Plastic Analysis of Shells

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


A shell is a thin-walled structure in which the material fills the space between two parallel or nearly parallel surfaces, the mean surface that halves the shell thickness being known as the middle surface of the shell. When the shell is loaded to the point of plastic collapse, the deformation proceeds in an unrestricted manner under constant load, provided geometry changes are disregarded and the material is considered as ideally plastic. As in the case of thin plates, the incipient deformation mode of the shell will be described in terms of that of the middle surface, and the associated stress field will be specified in terms of the stress resultants, acting across the shell thickness. The usual assumptions of rigid/plastic material will be made for the estimation of the collapse load.


Cylindrical Shell Yield Condition Yield Surface Flow Rule Face Sheet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Biron, A. and Hodge, P.G. (1967), Limit Analysis of Rotationally Symmetric Shells Under Central Boss Loading by a Numerical Method, J. Appl Mech, 34, 644.CrossRefGoogle Scholar
  2. Biron, A. and Sawczuk, A. (1967), Plastic Analysis of Rib-Reinforced Cylindrical Shells, J. Appl. Meck, 34, 37.CrossRefGoogle Scholar
  3. Brooks, G.M. (1987), Elastic-Plastic Ring Loaded Cylindrical Shells, J. Appl Mech, 54, 597.CrossRefzbMATHGoogle Scholar
  4. Brooks, G.M. (1988), Elastic-Plastic Ring Loaded Cylindrical Shells, J. Appl Mech, 55, 761.CrossRefGoogle Scholar
  5. Brooks, G.M. and Leung, C.P. (1989), Elastic-Plastic Analysis of a Radially Loaded Spherical Shell, J. Pressure Vessel Technol, Trans. ASME, 111, 39.Google Scholar
  6. Coon, M.D. and Gill, S.S. (1968), Effect of Change of Geometry on the Rigid/Plastic Limit Load of Cylinders, Int. J. Mech. Sci., 10, 355.CrossRefGoogle Scholar
  7. Demir, H.H. (1965), Cylindrical Shells Under Ring Loads, Proc. ASCE, Struct. Engng. Div.,91, 71.Google Scholar
  8. DeRuntz, J. A. and Hodge, P.G. (1966), Significance of the Concentrated Load on the Limit Analysis of Conical Shells, J. Appl. Mech., 33, 93.CrossRefGoogle Scholar
  9. Dinno, K.S. and Gill, S.S. (1965), The Limit Analysis of a Pressure Vessel Consisting of the Junction of a Cylindrical and a Spherical Shell, Int. J. Mech. Sci., 7, 21.CrossRefGoogle Scholar
  10. Dokmeci, M.C. (1966), A Shell of Constant Strength, Z. Angew Math. Phys., 17, 545.CrossRefGoogle Scholar
  11. Drucker, D.C. (1953), Limit Analysis of Cylindrical Shells Under Axially-Symmetric Loading, Proc. Ist Midwestern Conf. Solid Mech., Urbana, p. 158.Google Scholar
  12. Drucker, D.C. and Shield, R.T. (1957), Bounds on Minimum Weight Design, Quart. Appl. Math., 15, 269.MathSciNetzbMATHGoogle Scholar
  13. Drucker, D.C. and Shield, R.T. (1959), Limit Analysis of Symmetricaly Loaded Thin Shells of Revolution, J. Appl. Mech., 26, 61.MathSciNetzbMATHGoogle Scholar
  14. Drucker, D.C. and Shield, R.T. (1961), Design of Torispherical and Toriconical Pressure Vessel Heads, J. Appl. Mech., 28, 292.CrossRefGoogle Scholar
