Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Classical Plate Problems

  • Sébastien BrisardEmail author
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-3-662-53605-6_133-2

Synonyms

Definitions

Bending of plates refers to internal states of stress such that the membrane stress resultants vanish. From the kinematic point of view, it means that the in-plane displacements of the material points of the plate are null. In other words, the degrees of freedom of plates in bending are the out-of-plane displacement (thick and thin plates), together with the two material rotations (thick plates only) – see article “Direct Derivation of Plate Theories”.

The out-of-plane displacement is usually called deflection of the plate.

Introduction

This article discusses a few classical, closed-form solutions of plate problems. It is restricted to bending problems of isotropic, homogeneous, linearly elastic plates subjected to distributed forces only (no distributed couples).

The general field equations that govern the bending of such plates (both thick and thin) were...
This is a preview of subscription content, log in to check access.

References

  1. Arnold DN, Falk R (1989) Edge effects in the Reissner-Mindlin plate theory. In: Noor A, Belytschko T, Simo J (eds) Analytical and computational models for shells. American Society of Mechanical Engineers, New York pp 71–90Google Scholar
  2. Arnold DN, Falk RS (1990) The boundary layer for the Reissner–Mindlin plate model. SIAM J Math Anal 21(2):281–312. https://doi.org/10.1137/0521016 MathSciNetCrossRefGoogle Scholar
  3. Arnold DN, Falk RS (1996) Asymptotic analysis of the boundary layer for the Reissner–Mindlin plate model. SIAM J Math Anal 27(2):486–514. https://doi.org/10.1137/S0036141093245276 MathSciNetCrossRefGoogle Scholar
  4. Babuška I, Pitkäranta J (1990) The plate paradox for hard and soft simple support. SIAM J Math Anal 21(3):551–576. https://doi.org/10.1137/0521030 MathSciNetCrossRefGoogle Scholar
  5. Cooke DW, Levinson M (1983) Thick rectangular plates—II. Int Mech Sci 25(3):207–215. https://doi. org/10.1016/0020-7403(83)90094-2 CrossRefGoogle Scholar
  6. Dauge M, Yosibash Z (2000) Boundary layer realization in thin elastic three-dimensional domains and two-dimensional Hierarchic plate models. Int J Solids Struct 37(17):2443–2471. https://doi.org/10.1016/S0020-7683(99)00004-9 CrossRefGoogle Scholar
  7. Dauge M, Gruais I, Rössle A (2000) The influence of lateral boundary conditions on the asymptotics in thin elastic plates. SIAM J Math Anal 31(2):305–345. https://doi.org/10.1137/S0036141098333025 MathSciNetCrossRefGoogle Scholar
  8. Kirchhoff G (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik (Crelles Journal) 1850(40):51–88.  https://doi.org/10.1515/crll.1850.40.51 CrossRefGoogle Scholar
  9. Kromm A (1955) Über die Randquerkräfte bei gestützten Platten. ZAMM – J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 35(6–7):231–242.  https://doi.org/10.1002/zamm.19550350604 CrossRefGoogle Scholar
  10. Levinson M, Cooke DW (1983) Thick rectangular plates—I. Int J Mech Sci 25(3):199–205. https://doi. org/10.1016/0020-7403(83)90093-0 CrossRefGoogle Scholar
  11. Lévy M (1899) Sur l’équilibre élastique d’une plaque rectangulaire. Comptes Rendus des Séances de l’Académie des Sciences 129:535–539zbMATHGoogle Scholar
  12. Lim GT, Reddy JN (2003) On canonical bending relationships for plates. Int J Solids Struct 40(12):3039–3067. https://doi.org/10.1016/S0020-7683(03)00084-2 CrossRefGoogle Scholar
  13. Navier (1823) Extrait des recherches sur la flexion des plans élastiques. Bulletin des Sciences de la Société Philomatique de Paris, pp 92–102. http://philomathique.org/
  14. Reissner E (1947) On bending of elastic plates. Q Appl Math 5(1):55–68MathSciNetCrossRefGoogle Scholar
  15. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill classic textbook reissue series, 2nd edn. McGraw-Hill Publishing Company, New YorkGoogle Scholar
  16. Westphal T, de Barcellos CS, Tomás Pereira J (1996) On general fundamental solutions of some linear elliptic differential operators. Engineering Analysis with Boundary Elements 17(4):279–285. https://doi.org/10.1016/S0955-7997(96)00028-8 CrossRefGoogle Scholar
  17. Westphal T, Schnack E, de Barcellos C (1998) The general fundamental solution of the sixthorder Reissner and Mindlin plate bending models revisited. Computer Methods in Applied Mechanics and Engineering 166(3–4):363–378. https://doi.org/10.1016/S0045-7825(98)00101-7 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance

Section editors and affiliations

  • Karam Sab
    • 1
  1. 1.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance