Bending Of Rectangular Plates Research Articles

Bending of clamped orthotropic rectangular plates: a variational symbolic solution

The Galerkin method is herein applied to the classical bending problem of a uniformly-loaded orthotropic rectangular plate with clamped edges, a problem for which, to our knowledge, no exact analytical solution in its general form exists. The tedious and error-prone computations inherent in such an approach are facilitated through the use of a computer algebra system; and several solutions, based on different approximations for the infinite series representing the assumed deflection function for the plate, are worked out, thereby extending previous work in the literature. The accuracy and convergence of the present formulation are assessed on the basis of solutions corresponding to the special case of material isotropy; and one such existing, manually-derived, result is shown to be incorrect.

The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates

The Hadamard and SJT product of matrices are two types of special matrix product. The latter was first defined by Chen. In this study, they are applied to the differential quadrature (DQ) solution of geometrically nonlinear bending of isotropic and orthotropic rectangular plates. By using the Hadamard product, the nonlinear formulations are greatly simplified, while the SJT product approach minimizes the effort to evaluate the Jacobian derivative matrix in the Newton–Raphson method for solving the resultant nonlinear formulations. In addition, the coupled nonlinear formulations for the present problems can easily be decoupled by means of the Hadamard and SJT product. Therefore, the size of the simultaneous nonlinear algebraic equations is reduced by two-thirds and the computing effort and storage requirements are greatly alleviated. Two recent approaches applying the multiple boundary conditions are employed in the present DQ nonlinear computations. The solution accuracies are significantly improved in comparison to the previously given by Bert et al. The numerical results and detailed solution procedures are provided to demonstrate the superb efficiency, accuracy and simplicity of the new approaches in applying DQ method for nonlinear computations.

A three-dimensional solution for rectangular plate bending free of transversal normal stresses

A new Navier-type three-dimensionally exact solution for small deflections in bending of linear elastic isotropic homogeneous rectangular plates of constant thickness is presented. In this solution the transversal normal stresses identically vanish, and the loading is a volume force with a distribution across the plate thickness being a characteristic of the solution. The presented solution can be used for the evaluation of approximate plate theories and numerical solutions which comprise the assumption of transversal normal stresses vanishing in the whole plate. Copyright © 1999 John Wiley & Sons, Ltd.

VIBRATION CONTROL SIMULATION OF LAMINATED COMPOSITE PLATES WITH INTEGRATED PIEZOELECTRICS

A finite element formulation is presented to model the dynamic as well as static response of laminated composite plates containing integrated piezoelectric sensors and actuators subjected to both mechanical and electrical loadings. The formulation is based on the classical laminated plate theory and Hamilton's principle. In this formulation, the mass and stiffness of the piezo-layers have been taken into account. A four-node non-conforming rectangular plate bending element is implemented for the analysis. A simple negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through a closed control loop. The model is validated by comparison with existing results documented in the literature. Several numerical examples are presented. The influence of stacking sequence and position of sensors/actuators on the response of the plate is evaluated.

Experimental, numerical and analytical results for bending and buckling of rectangular orthotropic plates

Solid unstiffened, sandwich and hat-stiffened rectangular orthotropic fiber reinforced plastic (FRP) plates were tested for buckling by in-plane compression and for stresses and deflections under uniform out-of-plane pressure. The solid unstiffened and hat-stiffened plates were 154 × 77 cm (1 × w) (72 × 36 in), while the sandwich plates were 102 × 77 cm (1 × w) (48 × 36 in). Balsa core was used in the sandwich plates and in the hat-stiffeners. The two short edges of the unstiffened and sandwich plates were clamped, while the two long edges were simply supported. The two long edges of the hat-stiffened plates were free, while the short edges were clamped. The buckling load, as well as stresses and deflections from the tests, were then compared to those from finite element analysis (FEA) and analytic solutions. There was reasonably good agreement between FEA, analytic, and experimental buckling stresses for the unstiffened solid plates. There was reasonable agreement in buckling stresses between FEA and experimental results for the hat-stiffened plate. There was poor agreement between FEA, analytic, and experimental elastic buckling results for the sandwich plates because they failed in local buckling prior to global buckling. Under out-of-plane uniform pressure, FEA and analytic solutions of the stresses and deflections for the unstiffened solid plates agreed well with experimental results. There was poor agreement between FEA and experimental results for stresses and deflections of the hat-stiffened or sandwich plates. Experimental error could be traced, in part,to plate fabrication, the method of applying out-of-plane pressure, edge support, and instrumentation accuracy.

