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Showing papers on "Plate theory published in 1969"


Book
01 Jan 1969
TL;DR: The fundamental equation of classical plate theory can be found in this article, where anisotropic and variable-thickness versions of the classical plates are considered, as well as other considerations.
Abstract: : Contents: Fundamental Equations of Classical Plate Theory; Circular Plates; Elliptical Plates; Rectangular Plates; Parallelogram Plates; Other Quadrilateral Plates; Triangular Plates; Plates of Other Shapes; Anisotropic Plates; Plates With Inplane Forces; Plates With Variable Thickness; and Other Considerations.

2,137 citations


Journal ArticleDOI
TL;DR: In this article, the limitations of classical laminated plate theory are investigated by comparing solutions of several specific boundary value problems in this theory to the corresponding theory of elasticity solutions, and it is shown that conventional plate theory leads to a very poor description of laminate response at low span-to-depth ratios.
Abstract: Limitations of classical laminated plate theory are investigated by comparing solutions of several specific boundary value problems in this theory to the corresponding theory of elasticity solutions. The general class of problems treated involves the geometric configuration of any number of isotropic or orthotropic layers bonded together and subjected to cylindrical bending. In general it is found that conventional plate theory leads to a very poor description of laminate response at low span-to-depth ratios, but converges to the exact solution as this ratio increases. The analysis presented is also valid in the study of sandwich plates under cylindrical bending.

1,194 citations


Journal ArticleDOI
01 Oct 1969-Nature
TL;DR: The simple geometric ideas of plate theory are extended to include some forms of plate evolution as discussed by the authors The most important of these occurs where three plates meet such triple junctions are divided into two groups, stable and unstable, according to whether or not they can retain their geometry as the plates move.
Abstract: The simple geometric ideas of plate theory are extended to include some forms of plate evolution The most important of these occurs where three plates meet Such triple junctions are divided into two groups, stable and unstable, according to whether or not they can retain their geometry as the plates move These ideas suggest an explanation for some of the major changes which have occurred in the North Pacific during the Tertiary

596 citations



Journal ArticleDOI
TL;DR: In this article, the elastic buckling loads of shear diaphragms are derived from an analysis of light-gage, corrugated-metal diaphrasms of the type that occur in pre-engineered metal buildings.
Abstract: Formulas for the elastic buckling loads of shear diaphragms are derived. They are obtained from an analysis of light-gage, corrugated-metal diaphragms of the type that occur in pre-engineered metal buildings. The formulas are applicable to any rectangular, orthotropic plate loaded in pure shear, and they are derived using orthotropic plate theory and the Ritz energy method. Small deflection theory is used first to establish critical loads and buckling modes. Then, large deflection theory is used to predict post-buckling load versus lateral deflection relationships. The results of experiments that were designed to verify the accuracy of the formulas are presented in the form of load versus lateral deflection curves covering the pre and post-buckling ranges. Several light-gage, corrugated-metal diaphragms of different size and corrugation shape were tested to determine their buckling behavior. The results of the experiments compare favorably with the formulas derived.

148 citations


Journal ArticleDOI
TL;DR: In this paper, a compatible triangular plate element for normal and in-plane displacements was proposed for both inplane and out-plane displacement, and the nine degree of freedom element, strain energy, simply supported and clamped plates were discussed.
Abstract: Compatible triangular plate elements for normal and in-plane displacements, discussing nine degree of freedom element, strain energy, simply supported and clamped plates

134 citations


Journal ArticleDOI
TL;DR: Two dimensional natural convection boundary layer on finite isothermal horizontal plate, examining upward facing cold plate and downward facing hot plate as mentioned in this paper, examining upward and downward faces of cold and hot plate.

68 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate the effect of nonlinear plate stiffness and mutual interaction between the plate and external and/or internal (cavity) air flow on the results of the title problem.
Abstract: The title problem is investigated, emphasizing two effects that are usually ignored in the literature. These are (i) nonlinear plate stiffness and (ii) mutual interaction between the plate and external and/or internal (cavity) air flow. The method of analysis employs modal expansions for the plate and cavity (air) motions, with the solution effected in the time (rather than the frequency) domain. The external flow‐pressure perturbations due to the plate motion are assumed to be described adequately by classical acoustic theory including convection effects. A discussion is given as to when the two effects will be important and a practical, numerical example is worked out to illustrate the method.

