Plastic Buckling

  • J. Chakrabarty
Part of the Mechanical Engineering Series book series (MES)


In a typical boundary value problem, involving prescribed nominal traction rates on a part S F of the boundary surface, and prescribed velocities on the remainder S v , more than one mode of deformation may be possible when the applied load reaches a critical value. The lack of uniqueness of the deformation mode under given boundary conditions is commonly referred to as bifurcation, the current shape and mechanical state of the body being supposed to be given or previously determined. For a linear solid, in which the strain rate is a unique linear function of the stress rate during both loading and unloading, a bifurcation mode corresponds to an eigensolution of the field equations, and represents a mode quasi-statically possible under constant loads on S F and rigid constraints on S v In dealing with the conventional elastic/plastic solid, which is bilinear in the sense that the strain rate is related to the stress rate by separate linear functions for loading and unloading, it is convenient to introduce a linear comparison solid with identical boundary conditions (Section 1.5). While bifurcation in the linearized solid can occur under any given traction rates on S f and velocities on S v when the load becomes critical, bifurcation in the actual elastic/plastic solid would occur only under those traction rates for which there is no instantaneous unloading of the material that is currently plastic. The incremental theory of plasticity will be almost exclusively used in this chapter for the estimation of the critical load.


Cylindrical Shell Critical Stress Middle Surface Plastic Range Elastic Buckling 
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  1. Ades, CS. (1957), Bending Strength of Tubes in the Plastic Range, J. Aeronaut. Sci., 24, 505.Google Scholar
  2. Akserland, E.L. (1965), Refinement of Upper Critical Loading of Pipe Bending Taking Account of Geometric Nonlinearity (in Russian), Mekhanika: Machinostroenic, Izv. Akad Nauk, SSSR, 4 123.Google Scholar
  3. Alexander, J.M. (1960), An Approximate Analysis of the Collapse of Thin Cylindrical Shells Under Axial Loading, Quart. J. Mech. Appl. Math., 13, 10.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Ariaratnam, S.T. and Dubey, R.N. (1969), Instability in an Elastic-Plastic Cylindrical Shell Under Axial Compression, J. Appl. Mech., 36, 47.CrossRefGoogle Scholar
  5. Ashwell, D.G. (1959), On the Large Deflection of a Spherical Shell with an Inward Point Load, Proc. IUTAM on Theory of Thin Elastic Shells, Delft. Google Scholar
  6. Babcock, CD. (1983), Shell Stability, J. Appl. Mech., 50, 935.CrossRefGoogle Scholar
  7. Baker, J.F., Horne, M.R., and Roderick, J.W. (1949), The Behaviour of Continuous Stanchions, Proc. Roy. Soc. London Ser. A., 198, 493.zbMATHCrossRefGoogle Scholar
  8. Batdorf, S.R. (1949), Theories of Elastic-Plastic Buckling, J. Aeronaut. Sci., 16, 405.Google Scholar
  9. Batterman, S. (1964), Load-Deflection Behaviour of Shells of Revolution, J. Engng. Mech. Div., Proc. ASCE, 90, EM6, 4167.Google Scholar
  10. Batterman, S. (1965), Plastic Buckling of Axially Compressed Cylindrical Shells, AIAA J., 3, 316.zbMATHCrossRefGoogle Scholar
  11. Batterman, S. (1968), Free Edge Buckling of Axially Compressed Cylindrical Shells, J. Appl. Mech., 35, 73.CrossRefGoogle Scholar
  12. Batterman, S. (1969), Plastic Buckling of an Externally Pressurized Complete Spherical Shell, J. Engng. Mech. Div., Proc. ASCE, 95.Google Scholar
  13. Bijlaard, PP. (1949), Theory and Tests on the Plastic Stability of Plates and Shells, J. Aeronaut. Sci., 16, 529.MathSciNetGoogle Scholar
  14. Bijlaard, PP. (1956), Theory of Plastic Buckling of Plates and Application to Simply Supported Plates Subjected to Bending or Eccentric Compression in Their Plane, J. Appl. Mech., 23, 27.zbMATHGoogle Scholar
  15. Brazier, L.G. (1926), On the Flexure of Thin Cylindrical Shells and Other Sections, Proc. Roy. Soc. London Ser. A, 116, 104.Google Scholar
  16. Bushnell, D. (1982), Plastic Buckling of Various Shells, J. Pressure Vessels Technol., Trans. ASME, 104, 51.CrossRefGoogle Scholar
