Showing papers on "Plate theory published in 2006"
TL;DR: In this article, the static response of a simply supported rectangular plate subjected to a transverse uniform load is presented for a simple supported functionally graded rectangular plate, where material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents.
508 citations
TL;DR: In this paper, an elastic, rectangular, and simply supported, functionally graded material (FGM) plate of medium thickness subjected to transverse loading has been investigated, and the analytical solutions of P-, S- and E-FGM plates are proved by the numerical results of finite element method.
470 citations
TL;DR: Lim et al. as discussed by the authors proposed a general thin plate theory including surface effects, which can be used for size-dependent static and dynamic analysis of plate-like thin film structures, and some numerical examples are provided to verify the validities of the theory.
382 citations
TL;DR: In this paper, the first and third-order shear deformation plate theories, the Mori-Tanaka technique, and approximate the trial solution with multiquadric radial basis functions to analyze free vibrations of functionally graded plates.
327 citations
TL;DR: In this paper, a new theory for orthotropic plate analysis, which involves only two unknown functions and yet takes into account shear deformations, is presented, which gives rise to only two governing equations, which are completely uncoupled for static analysis, and are only inertially coupled (i.e., no elastic coupling at all) for dynamic analysis.
239 citations
01 Jan 2006
TL;DR: The Mechanics of Solids and Materials as mentioned in this paper provides a modern and integrated treatment of the foundations of solid mechanics as applied to the mathematical description of material behavior, including large strain, strain rate, temperature, time dependent deformation and localized plastic deformation.
Abstract: Mechanics of Solids and Materials intends to provide a modern and integrated treatment of the foundations of solid mechanics as applied to the mathematical description of material behavior. The 2006 book blends both innovative (large strain, strain rate, temperature, time dependent deformation and localized plastic deformation in crystalline solids, deformation of biological networks) and traditional (elastic theory of torsion, elastic beam and plate theories, contact mechanics) topics in a coherent theoretical framework. The extensive use of transform methods to generate solutions makes the book also of interest to structural, mechanical, and aerospace engineers. Plasticity theories, micromechanics, crystal plasticity, energetics of elastic systems, as well as an overall review of math and thermodynamics are also covered in the book.
235 citations
TL;DR: In this article, the authors proposed a method based on Mindlin's plate theory and the plane wave expansion method to study the propagation of Lamb waves in two-dimensional phononic-crystal plates.
Abstract: Based on Mindlin's plate theory and the plane wave expansion method, a formulation is proposed to study the propagation of Lamb waves in two-dimensional phononic-crystal plates. The method is applied to calculate the frequency band structure of a square array of crystalline gold cylinders in an epoxy matrix with a finite thickness. It is found that complete frequency band gaps for Lamb waves between different pass bands are opened up by tuning thickness of the phononic-crystal plate. The influence of plate thickness on the width of complete frequency band gap is calculated and discussed as well; the existence of frequency stop bands is sensitive to the variation of the thickness of the plate. Finally, we note that the proposed method provides a concise and efficient way in analyzing the frequency band structures of phononic-crystal plates in lower bands.
186 citations
TL;DR: In this article, the authors considered the problems of free vibrations, buckling, and static deflection of functionally graded plates in which material properties vary through the thickness of the plates.
185 citations
TL;DR: In this article, the authors used two variable refined plate theory for free vibration analysis of plate, which is variationally consistent and gives two governing equations, which are only inertially coupled and there is no elastic coupling at all.
185 citations
TL;DR: In this article, the equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory, assuming that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable.
150 citations
TL;DR: In this paper, the von Karman nonlinear strain-displacement relationship is used to account for the large deflection of the plate, and the incremental form considering the initial displacement and initial stress is adopted for the nonlinear temperature-dependent material properties of the functionally graded material.
TL;DR: In this paper, a two-dimensional boundary element method (BEM) is developed to simulate the water flow during the water impact of a horizontal circular cylinder, where a modal analysis is utilized for structural responses and the hydroelasticity.
TL;DR: In this paper, the nonlinear vibration and dynamic response of a functional graded material (FGM) plate with surface-bonded piezoelectric layers in thermal environments were investigated.
TL;DR: In this article, an analytical two-port, lumped-element model of a piezoelectric composite circular plate is presented, where the individual components of a unimorph transducer are modeled as lumped elements of an equivalent electrical circuit using conjugate power variables, and the transverse static deflection field as a function of pressure and voltage loading is determined to synthesize the twoport dynamic model.
