Page 72 f cirak the extension of timoshenko beam theory to plates is the reissner mindlin plate theory in reissner mindlin plate theory the outofplane shear deformations are nonzero in contrast to kirchhoff plate theory almost all commercial codes abaqus, lsdyna, ansys, use reissner mindlin type plate finite elements assumed displacements during loading. Vibration equations of thick rectangular plates using mindlin plate theory. W, l and t represent the width, length and thickness of the plate, respectively. Both theories are intended for thick plates in which the normal to the midsurface. Reissnermindlin plate theory an overview sciencedirect. The modal control is applied to study the active vibration control of the strip plates. In the mindlin reissner plate theory the transverse displacements and rotations of the mid surface normals are independent and obtained by employing an isoparametric interpolation respectively from the. Pdf introduction to plate bending theory nirajan paudel. Natural vibrations of thick circular plate based on the. In this study, we consider the problem of nonlinearly tapered annular plate with a free edge. The mindlin plate theory or moderately thick plate theory or shear deformation theory. The mindlinreissner theory of plates is an extension of kirchhofflove plate theory that takes into account shear deformations throughthethickness of a plate. However, the loworder standard isoparametric displacementbased plate elements without special treatments only. Theory of elasticity formulation of the mindlin plate equations nwoji, c.
Free vibration analysis of orthotropic rectangular mindlin. Plate bending theory bending materials science free 30. A study of the application of mindlin plate elements to thin. For the high order theories mindlin and reissner, which considers, shear deformations. However, elements based on the refined reissner mindlin plate theory are increasingly used in research and engineering practice, e. The effect of shear deformation is included which makes kirchhoff plate elements suitable for application in both thin and thick plates. Plate bending with transverse shear deformation mindlinreissner. The theory was developed in 1888 by love 2 using assumptions proposed by kirchhoff. Such kirchhoffmindlin relationships exhibit a onetoone mapping of the exact solutions in both plate models. Despite the simple formulation of this theory, discretization of the plate by means of the. Rectangular steel plates are widely used in variou s steel structures and steel.
The vibration of the mindlin plate and its solution according to the mindlin plate theory. Consider the thin plate in the xy plane of thickness t measured in the z direction shown in the figure below. The equilibrium equations for a reissnermindlin plate can also be similarly obtained as that of a thin plate. Free vibration analysis of orthotropic rectangular mindlin plates with general elastic boundary conditions. Floors, ceilings, windows, disc brakes, ship decks, truck beds the learning objectives understand the theory of thin plate bending, its limitations, and its applications in design and analysis. The reissner mindlin theory does not require the crosssection to be perpendicular to the axial axes after deformation, as shown in figure 2. Dynamics of the mindlin plate and its the authors 2011. Refined quadrilateral element based on mindlinreissner.
Therefore mindlin plate theory is applicable to modeling of classical thin plates as well as moderately. Computer program for the calculation of mindlin plates. Parametric earthquake analysis of thick plates using mindlin. The mindlin plate theory operates with three independent variables, i. That means that kirchhoff theory applies to thin plates, while mindlin theory applies to thick plates where shear deformation may be significant. These elements are attractive, because they are of low order only continuous interpolations are required and can model both thin and moderately thick plate situations. Floors, ceilings, windows, disc brakes, ship decks, truck beds the learning objectives understand the theory of thin plate bending, its limitations, and its applications in. They derived exact relationships between the two models in bending 5, buckling 6 and vibration 79. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. The original code went beyond an overhaul, and this final version will be detailed. Parametric earthquake analysis of thick plates using.
A study of the application of mindlin plate elements to. Bending solutions of sectorial thick plates based on. Faculty of civil engineering, state universiv at campinas, brazil abstract plates can be solved with classical or reissnermindlin plate theory in the same computer code with an appropriate treatment of the direct boundary element formulation. This study presents exact relationships between the bending solutions of sectorial plates based on the kirchhoff or classical thin plate theory and the mindlin plate theory. The reissnermindlin plate models the deformation of a thin plate under external forces. Reissner plate theory and assumed shear strain fieldspart i. The theory was proposed in 1951 by raymond mindlin. Kirchhoff plate elements studied in the previous chapter are restricted to thin plate situations only thicknessaverage side. Some of these plate theories developed in answer to the limitations of the kirchhoff plate theory are reissner plate theory, mindlin plate theory, reddy plate theory, levinsons plate theory, shimpis refined plate theory, etc. Pdf parametric earthquake analysis of thick plates using.
