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Showing papers on "Plate theory published in 1998"


Journal ArticleDOI
TL;DR: In this paper, the dynamic thermoelastic response of functionally graded cylinders and plates is studied, and a finite element model of the formulation is developed, where the heat conduction and the thermo-elastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading.
Abstract: The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.

1,196 citations


Journal ArticleDOI
TL;DR: In this paper, a method for calculating shear and normal stress concentration at the cutoff point of a composite plate is presented, which is based on linear elastic behavior of the materials.
Abstract: Epoxy-bonding a composite plate to the tension face is an effective technique for repair and retrofit of reinforced concrete beams. Experiments have indicated local failure of the concrete layer between the plate and longitudinal reinforcement in retrofitted beams. This mode of failure is caused by local stress concentration at the plate end as well as at the flexural cracks. This paper presents a method for calculating shear and normal stress concentration at the cutoff point of the plate. This method has been developed based on linear elastic behavior of the materials. The effect of the large flexural cracks along the beam has also been investigated. The model has been used to find the shear stress concentration at these cracks. The predicted results have been compared with both the finite element method and experimental results. The analytical models provide closed form solutions for calculating stresses at the plate ends that can easily be incorporated into design equations.

382 citations


Book
01 Jan 1998
TL;DR: In this article, the authors apply the Ritz method to the reduction of Mindlin theory to the Kirchhoff frequency relationship for a class of plates shear correction factor and apply it to annular and sectorial plates.
Abstract: Introduction: background of vibration plate vibration about this monograph. Mindlin plate theory and Ritz method: Mindlin plate theory energy functionals governing differential equations boundary conditions relations between Kirchhoff and Mindlin plates reduction of Mindlin theory to Kirchhoff frequency relationship for a class of plates shear correction factor Ritz method preliminary remarks application of Ritz method to Mindlin plates. Formulas in polar co-ordinates: energy functionals Eignevalue equation circular and annular plates sectorial and annular sectorial plates computer program software code -VPRITZP1 sample files for VPRITZP1 software code -VPRITZP2 sample files for VPRITZP2 benchmark checks annular plates sectorial plates annular sectorial plates. Formulation in rectangular co-ordinates: energy functionals Eigenvalue equation computer program software code - VPRITZE input file output file benchmark checks Isosceles triangular plates trapezoidal plates elliptical plates. Formulation in skew co-ordinates: skew co-ordinates transformation energy functionals Eigenvalue equation computer program software code -VPRITZSK sample files benchmark checks. Plates with complicating effects: initial inplane stress elastic foundations stiffeners nonuniform thickness line/curve/loop internal supports point supports mixed boundary conditions re-entrant corners perforated plates sandwich construction. Appendix: Gaussian quadrature subroutines subrutines for mathematical operations on polynomials.