Abstract: This paper presents an analytical two-port, lumped-element model of a piezoelectric composite circular plate. In particular, the individual components of a piezoelectric unimorph transducer are modeled as lumped elements of an equivalent electrical circuit using conjugate power variables. The transverse static deflection field as a function of pressure and voltage loading is determined to synthesize the two-port dynamic model. Classical laminated plate theory is used to derive the equations of equilibrium for clamped circular laminated plates containing one or more piezoelectric layers. A closed-form solution is obtained for a unimorph device in which the diameter of the piezoelectric layer is less than that of the shim. Methods to estimate the model parameters are discussed, and model verification via finite-element analyses and experiments is presented. The results indicate that the resulting lumped-element model provides a reasonable prediction (within 3%) of the measured response to voltage loading and the natural frequency, thus enabling design optimization of unimorph piezoelectric transducers.
Book•
01 Jan 2006
TL;DR: In this paper, the authors present a method for the integration of two-dimensional components in a finite element solution, based on the Green-Gauss Theorem, and evaluate its performance on a set of problems: 1.1 Lagrange Interpolation for Second-Order Problems, 2.2 Finite Element Formulation for Antisymmetric Loading Terms, 3.3 Multi-Factor Adjustment of Stresses, 4.4 Numerical Integration using Gauss Quadrature, and 5.5 Evaluation of Surface Integrals.
Abstract: CONTENTS OF THE BOOK WEB SITE. PREFACE. 1 ESSENTIAL BACKGROUND. 1.1 Steps in a Finite Element Solution. 1.1.1 Two-Node Uniform Bar Element. 1.2 Interpolation Functions. 1.2.1 Lagrange Interpolation for Second-Order Problems. 1.2.2 Hermite Interpolation for Fourth-Order Problems. 1.2.3 Lagrange Interpolation for Rectangular Elements. 1.2.4 Triangular Elements. 1.3 Integration by Parts. 1.3.1 Gauss's Divergence Theorem. 1.3.2 Green-Gauss Theorem. 1.3.3 Green-Gauss Theorem as Integration by Parts in Two Dimensions. 1.4 Numerical Integration Using Gauss Quadrature. 1.4.1 Gauss Quadrature for One-Dimensional Integrals. 1.4.2 Gauss Quadrature for Area Integrals. 1.4.3 Gauss Quadrature for Volume Integrals. 1.5 Mapped Elements. 1.5.1 Restrictions on Mapping of Areas. 1.5.2 Derivatives of the Assumed Solution. 1.5.3 Evaluation of Area Integrals. 1.5.4 Evaluation of Boundary Integrals. Problems. 2 ANALYSIS OF ELASTIC SOLIDS. 2.1 Governing Equations. 2.1.1 Stresses. 2.1.2 Strains. 2.1.3 Constitutive Equations. 2.1.4 Temperature Effects and Initial Strains. 2.1.5 Stress Equilibrium Equations. 2.2 General Form of Finite Element Equations. 2.2.1 Weak Form. 2.2.2 Finite Element Equations. 2.3 Tetrahedral Element. 2.3.1 Interpolation Functions for a Tetrahedral Element. 2.3.2 Tetrahedral Element for Three-Dimensional Elasticity. 2.4 Mapped Solid Elements. 2.4.1 Interpolation Functions for an Eight-Node Solid Element. 2.4.2 Interpolation Functions for a 20-Node Solid Element. 2.4.3 Evaluation of Derivatives. 2.4.4 Integration over Volume. 2.4.5 Evaluation of Surface Integrals. 2.4.6 Evaluation of Line Integrals. 2.4.7 Complete Mathematica/MATLAB Implementations. 2.5 Stress Calculations. 2.5.1 Optimal Locations for Calculating Element Stresses. 2.5.2 Interpolation-Extrapolation of Stresses. 2.5.3 Average Nodal Stresses. 2.5.4 Iterative Improvement in Stresses. 2.6 Static Condensation. 2.7 Substructuring. 2.8 Patch Test and Incompatible Elements. 2.8.1 Convergence Requirements. 2.8.2 Extra Zero-Energy Modes. 2.8.3 Patch Test for Plane Elasticity Problems. 2.8.4 Quadrilateral Element with Additional Bending Shape Functions. 2.9 Computer Implementation: fe2Quad. Problems. 3 SOLIDS OF REVOLUTION. 3.1 Equations of Elasticity in Cylindrical Coordinates. 3.2 Axisymmetric Analysis. 3.2.1 Potential Energy. 3.2.2 Finite Element Equations. 3.2.3 Three-Node Triangular Element. 3.2.4 Mapped Quadrilateral Elements. 3.3 Unsymmetrical Loading. 3.3.1 Fourier Series Representation of Loading. 3.3.2 Finite Element Formulation for Symmetric Loading Terms. 