Stability Analysis in Geomechanics by Linear Programming. I: Formulation
Publication: Journal of Geotechnical Engineering
Volume 118, Issue 11
Abstract
The limit analysis of stability problems in geomechanics is formulated as a pair of primal‐dual linear programs that encode, respectively, the kinematic and static limit theorems in a discrete version. The failure surface can take any arbitrary shape. The soil domain is divided, into rigid elements connected by interfacing Mohr‐Coulomb layers. For an assumed finite element mesh, the solution of either linear program identifies the critical collapse mechanism among all the possible failure mechanisms contained within the mesh, and gives the associated values of both the static and kinematic variables as well as the critical load parameter. This solution is both kinematically and statically admissible for the discretized system; for the continuum, it is an upper‐bound solution. The proposed method is able to deal with external forces acting on a soil mass with varying pore‐water pressure, and inhomogeneous materials having both cohesion and internal friction. An illustrative example is presented; this kinematic formulation accurately gives the upper‐bound solution.
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References
1.
Basudhar, P. K., Valsangkar, A. J., and Madhav, M. R. (1979). “Optimal lower bound of passive earth pressure using finite elements and non‐linear programming.” Int. J. Num. Ana. Meth. Geomechanics, 3, 367–379.
2.
Brinch Hansen, J. (1966). “Comparison of methods for stability analysis.” Bulletin No. 21 The Danish Geotechnical Institute, Copenhagen, Denmark.
3.
Casciaro, R., and Cascini, L. (1982). “A mixed formulation and mixed finite elements for limit analysis.” Int. J. Num. Meth. Eng., 18(2), 211–243.
4.
Chakrabarty, J. (1987). Theory of plasticity. McGraw‐Hill Book Co., Inc., New York, N.Y.
5.
Chen, W. F. (1975). Limit analysis and soil plasticity. Elsevier Scientific Publishing Co., Amsterdam, The Netherlands.
6.
Ching, R. K. H., and Fredlund, D. G. (1983). “Some difficulties associated with the limit equilibrium method of slices.” Canadian Geotech. J., 20(4), 661–672.
7.
Chuang, P.‐H., and Munro, J. (1983). “Linear programming with imprecise data.” Civ. Engrg. Systems, 1, (Sept.)37–41.
8.
Chuang, P.‐H., Munro, J., and Lloyd Smith, D. (1986). “Plastic limit design and analysis with imprecise data.” Steel structures: Recent research advances and their applications to design, M. N. Pavlovic, ed., Elsevier Applied Science, Amsterdam, The Netherlands, 71–84.
9.
Chuang, P.‐H. (1992). “Stability analysis in geomechanics by linear programming. II: Application.” J. Geotech. Engrg., ASCE, 118(11), 1716–1726.
10.
Coulomb, C. A. (1776). “Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l'architecture.” Royale Academie des Sciences, Paris, France, Vol. 7.
11.
Da Fonseca, A. M. A., Munro, J., and Smith, D. L. (1977). “Finite element limit analysis of plates by linear programming.” Proc., IASS Int. Conf. on Lightweight Shell and Space Structures for Normal and Seismic Zones, Alma‐Ata, U.S.S.R., MIR Publishers, 1, 95–106.
12.
Davis, E. H., and Booker, J. R. (1973). “Some adaptations of classical plasticity theory for soil stability problems.” Proc. Symp. on Role of Plasticity in Soil Mech., A. C. Palmer, ed., Cambridge, England, 24–41.
13.
Drucker, D. C. (1954). “Coulomb friction, plasticity and limit loads.” J. Appl. Mech., 21, ), 71–74.
14.
Drucker, D. C., and Prager, W. (1952). “Soil mechanics and plastic analysis or limit design.” Q. J. Appl. Math., 10(2), 157–165.
15.
Drucker, D. C., Prager, W., and Greenberg, H. J. (1952). “Extended limit design theorems for continuous media.” Q. J. Appl. Math., 9(4), 381–389.
16.
Engineering plasticity by mathematical programming. (1979). M. Z. Cohn and G. Maier, eds., Pergamon Press, New York, N.Y.
17.
