Finite elements for the reissnermindlin plate 1277 formulationsimilarly imposesthe kirchhoffconstraint onthefinite elementsubspaces in the limit. 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. Vibration equations of thick rectangular plates using mindlin. The approach is a generalization of the onedimensional euler bernoulli beam theory, which exploits the slender shape of a beam. Also the c1continuity requirement for kirchhoff elements poses. The purpose of this paper is to study shear lockingfree analysis of thick plates using mindlins theory and to determine the effects of the thicknessspan ratio, the aspect ratio and the boundary. In the reissnermindlin theory the transverse shear strains are assumed as constant throughout the plate thickness and thus some correction coef. Plate theory in continuum mechanics, plate theories are mathematical descriptions of mechanics for calculating deformations and stresses in flat 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 direct boundary element formulation.
Free vibration analysis of orthotropic rectangular mindlin. Mindlins problem for a halfspace indented by a flexible plate. On the performance of mindlin plate elements in modelling. A similar, but not identical, theory had been proposed earlier by eric reissner in 1945. Numerical results are obtained from the present theory for an example of a laminated plate under cylindrical bending. The aim of this paper is to give the title theory of shearable plates a precise and exact position with respect to threedimensional linear elasticity. An improved displacement methodology based on mindlin theory is developed and applied to a fournode, twelve degreesoffreedom quadrilateral element. We compare the results of the kirchhoff plate theory and the mindlin plate theory. Three numerical examples for thick plate with different boundary conditions are analyzed and the results are compared with those from relevant literature. In mindlinreissner plate theory assumption 4 of planes sections perpendicular to the plate mid sur face remain nearly perpendicular after deformation is dropped to account for shear. In 1214 using the classical theory of plate, free vibrations of asymmetric. An effective hybrid displacement function element method for. In recent years, several efficient plate bending elements have been reported in the liter ature.
The purpose of this paper is to study shear lockingfree analysis of thick plates using mindlin s theory and to determine the effects of the thicknessspan ratio, the aspect ratio and the boundary. The mindlin reissner theory of plates is an extension of kirchhofflove plate theory that takes into account shear deformations throughthethickness of a plate. Love theory, especially for moderately thin plates, this assertion is not yet fully substantiated. Pdf since 1960s, how to develop highperformance plate bending finite elements based on different plate theories has attracted a great deal. An improved transverse shear deformation theory for laminated. In general, exact solutions for cantilever plates using plate theory are quite involved and few exact solutions can be found in the literature. Twolevel multiscale enrichment methodology for modeling. The crosssection b t remains rectangular during bending when biaxial. In fact, the eulerbernoulli and timoshenko beam theories both have its counterpart in plate theory. Crosssection kinematics assumptions distributed load acts in the xz plane, which is also a plane of symmetry of a body. All structured data from the file and property namespaces is available under the creative commons cc0 license.
A new twolevel multiscale enrichment methodology for analysis of heterogeneous plates is presented. The migration technique, which is an advanced technique in geophysics to reverse the reflection wave field and to image the earth interior, is then used to backpropagate the recorded wave signals and to visually image the damage in the plate. Approximate solutions of plate equations based on energy principles. On the range of applicability of the reissnermindlin and. Over the years, researchers have tried to modify the classical plate theory to relax its restrictions. In this regard, scientists used twodimensional theories including classical theory of plate, first order shear theory and higherorder shear theory to exact solution of plates free vibration under simple boundary conditions 11. The state space concept has been used to develop levytype exact solutions for free vibration and buckling of laminated composite plates based on the first order and higher order theories 18, 19. Computer program for the calculation of mindlin plates 3 preface this thesis is about the making of a computer program capable of calculating rectangular mindlin plates resting on elastic winkler foundation.
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. Pdf shear lockingfree analysis of thick plates using. Modelling of inplane wave propagation in a plate using spectral element method and kanemindlin theory with application to damage detection magdalena rucka department of structural mechanics and bridge structures, faculty of civil and environmental engineering, gdansk university of technology narutowicza 1112, 80233 gdansk, poland. Later mindlin 1945 and reissener 1951 modified the thin plate theory to include transverse shear deformation, which is called moderately thick plate theory. On the peridynamic formulation for an orthotropic mindlin.
Files are available under licenses specified on their description page. The plate element obtained from our general 4node shell element is based on the mindlinreissner plate theory and represents an extension of the formulation given in reference 2, pp. An effective hybrid displacement function element method for solving the edge effect of mindlinreissner plate yan shang1,3, song cen1,3,4,, chenfeng li2 and junbin huang1 1department of engineering mechanics, school of aerospace engineering, tsinghua university, beijing, 84, china. Mindlin plate theory, theory of elasticity, shear deformation. Both theories are intended for thick plates in which the normal to the midsurface remains straight but not necessarily perpendicular to the midsurface. Abstract this work describes a nodally integrated finite element formulation for plates under the mindlinreissner theory. It is the purpose of this note to show that in the asymptotic regime usually assumed in. Due to symmetry, only a quarter of the square plate is considered. Mindlin theory for plates timoshenko theory for beams the kirchhoff theory assumes that a vertical line remains straight and perpendicular to the neutral plane of the plate during bending. The straindisplacement equations derived based on this theory contains no second derivatives of displacements.
Nov 30, 2006 in this study, we consider the problem of nonlinearly tapered annular plate with a free edge. The buckling load for a rectangular plate with four. The enrichments are applied in the displacement and strain levels. A new triangular hybrid displacement function element for static and free vibration analyses of mindlinreissner plate latin american journal of solids and structures 14 2017 765804 1 introduction the mindlinreissner plate theory is widely used to describe the deformation and resultant fields of an elastic plate subjected to transverse loads. Introduction to the theory of plates stanford university.