3.3.3 Finite Element Formulation for Antisymmetric Loading Terms. Problems. 4 MULTIFIELD FORMULATIONS FOR BEAM ELEMENTS. 4.1 Euler-Bernoulli Beam Theory. 4.2 Mixed Beam Element Based on EBT. 4.3 Timoshenko Beam Theory. 4.4 Displacement-Based Beam Element for TBT. 4.5 Shear Locking in Displacement-Based Beam Elements for TBT. 4.5.1 Possible Remedies for Shear Locking. 4.6 Mixed Beam Element Based on TBT. 4.7 Four-Field Beam Element for TBT. 4.8 Linked Interpolation Beam Element for TBT. 4.9 Concluding Remarks. Problems. 5 MULTIFIELD FORMULATIONS FOR ANALYSIS OF ELASTIC SOLIDS. 5.1 Governing Equations. 5.2 Displacement Formulation. 5.3 Stress Formulation. 5.4 Mixed Formulation. 5.5 Assumed Stress Field For Mixed Formulation. 5.5.1 Minimum Number of Stress Parameters. 5.5.2 Optimum Number of Stress Parameters. 5.5.3 Suggested Procedure for Determining Appropriate Stress Interpolation. 5.6 Analysis of Nearly Incompressible Solids. 5.6.1 Deviatoric and Volumetric Stresses and Strains. 5.6.2 Poisson Ratio Locking in the Displacement-Based Finite Elements. 5.6.3 Mixed Formulation for Nearly Incompressible Solids. 5.6.4 Finite Element Equations. 5.6.5 Assumed Pressure Solution. 5.6.6 Quadrilateral Elements for Planar Problems. Problems. 6 PLATES AND SHELLS. 6.1 Kirchhoff Plate Theory. 6.1.1 Equilibrium Equations. 6.1.2 Stress Computations. 6.1.3 Weak Form for Displacement-Based Formulation. 6.1.4 General Form of Kirchhoff Plate Element Equations. 6.2 Rectangular Kirchhoff Plate Elements. 6.2.1 MZC (Melosh, Zienkiewicz, and Cheung) Rectangular Plate Element. 6.2.2 Patch Test for Plate Elements. 6.2.3 BFS (Bogner, Fox, and Schmit) Rectangular Plate Element. 6.3 Triangular Kirchhoff Plate Elements. 6.3.1 BCIZ (Bazeley, Cheung, Irons, and Zienkiewicz) Triangular Plate Element. 6.3.2 Conforming Triangular Plate Elements. 6.4 Mixed Formulation for Kirchhoff Plates. 6.5 Mindlin Plate Theory. 6.6 Displacement-Based Finite Elements for Mindlin Plates. 6.6.1 Weak Form. 6.6.2 General Form of Mindlin Plate Element Equations. 6.6.3 Heterosis Element. 6.7 Multifield Elements for Mindlin Plates. 6.8 Analysis of Shell Structures. 6.8.1 Transformation Matrix. 6.8.2 Transformed Equations. Problems. 7 INTRODUCTION TO NONLINEAR PROBLEMS. 7.1 Nonlinear Differential Equation. 7.1.1 Approximate Solutions Using the Classical Form of the Galerkin Method. 7.1.2 Finite Element Solution. 7.2 Solution Procedures for Nonlinear Problems. 7.2.1 Constant Stiffness Iteration. 7.2.2 Load Increments. 7.2.3 Arc-Length Method. 7.3 Linearization and Directional Derivative. 7.3.1 Examples of Linearization. Problems. 8 MATERIAL NONLINEARITY. 8.1 Analysis of Axially Loaded Bars. 8.1.1 Weak Form. 8.1.2 Two-Node Finite Element. 8.1.3 One-Dimensional Plasticity. 8.1.4 Ramberg-Osgood Model. 8.2 Nonlinear Analysis of Trusses. 8.3 Material Nonlinearity in General Solids. 8.3.1 General Form of Finite Element Equations. 8.3.2 General Formulation for Incremental Stress-Strain Equations. 8.3.3 State Determination Procedure. 8.3.4 von Mises Yield Criterion and the Associated Hardening Models. Problems. 9 GEOMETRIC NONLINEARITY. 9.1 Basic Continuum Mechanics Concepts. 9.1.1 Deformation Gradient. 9.1.2 Green-Lagrange Strains. 9.1.3 Cauchy and Piola-Kirchhoff Stresses. 9.2 Governing Differential Equations and Weak Forms. 9.3 Linearization of the Weak Form. 9.4 General Form of Element Tangent Matrices. 9.4.1 State Determination and Check for Convergence. 9.5 Constitutive Equations. 9.5.1 Kirchhoff Material. 9.5.2 Compressible Neo-Hookean Material. 9.6 Computations For a Planar Analysis. 9.7 Deformation-Dependent Loading. 9.7.1 Linearized External Virtual Work for Pressure Loading: General Three-Dimensional Case. 9.7.2 Linearized External Virtual Work for Pressure Loading: Planar Case. 9.8 Linearized Buckling Analysis. 9.8.1 Buckling Load for Trusses. 9.9 Appendix: Double Contraction of Tensors. 9.9.1 Double Contraction of Two Second-Order Tensors. 9.9.2 Double Contraction of a Fourth-Order Tensor with a Second-Order Tensor. Problems. 10 CONTACT PROBLEMS. 10.1 Simple Normal Contact Example. 10.1.1 Direct Solution. 10.1.2 Solution Using Normal Contact Constraint. 10.2 Contact Example Involving Friction. 10.2.1 Solution of a Beam Problem with No Frictional Resistance. 10.2.2 Frictional Constraint Function. 10.2.3 Solution of a Beam Problem with Large Frictional Resistance. 10.2.4 Solution of a Beam Problem with Small Frictional Resistance. 10.3 General Contact Problems. 10.3.1 Contact Point and Gap Calculations. 10.3.2 Forces on the Contact Surface. 10.3.3 Lagrange Multiplier Weak Form. 10.3.4 Penalty Formulation. Problems. BIBLIOGRAPHY. INDEX.
