Showing papers on "Plate theory published in 2000"

Analysis of functionally graded plates

Abstract: Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright © 2000 John Wiley & Sons, Ltd.

Three-dimensional thermoelastic deformations of a functionally graded elliptic plate

Abstract: A new solution in closed form is obtained for the thermomechanical deformations of an isotropic linear thermoelastic functionally graded elliptic plate rigidly clamped at the edges. The through-thickness variation of the volume fraction of the ceramic phase in a metal–ceramic plate is assumed to be given by a power-law type function. The effective material properties at a point are computed by the Mori–Tanaka scheme. It is found that the through-thickness distributions of the in-plane displacements and transverse shear stresses in a functionally graded plate do not agree with those assumed in classical and shear deformation plate theories.

The Trefftz Finite and Boundary Element Method

Abstract: Part 1: finite element technique shape functions and element stiffness matrix brief historical background basic relationships in engineering problems modified variational principles the concept of T-complete solution comparison of T-elements with conventional finite elements comparison of T-elements with boundary elements. Part 2: potential problems - introduction statement of the problem T-complete functions assumed fields generation of element matrix equation rank condition special purpose functions sensitivity to mesh distortion orthotropic case the Helmholtz equation HT-element with boundary "traction" frame frameless T-elements. Part 3: linear elastostatics - introduction linear theory of elasticity assumed fields in plane elasticity T-complete functions variational formulations element stiffness equation special-purpose elements p-extension approach three-dimensional elasticity numerical examples. Part 4: thin plates - introduction thin plate theory assumed field T-complete functions and particular solutions variational formulations for plate bending generation of element stiffness matrix p-method elements special purpose functions Extension to thin plates on elastic foundation Two alternative plate bending p-elements Numerical examples and assessment. Chapter 5 - Thick Plates - Introduction Basic equations for Reissner-Mindlin plate theory Assumed fields and particular solution Variational formulation for HT thick plate elements Implementation of the new family of HT elements A 12 DOF quadrilateral element free of shear locking Extension to thick plates on elastic foundation Sensitivity to mesh distortion Numerical assessment. Chapter 6 - Transient Heat Conduction - Introduction Elements of heat conduction Time step formula Element matrix formulations T-complete functions and particular solutions Numerical examples. Chapter 7 - Geometrically Nonlinear Analysis of Plate Bending Problems - Introduction Basic equations of nonlinear thin plate bending Assumed fields and Trefftz functions Particular solutions Modified variational principle Element matrix Iterative scheme Extension to post-buckling thin plates on elastic foundation Geometrically nonlinear analysis of thick plates Numerical examples. Chapter 8 - Elastoplasticity - Introduction Time discretization Basic relations Assumed fields Constraints on the approximation functions Finite element equilibrium and compatibility equations Finite element equations Finite element governing system. Chapter 9 - Dynamics of Plate Bending Problems - Introduction Basic equations Time-stepping formulation Numerical examples. Chapter 10 - Trefftz Boundary Element Method - Introduction Potential problems Plane elasticity Thin plate bending Moderately thick plates.

Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories

Abstract: We derive field equations for a functionally graded plate whose deformations are governed by either the first-order shear deformation theory or the third-order shear deformation theory. These equations are further simplified for a simply supported polygonal plate. An exact relationship is established between the deflection of the functionally graded plate and that of an equivalent homogeneous Kirchhoff plate. This relationship is used to esplicitly express the displacements of a plate particle according to the first-order shear deformation theory in terms of the deflection of a homogeneous Kirchhoff plate. These relationships can readily be used to obtain similar correspondences between the deflections of a transversely laminated plate and a homogeneous Kirchhoff plate.

Effect of plate position relative to bending direction on the rigidity of a plate osteosynthesis. A theoretical analysis.

Abstract: Mechanical unloading of the plated bone segment is observed after plate osteosynthesis because the implant takes over a part of the physiological loading. Strain reduction in the bony tissue depends on the rigidity of the plate (cross-sectional area, geometrical form, and modulus of elasticity). The aim of the present study was to calculate theoretically the effect of plate position relative to bending direction on the overall bending stiffness of the composite system plate-bone. To calculate the rigidity, a cylindrical bone model with mechanical characteristics similar to a sheep tibia and a rectangular plate cross-section corresponding to a DC-plate with either a modulus of elasticity of steel or titanium was used. Calculations under different bending directions were performed according to the laws of the linear bending theory and the composite beam theory. The bending stiffness of a plate osteosynthesis reaches a minimum and a maximum respectively, in cases in which the bending moment acts in the direction of the main axis of the area moment of inertia of the plate. The minimum is present with the plate bent vertically, the maximum with the plate bent horizontally, e.g. on the tension side of the composite system--on the assumption that the bone structure opposite the plate is capable of withstanding compressive loading. For steel and titanium plates, factors of 2 and 2.25 respectively were calculated between the minimum and the maximum bending stiffnesses of the osteosynthesis. The bending rigidity of the plate alone has only a minimal effect on the total stiffness of the osteosynthesis. With a plate bent vertically, the difference between steel and titanium plates was 18%, with the plate bent horizontally (situated on the tension side), it was only 7%. The bending stiffness of a plate osteosynthesis depends on the cross-section, the geometrical form, and the modulus of elasticity of the plate, as well as on the plate position relative to the bending direction of the composite system. The modulus of elasticity of the plate is relatively unimportant, while with a given plate the individual plate position relative to the bending direction is of crucial importance. Thus, changing the modulus of elasticity of the plate cannot solve the problem of implant induced unloading of the bone cortex because the bending stiffness of the composite system depends much more on the plate position relative to the bending direction.

