Probabilistic slope stability analysis by a copula-based sampling method

Abstract

In probabilistic slope stability analysis, the influence of cross correlation of the soil strength parameters, cohesion and internal friction angle, on the reliability index has not been investigated fully. In this paper, an expedient technique is presented for probabilistic slope stability analysis that involves sampling a series of combinations of soil strength parameters through a copula as input to an existing conventional deterministic slope stability program. The approach organises the individual marginal probability density distributions of componential shear strength as a bivariate joint distribution by the copula function to characterise the dependence between shear strengths. The technique can be used to generate an arbitrarily large sample of soil strength parameters. Examples are provided to illustrate the use of the copula-based sampling method to estimate the reliability index of given slopes, and the computed results are compared with the first-order reliability method, considering the correlated random variables. A sensitivity study was conducted to assess the influence of correlational measurements on the reliability index. The approach is simple and can be applied in practice with little effort beyond what is necessary in a conventional analysis.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alonso, E.E.: Risk analysis of slopes and its application to slopes in Canadian sensitive clays. Geotechnique 26, 453–472 (1976)

    Article  Google Scholar 

  2. 2.

    Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control. AC-19(6), 16–722 (1974)

    Google Scholar 

  3. 3.

    Anderson, T.W., Darling, D.A.: A test of goodness-of-fit. J. Am. Stat. Assoc. 49, 765–769 (1954)

    Article  Google Scholar 

  4. 4.

    Ang, A.H.-S., Tang, W.H.: Probability Concepts in Engineering Planning and Design, p 562. Wiley, New York (1984)

    Google Scholar 

  5. 5.

    Baecher, G.B., Christian, J.T.: Reliability and statistics in geotechnical engineering. Wiley, New York (2003)

    Google Scholar 

  6. 6.

    Bergado, D.T., Anderson, L.R.: Stochastic analysis of pore pressure uncertainty for the probabilistic assessment of the safety of earth slopes. Soils and Found. 25(2), 87–105 (1985)

    Article  Google Scholar 

  7. 7.

    Bishop, A.W.: The use of the slip circle stability in analysis of slopes. Geotechnique 1, 7–17 (1955)

    Google Scholar 

  8. 8.

    Boyer, B., Gibson, M., Loretan, M.: Pitfalls in tests for changes in correlation. International Finance Discussion Paper 597, Board of Governors of the Federal Reserve System (1999)

  9. 9.

    Brejda, J., Moorman, J., Smith, T.B., Karlen, J.L., Allan, D.L., Dao, T.H.: Distribution and variability of surface soil properties at a regional scale. Soil Sci. Soc. Am. J. 64, 974–982 (2000)

    Article  Google Scholar 

  10. 10.

    Cherubini, C.: Data and considerations on the variability of geotechnical properties of soils. In: Proceedings of the International Conference on safety and reliability (ESREL 97) Lisbon, vol. 2, pp. 1583–1591 (1997)

  11. 11.

    Cho, S.E., Park, H.C.: Effect of spatial variability of cross-correlated soil properties on bearing capacity of strip footing. Int. J. Numer. Anal. Meth. Geomech. 34, 1–26 (2010)

    Google Scholar 

  12. 12.

    Chowdhury, R.N., Tang, W.H., Sidi, I.: Reliability model of progressive slope failure. Geotechnique 37(4), 467–481 (1987)

    Article  Google Scholar 

  13. 13.

    Chowdhury, R.N., Xu, D.: Reliability index for slope stability assessment-two methods compared. Reliab. Eng. Syst. Safe. 37(2), 99–108 (1992)

    Article  Google Scholar 

  14. 14.

    Chowdhury, R.N., Xu, D.W.: Rational polynomial technique in slope stability analysis. J. Geotech. Eng. Div. ASCE 119(12), 1910–28 (1993)

    Article  Google Scholar 

  15. 15.

    Christian, J.T., Ladd, C.C., Baecher, G.B.: Reliability applied to slope stability analysis. J. Geotech. Eng. ASCE 120(12), 2180–2207 (1994)

    Article  Google Scholar 

  16. 16.

    Clemen Robert, T., Reilly, T.: Correlations and copulas for decision and risk analysis. Manag. Sci. 45(2), 208–224 (1999)

    Article  Google Scholar 

  17. 17.

    Duncan, J.M.: Factors of safety and reliability in geotechnical engineering. J. Geotech. Geoenviron. ASCE 126, 307–316 (2000)

    Article  Google Scholar 

  18. 18.

    Dupuis, D.J.: Using Copulas in hydrology: benefits, cautions and issues. J. Hydrol. Eng. ASCE 12(4), 381–393 (2007)

    Article  Google Scholar 

  19. 19.

    El-Ramly, H., Morgenstern, N.R., Cruden, D.M.: Probabilistic slope stability analysis for practice. Can. Geotech. J. 39, 665–683 (2002)

    Article  Google Scholar 

  20. 20.