TL;DR: In this paper, a 4-node facet type quadrangular shell finite element based on a layerwise theory was developed for dynamic modeling of laminated structures with viscoelastic damping layers.
Abstract: This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.
TL;DR: In this paper, a finite element formulation based on the classical laminated plate theory for laminated structures with integrated piezoelectric layers or patches, acting as sensors and actuators, is presented.
TL;DR: In this paper, the sensitivity of the postbuckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes is investigated, and a generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometry imperfections such as sine type, local type, and global type imperfections.
TL;DR: In this paper, a model of a rectangular plate made of a porous material is presented, which bases on nonlinear displacement functions taking into account the hearing deformations of the material and the assumed linear geometrical and physical relationships.
Abstract: A rectangular plate made of a porous material is thernsubject of the work. Its mechanical properties varyrncontinuously on the thickness of a plate. Arnmathematical model of this plate, which bases onrnnonlinear displacement functions taking into accountrnshearing deformations, is presented. The assumedrndisplacement field, linear geometrical and physicalrnrelationships permit to describe the total potentialrnenergy of a plate. Using the principle ofrnstationarity of the total potential energy the set ofrnfive equilibrium equations for transversely and inrnplane loaded plates is obtained. The derivedrnequations are used for solving a problem of a bendingrnsimply supported plate loaded with transversernpressure. Moreover, the critical load of a bi-axiallyrnin-plane compressed plate is found. In both casesrninfluence of parameters on obtained solutions such asrna porosity coefficient or thickness ratio isrnanalysed. In order to compare analytical results arnfinite element model of a porous plate is built usingrnsystem ANSYS. Obtained numerical results are inrnagreement with analytical ones.
TL;DR: In this paper, the elastic buckling analysis of stiffened and un-stiffened corrugated plates via a mesh-free Galerkin method based on the first-order shear deformation theory (FSDT) is presented.
Abstract: This paper deals with elastic buckling analysis of stiffened and un-stiffened corrugated plates via a mesh-free Galerkin method based on the first-order shear deformation theory (FSDT). The corrugated plates are approximated by orthotropic plates of uniform thickness that have different elastic properties along the two perpendicular directions of the plates. The key to the approximation is that the equivalaent elastic properties of the orthotropic plates are derived by applying constant curvature conditions to the corrugated sheet. The stiffened corrugated plates are analyzed as stiffened orthotropic plates. The stiffeners are modelled as beams. The stiffness matrix of the stiffened corrugated plate is obtained by superimposing the strain energy of the equivalent orthotropic plate and the beams after implementing the displacement compatibility conditions between the plate and the beams. The mesh free characteristic of the proposed method guarantee that the stiffeners can be placed anywhere on the plate, and that remeshing is avoided when the stiffener positions change. A few selected examples are studied to demonstrate the accuracy and convergence of the proposed method. The results obtained for these examples, when possible, are compared with the ANSYS solutions or other available solutions in literature. Good agreement is evident for all cases. Some new results for both trapezoidally and sinusoidally corrugated plates are then reported.
