Advertisement

Plastic Bending of Plates

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

Abstract

In this chapter, we shall be concerned with the yield point state of perfectly plastic plates whose thickness is small compared to the dimensions of its plane faces. The load acting on the plate is normal to its surface, and is regarded as positive if it is pointing vertically downward. The vertical displacement of the middle surface is assumed to be generally small compared to the plate thickness, and plane vertical sections are assumed to remain plane during the bending. The deformation of the plate is therefore entirely defined by the vertical displacement of its middle surface, which remains effectively unstrained during the bending. A theory based on this model is found to be satisfactory not only in the elastic range but also in the plastic range of deflections. However, when the deflection of the plate exceeds the thickness, significant membrane forces are induced by the bending, the effect of which is to enhance the load carrying capacity of the plate.

Keywords

Circular Plate Face Sheet Annular Plate Collapse Load Hinge Line 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson C.A. and Shield, R.T. (1966), On the Validity of the Plastic Theory of Structures for Collapse Under Highly Localized Loading, J. Appl. Mech., 23, 629.CrossRefGoogle Scholar
  2. Boyce, W.E. (1959), A Note on Strain-Hardening Circular Plates, J. Mech. Phys. Solids, 7, 114.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Calladine, CR. (1968), Simple Ideas in the Large Deflection Plastic Theory of Plates and Slabs, in Engineering Plasticity (eds., J. Heyman and F. Leckie), p. 93, Cambridge University Press, UK.Google Scholar
  4. Chakrabarty, J. (1987), Theory of Plasticity, McGraw-Hill, New York.Google Scholar
  5. Chakrabarty, J. (1998), Large Deflections of a Clamped Circular Plate Pressed by a Hemispherical-Headed Punch, Metals and Materials, 4, 680.CrossRefGoogle Scholar
  6. Cinquini, C, Lamblin, D., and Guerlement, G. (1977), Variational Formulation of the Optimal Plastic Design of Circular Plates, Computer Methods Appl. Mech. Engng., 11, 19.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Cinquini, C and Zanon, P. (1985), Limit Analysis of Circular and Annular Plates, Ingenieur-Archiv, 55, 157.zbMATHCrossRefGoogle Scholar
  8. Collins, I.F. (1971), On the Analogy Between Plane Strain and Plate Bending Solutions in Rigid/perfectly Plasticity Theory, Int. J. Solids Struct., 7, 1037.CrossRefGoogle Scholar
  9. Collins, I.F. (1973), On the Theory of Rigid/Perfectly Plastic Plates Under Uniformly Distributed Loads, Acta Mech., 18, 233.zbMATHCrossRefGoogle Scholar
  10. Drucker, D.C and Hopkins, H.G. (1954), Combined Concentrated and Distributed Load on Ideally Plastic Circular Plates, Proc. 2nd US Nat. Congr. Appl. Mech. (Ann Arbor), p. 517.Google Scholar
  11. Drucker, D.C. and Shield, R.T. (1957), Bounds on Minimum Weight Design, Quart. Appl. Math., 15, 269.MathSciNetzbMATHGoogle Scholar
  12. Eason, G. (1958), Velocity Fields for Circular Plates with the von Mises Yield Condition, J. Mech. Phys. Solids, 6, 231.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Eason, G. (1960), The Minimum Weight Design of Circular Sandwich Plates, Z. Angew. Math. Phys., 11, 368.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Eason, G. (1961), The Elastic-Plastic Bending of a Simply Supported Plate, J. Appl. Mech., 28, 395.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Freiberger, W. and Tekinalp, B. (1956), Minimum Weight Design of Circular Plates, J. Mech. Phys. Solids, 4, 294.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Haddow, J.B. (1969), Yield Point Loading Curves for Circular Plates, Int. J. Mech. Sci., 11, 455.CrossRefGoogle Scholar
