A relook at Reissner’s theory of plates in bending
First Online: 08 February 2011 Received: 13 January 2010 Accepted: 24 January 2011 DOI:
Cite this article as: Vijayakumar, K. Arch Appl Mech (2011) 81: 1717. doi:10.1007/s00419-011-0513-4 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
This work is dedicated to Eric Reissner (1913–1996) whose works on plate theories have inspired several investigators on development of higher order theories of plates.
Reissner E.: On the theory of bending of elastic plates. J. Math. Phys.
, 184–191 (1944)
Reissner E.: Reflections on the theory of elastic plates. Appl. Mech. Rev.
, 1453–1464 (1985)
Lo K.H., Christensen R.M., Wu E.M.: A higher-order theory of plate deformation. J. Appl. Mech.
, 663–676 (1977)
Lo K.H., Christensen R.M., Wu E.M.: Stress determination for higher-order plate theory. Int. J. Solids Struct.
, 655–662 (1978)
Lewinski T.: A note on recent developments in the theory of elastic plates with moderate thickness. Eng. Trans.
(4), 531–542 (1986)
Lewinski T.: On refined plate models based on kinematical assumptions. Ingenieur Arch.
(2), 133–146 (1987)
Blocki J.: A higher-order linear theory for isotropic plates-i, theoretical considerations. Int. J. Solids Struct.
(7), 825–836 (1992)
MathSciNet MATH CrossRef
Kienzler R.: On consistent plate theories. Arch. Appl. Mech.
, 229–247 (2002). doi:
10-1007/s00419-002-0220-2 MATH CrossRef
Batista M.: The derivation of the equations of moderately thick plates by the method of successive approximations. Acta Mech.
, 159–168 (2010)
Reissner E.: The effect of transverse shear deformations on the bending of elastic plates. J. Appl. Mech.
, A69–A77 (1945)
Reissner E.: On bending of elastic plates. Q. Appl. Math.
(1), 55–68 (1947)
Reissner E.: On a variational theorem in elasticity. J. Math. Phys.
, 90–95 (1950)
Vijayakumar K.: Poisson–Kirchhoff paradox in flexure of plates. AIAA J.
(2), 247–249 (1988)
Vasiliev V.V.: Modern conceptions of plate theory. Compos. Struct.
, 39–48 (2000)
Love A.E.H.: A Treatise on Mathematical Theory of Elasticity, 4th edn, pp. 458–463. Cambridge University Press, Cambridge (1934)
Vijayakumar K.: A new look at Kirchhoff’s theory of plates. AIIA J.
, 1045–1046 (2009). doi:
Preusser G.: Eine systematische Herleitung verbesserter Plattengleichungen. Ingenieur Arch.
, 51–61 (1984)
Krenk S.: Theories for elastic plates via orthogonal polynomials. Trans. ASME
, 900–904 (1981)
Krishna Murthy A.V.: Higher-order theory of homogeneous plate flexure. AIIA J.
(6), 719–725 (1988)
Reddy J.N.: A simple higher order theory for laminated composite plates. J. Appl. Mech.
, 745–752 (1984)
Reissner E.: A twelvth order theory of transverse bending of transversely isotropic plates. ZAMM
, 285–289 (1983)
Lewinski T.: On the twelth-order theory of elastic plates. Mech. Res. Commun.
(6), 375–382 (1990)
Reissner E.: A mixed variational equation for a twelfth-order theory of bending of non-homogeneous transversely isotropic plates. Comput. Mech.
, 355–360 (1991)
Cheng S.: Elasticity theory of plates and a refined theory. J. Appl. Mech.
, 644–650 (1979)
Wang W., Shi M.X.: Thick plate theory based on general solutions of elasticity. Acta Mech.
, 27–36 (1997)
MathSciNet MATH CrossRef
Gao Y., Zhao B-.S.: A refined theory of elastic thick plates for extensional deformation
, 5–18 (2009). doi:
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)
Touratier M.: An efficient standard plate theory. Int. J. Eng. sci.
(8), 901–916 (1991)