Archive of Applied Mechanics

, Volume 81, Issue 11, pp 1717–1724 | Cite as

A relook at Reissner’s theory of plates in bending

Original

Abstract

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.

Keywords

Plates Bending Isotropy Elasticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Reissner E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–191 (1944)MathSciNetMATHGoogle Scholar
  2. 2.
    Reissner E.: Reflections on the theory of elastic plates. Appl. Mech. Rev. 38, 1453–1464 (1985)CrossRefGoogle Scholar
  3. 3.
    Lo K.H., Christensen R.M., Wu E.M.: A higher-order theory of plate deformation. J. Appl. Mech. 44, 663–676 (1977)MATHCrossRefGoogle Scholar
  4. 4.
    Lo K.H., Christensen R.M., Wu E.M.: Stress determination for higher-order plate theory. Int. J. Solids Struct. 14, 655–662 (1978)MATHCrossRefGoogle Scholar
  5. 5.
    Lewinski T.: A note on recent developments in the theory of elastic plates with moderate thickness. Eng. Trans. 34(4), 531–542 (1986)MathSciNetMATHGoogle Scholar
  6. 6.
    Lewinski T.: On refined plate models based on kinematical assumptions. Ingenieur Arch. 57(2), 133–146 (1987)MATHCrossRefGoogle Scholar
  7. 7.
    Blocki J.: A higher-order linear theory for isotropic plates-i, theoretical considerations. Int. J. Solids Struct. 29(7), 825–836 (1992)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kienzler R.: On consistent plate theories. Arch. Appl. Mech. 72, 229–247 (2002). doi:10-1007/s00419-002-0220-2 MATHCrossRefGoogle Scholar
  9. 9.
    Batista M.: The derivation of the equations of moderately thick plates by the method of successive approximations. Acta Mech. 210, 159–168 (2010)MATHCrossRefGoogle Scholar
  10. 10.
    Reissner E.: The effect of transverse shear deformations on the bending of elastic plates. J. Appl. Mech. 12, A69–A77 (1945)MathSciNetMATHGoogle Scholar
  11. 11.
    Reissner E.: On bending of elastic plates. Q. Appl. Math. 5(1), 55–68 (1947)MathSciNetMATHGoogle Scholar
  12. 12.
    Reissner E.: On a variational theorem in elasticity. J. Math. Phys. 29, 90–95 (1950)MathSciNetMATHGoogle Scholar
  13. 13.
    Vijayakumar K.: Poisson–Kirchhoff paradox in flexure of plates. AIAA J. 26(2), 247–249 (1988)CrossRefGoogle Scholar
  14. 14.
    Vasiliev V.V.: Modern conceptions of plate theory. Compos. Struct. 48, 39–48 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Love A.E.H.: A Treatise on Mathematical Theory of Elasticity, 4th edn, pp. 458–463. Cambridge University Press, Cambridge (1934)Google Scholar
  16. 16.
    Vijayakumar K.: A new look at Kirchhoff’s theory of plates. AIIA J. 47, 1045–1046 (2009). doi:10.2514/1.38471 CrossRefGoogle Scholar
  17. 17.
    Preusser G.: Eine systematische Herleitung verbesserter Plattengleichungen. Ingenieur Arch. 54, 51–61 (1984)MATHCrossRefGoogle Scholar
  18. 18.
    Krenk S.: Theories for elastic plates via orthogonal polynomials. Trans. ASME 48, 900–904 (1981)MATHCrossRefGoogle Scholar
  19. 19.
    Krishna Murthy A.V.: Higher-order theory of homogeneous plate flexure. AIIA J. 26(6), 719–725 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Reddy J.N.: A simple higher order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)MATHCrossRefGoogle Scholar
  21. 21.
    Reissner E.: A twelvth order theory of transverse bending of transversely isotropic plates. ZAMM 63, 285–289 (1983)MATHCrossRefGoogle Scholar
  22. 22.
    Lewinski T.: On the twelth-order theory of elastic plates. Mech. Res. Commun. 17(6), 375–382 (1990)MATHCrossRefGoogle Scholar
  23. 23.
    Reissner E.: A mixed variational equation for a twelfth-order theory of bending of non-homogeneous transversely isotropic plates. Comput. Mech. 7, 355–360 (1991)MATHCrossRefGoogle Scholar
  24. 24.
    Cheng S.: Elasticity theory of plates and a refined theory. J. Appl. Mech. 46, 644–650 (1979)MATHCrossRefGoogle Scholar
  25. 25.
    Wang W., Shi M.X.: Thick plate theory based on general solutions of elasticity. Acta Mech. 123, 27–36 (1997)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gao Y., Zhao B-.S.: A refined theory of elastic thick plates for extensional deformation 79, 5–18 (2009). doi:10.1007/s00419-008-0210-0 MATHGoogle Scholar
  27. 27.
    Gol’denveizer A.L., Kolos A.V.: On the derivation of two-dimensional equations in the theory of thin elastic plates. PMM 29(1), 141–155 (1965)Google Scholar
  28. 28.
    Touratier M.: An efficient standard plate theory. Int. J. Eng. sci. 29(8), 901–916 (1991)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

Personalised recommendations