  17. Haythornthwaite, R.M. (1954), The Deflection of Plates in the Elastic-Plastic Range, Proc. US Nat. Congr. Appl. Mech. (Ann Arbor), p. 521.Google Scholar
  18. Haythornthwaite, R.M. and Shield, R.T. (1958), A Note on the Deformable Region in a Rigid-Plastic Structure, J. Mech. Phys. Solids, 6, 127.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Hodge, P.G. (1963), Plastic Analysis of Rotationally Symmetric Plates and Shells, Prentice Hall, Englewood Cliffs, NJ.Google Scholar
  20. Hodge, P.G. (1964), Plastic Plate Theory, Quart. Appl. Math.,12, 74.MathSciNetGoogle Scholar
  21. Hodge, P.G. (1981), Plastic Analysis of Structures, Chap. 10, Krieger, New York.Google Scholar
  22. Hodge, P.G. and Belytschko, T. (1968), Numerical Methods for the Limit Analysis of Plates, J. Appl. Mech., 35, 196.Google Scholar
  23. Hodge, P.G. and Sankaranarayanan, S. (1960), Plastic Interaction Curves of Annular Plates in Tension and Bending, J. Mech. Phys. Solids, 8, 153.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Hopkins, H.G. (1957), On the Plastic Theory of Plates, Proc. Roy. Soc. London Ser. A 241, 153.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Hopkins, H.G. and Prager, W. (1953), The Load Carrying Capacity of Circular Plates, J. Mech. Phys. Solids, 2, 1.MathSciNetCrossRefGoogle Scholar
  26. Hopkins, H.G. and Prager, W. (1955), Limits of Economy of Material in Plates, J. Appl. Mech. y Trans. ASME, 22, 372.zbMATHGoogle Scholar
  27. Hopkins, H.G. and Wang, A.J. (1955), Load Carrying Capacities of Circular Plates of Perfectly-Plastic Material with Arbitrary Yield Condition, J. Mech. Phys. Solids, 3, 117.MathSciNetCrossRefGoogle Scholar
  28. Johansen, K.W. (1943), Brudlinieteorier, Gjellerup, Copenhagen.Google Scholar
  29. Johnson, W. (1969), Upper Bounds to the Load for the Transverse Bending of Flat Rigid Perfectly Plastic Plates, Int. J. Mech. Sci., 11, 913.CrossRefGoogle Scholar
  30. Johnson, W. and Mellor, P.B. (1983), Engineering Plasticity, Ellis Horwood, Chichester, UK.Google Scholar
  31. Jones, L.L. and Wood, R.H. (1967), Yield Line Analysis of Slabs, Thames and Hudson, London.Google Scholar
  32. Kondo, K. and Pian, T.H.H. (1981), Large Deformation of Rigid/Plastic Circular Plates, Int. J. Solids Struct., 17, 1043.zbMATHCrossRefGoogle Scholar
  33. König, J.A. and Rychlewsky, R. (1966), Limit Analysis of Circular Plates with Jump Nonhomogeneity, Int. J. Solids Struct, 2, 493.CrossRefGoogle Scholar
  34. Koopman, D.C.A. and Lance, R.H. (1965), On Linear Programming and Plastic Limit Analysis, J. Mech. Phys. Solids, 13, 77.CrossRefGoogle Scholar
  35. Kozlowski, W. and Mröz, Z. (1969), Optimal Design of Solid Plates, Int. J. Solids Struct., 5, 781.CrossRefGoogle Scholar
  36. Krajcinovic, D. (1976), Rigid-Plastic Circular Plates on Elastic Foundation, J. Engng. Mech. Div., Trans. ASCE, 102, 213.Google Scholar
  37. Lance, R.H. and Onat, E.T. (1962), A Comparison of Experiments and Theory in the Plastic Bending of Plates, J. Mech. Phys. Solids, 10, 301.CrossRefGoogle Scholar
  38. Maftolakos, D.E. and Mamalis, A.G. (1986), Upper and Lower Bounds for Rectangular Plates Transversely Loaded, Int. J. Mech. Sci., 12, 815.CrossRefGoogle Scholar
