim J, Salgado R, Yu HS (1999) Limit analysis of slopes subjected to pore-water pressures. J Geotech Geoenviron Eng 125(1):49–58CrossRefGoogle Scholar
  96. 96.
    Kim J, Salgado R, Lee J (2002) Stability analysis of complex soil slope using limit analysis. J Geotech Geoenviron Eng 128(7):546–557CrossRefGoogle Scholar
  97. 97.
    Pastor J, Thai TH, Francescato P (2003) Interior point optimization and limit analysis: an application. Commun Numer Method Eng 19:779–785.  https://doi.org/10.1002/cnm.619CrossRefGoogle Scholar
  98. 98.
    Liu FT, Fan YH, Yin JH (2011) The use of QP-free algorithm in the limit analysis of slope stability. J Comput Appl Math 235:3889–3897CrossRefGoogle Scholar
  99. 99.
    Jia C, Huang Q, Xia B (2015) Stability analysis of soil slope using discontinuity layout optimization. Adv Mater Res 1065–1069:190–198.  https://doi.org/10.4028/www.scientific.net/AMR.1065-1069.190CrossRefGoogle Scholar
  100. 100.
    Rongfu X, Gaopeng T (2015) Slope stability limit analysis based on inclined slices technique. Electron J Geotech Eng 20:1813–1832Google Scholar
  101. 101.
    Lingxi Q, Xiong Z (1995) Rigid finite element and its applications in engineering. Acta Mech Sinica 11(l):44–50CrossRefGoogle Scholar
  102. 102.
    Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Methods Geomech 12:61–77CrossRefGoogle Scholar
  103. 103.
    Sloan SW (1989) Upper bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 13(3):263–282CrossRefGoogle Scholar
  104. 104.
    Chen J, Yin JH, Lee CF (2005) The use of an SQP algorithm in slope stability analysis. Commun Numer Methods Eng 21:23–37.  https://doi.org/10.1002/cnm.723CrossRefGoogle Scholar
  105. 105.
    Liu F, Zhao J (2013) Limit analysis of slope stability by rigid finite-element method and linear programming considering rotational failure. Int J Geomech 13(6):827–839CrossRefGoogle Scholar
  106. 106.
    Yamagami T, Ueta Y (1988) Search for noncircular slip surfaces by the Morgenstern-Price method. In: Proceedings of the 6th international conference on numerical methods in geomechanics, Innsbruck, Austria, 11–15 April. A.A. Balkema, Rotterdam, pp 1335–1340Google Scholar
  107. 107.
    Kim J, Lee S (1997) An improved search strategy for the critical slip surface using finite element stress fields. Comput Geotech 21(4):295–313CrossRefGoogle Scholar
  108. 108.
    Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, New York. ISBN 978-0-471-91547-8Google Scholar
  109. 109.
    Lyamin AV, Sloan SW (2002) Lower bound limit analysis using non-linear programming. Int J Numer Methods Eng 55:573–611.  https://doi.org/10.1002/nme.511CrossRefGoogle Scholar
  110. 110.
    Lyamin AV, Sloan SW (2002) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Methods Geomech 26:181–216.  https://doi.org/10.1002/nag.198CrossRefGoogle Scholar
  111. 111.
    HTV P, Fredlund DG (2003) The application of dynamic programming to slope stability analysis. Can Geotech J 40(4):830–847CrossRefGoogle Scholar
  112. 112.
    Yang Y, Xing H, Yang X, Zhou J (2016) Determining the critical slip surface of three-dimensional soil slopes from the stress fields solved using the finite element method. Mathematical problems in Engineering, Hindawi Publishing Corporation, vol 2016, Article ID 7895615, 11 pages.  https://doi.org/10.1155/2016/7895615
  113. 113.
    Low B, Tang W (1997) Probabilistic slope analysis using Janbu’s generalized procedure of slices. Comput Geotech 21(2):121–142CrossRefGoogle Scholar
  114. 114.
    Bhattacharya G, Jana D, Ojha S, Chakraborty S (2003) Direct search for minimum reliability index of earth slopes. Comput Geotech 30(6):455–462CrossRefGoogle Scholar
  115. 115.
    Greco VR (1996) Efficient Monte Carlo technique for locating critical slip surface. J Geotech Eng ASCE 122(7):517–525CrossRefGoogle Scholar
  116. 116.
    Griffiths DV, Fenton GA (2004) Probabilistic slope stability analysis by finite elements. J Geotech Geoenviron 130(5):507–518CrossRefGoogle Scholar
  117. 117.
    Xue J, Gavin K (2007) Simultaneous determination of critical slip surface and reliability index for slopes. J Geotech Geoenviron 133:878–886CrossRefGoogle Scholar
  118. 118.
    Cho S (2007) Effects of spatial variability of soil properties on slope stability. Eng Geol 92(3–4):97–109CrossRefGoogle Scholar
  119. 119.
    Hong H, Roh G (2008) Reliability evaluation of earth slopes. J Geotech Geoenviron 134(12):1700–1705CrossRefGoogle Scholar
  120. 120.
    Schittkowski K (1986) NLPQL: a FORTRAN subroutine solving constrained nonlinear programming problems. Ann Oper Res 5(1):485–500CrossRefGoogle Scholar
  121. 121.
    Tan X, Wang J (2009) Finite element reliability analysis of slope stability. J Zhejiang Univ Sci A 10(5):645–652CrossRefGoogle Scholar
  122. 122.
    Khajehzadeh M, Taha M, El-Shafie A (2010) Harmony search algorithm for probabilistic analysis of earth slope. Electron J Geotech Eng 15:1647–1659Google Scholar
  123. 123.
    Khajehzadeh M, Taha M, El-Shafie A (2010) Modified particle swarm optimization for probabilistic slope stability analysis. Int J Phys Sci 5(15):2248–2258Google Scholar
  124. 124.
    Zhang H, Zhao Y (2010) Probabilistic slope stability analysis based on the upper bound theorem. In: International conference on E-Product, E-Service, and E-Entertainment (ICEEE), November 7–9. IEEE.  https://doi.org/10.1109/ICEEE.2010.5660372
  125. 125.
    Farah K, Ltifi M, Hassis H (2011) Reliability analysis of slope stability using stochastic finite element method. Proc Eng 10:1402–1407CrossRefGoogle Scholar
  126. 126.
    Celestino TB, Duncan JM (1981) Simplified search for non-circular slip surface. In: Proceedings of the 10th international conference on soil mechanics and foundation engineering, pp. 391–394Google Scholar
  127. 127.
    Zeng P, Jimenez R, Piña RJ (2015) System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms. Eng Geol 193:106–117CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Koushik Pandit
    • 1
  • Shantanu Sarkar
    • 1
  • Mahesh Sharma
    • 1
  1. 1.Geotechnical Engineering GroupCSIR-Central Building Research InstituteRoorkeeIndia

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