    Embrechts, P., McNeil, A.J., Straumann, D.: Correlation and dependence in risk management: properties and pitfalls. In: Dempster, M. (ed.) Risk Management: Value at Risk and Beyond, pp 176–223. Cambridge University Press, Cambridge (2002)

    Chapter  Google Scholar 

  21. 21.

    Fenton, G.A., Griffiths, D.V.: Bearing capacity prediction of spatially random - soils. Can. Geotech. J. 40(1), 54–65 (2003)

    Article  Google Scholar 

  22. 22.

    Ferson, S., Hajagos Janos, G.: Varying correlation coefficients can underestimate uncertainty in probabilistic models. Reliab. Eng. Syst. Saf. 91, 1461–1467 (2006)

    Article  Google Scholar 

  23. 23.

    Forrest William, S., Orr Trevor, L.L.: Reliability of shallow foundations designed to Eurocode 7, Vol. 4, pp 186-207 (2010)

  24. 24.

    Frees, E.W., Valdez, E.A.: Understanding relationships using copulas. North Amer. Actua. J. 2(1), 1–25 (1998)

    Article  Google Scholar 

  25. 25.

    Genest, C., MacKay, J.: The joy of copulas: bivariate distributions with uniform marginals. Am. Stat. 40, 280–283 (1986)

    Google Scholar 

  26. 26.

    Genest, C., Favre, A.C.: Everything you always wanted to know about copula modelling but were afraid to ask. J. Hydrol. Eng. ASCE 12(4), 347–368 (2007)

    Article  Google Scholar 

  27. 27.

    Genest, C., Remillard, B., Beaudoin, D.: Goodness-of-fit tests for copulas: a review and a power study. Insur. Math. Econ. 44, 199–214 (2009)

    Article  Google Scholar 

  28. 28.

    Harr, M.E.: Reliability based design in Civil Engineering. McGraw-Hill, New York (1987)

    Google Scholar 

  29. 29.

    Hasofer, A.A., Lind, A.M.: Exact and invariant second moment code format. J. Engrg. Mech. Div. ASCE 100(1), 111–121 (1974)

    Google Scholar 

  30. 30.

    Hassan, A., Wolff, T.: Search algorithm for minimum reliability index of earth slopes. J. Geotech. Geoenviron. Eng. ASCE 125, 301–308 (1999)

    Article  Google Scholar 

  31. 31.

    Hata, Y., Ichii, K., Tokida, K.-i.: A probabilistic evaluation of the size of earthquake induced slope failure for an embankment. In: Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards (2011). doi:10.1080/17499518.2011.604583

  32. 32.

    Honjo, Y., Suzuki, M., Matsuo, M.: Reliability analysis of shallow foundations in reference to design codes development. Comput. Geotech. 26, 331–346 (2000)

    Article  Google Scholar 

  33. 33.

    Husein Malkawi, A.I., Hassan, W.F., Abdulla, F.: Uncertainty and reliability analysis applied to slope stability. Struct. Saf. J. 22, 161–187 (2000)

    Article  Google Scholar 

  34. 34.

    Joe, H.: Multivariate models and dependence concept. Chapman and Hall, New York (1997)

    Book  Google Scholar 

  35. 35.

    Kojadinovic, I., Yan, J.: Modeling multivariate distributions with continuous margins using the copula R Package. J. Stat. Soft. 34(9), 1–20 (2010)

    Google Scholar 

  36. 36.

    Kotz, S., Balakrishnan, N., Johnson, N.: Continuous multivariate distributions. Wiley, New York (2000)

    Book  Google Scholar 

  37. 37.

    Lambert, P., Vandenhende, F.: A copula-based model for multivariate nonnormal longitudinal data: analysis of a dose titration safety study on a new antidepressant. Stat. Med. 21, 3197–3217 (2002)

    Article  Google Scholar 

  38. 38.

    Li, K.S., Lumb, P.: Probabilistic design of slopes. Can. Geotech. J. 24, 520–535 (1987)

    Article  Google Scholar 

  39. 39.

    Lumb, P.: Safety factors and the probability distribution of soil strength. Can. Geotech. J. 7, 225–242 (1970)

    Article  Google Scholar 

  40. 40.

    Qing, L., Kong, LB.: Probabilistic analysis of underground rock excavations using response surface method and SORM. Comput. Geotech. 38, 1008–1021 (2011)

    Article  Google Scholar 

  41. 41.

    Marshall, A.W., Olkin, I.: Families of multivariate distributions. J. Am. Stat. Assoc. 83, 834–841 (1988)

    Article  Google Scholar 

  42. 42.

    McNeil, A.J., Frey, R., Embrechts, P.: Quantitative risk management: concepts, techniques and tools. Princeton University Press, Princeton (2005)

    Google Scholar 

  43. 43.

    Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Wiley, New York (1999)

    Google Scholar 

  44. 44.

    Nadim, F.: Tools and strategies for dealing with uncertainty in geotechnics. In: Griffiths, D.V., Fenton, G.A. (eds.) Probabilistic methods in geotechnical engineering, pp 71–96. Springer, Wien (2007)

    Chapter  Google Scholar 

  45. 45.