172 citations


Book
22 Dec 1998
TL;DR: In this article, the authors proposed a structural topology optimization method for media with a periodic structure based on the homogenization of the plate model microstructure, and applied it to a cellular body with rectangular holes.
Abstract: Preface vii.- Acknowledgements xi.- Table of Contents xiii.- Notation xvii.- 1. Introduction.- 1.1 Overview.- 1.2 Mathematical description of optimization problem.- 1.3 Types of structural optimization.- 1.4 Aspects of topology optimization.- 1.5 Layout of the book.- References.- I: Homogenization.- 2. Homogenization Theory for Media with a Periodic Structure.- 2.1 Introduction.- 2.2 Periodicity and asymptotic expansion.- 2.3 One dimensional elasticity problem.- 2.4 General boundary value problem.- 2.5 Elasticity problem in cellular bodies.- References.- 3. Solution of Homogenization Equations for Topology Optimization.- 3.1 Introduction.- 3.2 Material models.- 3.2.1 Rectangular microscale voids.- 3.2.2 Ranked layered material cells.- 3.2.3 Artificial materials.- 3.3 Analytical Solution of the homogenization equation for rank laminate composites.- 3.3.1 Rank-1 materials.- 3.3.2 Rank-2 materials.- 3.3.3 Bi-material rank-2 composites.- 3.4 Numerical Solution of the homogenization equation for a cellular body with rectangular holes.- 3.4.1 Finite element formulation.- 3.4.2 Derivation of the boundary conditions from periodicity.- 3.4.3 Examples.- 3.4.4 Homogenization constitutive matrix for Square microcells with rectangular voids.- 3.4.5 Least squares smoothing.- References.- II: Topology Optimization.- 4. Structural Topology Optimization using Optimality Critieria Methods.- 4.1 Introduction.- 4.2 Kuhn-Tucker condition.- 4.3 Analytical optimality criteria.- 4.3.1 An illustrative example of variational analysis.- 4.3.2 An illustrative example of derivation of optimality criteria.- 4.4 Mathematical model for the topological structural optimization.- 4.5 Optimality criteria for the topological structural optimization.- 4.5.1 Optimality conditions.- 4.5.2 Updating scheme.- 4.5.3 A modified resizing scheme.- 4.6 Optimal Orientation.- 4.7 Algorithm.- 4.8 Examples.- References.- 5. Experiences in Topology Optimization of Plane Stress Problems.- 5.1 Introduction.- 5.2 Effect of material model.- 5.2.1 Material model with rectangular holes.- 5.2.2 Artificial material model.- 5.2.3 Rank-2 material model.- 5.3 Effect of resizing scheme.- 5.4 Effect of the orientation variable.- 5.5 Effect of finite element discretization.- 5.5.1 Continuation method.- 5.5.2 Unstructured mesh.- 5.7 Effect of material volume.- 5.8 Effect of resizing parameters.- 5.9 Examples.- 5.9.1 Bridge with support layout 1.- 5.9.2 Bridge with support layout 2.- 5.9.3 Bracket with a hole.- 5.9.4 Shear wall with openings.- References.- 6. Topological layout and Reinforcement Optimization of Plate Structures.- 6.1 Introduction.- 6.2 Selection of plate base cell model.- 6.3 A brief review of Mindlin-Reissner plate theory.- 6.4 Homogenization of the plate model microstructure.- 6.5 Optimization problem.- 6.6 The finite element method.- 6.7 Optimal rotation.- 6.8 Examples.- 6.8.1 Simple supported Square plate with a central point load.- 6.8.2 Simple supported Square plate subject to a uniform load.- 6.8.3 Square plate subject to four point loads.- 6.8.4 Square slab with a circular holes.- 6.8.5 Fiat slab of a multi-span floor.- References.- III: Other Methods and Integrated Structural Optimization.- 7. Alternative Approaches to Structural Topology Optimization.- 7.1 Introduction.- 7.2 Simulation of functional adaptation of bone mineralization.- 7.2.1 A remodelling scheme based on effective strain energy density.- 7.2.2 A scheme based on effective stresses.- 7.3 Evolutionary fully stressed design method.- References.- 8. Integrated Structural Optimization.- 8.1 Introduction.- 8.2 Overview of integrated structural optimization.- 8.3 Topology optimization module.- 8.3.1 Ground structure method.- 8.3.2 Bubble method.- 8.4 Image processing module.- 8.4.1 Elimination of mesh dependency and checkerboard Problems using noise cleaning techniques.- 8.5 Shape optimization module.- 8.5.1 Boundary Variation method.- 8.5.2 Adaptive growth method.- 8.6 Integrated adaptive topology and shape optimization.- 8.7 Final thoughts.- References.- Appendix A.- Appendix B.- Appendix C.- Appendix D HOMOG Manual.- Appendix E PLATO Manual.- Author Index.

116 citations


Journal ArticleDOI
Jae-Hung Han1, In Lee1
TL;DR: In this paper, a refined analysis of composite plates with distributed piezoelectric actuators for vibration control has been performed, where the in-plane displacements through the thickness have been modeled using the layerwise theory.
Abstract: In order to evaluate closed loop performances of the composite plates with distributed piezoelectric actuators, it is essential to obtain more exact system parameters such as natural modes, damping ratios, and modal actuation forces. In this paper a refined analysis of composite plates with distributed piezoelectric actuators for vibration control has been performed. The in-plane displacements through the thickness have been modeled using the layerwise theory. This layerwise model can describe more refined strain distributions and has the capability of more realistic modeling of boundary conditions. The finite element method based on the developed mechanics has been formulated. The constitutive equations for piezoelectric materials have been used to determine piezoelectric actuation forces, and the modal strain energy method has been applied to analyze the damping capacity of the structures. Through the comparison of present results with those available, the accuracy of the present method was verified. The closed loop performances have been evaluated using the simple control algorithms. Through the comparison of present results with those based on shear deformation plate theory, it is concluded that the developed model can describe more realistic smart composite plates with distributed piezoelectric actuators.