  17. Bushnell, D. and Galletly, G.D. (1974), Comparison of Test and Theory for Nonsymmetric Elastic-Plastic Buckling of Shells of Revolution, Int. J. Solids Struct., 10, 1271.CrossRefGoogle Scholar
  18. Chakrabarty, J. (1973), Plastic Buckling of Cylindrical Shells Under Uniform External Pressure, Z. Angew. Math. Phys., 24, 270.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Chakrabarty, J. (1998), Theory of Plasticity, 2nd ed., McGraw-Hill, Singapore.Google Scholar
  20. Chawalla, E. (1937), Aussermittig Gedruckte Baustahlstabe mit Elastisch Eingespannten Enden und Verschieden Grossen Angriffschebeln, Der Stahlbau, 15, 49.Google Scholar
  21. Chen, W.F. (1970), General Solution of Inelastic Beam-Column Problems, Engng. J. Mech. Div., Proc. ASCE, 96, EM4, 421.Google Scholar
  22. Chen, W.F. and Astuta, T. (1976), Theory of Beam Columns, Vol. 1, McGraw-Hill, New York.Google Scholar
  23. Corona, E. and Kyriakides, S. (1988), On the Collapse of Inelastic Tubes Under Combined Bending and Pressure, Int. J. Solids Struct., 24, 505.CrossRefGoogle Scholar
  24. Donneil, L.H. (1933), Stability of Thin Walled Tubes Under Torsion, NACA Report 479, pp. 1–24.Google Scholar
  25. Dubey, R.N. (1978), On Bifurcation in Elastic-Plastic Solids, Nucl, Engng. Des., 49, 217.CrossRefGoogle Scholar
  26. El-Ghazaly, H.A. and Sherbourne, A.N. (1986), Deformation Theory for Elastic-Plastic Buckling Analysis of Plates Under Nonproportional Planar Loading, Computers and Structures, 22, 131.CrossRefGoogle Scholar
  27. Engesser, F. (1898), Z Ver. Deut. Ingr., 42, 927.Google Scholar
  28. Flügge, W. (1932), Die Stabilität der Kreiszylinderschale, Ingenieur-Archiv, 3, 24.CrossRefGoogle Scholar
  29. Gellin, S. (1979), Effect of an Axisymmetric Imperfection on the Plastic Buckling of an Axially Compressed Cylindrical Shell, J. Appl. Mech., 46, 125.zbMATHCrossRefGoogle Scholar
  30. Gellin, S. (1980), The Plastic Buckling of Long Cylindrical Shells Under Pure Bending, Int. J. Solids Struct., 16, 397.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Gerard, G. (1957), Handbook of Structural Stability, Part I, Buckling of Flat Plates, NACA TN3781.Google Scholar
  32. Gerard, G. (1962), Introduction to Structural Stability Theory, McGraw-Hill, New York.Google Scholar
  33. Gjelsvik, A. and Lin, G.S. (1985), Plastic Buckling of Plates with Edge Frictional Shear Effects, J. Engng. Mech. Div., Trans. ASCE, 113, 953.CrossRefGoogle Scholar
  34. Hamada, H. (1985), In-Plane Buckling of Circular Plates, Proc. J. Soc. Mech. Engrs., 51, 1928.Google Scholar
  35. Hill, R. and Sewell, M.J. (1960), A General Theory of Inelastic Column Failure—I and II, J. Mech. Phys. Solids, 8, 105 and 112.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Hill, R. and Sewell, M.J. (1962), A General Theory of Inelastic Column Failure—III, J. Mech. Phys. Solids, 10, 185.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Horne, M.R. (1956), The Elastic-Plastic Theory of Compression Members, J. Mech. Phys. Solids, 4, 104.zbMATHCrossRefGoogle Scholar
  38. Home, M.R. and Merchant, W. (1965), The Stability of Frames, Pergamon Press, Oxford, UK.Google Scholar
  39. Hutchinson, J.W. (1972), On the Postbuckling Behavior of Imperfection Sensitive Structures in Plastic Range, J. Appl. Mech., 39, 155.CrossRefGoogle Scholar
  40. Hutchinson, J.W. (1973), Imperfection Sensitivity in the Plastic Range, J. Mech. Phys. Solids, 21, 163.zbMATHCrossRefGoogle Scholar
  41. Hutchinson, J.W. (1974), Plastic Buckling, Adv. in Appl. Mech., 14, 67.Google Scholar
  42. Inoue, T. and Kato, B. (1993), Analysis of Plastic Buckling of Steel Plates, Int. J. Solids Struct., 15, 567.Google Scholar
  43. Ketter, R.L. (1961), Further Studies of the Strength of Beam Columns, J. Struct. Div., Proc. ASCE, 87, ST6, 135.Google Scholar
  44. Leckie, RA. and Penny, R.K. (1968), Plastic Instability of Spherical Shells, in Engineering Plasticity (eds., J. Heyman and F.A. Leckie), Cambridge University Press, UK, p. 401.Google Scholar
  45. Lee, L.H.N. (1962), Inelastic Buckling of Initially Imperfect Cylindrical Shells Subject to Axial Compression, J. Aeronaut. Sci., 29, 87.zbMATHGoogle Scholar