  15. Eason, G. (1959), The Load Carrying Capacity of Cylindrical Shells Subjected to a Ring of Force, J. Mech. Phys. Solids, 7, 169.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Eason, G. and Shield, R.T. (1955), The Influence of Free Ends on the Load Carrying Capacities of Cylindrical Shells, J. Mech. Phys. Solids, 4, 17.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Flügge, W. (1960), Stresses in Shells, Springer-Verlag, Berlin.CrossRefzbMATHGoogle Scholar
  18. Flügge, W. and Nakamura, T. (1965), Plastic Analysis of Shells of Revolution Under Axisymmetric Loads, Ingenieur-Archiv, 34, 238.CrossRefGoogle Scholar
  19. Freiberger, W. (1956), Minimum Weight Design of Cylindrical Shells, J. Appl. Mech., 23, 576.MathSciNetzbMATHGoogle Scholar
  20. Gill, S.S. (1964), The Limit Pressure for a Flush Cylindrical Nozzle in a Spherical Pressure Vessel, Int. J. Mech. Sci, 6, 105.CrossRefGoogle Scholar
  21. Gill, S.S. (1970), The Stress Analysis of Pressure Vessels and Pressure Vessel Components, Chapter 3, Pergamon Press, Oxford, U.K.Google Scholar
  22. Gill, S.S. and Leckie, FA. (1968), The Effect of Geometry Changes on the Application of Limit Analysis to the Design of Pressure Vessel Nozzles, Int. J. Mech. Sci., 10, 989.CrossRefGoogle Scholar
  23. Haydi, H.A. and Sherbourne, A.N. (1974), Plastic Analysis of Shallow Spherical Shells, J. Appl. Mech., 41, 593.CrossRefGoogle Scholar
  24. Hodge, P.G. (1954), The Rigid-Plastic Analysis of Symmetrically Loaded Cylindrical Shells, J. Appl. Mech., 21, 336.MathSciNetzbMATHGoogle Scholar
  25. Hodge, P.G. (1956), Displacements in an Elastic-Plastic Cylindrical Shell, J. Appl. Mech., 23, 73.zbMATHGoogle Scholar
  26. Hodge, P.G. (1960a), Yield Conditions for Rotationally Symmetric Shells Under Axisym-metric Loading, J. Appl. Mech., 27, 323.MathSciNetCrossRefGoogle Scholar
  27. Hodge, P.G. (1960b), Plastic Analysis of Circular Conical Shells, J. Appl. Mech., 27, 696.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Hodge, P.G. (1961), The Mises Yield Condition for Rotationally Symmetric Shells, Quart. Appl. Math., 18, 305MathSciNetzbMATHGoogle Scholar
  29. Hodge, P.G. (1963), Limit Analysis of Rotationally Symmetric Plates and Shells, Chap. 7, Prentice Hall, Englewood Cliffs, NJ.zbMATHGoogle Scholar
  30. Hodge, P.G. (1964a), Rigid-Plastic Analysis of Spherical Caps with Cutouts, Int. J. Mech. Sci., 6, 177.CrossRefGoogle Scholar
  31. Hodge, P.G. (1964b), Plastic Design of a Closed Cylindrical Structure, J. Mech. Phys. Solids, 12, 1.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Hodge, P.G. (1964c), Full Strength Reinforcement of a Cutout in a Cylindrical Shell, J. Appl. Mech., 31, 667.MathSciNetCrossRefzbMATHGoogle Scholar
  33. Hodge, P.G. (1981), Plastic Analysis of Structures, Chap. 11, Krieger, Florida.Google Scholar
  34. Hodge, P.G. and Lakshmikantham, C. (1962), Yield Loads of Spherical Caps with Cutouts, Proc. 4th US Nat. Congr. Appl. Mech. Berkeley, 2, 951.Google Scholar
  35. Hodge, P.G. and Nardo, S.V.N. (1958), Carrying Capacity of an Elastic-Plastic Cylindrical Shell with Linear Strain-Hardening, J. Appl. Mech., 25, 79.zbMATHGoogle Scholar
  36. Hodge, P.G. and Panarelli, J. (1962), Interaction Curve for Circular Cylindrical Shells According to the Mises or Tresca Yield Criteria, J. Appl. Mech., 29, 375CrossRefzbMATHGoogle Scholar