Frémond, M., and Salencon, J. (1973). “Limit analysis by finite‐element methods.” Proc., Symp. on Role of Plasticity in Soil Mech., A. C. Palmer, ed., Cambridge, England.
18.
Lysmer, J. (1970). “Limit analysis of plane problems in soil mechanics.” J. Soil Mech. Found. Div., ASCE, 96(4), 1311–1334.
19.
Maier, G., and Munro, J. (1982). “Mathematical programming applications to engineering plastic analysis.” Appl. Mech. Review, 5(12), 1631–1643.
20.
Maier, G., and Lloyd Smith, D. (1986). “Update to Mathematical programming applications to engineering plastic analysis.” Appl. Mech. Update., 377–383.
21.
Martins, J. B., Reis, E. B., and Matos, A. C. (1981). “New methods of analysis for stability of slopes.” Proc. 10th Int. Conf. Soil Mech. and Found. Engrg., Stockholm, Sweden, 3, 463–467.
22.
Martins, J. B. (1982). “Embankments and slopes by mathematical programming.” Numerical Methods in geomechanics; Proc., NATO Advanced Study Institute, J. B. Martins, ed., Vimeiro, Lisbon, Portugal, 305–334.
23.
Munro, J. (1979). “Optimal plastic design of frames.” Engineering plasticity by mathematical programming, M. Z. Cohn and G. Maier, eds., Pergamon Press, New York, N.Y., 135–171.
24.
Munro, J. (1982). “Plastic analysis in geomechanics by mathematical programming.” Numerical methods in geomechanics; Proc., NATO Advanced Study Institute, J. B. Martins, ed., Vimeiro, Lisbon, Portugal, 247–272.
25.
Munro, J., and Chuang, P.‐H. (1986). “Optimal plastic design with imprecise data.” J. Engrg. Mech., ASCE, 112(9), 888–903.
26.
Munro, J., and Lloyd Smith, D. (1972). “Linear programming duality in plastic analysis and synthesis.” Proc., Int. Symp. Computer‐Aided Struct. Design, Univ. of Warwick, England, 1, A1.22‐A1.54.
27.
Palmer, A. C. (1966). “A limit theorem for materials with non‐associated flow laws.” J. Mecanique, Paris, France, 5(2), 217–222.
28.
Proc., Symp. on Role of Plasticity in Soil Mech. (1973). A. C. Palmer, ed., Cambridge, England.
29.
Rankine, W. J. M. (1857). “On stability of loose earth.” Proc., Phil. Trans. Roy Soc., London, England, 147, 9–27.
30.
Salencon, J. (1977). Application of the theory of plasticity in soil mechanics. English translation by R. W. Lewis and H. Virlogeux, John Wiley and Sons, New York, N.Y.
31.
Skempton, A. W., and Hutchinson, J. (1969). “Stability of natural slopes and embankment foundations.” Proc., 7th Int. Conf. Soil Mech. Found. Engrg., Sociedad Mexicana de Mecánica de Suelos, A.C., Mexico, 291–340.
32.
Taylor, D. W. (1948). Fundamentals of soil mechanics. John Wiley and Sons, New York, N.Y.
33.
Whitman, R. V., and Bailey, W. A. (1967). “Use of computers for slope stability analysis.” J. Soil Mech. Found. Div., ASCE, 93(4), 475–498.
34.
Zangwili, W. S. (1969). Nonlinear programming. Prentice‐Hall, Englewood Cliffs, N.J.
35.
Zavelani‐Rossi, A., Gatti, G., and Gioda, G. (1974). “On finite‐element stability analysis in soil mechanics.” Int. Symp. on Discrete Methods in Engrg., CISE, Segrate, Milan, Italy.
36.
Zienkiewicz, O. C., Humpheson, C., and Lewis, R. W. (1975). “Associated and non‐associated viscoplasticity and plasticity in soil mechanics.” Geotechnique, London, England, 25(4), 671–689.
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Published In
Journal of Geotechnical Engineering
Volume 118 • Issue 11 • November 1992
Pages: 1696 - 1715
Copyright
Copyright © 1992 ASCE.
History
Published online: Nov 1, 1992
Published in print: Nov 1992
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