100 citations


Journal ArticleDOI
TL;DR: An approximate parametric variational principle for Reissner--Mindlin elements as the plate thickness approaches zero is formulated and proved, which makes the results applicable to a large class of nonlocking elements in everyday engineering use.
Abstract: We present a new Neumann--Neumann-type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier methods of the authors; this improves parallel scalability. A new abstract framework for Neumann--Neumann preconditioners is used to prove almost optimal convergence properties of the method. The convergence estimates are independent of the number of subdomains, coefficient jumps between subdomains, and depend only polylogarithmically on the number of elements per subdomain. We formulate and prove an approximate parametric variational principle for Reissner--Mindlin elements as the plate thickness approaches zero, which makes the results applicable to a large class of nonlocking elements in everyday engineering use. The theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice.

91 citations


Journal ArticleDOI
TL;DR: A unified third-order laminate plate theory that contains classical, first-order and thirdorder theories as special cases is presented in this paper, where analytical solutions using the Navier and Levy solution procedures are presented.
Abstract: A unified third-order laminate plate theory that contains classical, first-order and third-order theories as special cases is presented. Analytical solutions using the Navier and Levy solution procedures are presented. The Navier solutions are limited to simply supported rectangular plates while the Levy solutions are restricted to rectangular plates with two parallel edges simply supported and other two edges having arbitrary combination of simply supported, clamped, and free boundary conditions. Numerical results of bending and vibration for a number of problems are discussed in the second part of the paper.

82 citations


Journal ArticleDOI
TL;DR: In this article, a piecewise cubic spline interpolation scheme has been used to represent the basic material properties along the lamina material axes and a laminated plate theory that includes an iterative incremental constitutive law to account for the non-linear behavior of the lamine is combined with a strain-energy based failure criterion for orthotropic materials.

76 citations


Journal ArticleDOI
TL;DR: In this article, a high-order theory for bending of structural sandwich plates with inserts and other hard points is introduced, which accounts for the transverse flexibility of the core material, including separate descriptions of the face sheets and the core materials as well as general specification of loads and boundary conditions.
Abstract: Sandwich structures are very susceptible to failure due to local stress concentrations induced in areas of load introduction, supports, geometrical and material discontinuities. These local stress concentrations are caused by localised bending effects, where the individual face sheets tend to bend about their own middle surface rather than about the middle surface of the sandwich. This paper deals with such local effects seen around inserts in structural sandwich plates. A high-order theory for bending of sandwich plates, developed and adapted especially for the purpose of studying sandwich plates with inserts and other “hard points”, is introduced. The theory, which accounts for the transverse flexibility of the core material, includes separate descriptions of the face sheets and the core materials as well as general specification of loads and boundary conditions. The theory is formulated in terms of first-order partial differential equations, which are solved numerically using the “multi-segment method of integration”. Examples involving sandwich plates with “through-the-thickness” inserts subjected to axisymmetric and non-axisymmetric external loading are presented. The paper is concluded by a discussion of design aspects.

67 citations


Journal ArticleDOI
TL;DR: In this paper, a plate consisting of one or more layers, resting on a general elastic foundation, is subjected to a nonlinear through-the-thickness temperature distribution, together with arbitrary wheel loads.
Abstract: This study considers a plate consisting of one or more layers, resting on a general elastic foundation, and subjected to a nonlinear through-the-thickness temperature distribution, together with arbitrary wheel loads. The resultant bending stress distribution in the plate is presented as the sum of the bending stresses due to the applied loading and to an equivalent linear temperature gradient, plus the pure thermal stresses due to the nonlinear part of temperature distribution. The resulting algorithm has been implemented into finite-element code ILSL2, capable of accommodating one- or two-layered slabs with a temperature distribution through the individual layers described by a linear function, a quadratic function, or explicitly provided by the user at a number of individual depths. Several examples are used to illustrate application of the method.