The variationalasymptotic method, a powerful mathematical approach, has been used to construct a highly accurate reissnermindlin plate theory for composite laminated plates. Linear free vibration analysis of rectangular mindlin plates. The results are given as power series in terms of deflection, rotation and stresses. Exact expressions of deflection, momentresultants, and stresses are presented for nonuniform thickness. Mindlins problem for a halfspace indented by a flexible plate 255 fig. A new triangular hybrid displacement function element for. The reissner or the mindlin plate theories include approximately the effect of transverse shear strains and yields a simple refined plate theory by introducing. Plate analysis using classical or reissner mindlin theories.
Mathematical construction of a reissnermindlin plate theory. Plate analysis using classical or reissner mindlin theories l. Thin plate b smallest side thick plate t 20 b 2000 1 10 1. Closed form solutions for free vibrations of rectangular. We will develop a twodimensional plate theory which employs the inplane coordinates x and y in see plate and associated x, y, z coordinate system as independent variables. Structural health monitoring using geophysical migration. Comparison with results from exact threedimensional analysis shows that the present theory is more accurate than. The equations of reissnermindlin plates obtained by the method of.
Mindlin plate theory and abaqus uel implementation mit. Navier and levy solutions for shear deformable plates. For the limiting case of isotropy, the present theory reduces to an improved version of mindlin s theory. Both mindlins and reissners theories are sixth order. Eigenfunctions of crack problems in the mindlin plate theory. Bending analyses of 1d orthorhombic quasicrystal plates. Forthisreasonmanyefforts were devoted to approximate solutions with a high level of accuracy. Modified mindlin plate theory and shear lockingfree finite.
The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stressfree boundary conditions on the plate boundary surface, thus a shear. Likewise, the generalization of the timoshenkoehrenfest beam theory to plates was given by uflyand and mindlin in the years 19481951. 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. What links here related changes upload file special pages permanent link page. An important issue in the numerical analysis of reissnermindlin plates is the shear locking. In this study, we consider the problem of nonlinearly tapered annular plate with a free edge. Analysis of free and forced vibration of fgm rectangular. For the limiting case of isotropy, the present theory reduces to an improved version of mindlins theory. Conclusion the mindlin plate theory or thick plate theory or shear deformation theory allow for possible transverse shear strains. Reissner and stein provide a simplified theory for cantilever plates that is an improvement over older theories such as saintvenant plate theory.
Mindlin plate theory, vibration, thick plate, mode shape introduction the classic plate theory cpt provides a theoretical model of plate behavior which has some considerable advantages, which cab be employed with confidence over a reasonable range of applications, but which also has significant limitations. Modelling of inplane wave propagation in a plate using. Nodally integrated finite element formulation for mindlin. The theory is applicable to plates for which each layer is made with a monoclinic material. Theory of elasticity formulation of the mindlin plate equations. Considering the significance of this matter, later al vibration of thick rectangular plates was studied on the basis of mind lin plate theory. Mindlin plate theory is adopted to model the propagating waves, and a twodimensional 26 order. The reissner mindlin theory does not require the crosssection to be perpendicular to the axial axes after deformation, as shown in figure 2.
A fournode plate bending element based on mindlinreissner. The mindlinreissner theory of plates is an extension of kirchhofflove plate theory that takes. Kirchhoff plate elements studied in the previous chapter are restricted to thin plate situations only thicknessaverage side. W, l and t represent the width, length and thickness of the plate, respectively. Mindlin plate theory predicts three frequency spectra or, equivalently, three branches to a phase velocity dispersion diagram, the lowest of whichthew1modeprovides rotatory inertia and shear deformation corrections to classical thin plate theory. Among these elements, the most popular models are usually formulated by two theories. The importance of these theories stems from the fact that. 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. Therefore, if afinite element methodbasedonthe standard formulation is to approximate well uniformly with respect to plate thickness, then the subspace.
I devote the program to be helpful to the students and be a useful tool in the preliminary design of simple structures. 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. In mindlin reissner plate theory assumption 4 of planes sections perpendicular to the plate mid sur face remain nearly perpendicular after deformation is dropped to account for shear. The the formulation makes use of the weighted residual method and nodal integration to derive the assumed strain relations.
The supported edge may be simply supported, clamped or elastically restrained against rotation. A new triangular hybrid displacement function element for static and free vibration analyses of mindlin reissner plate latin american journal of solids and structures 14 2017 765804 1 introduction the mindlin reissner plate theory is widely used to describe the deformation and resultant fields of an elastic plate subjected to transverse loads. Pdf vibration equations of thick rectangular plates using. Mindlin plate element is carried out for the uel module of abaqus. Twolevel multiscale enrichment methodology for modeling of. Following the modified mindlin theory shear lockingfree finite element formulation is given. To observe the locking behaviour in the krigingbased finite element analyses kfea, a simplysupported square plate under uniform load is analysed. Kirchhoff theory for plates eulerbernoulli theory for beams mindlin theory for plates timoshenko theory for beams. The theory was proposed in 1951 by raymond mindlin. An improved transverse shear deformation theory for. Several alternative plate theories have appeared in the literature, among which those of mindlin 1951 and reissner 1945 are the most well known. Computer program for the calculation of mindlin plates. As presented there, the variational indicator of a mindlinreissner plate is, in linear elastic static analysis, where. Ferreira, matlab codes for finite element analysis.
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