  46. Li, S. and Reid, S.R. (1992), The Plastic Buckling of Axial Compressed Square Tubes, J. Appl. Mech., 59, 276.zbMATHCrossRefGoogle Scholar
  47. Lu, L.W. and Kamalvand, H. (1968), Ultimate Strength of Laterally Loaded Columns, J. Struct. Div., Proc. ASCE, 94, ST6, 1505.Google Scholar
  48. Mamalis, A.G. and Johnson, W. (1983), Quasi-Static Crumpling of Thin-Walled Circular Cylinders and Frusta Under Axial Compression, Int. J. Mech. Sci., 25, 713.CrossRefGoogle Scholar
  49. Needleman, A. (1975), Post-Bifurcation Behavior and Imperfection Sensitivity of Elastic/Plastic Circular Plates, Int. J. Mech. Sci., 17, 1.CrossRefGoogle Scholar
  50. Onat, E.T. and Drucker, D.C. (1953), Inelastic Instability and Incremental Theories of Plasticity, J. Aeronaut. Sci., 20, 181.MathSciNetGoogle Scholar
  51. Pearson, CE. (1950), Bifurcation Criteria and Plastic Buckling of Plates and Columns, J. Aeronaut. Sci., 17, 417.MathSciNetGoogle Scholar
  52. Pearson, C.E. (1956), A General Theory of Elastic Stability, Quart. Appl. Math., 14, 133.MathSciNetzbMATHGoogle Scholar
  53. Petryk, H. (1983), A Stability Postulate for Quasi-Static Processes of Plastic Deformation, Arch. Mech., 35, 753.MathSciNetzbMATHGoogle Scholar
  54. Pugsley, A. (1979), On the Crumpling of Thin Cylindrical Tubes, Quart. J. Mech. Appl. Math., 32, 1.MathSciNetCrossRefGoogle Scholar
  55. Pugsley, A. and Macaulay, M. (1960), Large Scale Crumpling of Thin Cylindrical Columns, Quart. J. Mech. Appl. Math., 13, 1.MathSciNetCrossRefGoogle Scholar
  56. Reddy, B.D. (1979), Plastic Buckling of a cylindrical Shell in Pure Bending, Int. J. Mech. Sci., 21, 671.zbMATHCrossRefGoogle Scholar
  57. Reddy, B.D. (1980), Buckling of Elastic-Plastic Discretely Stiffened Cylinders in Axial Compression, Int. J. Solids Struct., 16, 313.zbMATHCrossRefGoogle Scholar
  58. Rees, D.W.A. (1982), Plastic Torsional Buckling of Thin-Walled Cylinders, J. Appl. Mech., 49, 663.CrossRefGoogle Scholar
  59. Seide, P. and Weingarten, V.I. (1961), On the Buckling of Circular Cylindrical Shells Under Pure Bending, J. Appl. Mech., 28, 112.MathSciNetCrossRefGoogle Scholar
  60. Sewell, M.J. (1963), A General Theory of Elastic and Inelastic Plate Failure, Part I, J. Mech. Phys. Solids, 11, 377.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Sewell, M.J. (1964), A General Theory of Elastic and Inelastic Plate Failure, Part II, J. Mech. Phys. Solids, 12, 279.MathSciNetCrossRefGoogle Scholar
  62. Sewell, M.J. (1973), A Yield Surface Corner Lowers the Buckling Stress of an Elastic/Plastic Plate, J. Mech. Phys. Solids, 21, 19.zbMATHCrossRefGoogle Scholar
  63. Shanley, F.R. (1947), Inelastic Column Theory, J. Aeronaut. Sci., 14, 251.zbMATHGoogle Scholar
  64. Shrivastava, H.P. (1979), Inelastic Buckling of Axially Compressed Cylindrical Shells, Int. J. Solids Struct., 15, 567.MathSciNetzbMATHCrossRefGoogle Scholar
  65. Southwell, R.V. (1913), On the Collapse of Tubes by External Pressure, Phil. Mag., 25, 687.zbMATHGoogle Scholar
  66. Timoshenko, S.P. and Gere, J.M. (1961), Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York.Google Scholar
  67. Tomita, Y. (1994), Simulations of Plastic Instabilities in Solid Mechanics, Appl. Mech. Rev., 47, 171.CrossRefGoogle Scholar
  68. Tugcu, P. (1991), Plate Buckling in the Plastic Range, Int. J. Mech. Sci., 33, 1.CrossRefGoogle Scholar
  69. von Karman, Th. (1910), Untersuchungen Über Knickfestigkeit, Mitteilungen Über Forschungsarbeit, Ver. Deut. Ing., Vol. 81, Springer-Verlag, Berlin.Google Scholar
  70. Wierzbicki, T. and Abramowicz, W. (1983), On the Crushing Mechanism of Thin-Walled Structures, J. Appl. Mech., 50, 727.zbMATHCrossRefGoogle Scholar
  71. Zhang, L.C. and Yu, T.X. (1987), An Investigation of the Brazier Effect of a Cylindrical Tube Under Pure Elastic-Plastic Bending, Int, J. Pres. Ves. Piping, 30, 77.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2000

Authors and Affiliations

  • J. Chakrabarty
    • 1
  1. 1.Department of Mechanical EngineeringNational Taiwan UniversityTaipeiTaiwan, R.O.C.

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