58 citations


Journal ArticleDOI
TL;DR: In this paper, the accuracy of p-version finite element formulations for Reissner-Mindlin plate problems was investigated and it was shown that significantly improved results can be obtained, if shear forces are computed from equilibrium equations instead.
Abstract: This paper addresses the question of accuracy of p-version finite element formulations for Reissner–Mindlin plate problems. Three model problems, a circular arc, a rhombic plate and a geometrically complex structure are investigated. Whereas displacements and bending moments turn out to be very accurate without any post-processing even for very coarse meshes, the quality of shear forces computed from constitutive equations is poor. It is shown that significantly improved results can be obtained, if shear forces are computed from equilibrium equations instead. A consistent computation of second derivatives of the shape functions is derived. © 1998 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, the optimum microstructures derived in explicit analytical form by Gibianski and Cherkaev are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading.
Abstract: In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium. The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem. Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.

Journal ArticleDOI
TL;DR: In this article, an analysis and experiments on quasi-unidirectional and angle-ply laminate end-notched flexure specimens are presented based on laminated beam theory incorporating first-order shear deformation theory.

Journal ArticleDOI
TL;DR: In this article, the basic mechanics and mechanism concerning compressive stability of composite laminates with multiple circular delaminations is studied analytically and experimentally, and a buckling equation is derived using the Rayleigh-Ritz method based on classical plate theory and solved as an eigenvalue problem.
Abstract: The basic mechanics and mechanism concerning compressive stability of composite laminates with multiple circular delaminations is studied analytically and experimentally. An experimental program, employing two types of quasi-isotropic laminates with a conventional and toughened epoxy resin, is used to evaluate the validity of the mechanistic model and further demonstrate the accuracy of finite element analysis conducted in the associated paper. Embedded delaminations are introduced at regular intervals in the thickness direction. The loading edges are fixed, and the side edges are simply supported. Although the buckling load does not depend on the matrix resin toughness, the strength is affected by the toughness. In the analysis, a buckling equation is derived using the Rayleigh-Ritz method, based on classical plate theory and solved as an eigenvalue problem. This method is chosen due to its efficiency. As the buckling mode of the lowest buckling load becomes physically admissible due to the assumptions of equally spaced delaminations and the classical plate theory, the contact problem does not need to be considered, that is, all of the delaminated portions deform by the same amount and do not overlap one another even without any constraints. The buckling loads analytically obtained agree well with experimental and finite element results described in the associated paper. The effects of size and number of circular delaminations on the buckling and failure load are also discussed in detail.

Journal ArticleDOI
TL;DR: An unsymmetric end-notched flexure test is described and its suitability for interfacial fracture toughness testing is evaluated in this article, where the test specimen consists of a beam-type geometry that is comprised of two materials, one "top" and one "bottom" with a crack at one end along the bimaterial interface.

Journal ArticleDOI
TL;DR: In this article, the authors present finite element models and numerical results for bending and natural vibration using the unified third-order plate theory developed in Part 1 of this paper, which contains the classical, first-order, and other thirdorder plate theories as special cases.
Abstract: Finite element models and numerical results are presented for bending and natural vibration using the unified third-order plate theory developed in Part 1 of this paper. The unified third-order theory contains the classical, first-order, and other third-order plate theories as special cases. Analytical solutions are developed using the Navier and Levy solution procedures (see Part 1 of the paper). Displacement finite element models of the unified third-order theory are developed herein. The finite elelment models are based on C-0 interpolation of the inplane displacements and rotation functions and C-1 interpolation of the transverse deflection. Numerical results of bending and natural vibration are presented to evaluate the accuracy of various plate theories.

Journal ArticleDOI
TL;DR: In this article, the differential relationship between the deflection of the classical Kirchhoff and third-order Reddy plate theories is developed, and it is used to determine the relationship between deflections of polygonal plates with simply supported boundary conditions.
Abstract: In this paper the differential relationship between the deflections of the classical Kirchhoff and third-order Reddy plate theories is developed, and it is used to determine the relationship between the deflections of polygonal plates with simply supported boundary conditions. As an example, the deflection of a simply supported rectangular plate using the third-order plate theory is obtained from the relationship developed herein. The relationship indicates that the third-order theory yields virtually the same solution as the first-order theory for simply supported rectangular plates.

Journal ArticleDOI
TL;DR: In this article, the effect of transverse shear strain has been considered by using a higher-order plate theory without the need for a shear correction factor, and a set of boundary-compliant admissible shape functions is developed to obtain solutions which are unattainable using the classical trigonometric admissible functions.