The firstorder shear deformation plate theory fsdpt, described in detail in chapter 4, is an extension of the wellknown timoshenko beam theory andor the mindlinreissner plate theory that is applied to laminated composite plates. Faculty of civil engineering, state universiv at campinas, brazil abstract plates can be solved with classical or reissnermindlin plate theory in the same computer code with an appropriate treatment of. The implementation of the mindlin plate element is carried out for the uel module of abaqus. Bending solutions of sectorial thick plates based on reissner. Welch abstract this thesis is a study of mindlin plate theory in finite element analysis with particular emphasis on locking and instability behaviors of different element formulations. In the mindlinreissner plate theory the transverse displacements and rotations of the mid surface normals are independent and obtained by employing an isoparametric interpolation respectively from the. The same is true of the asymptotic expansion of the reissnermindlin solution 2. A similar, but not identical, theory had been proposed earlier by eric reissner in 1945. Another theory, that has been subject of most recent research in finite plate elements, is the mindlinireissner theory refs. Stress distribution through the thickness of a plate and resultant bending moment plate theory and beam theory. Application of mindlin plate theory to analysis of. Page 54 f cirak overview of plate theories in analogy to beams there are several different plate theories the extension of the eulerbernoulli beam theory to plates is the kirchhoff plate theory suitable only for thin plates the extension of timoshenko beam theory to plates is the reissnermindlin plate theory suitable for thick and thin plates.
Exact expressions of deflection, momentresultants, and stresses are presented for nonuniform thickness. Analysis of thin plates by the elementfree galerkin method. For a unit width beam, flexural rigidity dei et 312. Kirchhoff plate theory, also called the classical small deflection theory of thin plates is the simplest and most commonly. Hence, the construction of thick plate elements is indeed easier than that of the thin plate elements, and more and more attention was paid to develop plate bending elements based on the mindlinreissner plate theory 886. The standard expansion for linear elasticity on a thin plate begins with a term coming from the kirchho.
Finally, numerical examples are given to analyze and discuss the results. That means that kirchhoff theory applies to thin plates, while mindlin theory applies to thick plates where shear deformation may. This thin plate theory is also called the kirchhoff. In contrast, mindlin theory retains the assumption that the line remains straight, but no longer perpendicular to the neutral plane. We compare the results of the kirchhoff plate theory and the mindlin plate theory. Plate analysis using classical or reissner mindlin theories l.
The extension of the eulerbernoulli beam theory to plates is the kirchhoff plate theory suitable only for thin plates the extension of timoshenko beam theory to plates is the reissner mindlin plate theory suitable for thick and thin plates as discussed for beams the related finite elements have problems if applied to thin problems. Princeton class in german thinshell structures yields. The exact displacement function of the timoshenkos beam is used to derive the element displacements of the refined element rdkqm. Introduction to the theory of plates stanford university.
However, elements based on the refined reissnermindlin plate theory are increasingly used in research and engineering practice, e. The supported edge may be simply supported, clamped or elastically restrained against rotation. The firstorder shear deformation plate theory fsdpt, described in detail in chapter 4, is an extension of the wellknown timoshenko beam theory andor the mindlin reissner plate theory that is applied to laminated composite plates. However, because transverse shear deformation is neglected, the kirchho plate theory is only appropriate for modeling thin structures. In the mindlin theory, transverse shear deformation is included. Reissnermindlin plate theory is very adequate for studying composite laminate plates for which shear deformation effects are important. Page 72 f cirak the extension of timoshenko beam theory to plates is the reissnermindlin plate theory in reissnermindlin plate theory the outofplane shear deformations are nonzero in contrast to kirchhoff plate theory almost all commercial codes abaqus, lsdyna, ansys, use reissner mindlin type plate finite elements assumed displacements during loading. Reissnermindlin plate theory an overview sciencedirect topics.