60 citations


Journal ArticleDOI
TL;DR: In this paper, the main computational problem is that of testing whether or not, for a specified wavelength of buckle, any chosen value of the compressive stress is less than or greater than the lowest buckling stress of the structure.

51 citations


Journal ArticleDOI
TL;DR: In this paper, the authors compared results obtained from using nine approximate analytical methods: point matching, boundary point least square, TrefftzMorley, interior collocation, interior least squares, subdomain, Galerkin, Ritz, and Kantorovich.
Abstract: It is well known that classical, exact methods of analytical solution cannot be applied to the plate bending problem in cases of irregular boundary shape and arbitrary transverse loading and constraint. The paper summarizes and compares results obtained from using nine approximate analytical methods: point matching, boundary point least squares, TrefftzMorley, interior collocation, interior least squares, subdomain, Galerkin, Ritz, and Kantorovich. For purposes of concrete comparison, each of the methods is applied to two problems for which exact, although intricate, solutions are known: 1) a uniformly loaded, simply supported elliptical plate; 2) a square plate having free edges, supported at four asymmetrically located interior points, and loaded by its own weight. Comparative results are presented. Each technique is rated good, fair, or poor according to 11 important technical criteria and the underlying rationale is explained.

49 citations


Journal ArticleDOI
TL;DR: In this paper, the dynamic response of a simply supported rectangular plate is treated within the framework of classical, three-dimensional elasticity theory, and an exact solution in series form is given, and a comparison of this solution with previously obtained corresponding solutions within the classical and improved plate theories is also presented.

Book ChapterDOI
01 Jan 1969
TL;DR: In this paper, it was shown that for large values of the ratio (applied shear strain)/(shear strain at initial buckling) the relation between applied loads and planar displacements and stresses again approaches linearity, and it is this asymptotic regime for which tension field theory is applicable.
Abstract: Tension field theory describes the highly buckled (wrinkled) state of membranes or very thin plates whose boundaries are subjected to certain planar displacements well in excess of those necessary to initiate buckling. The present interest in tension field theory is because lightweight structures with stretched membrane components have potential applications in space. In addition, membrane structures which are pretensioned by internal pressure have application to lightweight portable bridges, protective coverings and various air cushion devices. The theory was conceived by Wagner (1929) [1] whose primary concern was to explain the behaviour of thin metal webs in beams and spars carrying a shear load well in excess of the initial buckling value. Such webs offer little resistance to the compressive strain component of the shear and the spar flanges must be held apart by struts to prevent collapse. In the simple case of rigid spar flanges and rigid perpendicular struts the stress field in the web in the highly buckled state is primarily that of tension at 45°. As the shear load increases so does the magnitude of this tensile stress field and, just as a taut string resists a kinking action, so too does this tensile stress field resist the out-of-plane displacements engendered by the buckling action of the compressive stresses; these opposing actions result in a decreasing wavelength along the compressive buckles which form at right angles to the tension field. Strictly speaking such problems are non-linear and their exact analysis presents formidable difficulties. However, within the framework of large-deflexion plate theory it may be shown that for large values of the ratio (applied shear strain)/(shear strain at initial buckling) the relation between applied loads and planar displacements and stresses again approaches linearity, and it is this asymptotic regime for which tension field theory is applicable. In this regime the flexural stresses and the planar compressive (post-buckling) stresses are negligible compared with the tensile stresses; the assumption that their magnitude is zero is physically equivalent to the assumption of zero flexural membrane stiffness, and it is this which characterises tension field theory: the membrane is envisaged as being finely wrinkled at right angles to the lines of tension. In general these “tension rays” are not necessarily parallel and the boundary conditions need not be those of pure shear, as in our previous example, but shear must play a dominant role in the boundary deformation because of the requirement that the principal strains at any point are of opposite sign. This requirement will be considered in greater detail later but it is clear that if the principal strains are both positive so too are the principal stresses, and if the principal strains are both negative the membrane is ineffective in carrying load.