FAST CONVERGENT SERIES SOLUTIONS FOR THERMAL BENDING OF THIN RECTANGULAR PLATES

Thermally induced bending of thin rectangular plates with one clamped and three simply supported edges is studied in detail for the case of a spacewise constant thermal moment. Using this sample problem, it is demonstrated that classical series representations for thermally induced bending moments and shear forces may exhibit numerical instabilities, slow convergence, and divergence. Fast convergent solutions are developed by replacing hyperbolic functions in the classical series representations by means of exponential functions with a negative argument and by utilizing Kummer's transformation. Divergence is overcome using Cesaro's generalized C1-summation method. The presented series solutions are checked numerically via finite element computations. Symbolic computation is used to derive and to evaluate the series solutions and to derive limiting values at the plate corners. For practical use, tables and graphical representations of results are presented in the form ofCzerny's Tables for force-loaded rectangular plates.

Magnetoelastic bending and stability of soft ferromagnetic rectangular plates

In this paper, the phenomenon of magnetoelastic bending is theoretically simulated for soft ferromagnetic rectangular thin plates in applied magnetic fields. A numerical program of 3d FEM is established to capture the nonlinear coupling intercation between magnetic fields and bending deflection. After the nonlinear characteristic of the bending deflection and the magnetic (field) force is quantitatively displayed, we discuss the critical magnetic field and the effect of the incident angle of obliquely applied magnetic field on the critical field to the phenomenon of magnetoelastic instability.

On a method of solving physically nonlinear problems in the bending of thick rectangular plates

We study the three-dimensional boundary-value problem of the physically nonlinear theory of elasticity for bending of a homogeneous isotropic thick plate by transverse forces. We assume that the intensity of the load is such that the connection between the stresses and small strains within the elastic limit can be described by nonlinear relations in the form of H. Kauderer. We develop an approximate analytic method that makes it possible to reduce the original nonlinear boundary-value problem to a recursive sequence of corresponding linear boundary-value problems.

A New Rectangular Plate Element for Vibration Analysis of Laminated Composites

In this paper, a higher order rectangular plate bending element based on a Higher Order Shear Deformation Theory (HSDT) is developed. The element has 4 nodes and 20 degrees of freedom. The transverse displacement is interpolated by using an optimized interpolation function while the additional rotation degrees of freedom are approximated by linear Lagrange interpolation. The consistent element mass matrix is used. A damped element is introduced to the finite element model. The proposed FEM is used to calculate eigenfrequencies and modal damping of composite plates with various boundary conditions and different thicknesses. The results show that the present FEM gives excellent results when compared to other methods and experiment results, and is efficient and reliable for both thick and thin plates. The proposed finite element model does not lock in the thin plate situation and does not contain any spurious vibration mode, and converges rapidly. It will provide a good basis for the inverse analysis of vibration of a structure.

Infinite systems for a biharmonic problem in a rectangle

This paper addresses a general analytical method based upon Mellin's transforms technique of investigating asymptotic behaviour of infinite systems of linear algebraic equations occurring in some basic two–dimensional biharmonic problems in a rectangular domain. The object of this paper is to prove the advantages of the asymptotic analysis when studying the concrete problems of an equilibrium of an elastic rectangle, creeping flow of viscous fluid set up in a rectangular cavity by tangential velocities applied along its walls, and bending of a clamped thin rectangular plate by a normal load at one surface. The method is illustrated by several numerical examples; the rapidity of convergence and the accuracy of results are studied.

Analytic and finite element solutions for bending and buckling of orthotropic rectangular plates

This paper presents a critical review of analytic solutions for bending and buckling of flat, rectangular, orthotropic thin plates. Considered are plates with all edges simply supported, two edges simply supported and two edges clamped, and all edges clamped. An orthotropy resealing technique is employed to simplify the analysis. The material orthotropy is characterized by two non-dimensional parameters, λ = (D22/D11 and η = (D12 + 2D26)/√D11D22. When η ≈ 1, many solutions for orthotropic plates can be obtained directly from the corresponding isotropic results. Systematic comparisons with finite element solutions are made for the critical buckling load of a plate under in-plane compression, and for deflection and stresses in a plate under out-of-plane uniform pressure. It is found that for plates with all edges clamped, the analytic solution for critical buckling load is neither accurate nor conservative; a better solution needs to be developed for design purposes. The validity of the thin plate theory solutions over a range of plate thicknesses is also examined.

