Eigenvalue Problem from the Stability Analysis of Slopes
Publication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 135, Issue 5
Abstract
Of the existing methods for the three-dimensional (3D) limit equilibrium analysis of slopes, none can simultaneously satisfy all six equilibrium equations. Except for Fellenius’ method that satisfies only one condition of moment equilibrium, all these methods could encounter numerical problems in their applications. Based on the global analysis procedure that considers the whole sliding body instead of individual columns as the loaded body, it is shown that the 3D limit equilibrium analysis of slopes simply reduces to the solution of a generalized eigenvalue problem in which the largest real eigenvalue is just the factor of safety (FOS). The proposed solution is rigorous and can accommodate any shape of slip surfaces. Under undrained conditions, the problem has a unique solution and the FOS has an explicit expression. In addition, through transforming the volume integrals over the sliding body into the boundary integrals, the proposed method does not need to partition the sliding body into columns.
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Acknowledgments
The writer is very grateful to the reviewers who have carefully reviewed the manuscript and offered valuable comments and constructive suggestions. This work is funded by the National Natural Science Foundation of China (NSFC), Project Nos. NNSFC50779031 and NNSFC90715028.
References
Bell, J. M. (1968). “General slope stability analysis.” J. Soil Mech. and Found. Div., 94, 1253–1270.
Chang, M. (2005). “Three-dimensional stability analysis of the Kettleman Hills landfill slope failure based on observed sliding-block mechanism.” Comput. Geotech., 32, 587–599.
Chen, R. H., and Chameau, J. L. (1982). “Three-dimensional limit equilibrium analysis of slopes.” Geotechnique, 33, 31–40.
Chen, W. F. (1975). Limit analysis and soil plasticity, Elsevier, Amsterdam, The Netherlands.
Chen, Z. Y. (2003). Soil slope stability analysis—Theory, methods, and programs, China Water Power Press, Beijing.
Chen, Z. Y. (2004). “A generalized solution for tetrahedral rock wedge stability analysis.” Int. J. Rock Mech. Min. Sci., 41, 613–628.
Cheng, Y. M., Lansivaara, T., and Siu, J. (2008). “Impact of convergence on slope stability analysis and design.” Comput. Geotech.,35, 105–113.
Cheng, Y. M., Liu, H. T., Wei, W. B., and Au, S. K. (2005). “Location of critical three-dimensional non-spherical failure surface by NURBS functions and ellipsoid with applications to highway slopes.” Comput. Geotech., 32, 387–399.
Chugh, A. K. (1981). “Multiplicity of numerical solutions for slope stability problems.” Int. J. Numer. Analyt. Meth. Geomech., 5, 313–322.
Duncan, J. M. (1996). “State of the art: Limit equilibrium and finite element analysis of slopes.” J. Geotech. Engrg., 122, 577–596.
Gens, A., Hutchinson, J. N., and Cavounidis, S. (1988). “Three dimensional analysis of slides in cohesive soils.” Geotechnique, 38, 1–23.
Hoek, E., and Bray, J. W. (1977). Rock slope engineering, The Institute of Mining and Metallurgy, London.
Huang, C. C., and Tsai, C. C. (2000). “New method for 3D and asymmetric slope stability analysis.” J. Geotech. Geoenviron. Eng., 126, 917–927.
Hungr, O., Salgado, F. M., and Byrne, P. M. (1989). “Evaluation of a three-dimensional method of slope stability analysis.” Can. Geotech. J., 26, 679–686.
Kressner, D. (2005). Numerical methods for general and structured eigenvalue problems, Springer, Berlin.
Lam, L., and Fredlund, D. G. (1993). “A general limit equilibrium model for three-dimensional slope stability analysis.” Cognition, 30, 905–919.
Leshchinsky, D., and Huang, C. C. (1992). “Generalized three dimensional slope stability analysis.” J. Geotech. Engrg., 118, 1748–1764.
MathWorks. (2002). MATLAB 6.5, Natick, Mass.
Seed, R. M., et al. (1990). “Kettleman Hills waste landfill slope failure. II: Stability analysis.” J. Geotech. Engrg., 116, 669–690.
Stark, T. D., and Eid, H. T. (1998). “Performance of three-dimensional slope stability methods in practice.” J. Geotech. Geoenviron. Eng., 124, 1049–1060.
Ugai, K. (1985). “Three-dimensional stability analysis of vertical cohesive slopes.” Soils Found., 25, 41–48.
Wang, Y. J., Yin, J. H., Chen, Z., and Lee, C. F. (2004). “Analysis of wedge stability using different methods.” Rock Mech. Rock Eng., 37, 127–150.
Xie, M. W., Esaki, T., Cheng Qiu, C., and Wang, C. (2006). “Geographical information system-based computational implementation and application of spatial three-dimensional slope stability analysis.” Can. Geotech. J., 33, 260–274.
Zhang, X. (1988). “Three-dimensional stability analysis of concave slopes in plan view.” J. Geotech. Engrg., 114, 658–671.
Zienkiewicz, O. C., and Taylor, R. L. (1989). The finite element methods, 4th Ed., Vol. 1, McGraw-Hill, London.
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Published In
Journal of Geotechnical and Geoenvironmental Engineering
Volume 135 • Issue 5 • May 2009
Pages: 647 - 656
Copyright
© 2009 ASCE.
History
Received: Oct 17, 2007
Accepted: Aug 6, 2008
Published online: Feb 27, 2009
Published in print: May 2009
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