Journal ArticleDOI
TL;DR: In this paper, post-buckling analysis for a simply supported, shear-deformable composite laminated plate subjected to combined axial and thermal loads is presented, where the initial geometrical imperfection of the plate is taken into account.

Book
01 Jan 1998
TL;DR: The boundary element method for Reissner plates resting on elastic foundations has been used for plate bending analysis as discussed by the authors, and an analog equation solution has been proposed to solve the plate bending problem.
Abstract: The boundary element method for Reissner plates resting on elastic foundations Boundary element analysis of thick Reissner plates in bending Elastoplastic analysis of Reissner's plates using the boundary element method Nonlinear material analysis of Reissner's plates Stress resultant based integral equation formulation for plate bending analysis Fracture analysis of plate bending problems using the boundary element method Adaptive boundary element formulations for plate bending analysis Nonlinear analysis of plate bending by boundary element method Analysis of plates with variable thickness. An analog equation solution Stability.

Journal Article
TL;DR: In this article, a versatile spline finite strip method for analyzing the geometrically non-linear response of rectangular, composite laminated plates of arbitrary lay-up to progressive end shortening in their plane is presented.
Abstract: Description is given of a versatile spline finite strip method for analysing the geometrically non-linear response of rectangular, composite laminated plates of arbitrary lay-up to progressive end shortening in their plane. The plates are assumed to be thin, thus allowing the analysis to be based on the use of classical plate theory, and the non-linearity is introduced in the strain-displacement equations in the manner of the von Karman assumption. A number of finite strip models have been developed but attention is concentrated on a particular model whose displacement field uses cubic B-splines longitudinally and quadratic crosswise interpolation of the in-plane displacements. Description is given of the use of this model in applications involving plates which have simply supported ends and which either are made of homogeneous, isotropic material or of anistropic material or are laminates with unbalanced cross-ply or angle-ply lamination.

Journal ArticleDOI
TL;DR: In this paper, a closed-form analytical solution is presented for a three-layer pavement system, subjected to a periodic variation of either ambient air temperature or pavement surface temperature, coupled with a plate theory with Winkler foundation to calculate curling stresses and bending moments.
Abstract: Temperature is an important factor influencing the functioning of concrete pavements. In the past, analysis of temperature distribution in pavement has been done using numerical techniques such as finite difference or finite element method; both require significant computational efforts. In this paper, a closed-form analytical solution is presented for a three-layer pavement system, subjected to a periodic variation of either ambient air temperature or pavement surface temperature. The thermal analysis is coupled with a plate theory with Winkler foundation to allow for calculation of curling stresses and bending moments. The methods for characterizing periodic temperature variations are also described. The main findings from the numerical studies show that the temperature distribution with depth can be highly nonlinear, particularly when daily temperature variation is considered. Thus, the frequency of temperature variation, rather than the amplitude, has the most significant effect on the calculated temperature distribution with depth in the concrete pavement layer. A frequency of 2 pi rad/d or higher can be expected to cause nonlinear temperature distribution and require that calculation of curling stresses be based on a nonlinear distribution model.