Steady-State Thermal Bending of Thick Rectangular Plates

An analysis of the steady-state thermal bending of a thick rectangular plate is carried out by a tenth-order theory of moderately thick plates. The boundary conditions for a temperature field are considered to be a partially prescribed constant temperature on the top face, zero temperature at the bottom face and two edges, and no flux at the two edges. The temperature field is rigorously analyzed by the theory of three-dimensional heat conduction. The mechanical boundary conditions of the plate are considered traction-free at the top and bottom faces, the two opposite edges are simply supported, and the other two edges are free. The displacement and stress field is analyzed exactly by a particular solution deduced from part of a three-dimensional elasticity solution and a homogeneous solution consisting of three solutions to plane stress, generalized plane stress, and antisymmetric stress. Numerical results for thermal stresses and displacements are presented. One of the normal stresses at the top face becomes discontinuous at the boundary of the prescribed surface temperature.

Bending and Vibration Analysis of Composite Rectangular Sandwich Plates. Effect of Transverse Shear Deformation on Face Layers.

The analysis of composite rectangular sandwich plates composed of an orthotropic core and composite laminated faces is presented for bending and free vibration. A method of solution based upon the Ritz method is used. A displacement field which includes transverse shear deformation of the face layer is applied to the honeycomb sandwich plate. The effect of transverse shear deformation on the center deflection and natural frequencies is studied, and the applicability of the displacement field is discussed. Also, the effects of fiber angle on the face layer, core thickness and transverse shear modulus of the core on natural frequencies of the honeycomb sandwich plate are discussed.

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example.

Reciprocal theorem method for solving the problems of bending of thick rectangular plates

In this paper, reciprocal theorem method (RTM) is generalized to solve the problems of bending of thick rectangular plates based on Reissner’s theory. First, the paper gives the basic solution of the bending of thick rectangular plates: and then the exact analytical solution of the bending of thick rectangular plate with three clamped edges and one free edge under uniformly distributed load is found by RTM: finally, we analyze numerical results of the solution.

The bending of thin rectangular plates with mixed supported segments of straight edges

In this paper, the exact analytical solution of the rectangular plate having simply supported segments mixed with free segments of straight edges are first given by means of the method of reciprocal theorem.

An improved in-plane thermoelastic theory for laminated composite plates

A new laminated plate thermoelastic theory based upon a new mixed variational principle proposed by Reissner [Int. J. Numer. Meth. Engng20, 1366–1368 (1984)] is developed to study the thermal stresses of laminated composite plates. Across each individual layer, piecewise linear continuous displacement and quadratic transverse shear stress distributions are assumed. The temperature variation through the thickness is considered to be linear. The theory is examined by applying it to the problem of rectangular plate bending. Some numerical results for thermal stresses and deflections are compared with those obtained using the high-order and the first-order zig-zag thermoelastic theories.

The derivation of exact static conditions at the corner points for the bending of thick rectangular plates

In this paper, exact static conditions at the corner points for the bending of thick rectangular plates are strictly derived from the theorem of minimum potential energy.

Numerical implementation of the spline-collocation method for bending of rectangular plates

Different approaches are considered to the application of the spline-collocation method to bending of rectangular plates. The effect of the choice of collocation nodes on the order of accuracy of the method is investigated. Numerical results obtained with different arrangements of collocation points are reported.

New rectangular plate bending elements and their performance

Three rectangular plate bending elements have been developed: E14A, E14B and E16. They have 14 and 16 degrees of freedom and are displacement-based Mindlin plate theory elements. These elements behave well for a wide range of aspect ratios. None of these elements show any sign of the ‘shear locking’ phenomenon for thin plates. They do not have any additional zero energy modes when an appropriate order of integration is applied. The convergence characteristics of E16 proved to be very good—better than E14A and E14B. An explanation of why some of these elements are better than the others is given.