TL;DR: In this paper, the authors presented accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions The thickness variation in one or two directions of the plate is taken in polynomial form.
TL;DR: In this paper, a mesh-free computational method is proposed to solve the Kirchhoff plate problems of various geometries, where the deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique.
Abstract: A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: Zhang et al. as discussed by the authors presented an alternative method for the derivation of exact solutions of a simply supported rectangular functionally graded piezoelectric plate or laminate, which can be expressed in more concise and easier handling forms comparing to the approach used in literature for the problems.
TL;DR: In this article, a spectral plate element (SPE) is developed to analyze wave propagation in anisotropic laminated composite media, which is based on first-order laminated plate theory, which takes shear deformation into consideration.
Abstract: A new spectral plate element (SPE) is developed to analyze wave propagation in anisotropic laminated composite media. The element is based on the first-order laminated plate theory, which takes shear deformation into consideration. The element is formulated using the recently developed methodology of spectral finite element formulation based on the solution of a polynomial eigenvalue problem. By virtue of its frequency-wave number domain formulation, single element is sufficient to model large structures, where conventional finite element method will incur heavy cost of computation. The variation of the wave numbers with frequency is shown, which illustrates the inhomogeneous nature of the wave. The element is used to demonstrate the nature of the wave propagating in laminated composite due to mechanical impact and the effect of shear deformation on the mechanical response is demonstrated. The element is also upgraded to an active spectral plate clement for modeling open and closed loop vibration control of plate structures. Further, delamination is introduced in the SPE and scattered wave is captured for both broadband and modulated pulse loading.
TL;DR: In this paper, a computational analysis of an elastic plate dropped against regular long waves is presented within the linear potential-flow theory and the contributions of kinetic and potential energies to the total energy are analyzed.
Abstract: A computational analysis of an elastic plate dropped against regular long waves is presented. The problem is considered within the linear potential-flow theory. The liquid flow is two-dimensional and the plate is modelled by an Euler beam. The analysis is based on the normal-mode method with hydroelastic behavior of the plate being of main interest. Different impact conditions are considered to study the dependence of the total energy of the plate–liquid system on impact geometry and plate properties. The contributions of kinetic and potential energies to the total energy are analyzed. It is shown that the kinetic part of the system energy is small at the instant of time when bending stresses in the beam approach their maximum values. Estimations of both the total energy and the maximum of bending stresses are presented. Most of the calculations are performed for the conditions of experiments carried out in MARINTEK. A range of the problem parameters is also considered, to reveal peculiarities of the unsteady interaction between a falling elastic plate and surface waves.
TL;DR: In this paper, an experimental delta-wing/store model with freeplay has been designed and tested in the Duke wind tunnel and the wing structure is modeled theoretically using von Karman plate theory that accounts for geometric strain displacement nonlinearities in the plate wing structure.
Abstract: An experimental delta-wing/store model with freeplay has been designed and tested in the Duke wind tunnel. The wing structure is modeled theoretically using von Karman plate theory that accounts for geometric strain- displacement nonlinearities in the plate wing structure. A component modal analysis is used to derive the full structural equations of motion for the wing/store system. A linear three-dimensional time-domain vortex lattice aerodynamic model including a reduced-order model aerodynamic technique and a slender-body aerodynamic theory for the store are also used to investigate the nonlinear aeroelastic system. The effects of the freeplay gap, the span location of the store, and the initial conditions on the limit-cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for the smaller LCO amplitudes, that is, for flow velocities slightly higher than the flutter velocity, but are not good for the larger LCO amplitudes, that is, higher flow velocities. The theoretical model needs to be improved to determine LCO response for larger-amplitude motions.
TL;DR: In this paper, a set of characteristic equations for determining the structure and orders of singularities of the stress resultants in the vicinity of the vertex is developed. But the singularities in angular corners of plates of arbitrary thickness with various boundary conditions subjected to in-plane loading are studied within first-order plate theory.
TL;DR: In this article, free axisymmetric vibrations of non-homogeneous isotropic circular plates of nonlinear thickness variation have been analyzed on the basis of classical plate theory employing the DQ method.
TL;DR: In this paper, a modal solution to the coupled buckling problem within the first-order approximation of Koiter's asymptotic theory, using the transition matrix method, has been presented.
TL;DR: In this paper, a generalized layer-wise stochastic finite element formulation is developed for the buckling analysis of both homogeneous and laminated plates with random material properties, and the effect of variation in these stresses on the mean and coefficient of variation of buckling strength is studied.