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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 derived in article “Direct Derivation of Plate...
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References
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 71–90
Arnold DN, Falk RS (1990) The boundary layer for the Reissner–Mindlin plate model. SIAM J Math Anal 21(2):281–312. http://epubs.siam.org/doi/abs/10.1137/0521016
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. http://epubs.siam.org/doi/abs/10.1137/S0036141093245276
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. http://epubs.siam.org/doi/abs/10.1137/0521030?journal Code=sjmaah
Cooke DW, Levinson M (1983) Thick rectangular plates—II. Int Mech Sci 25(3):207–215. https://doi.org/10.1016/0020-7403xxxangopen(angclosexxx83xxxangopen)angclosexxx90094-2. http://www. sciencedirect.com/science/article/pii/0020740383900942
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. http://www.sciencedirect.com/science/article/pii/S0020768399000049
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. http://epubs.siam.org/doi/abs/10.1137/S0036141098333025
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. http://www.degruyter.com/view/j/crll.1850.issue-40/crll.1850.40.51/crll.1850.40.51.xml
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. http://onlinelibrary.wiley.com/doi/10.1002/zamm.19550350604/abstract
Levinson M, Cooke DW (1983) Thick rectangular plates—I. Int J Mech Sci 25(3):199–205. https://doi.org/10.1016/0020-7403xxxangopen(angclosexxx83xxxangopen)angclosexxx90093-0. http://www.scie- nce direct.com/science/article/pii/0020740383900930
Lévy M (1899) Sur l’équilibre élastique d’une plaque rectangulaire. Comptes Rendus des Séances de l’Académie des Sciences 129:535–539
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. http://www.sciencedirect.com/science/article/pii/S0020768303000842
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/
Reissner E (1947) On bending of elastic plates. Q Appl Math 5(1):55–68
Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill classic textbook reissue series, 2nd edn. McGraw-Hill Publishing Company, New York
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Brisard, S. (2018). Classical Plate Problems. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_133-1
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DOI: https://doi.org/10.1007/978-3-662-53605-6_133-1
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Chapter history
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Latest
Classical Plate Problems- Published:
- 01 March 2019
DOI: https://doi.org/10.1007/978-3-662-53605-6_133-2
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Original
Classical Plate Problems- Published:
- 23 November 2017
DOI: https://doi.org/10.1007/978-3-662-53605-6_133-1