  39. Mansfield, E.H. (1957), Studies in Collapse Analysis of Rigid-Plastic Plates with a Square Yield Diagram, Proc. Roy. Soc. London Sen A 241, 311.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Marcal, P.V. (1967), Optimal Plastic Design of Circular Plates, Int. J. Solids Struct., 3, 427.CrossRefGoogle Scholar
  41. Massonnet, C.E. (1967), Complete Solutions Describing the Limit State in Reinforced Concrete Slabs, Mag. Conc. Res., 19, 58.CrossRefGoogle Scholar
  42. Mazumdar, J. and Jain, R.K. (1989), Elastic-Plastic Bending of Plates of Arbitrary Shape— A New Approach, Int. J. Plasticity, 5, 463.zbMATHCrossRefGoogle Scholar
  43. Mröz, Z. (1961), On a Problem of Minimum Weight Design, Quart. Appl Math., 19, 3.Google Scholar
  44. Myszkowsky, J. (1971), Endliche Durchbiegungen Beliebig Eigenspannter Dünner Kreis, Ingenieur-Archiv, 40, 1.CrossRefGoogle Scholar
  45. Naghdi, P.M. (1952), Bending of Elastoplastic Circular Plate with Large Deflection, J. Appl Mech., 19, 293.zbMATHGoogle Scholar
  46. Nemirovsky, U.V. (1962), Carrying Capacity of Rib-Reinforced Circular Plates (in Russian), Izv. Nauk. USSR, Mekh. Mack, 2, 163.Google Scholar
  47. Oblak, M. (1986), Elastoplastic Bending Analysis for Thick Plate, Z. Angew. Math. Mech., 66, 320.CrossRefGoogle Scholar
  48. Ohashi, Y, and Kamiya, N. (1967), Bending of Thin Plates of Material with a Nonlinear Stress-Strain Relation, Int. J. Mech. Sci., 9, 183.CrossRefGoogle Scholar
  49. Ohashi, Y and Kawashima, I. (1969), On The Residual Deformation of Elastoplastically Bent Circular Plate After Perfect Unloading, Z. Angew. Math. Mech., 49, 275.zbMATHCrossRefGoogle Scholar
  50. Ohashi, Y and Murakami, S. (1964), The Elasto-Plastic Bending of a Clamped Thin Circular Plate, Proc. 11th Int. Conf. Appl. Mech., p. 212.Google Scholar
  51. Ohashi, Y and Murakami, S. (1966), Large Deflection of Elastoplastic Bending of a Simply Supported Circular Plate Under a Uniform Load, J. Appl. Mech., Trans. ASME, 33, 866.CrossRefGoogle Scholar
  52. Onat, E.T. and Haythornthwaite, R.M. (1956), The Load Carrying Capacity of Circular Plates at Large Deflections, J. Appl. Mech., 23, 49.zbMATHGoogle Scholar
  53. Onat, E.T., Schumann, W., and Shield, R.T. (1957), Design of Circular Plates for Minimum Weight, Z Angew. Math. Phys., 8, 485.MathSciNetzbMATHCrossRefGoogle Scholar
  54. Pell, W.H. and Prager, W. (1951), Limit Design of Plates, Proc. 1st US Nat. Congr. Appl. Mech. (Chicago), p. 547.Google Scholar
  55. Popov, E.P., Khojestch-Bakht, M, and Yaghmai, S. (1967), Analysis of Elastic-Plastic Circular Plates, J. Engng. Mech. Div., Trans. ASCE, 93, 49.Google Scholar
  56. Prager, W. (1955), Minimum Weight Design of Plates, De. Ingenieur (Amsterdam), 67, 141.Google Scholar
  57. Prager, W. (1956a), The General Theory of Limit Design, Proc. 8th Int. Congr. Appl. Mech. (Istanbul, 1952), Vol. 2, p. 65.MathSciNetGoogle Scholar
  58. Prager, W. (1956b), A New Method of Analyzing Stresses and Strains in Work-Hardening Plastic Solids, J. Appl Mech., 23, 493.MathSciNetzbMATHGoogle Scholar
  59. Prager, W. (1959), An Introduction to Plasticity, Chap. 3, Addison-Wesley, Reading, MA.Google Scholar
  60. Prager, W. and Shield, R.T. (1959), Minimum Weight Design of Circular Plates Under Arbitrary Loading, Z. Angew. Math. Phys., 10, 421.MathSciNetCrossRefGoogle Scholar
  61. Save, M.A. and Massonnet, C.E. (1972), Plastic Analysis and Design of Plates, Shells and Disks, North-Holland, Amsterdam.zbMATHGoogle Scholar
  62. Sawczuk, A. (1989), Mechanics and Plasticity of Structures, Ellis Horwood, Chichester.zbMATHGoogle Scholar
  63. Sawczuk, A. and Duszek, M. (1963), A Note on the Interaction of Shear and Bending in Plastic Plates, Arch. Mech. Stos., 15, 411.MathSciNetzbMATHGoogle Scholar
  64. Sawczuk, A. and Hodge, P.G. (1968), Limit Analysis and Yield Line Theory, J. Appl. Mech., Trans. ASME, 35, 357.CrossRefGoogle Scholar
  65. Sawczuk, A. and Jaeger, T. (1963), Grenztragfähigkeits Theorie der Platten, Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  66. Sheu, C.Y. and Prager, W. (1969), Optimal Plastic Design of Circular and Annular Plates with Piecewise Constant Cross Section, J. Mech. Phys. Solids, 17, 11.zbMATHCrossRefGoogle Scholar
  67. Schumann, W (1958), On Limit Analysis of Plates, Quart. Appl. Math., 16, 61.MathSciNetzbMATHGoogle Scholar
  68. Sherbourne, A.N. and Srivastava, A. (1971), Elastic-Plastic Bending of Restrained Pin-Ended Circular Plates, Int. J. Mech. Sci., 13, 231.CrossRefGoogle Scholar
  69. Shield, R.T. (1960), Plate Design for Minimum Weight, Quart. Appl. Math., 18, 131.MathSciNetzbMATHGoogle Scholar
  70. Shield, R.T. (1963), Optimum Design Methods for Multiple Loading, Z. Angew. Math. Phys.,14, 38.MathSciNetzbMATHCrossRefGoogle Scholar
  71. Shull, H.E. and Hu, L.W. (1963), Load Carrying Capacity of Simply Supported Rectangular Plates, J. Appl. Mech., 30, 617.CrossRefGoogle Scholar
  72. Skrzypek, J.J. and Hetnarski, R.B. (1993), Plasticity and Creep: Theory, Examples and Problems, CRC Press, Boca Raton, Florida.zbMATHGoogle Scholar
  73. Sobotka, Z. (1989), Theory of Plasticity and Limit Design of Plates, Academia, Prague.zbMATHGoogle Scholar
  74. Sokolovsky, W.W. (1948), Elastic-Plastic Bending of Circular and Annular Plates (in Russian), Prikl. Mat. Mekh., 8, 141.MathSciNetGoogle Scholar
  75. Sokolovsky, W.W. (1969), Theory of Plasticity (in Russian), 3rd ed., Nauka, Moscow.Google Scholar
  76. Tanaka, M. (1972), Large Deflection Analysis of Elastic-Plastic Circular Plates with Combined Isotropic and Kinematic Hardening, Ingenieur-Archiv, 41, 342.zbMATHCrossRefGoogle Scholar
  77. Tekinalp, B. (1957), Elastic-Plastic Bending of a Built-in Circular Plate Under Uniformly Distributed Load, J. Mech. Phys. Solids, 5, 135.MathSciNetCrossRefGoogle Scholar
  78. Turvey, G.J. (1979), Thickness-Tapered Circular Plates—An Elastic-Plastic Large Deflection Analysis, J. Struct. Mech., Trans. ASCE, 7, 247.CrossRefGoogle Scholar
  79. Yu, T.X. and Johnson, W (1982), The Large Elastic-Plastic Deflection with Springback of a Circular Plate Subjected to Circumferential Moments, J. Appl. Mech., 49, 507.zbMATHCrossRefGoogle Scholar
  80. Yu, T.X., Johnson, W, and Stronge, W.J. (1984), Stamping and Springback of Circular Plates Deformed in Hemispherical Dies, Int. J. Mech. Sci., 26, 131.CrossRefGoogle Scholar
  81. Yu, T.X. and Stronge, W.J. (1985), Wrinkling of a Circular Plate Stamped by a Spherical Punch, Int. J. Solid Struct., 21, 995.CrossRefGoogle Scholar
  82. Zaid, M. (1959), On the Carrying Capacity of Plates of Arbitrary Shape and Variable Fixity Under a Concentrated Load, J. Appl. Mech., 26.Google Scholar
  83. Zhang, L.C. and Yu, T.X. (1991), An Experimental Investigation on Stamping of Elastic-Plastic Circular Plates, J. Mater. Process. Technol, 28, 321.CrossRefGoogle 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