Journal ArticleDOI
W. Visser1
TL;DR: Triangular plate bending element using Herrmann variational method, deriving matrices for finite elements as discussed by the authors, is the only known example of a finite element bending element that can be represented as a matrices.
Abstract: Triangular plate bending element using Herrmann variational method, deriving matrices for finite elements

Journal ArticleDOI
TL;DR: In this article, the singular-perturbation method was used to obtain a flutter boundary for thin plates exposed to supersonic flow, and a two-variable expansion in terms of the space coordinate was proposed.
Abstract: The paradox concerning aeroelastic stability of thin plates exposed to supersonic flow is resolved by the use of singular-perturbation methods. The form of the known solution for a membrane suggests a particular limit process, in which the plate thickness and certain other parameters approach zero, and also provides motivation for two methods of obtaining the desired asymptotic expansions. One procedure involves removal of an appropriate exponential factor from the solution and subsequent use of a boundary-layer approach because of the importance of bending stresses near the edges. The second method is based on a two-variable expansion in terms of the space coordinate. In each case, a Poincare type of distorted time coordinate is introduced. A flutter boundary is obtained for a range of the parameters which has not been studied previously, and the significance of the membrane solution is explained in relation to this result. The approximations used permit substantial simplification in the computations.

ReportDOI
01 Dec 1969
TL;DR: In this paper, a generalized Duhamel- Neumann form of Hooke's law is employed to treat a wide variety of environmental problems by the joint application of solid mechanics and elementary physical chemistry.
Abstract: : The deformation of a solid induced by swelling is equivalent to that caused by a temperature change. A generalized Duhamel- Neumann form of Hooke's law is employed to treat a wide variety of environmental problems by the joint application of solid mechanics and elementary physical chemistry. This approach is illustrated for a swollen fiber reinforced material, employing physical chemistry concepts, micromechanics, and laminated anisotropic plate theory. The specific results are applicable to the design of dimensionally stable composite materials invariable thermal or swelling environments. A new strain invariant for laminates under these types of environments is also introduced.

Journal ArticleDOI
TL;DR: In this article, a specialized form of Reissner's variational principle for stresses and displacements suitable to formulate a finite element version of the governing equations for thin shells is presented.

Proceedings ArticleDOI
01 Jan 1969
TL;DR: Plate theories of linearly elastic materials under free undamped vibration, giving 500 references are given in this paper, with 500 references given in the references section and 500 references in the Appendix.
Abstract: Plate theories of linearly elastic materials under free undamped vibration, giving 500 references

Journal ArticleDOI
TL;DR: In this article, a method for determining a finite integral transform which will remove the presence of one of the independent variables in a fourth-order partial differential equation is applied to the equation of motion of classical plate theory for complete and annular circular plates subjected to various boundary conditions.

Journal ArticleDOI
TL;DR: In this paper, the six governing, coupled, linear partial-differential equations of plates can be derived using a method suggested by J. N. Goodier and E. Reissner using the calculus of variations.
Abstract: The six governing, coupled, linear partial-differential equations of plates can be derived using a method suggested by J. N. Goodier. These equations, which are only slightly different from those derived by E. Reissner using the calculus of variations, are solved for a wide class of problems. Any loading that satisfies the Dirichlet conditions of Fourier analysis and virtually any boundary conditions may be handled. Most problems that can be solved using classical theory are also solvable by the methods presented with only a slightly greater amount of effort. The solution obtained for the Reissner-Goodier theory is comparable in method to the solution of the biharmonic equation in polar coordinates of classical plate theory given by A. Clebsch in his "Theorie der Elasticitat fester Korper," 1862, Leipzig. Two examples are considered in detail. Both solutions are obtained by specialization of the more general solution with the first example solved in closed form. Numerical results are displayed in a series of nondimensiona l curves. These curves show that the effects of transverse shear are most significant on the plate boundaries and in regions of high stress gradients.

