Reliability Analysis of Rock Wedges
This article has a reply.
VIEW THE REPLYPublication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 123, Issue 6
Abstract
The stability analyses of tetrahedral wedges in rock slopes are commonly conducted using the versatile sequential vectorial operations or stereographic projection methods. This paper presents an alternative approach using closed-form equations, for cases where the upper ground surface dips in the same direction as the slope face. The equations allow determination of kinematic admissibility, the modes of sliding (whether along two planes or a single plane), and the corresponding factor of safety. Since the factor of safety does not reflect the uncertainty of parameters, the Hasofer-Lind second moment reliability index is explored. Useful insights are obtained based on the perspective of an ellipsoid expanding in the original space of the random variables. The proposed perspective also suggests a practical method of computing the reliability index using the Solver tool available in a spreadsheet software. First-order probability bounds based on reliability indices of a tetrahedral wedge are in good agreement with the results of Monte Carlo simulation. The need to account for multiple failure modes and to consider all surfaces bounding the failure domain is demonstrated.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Ang, H. S., and Tang, W. H. (1984). Probability concepts in engineering planning and design, vol. II—decision, risk, and reliability. John Wiley & Sons, Inc., New York, N.Y.
2.
Chowdhury, R. N., and Xu, D. W.(1995). “Geotechnical system reliability of slopes.”Reliability Engrg. and Sys. Safety, Northern Ireland, 47, 141–151.
3.
Christian, J. T., Ladd, C. C., and Baecher, G. B.(1994). “Reliability applied to slope stability analysis.”J. Geotech. Engrg., ASCE, 120(12), 2180–2207.
4.
Cornell, C. A.(1967). “Bounds on the reliability of structural systems.”J. Struct. Div., ASCE, 93(1), 171–200.
5.
Dai, S. H., and Wang, M. O. (1992). Reliability analysis in engineering applications. Van Nostrand Reinhold, New York, N.Y.
6.
Ditlevsen, O. (1981). Uncertainty modeling: with applications to multidimensional civil engineering systems. McGraw–Hill Book Co., Inc., New York, N.Y.
7.
Einstein, H. H.(1991). “Reliability in rock engineering.”Proc., Geotech. Engrg. Congr., Geotech. Spec. Publ. #27, ASCE, New York, N.Y., 1, 608–633.
8.
Einstein, H. H., and Baecher, G. B. (1983). “Probabilistic and statistical methods in engineering geology—part 1: exploration.”Rock Mech. and Rock Engrg., 16, Springer-Verlag New York, Inc., New York, N.Y., 39–72.
9.
Goodman, R. E. (1976). Methods of geological engineering in discontinuous rocks. West Publishing Co., St. Paul, Minn.
10.
Goodman, R. E. (1989). Introduction to rock mechanics, 2nd Ed., John Wiley & Sons, Inc., New York, N.Y.
11.
Goodman, R. E.(1995). “Thirty-fifth Rankine lecture: `Block theory and its application.”' Geotechnique, London, 45(3), 383–423.
12.
Goodman, R. E., and Taylor, R. L. (1967). “Methods of analysis for rock slopes and abutments: a review of recent developments.”Failure and breakage of rocks, C. Fairhurst, ed., Am. Inst. of Min., Metallurgical and Petr. Engrs., New York, N.Y., 303–320.
13.
Hasofer, A. M., and Lind, N. C.(1974). “Exact and invariant second-moment code format.”J. Engrg. Mech., ASCE, 100(1), 111–121.
14.
Hendron, A. J., Cording, E. J., and Aiyer, A. K. (1971). “Analytical and graphical methods for the analysis of slopes in rock masses.”U.S. Army Engrg. Nuclear Cratering Group Tech. Rep. no. 36, U.S. Army Engineers Waterways Experiment Station, Vicksburg, Miss.
15.
Hoek, E., and Bray, J. (1977). Rock slope engineering. Inst. Min. Metall., London.
16.
Hoek, E., and Bray, J. (1981). Rock slope engineering, 3rd Ed., Inst. Min. Metall., London.
17.
Hoek, E., Bray, J. W., and Boyd, J. M.(1973). “The stability of a rock slope containing a wedge resting on two intersecting discontinuities.”Quarterly J. Engrg. Geol., 6(1), 1–55.
18.
Hoek, E., Kaiser, P. K., and Bawden, W. F. (1995). Support of underground excavations in hard rock. A. A. Balkema, Rotterdam, The Netherlands.
19.
Jaeger, J. C.(1971). “Friction of rocks and the stability of rock slopes.”Geotechnique, London, 21(2), 97–134.
20.
John, K. W.(1968). “Graphical stability analysis of slopes in jointed rock.”J. Soil Mech. and Found. Div., ASCE, 94(2), 497–526.
21.
Johnson, R. A., and Wichern, D. W. (1992). Applied multivariate statistical analysis, 3rd Ed., Prentice-Hall, Inc., Englewood Cliffs, N.J.
22.
Kitch, W. A., Wright, S. G., and Gilbert, R. B.(1995). “Probabilistic analysis of reinforced soil slopes.”Proc., 10th Conf. on Engrg. Mech., ASCE, New York, N.Y., 1, 325–328.