  37. Hoffman, G.A. (1962), Minimum Weight Proportions of Pressure Vessel Heads, J. Appl. Mech., 29, 662.CrossRefGoogle Scholar
  38. Hu, T.C. and Shield, R.T. (1961), Uniqueness in the Optimum Design of Structures, J. Appl. Mech., 28, 284.MathSciNetCrossRefzbMATHGoogle Scholar
  39. Ilyushin, A.A. (1948), Pasticity (in Russian), Gesteckhizdat, Moscow.Google Scholar
  40. Issler, W. (1964), Membranschalen Gleicher Festigkeit, Ingenieur-Archiv, 33, 330.CrossRefzbMATHGoogle Scholar
  41. Ivanov, G.V (1967), Inzh. Zh. Mekh. Tuerdogo, Tele, 6, 74.Google Scholar
  42. Kuech, R.W. and Lee, S.L. (1965), Limit Analysis of Simply Supported Conical Shells Subjected to Uniform Internal Pressure, J. Franklin Inst., 280, 71.CrossRefGoogle Scholar
  43. Lakshmikantham, C. and Hodge, P.G. (1963), Limit Analysis of Shallow Shells, J. Appl. Mech., 30, 215.MathSciNetCrossRefzbMATHGoogle Scholar
  44. Lance, R.H. and Lee, C.H. (1969), The Yield Point Load of a Conical Shell, Int. J. Mech. Sci., 11, 129.CrossRefzbMATHGoogle Scholar
  45. Lance, R.H. and Onat, E.T (1963), Analysis of Plastic Shallow Conical Shells, J. Appl Mech., 30, 199.CrossRefGoogle Scholar
  46. Leckie, F.A. and Payne, D.J. (1965), Some Observation on the Design of Spherical Pressure Vessels with Flush Cylindrical Nozzles, Proc. Instn. Mech. Engrs., 180, 497.CrossRefGoogle Scholar
  47. Lee, L.C. and Onat, E.T. (1968), Analysis of Plastic Spherical Shells, Engineering Plasticity (Eds., J. Heyman and F.A. Leckie), Cambridge University Press, U.K., p. 413.Google Scholar
  48. Lee, S.L. and Thorn, B.J. (1964), Deformation of Axisymmetrically Loaded Cylindrical Shells at Incipient Plastic Collapse, Int. J. Mech. Sci., 6, 247.CrossRefGoogle Scholar
  49. Liu, Y.H., Cen, Z.Z., and Xu, B.Y. (1995), Numerical Method for Plastic Limit Analysis of 3-D Structures, Int. J. Solids Struct., 32, 1645.CrossRefzbMATHGoogle Scholar
  50. Montague, P. and Horner, M.R. (1968), Elastic-Plastic Axisymmetric Analysis of a Thin-walled Cylindrical Shell Subjected to Radial Pressure Differences, J. Appl. Mech., 35, 787.CrossRefzbMATHGoogle Scholar
  51. Ohashi, Y. and Okouchi, T. (1975), The Elastic-Plastic Deformation of a Supported Short Cylindrical Shell of Mild Steel Under Internal Pressure, Int. J. Mech. Sci., 17, 267.CrossRefzbMATHGoogle Scholar
  52. Olszak, W. and Sawczuk, A. (1967), Inelastic Behaviour in Shells, Noordhoff, Groningen.zbMATHGoogle Scholar
  53. Onat, E.T. (1955), Plastic Collapse of Cylindrical Shells Under Axially Symmetrical Loading, Quart. Appl Math., 13, 63.MathSciNetzbMATHGoogle Scholar
  54. Onat, E.T. (1960), Plastic Analysis of Shallow Conical Shells, Proc. ASCE, 86, EM6–1.Google Scholar
  55. Onat, E.T. and Prager, W. (1954), Limit Analysis of Shells of Revolution, Proc. Roy. Netherlands Acad. Sci., B, 57, 534.MathSciNetzbMATHGoogle Scholar
  56. Onat, E.T. and Prager, W. (1955), Limits of Economy of Material in Shells, De Ingenieur, 67, 46.Google Scholar
  57. Palusamy, S. (1971), Limit Analysis of Spherical Shells Subjected to External Axial Force, Nuclear Engng. Design, 16, 13.CrossRefGoogle Scholar
  58. Palusamy, S. and Luid, N.C. (1972), Limit Analysis of Non-Axisymmetrically Loaded Spherical Shells, J. Appl. Mech., 39, 422.CrossRefzbMATHGoogle Scholar