Journal ArticleDOI
TL;DR: In this paper, the authors derived analytical solutions for the three-dimensional time-dependent buckling of a non-Newtonian viscous plate in a less viscous medium.
Abstract: We derive analytical solutions for the three-dimensional time-dependent buckling of a non-Newtonian viscous plate in a less viscous medium. For the plate we assume a power-law rheology. The principal, axes of the stretching D-ij in the homogeneously deformed ground state are parallel and orthogonal to the bounding surfaces of the plate in the flat state. In the model formulation the action of the less viscous medium is replaced by equivalent reaction forces. The reaction forces are assumed to be parallel to the normal vector of the deformed plate surfaces. As a consequence, the buckling process is driven by the differences between the in-plane stresses and out of plane stress, and not by the in-plane stresses alone as assumed in previous models. The governing differential equation is essentially an orthotropic plate equation for rate dependent material, under biaxial pre-stress, supported by a viscous medium. The differential problem is solved by means of Fourier transformation and largest growth coefficients and corresponding wavenumbers are evaluated. We discuss in detail fold evolutions for isotropic in-plane stretching (D-11 = D-22), uniaxial plane straining (D-22 = 0) and in-plane flattening (D-11 = -2D(22)). Three-dimensional plots illustrate the stages of fold evolution for random initial perturbations or initial embryonic folds with axes non-parallel to the maximum compression axis. For all situations, one dominant set of folds develops normal to D-11, although the dominant wavelength differs from the Biot dominant wavelength except when the plate has a purely Newtonian viscosity. However, in the direction parallel to D-22, there exist infinitely many modes in the vicinity of the dominant wavelength which grow only marginally slower than the one corresponding to the dominant wavelength. This means that, except for very special initial conditions, the appearance of a three-dimensional fold will always be governed by at least two wavelengths. The wavelength in the direction parallel to D-11 is the dominant wavelength, and the wavelength(s) in the direction parallel to D-22 is determined essentially by the statistics of the initial state. A comparable sensitivity to the initial geometry does not exist in the classic two-dimensional folding models. In conformity with tradition we have applied Kirchhoff's hypothesis to constrain the cross-sectional rotations of the plate. We investigate the validity of this hypothesis within the framework of Reissner's plate theory. We also include a discussion of the effects of adding elasticity into the constitutive relations and show that there exist critical ratios of the relaxation times of the plate and the embedding medium for which two dominant wavelengths develop, one at ca. 2.5 of the classical Biot dominant wavelength and the other at ca. 0.45 of this wavelength. We propose that herein lies the origin of parasitic folds well known in natural examples.

Journal ArticleDOI
TL;DR: In this paper, a least squares method, using the minus one norm developed by Bramble, Lazarov, and Pasciak, is introduced to approximate the solution of the Reissner-Mindlin plate problem with small parameter t, the thickness of the plate.
Abstract: In this paper a least squares method, using the minus one norm developed by Bramble, Lazarov, and Pasciak, is introduced to approximate the solution of the Reissner-Mindlin plate problem with small parameter t, the thickness of the plate. The reformulation of Brezzi and Fortin is employed to prevent locking. Taking advantage of the least squares approach, we use only continuous finite elements for all the unknowns. In particular, we may use continuous linear finite elements. The difficulty of satisfying the inf-sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form, which is very cheap computationally. It is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter t. Apart from the simplicity of the elements, the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system. For each diagonal block, one only needs a preconditioner for a second order elliptic problem.

Journal ArticleDOI
TL;DR: In this paper, an approximate solution of the problem has been obtained by the Ritz method, which employs functions based upon the static deflection of polar orthotropic plates and has a faster rate of convergence as compared to the polynomial co-ordinate functions.

Journal ArticleDOI
TL;DR: In this paper, the Ritz method is applied in a 3D analysis to obtain accurate frequencies for annular plates having any combination of free or fixed boundaries at both inner and outer edges.

Journal ArticleDOI
J. C. Klug1, C.T. Sun1
TL;DR: In this paper, a large deflection theory is used to study the effect of different variables including the thermal residual stresses and host and repair plate thicknesses on a metallic panel under in-plane loading.

Journal ArticleDOI
TL;DR: Based on first-order plate theory, a hybrid stress bimodulus Mindlin plate element is developed in this article, where both the displacement and stress distributions of laminated plates are determined.

Journal ArticleDOI
TL;DR: In this paper, a thermal postbuckling analysis for a simply supported, moderately thick rectangular plate subjected to uniform or nonuniform tent-like temperature loading and resting on a softening nonlinear elastic foundation is presented.
Abstract: A thermal postbuckling analysis is presented for a simply supported, moderately thick rectangular plate subjected to uniform or nonuniform tent-like temperature loading and resting on a softening nonlinear elastic foundation The initial geometrical imperfection of the plate is taken into account The formulations are based on the Reissner-Mindlin plate theory considering the first-order shear-deformation effect, and including plate-foundation interaction and thermal effects The analysis uses a deflection-type perturbation technique to determine the thermal buckling loads and postbuckling equilibrium paths Numerical examples are presented that relate to the performances of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, thermal load ratio and initial geometrical imperfections are studied Typical results are presented in dimensionless graphical form and exhibit interesting imperfection sensitivity

Journal ArticleDOI
TL;DR: In this article, an approximate solution of the problem using functions based on static deflection of polar orthotropic circular plates was obtained by the Rayleigh-Ritz method using functions with a faster rate of convergence as compared to the polynomial co-ordinate functions.