23.
Kulatilake, P. H. S. W., Chen, J., Teng, J., Shufang, X., and Pan, G.(1996). “Discontinuity geometry characterization in a tunnel close to the proposed permanent shiplock area of the Three Gorges Dam site in China.”Int. J. Rock Mech. Min. Sci. and Geomech. Abstr., Oxford, England, 33(3), 255–277.
24.
Kulatilake, P. H. S. W., Finley, R. E., and Ghosh, A. (1985). “Effect of variability of joint orientation and strength on factor of safety of wedge stability.”Proc., Int. Symp. Fundamentals of Rock Joints, O. Stephansson, ed., Bjorkliden, Sweden, 25–33.
25.
Li, K. S., and Lumb, P.(1987). “Probabilistic design of slopes.”Can. Geotech. J., 24(4), 520–535.
26.
Londe, P., Vigier, G., and Vormeringer, R.(1969). “The stability of rock slopes, a three-dimensional study.”J. Soil Mech. and Found. Div., ASCE, 95(1), 235–262.
27.
Low, B. K. (1979). “Reliability of rock slopes with wedge mechanisms,” MS thesis, Massachusetts Inst. of Technol., Cambridge, Mass.
28.
Low, B. K.(1996). “Practical probabilistic approach using spreadsheet.”ASCE Geotech. Spec. Publ. no. 58, Proc., Uncertainty in Geologic Environment: from Theory to Practice, ASCE Spec. Conf., ASCE, New York, N.Y., 2, 1284–1302.
29.
Low, B. K., and Einstein, H. H. (1992). “Simplified reliability analysis for wedge mechanisms in rock slopes.”Proc., 6th Int. Symp. on Landslides, A. A. Balkema, Rotterdam, The Netherlands, 499–507.
30.
Low, B. K., and Tang, W. H. (1997). “Efficient reliability evaluation using spreadsheet.”J. Engrg. Mech., ASCE, 123(7).
31.
Luckman, P. G., Der-Kiureghian, A., and Sitar, N. (1987). “Use of stochastic stability analysis for Bayesian back calculation of pore pressures acting in a cut slope at failure.”Proc., ICASP5, 5th Int. Conf. on Applications of Statistics and Probability in Soil and Struct. Engrg., N. C. Lind, ed., 2, Inst. for Risk Res., Univ. of Waterloo, Ontario, Canada, 922–929.
32.
Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of structural safety. Prentice-Hall, Inc., Englewood Cliffs, N.J.
33.
Melchers, R. E. (1987). Structural reliability: analysis and prediction. Ellis Horwood Ltd., Chichester, West Sussex, England.
34.
Mostyn, G. R., and Li, K. S. (1993). “Probabilistic slope analysis—state of play.”Proc., Conf. on Probabilistic Methods in Geotech. Engrg., A. A. Balkema, Rotterdam, The Netherlands, 89–109.
35.
National Research Council. (1995). Probabilistic methods in geotechnical engineering. National Academy Press, Washington, D.C.
36.
Oka, Y., and Wu, T. H.(1990). “System reliability of slope stability.”J. Geotech. Engrg., ASCE, 116(8), 1185–1189.
37.
Rackwitz, R. (1976). “Practical probabilistic approach to design.”Bull. 112, Comite European du Beton, Paris.
38.
Shanley, R. J., and Mahtab, M. A.(1976). “Delineation and analysis of clusters in orientation data.”Math. Geol., 8(1), 9–23.
39.
Shinozuka, M.(1983). “Basic analysis of structural safety.”J. Struct. Engrg., ASCE, 109(3), 721–740.
40.
Tang, W. H. (1993). “Recent developments in geotechnical reliability.”Probabilistic methods in geotechnical engineering, K. S. Li and S-C. R. Lo, eds., A. A. Balkema, Rotterdam, The Netherlands, 3–27.
41.
Tichy, M. (1993). Applied methods of structural reliability. Kluwer Academic, Dordrecht, Boston.
42.
Vanmarcke, E. H.(1977). “Reliability of earth slopes.”J. Geotech. Engrg., ASCE, 103(11), 1247–1266.
43.
Veneziano, D. (1974). “Contributions to second moment reliability.”Res. Rep. no. R74-33, Dept. of Civ. Engrg., Massachusetts Inst. of Technol., Cambridge, Mass.
44.
Wathugala, D. N., Kulatilake, P. H. S. W., Wathugala, G. W., and Stephansson, O.(1990). “A general procedure to correct sampling bias on joint orientation using a vector approach.”Comp. and Geotechnics, Oxford, England, 10, 1–31.
45.
Whitman, R. V.(1984). “Evaluating calculated risk in geotechnical engineering.”J. Geotech. Engrg., ASCE, 110(2), 143–188.
46.
Wittke, W. (1990). Rock mechanics: theory and applications with case histories. Springer-Verlag New York, Inc., New York, N.Y.
Information & Authors
Information
Published In
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Jun 1, 1997
Published in print: Jun 1997
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.