Improved finite strip mindlin plate bending element using. Reissnermindlin theory kirchhoff plate elements studied in the previous chapter are restricted to thin plate situations only. Analysis of thin plates by the elementfree galerkin method petr krysl and ted belytschko october 7, 1999. Specifications of boundary conditions for reissnermindlin. Balch division of mechanics and computation department of mecanical engineering stanford university stretching and bending of plates fundamentals introduction a plate is a structural element which is thin and. A new quadrilateral thinthick plate element rdkqm based on the mindlinreissner plate theory is proposed. It is an extension to arbitrarily layered plates of the reissnermindlin plate theory which appears as a special case of the bendinggradient plate theory when the plate is homogeneous. Kirchhoff and mindlin plates university of british columbia. We will begin with general background theory of kirchhoff plate bending and mindlin plate bending in chapter 1. If the basis functionsn a nare chosen as piecewise. This leads to using the thin plate theory, which neglects transverse shear. Cho2 1faculty of mechanical engineering and naval architecture university of zagreb ivana lucica 5 0 zagreb, croatia email.
The thin plate is assumed to be of thickness t and its deformation is described by using two variables, the displacement, w, and the rotation, in the limit case of t 0, the reissnermindlin plate model is. Also the c 1 continuity requirement for kirchhoff elements poses severe difficulties for deriving a conforming deflection field. Classical thin plate theory is based upon the assumption which were initiated for beam by bernoulli. Nikhil padhye, subodh kalia implementation of mindlin plate element 1 mindlin plate theory l w t mid surface top surface x y z figure 1. This requirement in the thin plate theory is solved easily if mindlin theory is used in the analysis of thin plates. We will see that by the use of special techniques, mindlin plate elements can be made accurate for thin plate problems. Review article developments of mindlinreissner plate elements. Mindlin, 1951 is applied for thick plates, where the shear deformation and rotary inertia effects are included. The research aim is to present a theory of elasticity formulation of the mindlin plate equations for static. Dr abhay bambole sharvari dhepe contents introduction kirchhoffs plate theory mindlin plate theory. Parametric earthquake analysis of thick plates using mindlins theory article pdf available in shock and vibration 176. The mindlin plate theory or moderately thick plate theory or shear deformation theory was developed in the mid1900s to allow for possible transverse shear strains.
In this report, we present a model for plate bending based on the mindlin plate theory for small elastic deformations. Theory of elasticity formulation of the mindlin plate equations. Thus, the context of this study is the application of mindlin plate elements to the bending of thin plates. In this theory, there is the added complication that vertical line elements before deformation do not have to remain perpendicular to the midsurface after deformation, although. Furthermore, since 1 continuity between adjacent elements is required, the construction procedures of kirchho plate elements are quite di cult. If t is more than about onetenth the span of the plate, then transverse shear deformation must be accounted for and the plate is then said to be thick. Finally, since the region is thin in the z7 directionitcanbearguedthattheothernonzerostresseswillhavelile variationwith z. On the range of applicability of the reissnermindlin and. This paper presents an overview of the governing equations for the bending study of the types of plates, with several known plate theories from the literature. A comparative analysis of mindlin and kirchhoff bending. Professor ernest hinton for rectangular isotropic plates based on mindlin plate theory. The mindlin reissner theory of plates is an extension of kirchhofflove plate theory that takes into account shear deformations throughthethickness of a plate. The kirchhofflove theory is an extension of eulerbernoulli beam theory to thin plates.
Plate analysis using classical or reissner mindlin theories. One of the differences between plane stress and plate theory is that in the plate theory the stress components are allowed to vary through the thickness of the plate, so that there can be bending moments, fig. The thickness is usually constant but may be variable and is. Pdf vibration equations of thick rectangular plates using.
The extension of timoshenko beam theory to plates is the reissnermindlin plate theory in reissnermindlin plate theory the outofplane shear deformations are nonzero in contrast to kirchhoff plate theory almost all commercial codes abaqus, lsdyna, ansys, use reissnermindlin type plate finite elements. The reissnermindlin theory does not require the crosssection to be perpendicular to the axial axes after deformation, as shown in figure 2. On the missing modes when using the exact frequency. Such kirchhoff mindlin relationships exhibit a onetoone mapping of the exact solutions in both plate models. For thick plates,mindlinreissner plate bending theory can be used. Thickness is very small compared to other two dimensions ie width and length. Plate bending theory bending materials science free. The program itself is constructed in fortran computational language and provides solution for simply. Natural vibrations of thick circular plate based on the modi.
Jun 22, 2014 2012 10 10 mostafa abdalla lecture 1, about plate theory. A plate is a flat solid body whose thickness is small compared to the other dimensions and is subjected to bending loads. A study of the application of mindlin plate elements to thin plates thesis directed by assistant professor samuel w. Thin plate formulation this is similar to the beam formula, but since the plate is very wide we have a situation similar to plain strain.
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