A relook at Reissner’s theory of plates in bending Original

First Online: 08 February 2011 Received: 13 January 2010 Accepted: 24 January 2011 DOI :
10.1007/s00419-011-0513-4

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.

References 1.

Reissner E.: On the theory of bending of elastic plates. J. Math. Phys.

23 , 184–191 (1944)

MathSciNet MATH 2.

Reissner E.: Reflections on the theory of elastic plates. Appl. Mech. Rev.

38 , 1453–1464 (1985)

CrossRef 3.

Lo K.H., Christensen R.M., Wu E.M.: A higher-order theory of plate deformation. J. Appl. Mech.

44 , 663–676 (1977)

MATH CrossRef 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)

MATH CrossRef 5.

Lewinski T.: A note on recent developments in the theory of elastic plates with moderate thickness. Eng. Trans.

34 (4), 531–542 (1986)

MathSciNet MATH 6.

Lewinski T.: On refined plate models based on kinematical assumptions. Ingenieur Arch.

57 (2), 133–146 (1987)

MATH CrossRef 7.

Blocki J.: A higher-order linear theory for isotropic plates-i, theoretical considerations. Int. J. Solids Struct.

29 (7), 825–836 (1992)

MathSciNet MATH CrossRef 8.

Kienzler R.: On consistent plate theories. Arch. Appl. Mech.

72 , 229–247 (2002). doi:

10-1007/s00419-002-0220-2
MATH CrossRef 9.

Batista M.: The derivation of the equations of moderately thick plates by the method of successive approximations. Acta Mech.

210 , 159–168 (2010)

MATH CrossRef 10.

Reissner E.: The effect of transverse shear deformations on the bending of elastic plates. J. Appl. Mech.

12 , A69–A77 (1945)

MathSciNet MATH 11.

Reissner E.: On bending of elastic plates. Q. Appl. Math.

5 (1), 55–68 (1947)

MathSciNet MATH 12.

Reissner E.: On a variational theorem in elasticity. J. Math. Phys.

29 , 90–95 (1950)

MathSciNet MATH 13.

Vijayakumar K.: Poisson–Kirchhoff paradox in flexure of plates. AIAA J.

26 (2), 247–249 (1988)

CrossRef 14.

Vasiliev V.V.: Modern conceptions of plate theory. Compos. Struct.

48 , 39–48 (2000)

MathSciNet CrossRef 15.

Love A.E.H.: A Treatise on Mathematical Theory of Elasticity, 4th edn, pp. 458–463. Cambridge University Press, Cambridge (1934)

16.

Vijayakumar K.: A new look at Kirchhoff’s theory of plates. AIIA J.

47 , 1045–1046 (2009). doi:

10.2514/1.38471
CrossRef 17.

Preusser G.: Eine systematische Herleitung verbesserter Plattengleichungen. Ingenieur Arch.

54 , 51–61 (1984)

MATH CrossRef 18.

Krenk S.: Theories for elastic plates via orthogonal polynomials. Trans. ASME

48 , 900–904 (1981)

MATH CrossRef 19.

Krishna Murthy A.V.: Higher-order theory of homogeneous plate flexure. AIIA J.

26 (6), 719–725 (1988)

MathSciNet CrossRef 20.

Reddy J.N.: A simple higher order theory for laminated composite plates. J. Appl. Mech.

51 , 745–752 (1984)

MATH CrossRef 21.

Reissner E.: A twelvth order theory of transverse bending of transversely isotropic plates. ZAMM

63 , 285–289 (1983)

MATH CrossRef 22.

Lewinski T.: On the twelth-order theory of elastic plates. Mech. Res. Commun.

17 (6), 375–382 (1990)

MATH CrossRef 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)

MATH CrossRef 24.

Cheng S.: Elasticity theory of plates and a refined theory. J. Appl. Mech.

46 , 644–650 (1979)

MATH CrossRef 25.

Wang W., Shi M.X.: Thick plate theory based on general solutions of elasticity. Acta Mech.

123 , 27–36 (1997)

MathSciNet MATH CrossRef 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
MATH 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)

28.

Touratier M.: An efficient standard plate theory. Int. J. Eng. sci.

29 (8), 901–916 (1991)

MATH CrossRef Authors and Affiliations 1. Department of Aerospace Engineering Indian Institute of Science Bangalore India