Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates
01 Mar 1951-Journal of Applied Mechanics-transactions of The Asme (JOURNAL OF APPLIED MECHANICS (AMERICAN SOCIETY OF MECHANICAL ENGINEERS: ASME))-Vol. 18, Iss: 1, pp 31-38
About: This article is published in Journal of Applied Mechanics-transactions of The Asme.The article was published on 1951-03-01. It has received 4554 citations till now. The article focuses on the topics: Elasticity (economics) & Rotary inertia.
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TL;DR: In this paper, a higher-order shear deformation theory of laminated composite plates is developed, which accounts for parabolic distribution of the transverse shear strains through the thickness of the plate.
Abstract: A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano (1970), but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.
3,504 citations
3,323 citations
Cites background from "Influence of rotary inertia and she..."
...In both flexure and extension of homogeneous plates, the thickness velocities are set equal to zero in the kinetic energy, for the low frequency approximation, because their contributions are negligibly small at the low frequencies to which the resulting equations are restricted owing to the suppression of the thicknessshear deformation and the omission of the thickness-stretch stress [18, 19 ]....
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Book•
24 Aug 2001TL;DR: In this paper, the authors introduce the theory of thin plates and thin shells, and apply it to the analysis of shell structures, including the moment theory of circular cylindrical shells.
Abstract: Part 1 Thin plates: introduction the fundamentals of the small-deflection plate bending theory rectangular plates circular plates bending of plates of various shapes plate bending by approximate and numerical methods advanced topics buckling of plates vibration of plates. Part 2 Thin shells: introduction to the general linear shell theory geometry of the middle surface the general linear theory of thin shells the membrane theory of shells applications of the membrane theory to the analysis of shell structures moment theory of circular cylindrical shells the moment theory of shells of revolution approximate theories of shell analysis and their application advanced topics buckling of shells vibration of shells. Appendices: some reference data Fourier series expansion verification of relations of the theory of surfaces derivation of the strain-displacement relations verification of equilibrium equations.
980 citations
Cites background from "Influence of rotary inertia and she..."
...Reissner [10,11], Timoshenko [4], Vasil’ev [12], Green [14], Mindlin [15], etc....
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...Galerkin collected numerous bending problems for plates of arbitrary shape in a monograph [15]....
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TL;DR: A review of the Zig-Zag theories for multilayered structures can be found in this article, where the authors refer to these three theories by using the following three names: Lekhnitskii Multi-layered Theory, ~LMT!, Ambartsumian Multi-Layered Theory ~AMT!, and Reissner Multilayed Theory ~RMT.
Abstract: This paper gives a historical review of the theories that have been developed for the analysis of multilayered structures. Attention has been restricted to the so-called Zig-Zag theories, which describe a piecewise continuous displacement field in the plate thickness direction and fulfill interlaminar continuity of transverse stresses at each layer interface. Basically, plate and shell geometries are addressed, even though beams are also considered in some cases. Models in which the number of displacement variables is kept independent of the number of constitutive layers are discussed to the greatest extent. Attention has been restricted to those plate and shell theories which are based on the so-called method of hypotheses or axiomatic approach in which assumptions are introduced for displacements and/or transverse stresses. Mostly, the work published in the English language is reviewed. However, an account of a few articles originally written in Russian is also given. The historical review conducted has led to the following main conclusions. 1! Lekhnitskii ~1935! was the first to propose a Zig-Zag theory, which was obtained by solving an elasticity problem involving a layered beam. 2! Two other different and independent Zig-Zag theories have been singled out. One was developed by Ambartsumian ~1958!, who extended the well-known Reissner-Mindlin theory to layered, anisotropic plates and shells; the other approach was introduced by Reissner ~1984!, who proposed a variational theorem that permits both displacements and transverse stress assumptions. 3 ! On the basis of historical considerations, which are detailed in the paper, it is proposed to refer to these three theories by using the following three names: Lekhnitskii Multilayered Theory, ~LMT!, Ambartsumian Multilayered Theory ~AMT!, and Reissner Multilayered Theory ~RMT!. As far as subsequent contributions to these three theories are concerned, it can be remarked that: 4! LMT although very promising, has almost been ignored in the open literature. 5! Dozens of papers have instead been presented which consist of direct applications or particular cases of the original AMT. The contents of the original works have very often been ignored, not recognized, or not mentioned in the large number of articles that were published in journals written in the English language. Such historical unfairness is detailed in Section 3.2. 6! RMT seems to be the most natural and powerful method to analyze multilayered structures. Compared to other theories, the RMT approach has allowed from the beginning development of models which retain the fundamental effect related to transverse normal stresses and strains. This review article cites 138 references. @DOI: 10.1115/1.1557614#
972 citations
TL;DR: In this paper, a new standard plate theory, which accounts for cosine shear stress distribution and free boundary conditions for shear stresses upon the top and bottom surfaces of the plate, is presented.
932 citations
References
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01 Jan 1934
TL;DR: The theory of the slipline field is used in this article to solve the problem of stable and non-stressed problems in plane strains in a plane-strain scenario.
Abstract: Chapter 1: Stresses and Strains Chapter 2: Foundations of Plasticity Chapter 3: Elasto-Plastic Bending and Torsion Chapter 4: Plastic Analysis of Beams and Frames Chapter 5: Further Solutions of Elasto-Plastic Problems Chapter 6: Theory of the Slipline Field Chapter 7: Steady Problems in Plane Strain Chapter 8: Non-Steady Problems in Plane Strain
20,724 citations
TL;DR: In this article, the correction for shear of the differential equation for transverse vibrations of prismatic bars is discussed, where the correction is based on the correction of the transverse vibration of a prismatic bar.
Abstract: (1921). LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 41, No. 245, pp. 744-746.
2,255 citations
TL;DR: In this article, the transverse vibrations of bars of uniform cross-section were studied and the authors proposed a method to measure the transversal vibrations of a bar of uniform shape.
Abstract: (1922). X. On the transverse vibrations of bars of uniform cross-section. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 43, No. 253, pp. 125-131.
917 citations
580 citations
01 Jan 1947
TL;DR: Ausgehend von einem kinematischen Schema wird eine Methode entwickelt, welche es moglich macht, den Anschlus der technischen Festigkeitslehre an die mathematische Elastizitatstheorie zu bewerkstelligen.
Abstract: Ausgehend von einem kinematischen Schema wird eine Methode entwickelt, welche es moglich macht, den Anschlus der technischen Festigkeitslehre an die mathematische Elastizitatstheorie zu bewerkstelligen. Ganz wie in der Methode von Lagrange nur mit Kinematik und Energiebegriff gearbeitet wird, werden die Gleichungen aus dem Prinzip der virtuellen Verschiebungen hergeleitet. Die Randbedingungen brauchen dabei nicht von vornherein im kinematischen Ansatz schon befriedigt zu sein; das Verfahren ist so geartet, das man das Optimum der Naherung, das sich mit dem gewahlten Ansatz uberhaupt erreichen last, auch wirklich erhalt. Von groser praktischer Bedeutung wird die Schubkorrektion dann, wenn es sich um I-Profile handelt. Der Einflus kann bei handelsublichen Formen 10 bis 30 % betragen.
189 citations