    Nelsen, R.B.: An introduction to copulas, 2nd edition. Springer, New York (2006)

    Google Scholar 

  46. 46.

    Nguyen, V.U., Chowdhury, R.N.: Probabilistic study of spoil pile stability in strip coal mines-two techniques compared. Rock. Mech. Min. Sci. Geomech. Abstr., 303–212 (1984)

  47. 47.

    Phoon, K.K., Nadim, F.: Modelling non-Gaussian random vectors for FORM: state-of-the-art review. International workshop on risk assessment in site characterization and geotechnical design. India Institute of Science, Bangalore, India, pp 26–27 (2004)

  48. 48.

    Phoon, K.K., Kulhanny, F.H.: Evaluation of geotechnical property variability. Can. Geotech. J. 36, 625–639 (1999)

    Article  Google Scholar 

  49. 49.

    Poulin, A., Huard, D., Favre, A., Pugin, S.: Importance of tail dependence in bivariate frequency analysis. J. Hydrol. Eng. ASCE 12(4), 394–403 (2007)

    Article  Google Scholar 

  50. 50.

    Pouillot, R., Delignette-Muller, M.L.: Evaluating variability and uncertainty separately in microbial quantitative risk assessment using two R packages. Int. J. Food Microbiol. 142, 330–340 (2010)

    Article  Google Scholar 

  51. 51.

    R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN:3-900051-07-0. http://www.R-project.org (2008)

  52. 52.

    Rackwitz, R.: Reviewing probabilistic soils modeling. Comput. Geotech. 26(3), 199–223 (2000)

    Article  Google Scholar 

  53. 53.

    Rackwitz, R., Fiessler, B.: An algorithm for calculation of structural reliability under combined loading. P Berichte zur Sicherheitstheorie der Bauwerke, Lab. f. Konstr Ingb. Munchen, Germany (1977)

    Google Scholar 

  54. 54.

    Salvadori, G., De Michele, C., Kottegoda, N.T., Rosso, R.: Extremes in Nature. Springer, Dordecht (2007)

    Google Scholar 

  55. 55.

    Schweizer, B.: Thirty years of copulas. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (eds.) Advances in probability distributions with given marginals. Kluwer, Dordrecht (1991)

    Google Scholar 

  56. 56.

    Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique Université de Paris 8, 229–231 (1959)

    Google Scholar 

  57. 57.

    Tamimi, S., Amadei, B., Frangopol, D.M.: Monte Carlo simulation of rock slope stability. Comput. Struct. 33(6), 1495–1505 (1989)

    Article  Google Scholar 

  58. 58.

    Tang, W.H., Yucemen, M.S., Ang, A.H.-S.: Probability-based short term design of slopes. Can. Geotech. J. 13(3), 201–215 (1976)

    Article  Google Scholar 

  59. 59.

    Thoft-Christensen, P., Baker, M.J.: Structural reliability theory and its applications. Springer, New York (1982)

    Book  Google Scholar 

  60. 60.

    Tobutt, D.C.: Monte Carlo simulation methods for slope stability. Comput. Geosci. 8(2), 199–208 (1982)

    Article  Google Scholar 

  61. 61.

    Venables, W.N., Ripley, B.D.: Modern applied statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0 (2002)

  62. 62.

    Wolff, T.H.: Analysis and design of embankment dam slopes: a probabilistic approach, Ph.D. Thesis. Purdue University, Lafayette, Indiana (1985)

    Google Scholar 

  63. 63.

    Wolff, T.F., Hassan, A., Khan, R., Ur-Rasul, I., Miller, M.: Geotechnical reliability of dam and levee embankments. Technical report prepared for U.S. Army Engineer Waterways Experiment Station. Geotechnical Laboratory, Vicksburg, MS (1995)

    Google Scholar 

  64. 64.

    Yan, J.: Enjoy the joy of copulas: with a package copula. J. Stat. Softw. 21(4), 1–21 (2007)

    Google Scholar 

  65. 65.

    Yan, J., Kojadinovic, I.: Copula: multivariate dependence with copulas. R package version 0.9-5. http://CRAN.R-project.org/package=copula (2010)

  66. 66.

    Zahn, J.J.: Empirical failure criteria with correlated resistance variables. J. Struct. Eng. 116(11), 3122–3137 (1989)

    Article  Google Scholar 

  67. 67.

    Zhang, L., Singh, V.P.: Trivariate flood frequency analysis using the Gumbel-Hougaard copula. J. Hydrol. Eng. 12, 431–439 (2007)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xing Zheng Wu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wu, X.Z. Probabilistic slope stability analysis by a copula-based sampling method. Comput Geosci 17, 739–755 (2013). https://doi.org/10.1007/s10596-013-9353-3

Download citation

Keywords

  • Probabilistic analysis
  • Slope stability
  • Monte Carlo simulation
  • Copula
  • Cross correlation
  • Cohesion
  • Friction angle