Vibration and stability of thick plates on elastic foundations

Abstract: Natural frequencies and buckling stresses of a thick isotropic plate on two-parameter elastic foundations are analyzed by taking into account the effect of shear deformation, thickness change, and rotatory inertia. Using the method of power series expansion of the displacement components, a set of fundamental dynamic equations of a two-dimensional, higher-order theory for thick rectangular plates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are used to solve the eigenvalue problems of a simply supported thick elastic plate. To assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress are examined in detail. The distribution of modal transverse stresses are obtained by integrating the three-dimensional equations of motion in the thickness direction. The present approximate theories can accurately predict the natural frequencies and buckling stresses of thick plates on elastic foundations as compared with Mindlin plate theory and classical plate theory.

Refined quadrilateral element based on Mindlin/Reissner plate theory

Abstract: A new quadrilateral thin/thick plate element RDKQM based on the Mindlin/Reissner plate theory is proposed. The exact displacement function of the Timoshenko's beam is used to derive the element displacements of the refined element RDKQM. The convergence for the very thin plate can be ensured theoretically. Numerical examples presented show that the proposed model indeed possesses higher accuracy in the analysis of thin/thick plates. It can pass the patch test required for the Kirchhoff thin plate elements, and most important of all, it is free from locking phenomenon for extremely thin plates. Copyright © 2000 John Wiley & Sons, Ltd.

Cylindrical Bending of Laminated Plates with Distributed and Segmented Piezoelectric Actuators/Sensors

Abstract: The generalized plane quasistatic deformations of linear piezoelectric laminated plates are analyzed by the Eshelby‐Stroh formalism. The laminate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. The three-dimensional differential equations of equilibrium for a piezoelectric body are exactly satiseed at every point in the body. The analytical solution is in terms of an ine nite series; the continuity conditions at the interfaces between adjoining laminae and boundary conditions at the edges are satise ed in the sense of Fourier series. The formulation admits different boundary conditions at the edges and is applicable to thick and thin laminated plates. Results are presented for laminated elastic plates with a distributed piezoelectric actuator on the upper surface and a sensor on the lower surface and subjected to different sets of boundary conditions at the edges. Results are also provided for a piezoelectric bimorph and an elastic plate with segmented piezoelectric actuators bonded to its upper and lower surfaces.

Refined higher order finite element models for thermal buckling of laminated composite and sandwich plates

Abstract: Two refined higher order theories, one that neglects and the other that takes into account the effect of transverse normal deformation, are used to develop two discrete finite element models for the thermal buckling analysis of composite laminates and sandwiches. The two models, one with nine degrees of freedom per node and the other with eleven degrees of freedom, are based on a nine-node Lagrangian isoparametric element. The geometric stiffness matrices are developed by taking into consideration the effects of the higher order terms on the initial in-plane and transverse shear stresses. The accuracy of the present formulations is first evaluated by analyzing sample problems for which analytical three-dimensional solutions exist in the literature. Numerical results are presented for the first time for sandwich plates, demonstrating the importance and accuracy of the higher order theory in comparison to first-order theory. Some new results are also given for sandwich plates with angle-ply composite face s...

Single- vs Multilayer Plate Modelings on the Basis of Reissner's Mixed Theorem

Abstract: The use of Reissner's mixed variational theorem (Reissner, E., On a Certain Mixed Variational Theory and a Proposed Applications, International Journal for Numerical Methods in Engineering, Vol. 20, 1984, pp. 1366-1368; Reissner, E.,On a Mixed Variational Theorem and on a Shear Deformable Plate Theory, International Journal of Numerical Methods in Engineering, Vol. 23, 1986, pp. 193-198) to analyze laminated plate structures is examined. The two cases of single-layer and multilayer models have been compared. Governing equilibrium and constitutive equations have been derived in a unified manner. Navier-type closed-form solutions are presented for the particular case of cross-ply simply supported plates. Thin and thick, as well as symmetrically and asymmetrically laminated plates, have been investigated. Displacements and transverse stresses have been evaluated and compared with available mixed two-dimensional results and three-dimensional solutions. The following have been concluded: 1) Reissner's mixed theorem is a very suitable tool to analyze laminated structures. 2) Multilayer modelings lead to an excellent agreement with exact solution for both displacement and transverse stress evaluations. Such an agreement, which has been confirmed for very thick geometries (a/h ≤ 4), does not depend on laminate layouts. No remarkable differences have been found for stresses evaluated a priori by the assumed model with respect to exact results. 3) Single-layer analyses lead to an accurate description of the response of thick plates. Major discrepancies have been found for very thick plate geometries with exact solutions. Nevertheless, their accuracy is very much subordinate to the order of the used expansion as well as to laminated layouts. Better transverse stress evaluations are obtained upon integration of three-dimensional equilibrium equations a posteriori than those furnished a priori. This trend has been confirmed for both thick and thin plates.

A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect

Abstract: This paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.

Analysis of plates reinforced with beams

Abstract: In this paper a solution to the problem of plates reinforced with beams is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems which are nonlinearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate–beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. The resulting deflections are considerably smaller than those obtained by other models.

Flexural edge waves and comments on 'a new bending wave solution for the classical plate equation' [J. Acoust. Soc. Am. 104, 2220-2222 (1998)]

Abstract: A brief review is presented of the theory of flexural edge waves, first predicted in 1960 by Yu K. Konenkov using Kirchhoff plate theory. It is demonstrated that the flexural edge wave is also predicted by Mindlin’s plate theory, and that the prediction agrees with measured data. It is noted that the edge wave was erroneously presented as a new type of bending wave solution in a recently published paper in this journal.