  59. Paul, B. (1959), Carrying Capacity of Elastic-Plastic Shells with Various End Conditions Under Hydrostatic Compression, J. Appl. Mech., 26, 553.MathSciNetGoogle Scholar
  60. Paul, B. and Hodge, P.G. (1958), Carrying Capacity of Elastic-Plastic Shells Under Hydrostatic Pressure, Proc. 3rd US Nat. Congr. Appl. Mech, Providence, p. 631.Google Scholar
  61. Prager, W. (1959), An Introduction to Plasticity, Chap. 3, Addison-Wesley, Reading, MA.zbMATHGoogle Scholar
  62. Prager, W. and Shield, R.T. (1967), A General Theory of Optimal Plastic Design, J. Appl. Mech., 34, 184.CrossRefzbMATHGoogle Scholar
  63. Reiss, R. (1974), Minimum Weight Design of Conical Shells, J. Appl. Mech., 41, 599.CrossRefzbMATHGoogle Scholar
  64. Reiss, R. (1979), Minimum Weight Design of Shallow Conical Shells, J. Appl. Mech., 46, 599.Google Scholar
  65. Reiss, R. and Megarefs, G.J. (1969), Minimal Design of Axisymmetric Cylindrical Shells Obeying Mises Criterion, Acta Mech., 7, 72.CrossRefzbMATHGoogle Scholar
  66. Robinson, M. (1971), A Comparison of Yield Surfaces for Thin Shells, Int. J. Mech. Sci., 13, 345.CrossRefGoogle Scholar
  67. Ruiz, C. and Chukwujckwu, S.E. (1967), Limit Analysis and Design of Ring-Reinforced Radial Branches in Cylindrical and Spherical Vessels, Int. J. Mech. Sci., 9, 11.CrossRefGoogle Scholar
  68. Sankaranarayanan, R. (1960), Plastic Interaction Curves for Circular Cylindrical Shells Under Combined Lateral and Axial Pressures, J. Franklin Inst., 270, 5.MathSciNetCrossRefGoogle Scholar
  69. Save, M. (1961), On Yield Conditions in Generalized Stresses, Quart. Appl. Math., 19, 3.MathSciNetGoogle Scholar
  70. Save, M.A. and Massonnet, C.E. (1972), Plastic Analysis and Design of Plates, Shells and Disks, North-Holland, Amsterdam.zbMATHGoogle Scholar
  71. Sawczuk, A. (1982), On Plastic Shell Theories at Large Strains and Displacements, Int. J. Mech. Sci., 24, 231.CrossRefzbMATHGoogle Scholar
  72. Sawczuk, A. (1989), Mechanics and Plasticity of Structures, Ellis Horwood, Chichester, UK.Google Scholar
  73. Sawczuk, A. and Hodge, P.G. (1960), Comparison of Yield Conditions for Circular Cylindrical Shells, J. Franklin Inst., 269, 362.MathSciNetCrossRefGoogle Scholar
  74. Sayir, M. (1966), Kollapsbelastung von Rotationsymmetrischen Zylinderschalen, Z Angew. Math. Phys., 17, 353.CrossRefGoogle Scholar
  75. Shapiro, G.S. (1961), On Yield Surfaces for Ideally Plastic Shells, Problems of Continuum Mechanics, SIAM, Philadelphia, PA.Google Scholar
  76. Shield, R.T. (1960), Optimum Design of Shells, J. Appl. Mech., 27, 316.MathSciNetCrossRefzbMATHGoogle Scholar
  77. Timoshenko, S. and Woinowsky-Krieger, S. (1959), Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York.Google Scholar
  78. Xu, B.Y, Liu, YH. and Cen, Z.Z. (1998), Some Developments in Limit Analysis Solutions of Structures, Metals and Materials, 4, 329.CrossRefGoogle Scholar
  79. Yeom, D.J. and Robinson, M. (1996), Limit Analysis of a Spherical Shell Under Axial Loading on Central Boss, J. Pressure Vessel Technol, 118, 454.CrossRefGoogle Scholar
  80. Ziegler, H. (1958), Kuppeln Gleicher Festigkeit, Ingenieur-Archiv, 26, 378.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

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

Personalised recommendations