Journal ArticleDOI
TL;DR: In this article, exact deformation solutions for rationally based discrete-continuous models of rectangular ribbed plates having simple end supports and flexible side supports are found for general loadings and are improvements over those based on orthotropic plate theory in that the inconsistencies inherent in the use of an equivalent continuum are avoided.
Abstract: Exact deformation solutions are found for rationally based discrete-continuous models of rectangular ribbed plates having simple end supports and flexible side supports. The formulas are valid for general loadings and are improvements over those based on orthotropic plate theory in that the inconsistencies inherent in the use of an equivalent continuum are avoided. The solutions are double Fourier series (infinite with respect to the continuous variable along the rib line and finite with respect to the discrete variable denoting the rib) plus simple corrective terms where required to satisfy special boundary conditions. Fourth order mathematical models are used for a Composite Membrane Analysis and a Non-Composite Flexural Analysis and a general eighth order model is used for Composite Membrane-Flexural Analysis. The major results are numerically illustrated.

Journal ArticleDOI
TL;DR: In this paper, a theoretical and experimental study of solar radiation passing through a thin semi-transparent slab to heat a fluid is presented; the system of differential equations describing the temperature of the slab and the fluid as a function of time is derived and solved; the theoretical curves generated by the solution for the fluid temperature are compared with experimental readings obtained using water as the fluid and acrylic plastic (methyl-methacrylate) as the semitransparent material.

Journal ArticleDOI
TL;DR: In this article, the authors derived constitutive relations in a form suitable for use in stability problems for an elastic-plastic material undergoing a perturbation from a state of uniaxial stress, based on Green and Naghdi's general theory of an elasticplastic continuum.

Journal ArticleDOI
TL;DR: In this paper, a set of general governing equations which include the effects of transverse shear deformation and rotatory inertia are derived for the study of free vibrations of rectangular plates composed of an orthotropic material.

Journal ArticleDOI
TL;DR: An exact polynomial solution for the St Venant flexure problem of a pretwisted rectangular plate treated as a shallow hyperbolic paraboloidal shell is given in this paper.

Journal ArticleDOI
TL;DR: In this paper, the authors compare the results from testing two different bridge structures: a 200-FT Span THROUGH TRUSS and an 80-FT Span Compatible SLAB-STEEL GIRDER BRIDGE.
Abstract: RESULTING EXPERIMENTAL DATA, FROM TESTING TWO BRIDGE STRUCTURES, IS COMPARED TO RESULTS FROM FINITE DIFFERENCE PLATE THEORY. THE TWO BRIDGE STRUCTURES TESTED ARE A 200-FT SPAN THROUGH TRUSS, AND AN 80-FT SPAN COMPOSITE SLAB-STEEL GIRDER BRIDGE. STRAIN DATA WERE OBTAINED DURING THE TRUSS BRIDGE TESTS, AND BOTH STRAINS AND DEFLECTIONS WERE OBTAINED DURING THE STEEL GIRDER BRIDGE TESTS. RESULTS FROM THE COMPARISON OF EXPERIMENTAL AND ANALYTICAL DATA GIVE EXCELLENT AGREEMENT. THE ANALYTICAL TECHNIQUE IS COMPUTER ORIENTED, REQUIRING MINOR CALCULATIONS IN ORDER TO OBTAIN THE RESULTING DATA. /AUTHOR/

Journal ArticleDOI
E. Reissner1
TL;DR: In this article, a procedure for the derivation of two-dimensional shell theory from three-dimensional elasticity theory for the special case of a flat plate is considered explicitly, and the starting point of the work is a suitable version of elasticity theories including moment as well as force stresses.

Journal ArticleDOI
TL;DR: In this paper, the Von Karman equations in dimensionless and finite difference forms for deflections of elastic circular plate with central hole under load solved by iteration are applied to elastic circular plates.
Abstract: Von Karman equations in dimensionless and finite difference forms for deflections of elastic circular plate with central hole under load solved by iteration