Classical Thin Plate Theory Research Articles

Sound radiation from shear deformable stiffened laminated plates

Sound radiation from shear deformable stiffened laminated plates is studied theoretically. The equations of motion for the composite laminated plate are derived on the basis of the first-order shear deformation plate theory. Two sets of parallel stiffeners interact with the laminated plate only through the normal line forces. By using the Fourier wavenumber transform and the stationary phase method, the far-field sound pressure is described analytically. Sound pressure given by the first-order shear deformation plate theory and the classical thin plate theory is compared, and the differences of sound pressure are shown in the high frequency range for an isotropic plate. Sound pressure and the transverse displacement spectra are presented to illustrate the effects of force location, stiffeners and angle-ply layers. Sound radiation from symmetric and antisymmetric composite plates with multiple loadings is also investigated.

Three-dimensional vibrations of annular thick plates with linearly varying thickness

The free vibration of annular thick plates with linearly varying thickness along the radial direction is studied, based on the linear, small strain, three-dimensional (3-D) elasticity theory. Various boundary conditions, symmetrically and asymmetrically linear variations of upper and lower surfaces are considered in the analysis. The well-known Ritz method is used to derive the eigen-value equation. The trigonometric functions in the circumferential direction, the Chebyshev polynomials in the thickness direction, and the Chebyshev polynomials multiplied by the boundary functions in the radial direction are chosen as the trial functions. The present analysis includes full vibration modes, e.g., flexural, thickness-shear, extensive, and torsional. The first eight frequency parameters accurate to at least four significant figures for five vibration categories are obtained. Comparisons of present results for plates having symmetrically linearly varying thickness are made with others based on 2-D classical thin plate theory, 2-D moderate thickness plate theory, and 3-D elasticity theory. The first 35 natural frequencies for plates with asymmetrically linearly varying thickness are compared to the finite element solutions; excellent agreement has been achieved. The asymmetry effect of upper and lower surface variations on the frequency parameters of annular plates is discussed in detail. The first four modes of axisymmetric vibration for completely free circular plates with symmetrically and asymmetrically linearly varying thickness are plotted. The present results for 3-D vibration of annular plates with linearly varying thickness can be taken as benchmark data for validating results from various plate theories and numerical methods.

Stability plate with longitudinal constructive discontinuity

The influence of longitudinal constructive discontinuity on the stability of the plate in the domain of elastic stability is solved based on the classical thin plate theory. The constructive discontinuities divide the plate into fields of different thickness. The plate has two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The Levy method is used for the solution of the problem of stability, with the aim of developing an analytical approach when researching the stability of plates with longitudinal constructive discontinuities and also with the aim of obtaining exact solutions for plates with non-uniform thickness. The exact solutions for stability presented herein are very valuable as they may serve as benchmark results for researches in this area.

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.

Elasto-Electrostatic Analysis of Circular Microplates Used in Capacitive Micromachined Ultrasonic Transducers

The active structural component of a capacitive micromachined ultrasonic transducer (CMUT) is the top plate which vibrates under the influence of a time-varying electrostatic force thereby producing ultrasound waves of the desired frequency in the surrounding medium. Analysis of MEMS devices which rely on electrostatic actuation is complicated due to the fact that the structural deformations alter the electrostatic forces, which redistribute and modify the applied loads. Hence, it becomes imperative to consider the electrostatics-structure coupling aspect in the design of these devices. This paper presents an approximate analytical solution for the static deflection of a thin, clamped circular plate caused by electrostatic forces which are inherently nonlinear. Traditionally, finite element simulations using some commercial software such as ANSYS are employed to determine the structural deflections caused by electrostatic forces. Since the structural deformation alters the electrostatic field, a coupled-field simulation is required wherein the electrostatic mesh is continuously updated to coincide with the deflection of the structure. Such simulations are extremely time consuming, in addition to being nontransparent and somewhat hard to implement. We employ the classical thin-plate theory which is adequate when the ratio of the diameter to thickness of the plate is very large, a situation commonly prevalent in many MEMS devices, especially the CMUTs. We solve the thin-plate electrostatic-elastic equation using the Galerkin-weighted residual technique, under the assumption that the deflections are small in comparison to the thickness of the plate. The evaluation of the electrostatic force between the two plates is simplified due to the fact that the electrostatic gap is much smaller than the lateral dimensions of the device. The results obtained are compared to those found from ANSYS simulations and an excellent agreement is observed between the two. The pull-in voltage predicted by our model is close to the value predicted by ANSYS simulations.

Refined Buckling Analysis of Rectangular Plates Under Uniaxial and Biaxial Compression

In the traditional buckling analysis of rectangular plates the classical thin plate theory is generally applied, so neglecting the plating shear deformation. It seems quite clear that this method is not totally appropriate for the analysis of thick plates, so that in the following the two variable refined plate theory proposed by Shimpi (2006), that permits to take into account the transverse shear effects, is applied for the buckling analysis of simply supported isotropic rectangular plates, compressed in one and two orthogonal directions. The relevant results are compared with the classical ones and, for rectangular plates under uniaxial compression, a new direct expression, similar to the classical Bryan’s formula, is proposed for the Euler buckling stress. As the buckling analysis is a widely diffused topic for a variety of structures, such as ship ones, some applications for plates uniformly compressed in one and two orthogonal directions are presented and the relevant theoretical results are compared with those ones obtained by a FEM analysis, carried out by ANSYS, to show the feasibility of the presented method. Keywords—Buckling analysis, Thick plates, Biaxial stresses

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.

Vibration of isotropic and composite plates using computed shape function and its application to elastic support optimization

Vibration of plates with various boundary and internal support conditions is analyzed, based on classical thin-plate theory and the Rayleigh–Ritz approach. To satisfy the support conditions, a new set of admissible functions, namely the computed shape functions, is applied to each of the two orthogonal in-plane directions. Similar to conventional finite element shape functions, parameters associated with each term of the proposed functions represent the actual displacements of the plates, thus making the method easily applicable to a wide range of support conditions, including continuous or partial edge supports and discrete internal supports. The method can also be applied to plates consisting of rectangular segments, like an L-shape plate, which sub-domains can be formulated using the computed shape functions and subsequently assembled in the usual finite element manner. Unlike many other admissible functions proposed in the literature, however, the computed shape functions presented herein are C 1—continuous and involve no complicated mathematical functions; they can be easily computed a priori by means of a continuous beam computer program and only the conventional third-order beam shape functions are involved in subsequent formulation. In all the examples given herein, only a few terms of these functions are sufficient to obtain accurate frequencies, thus demonstrating its computational effectiveness and accuracy. The method is further extended to the study of optimal location and stiffness of discrete elastic supports for maximizing the fundamental frequency of plates. Unlike rigid point supports with infinite stiffness, which optimal locations have been studied by many researchers, only discrete supports with a finite stiffness is considered in this paper. The optimal location and stiffness of discrete supports are determined for isotropic plates and laminated plates with various stacking sequences, which results are presented for the first time in literature.

A Novel Stress Characterization Technique for the Development of Low-Stress Ohmic Contacts to HgCdTe

HgCdTe material intended for long-wavelength infrared detection is particularly susceptible to damage from stress. As a result, an ideal ohmic contact needs to have good adhesion and low specific contact resistance. The contact should act as a diffusion barrier and induce the least amount of stress in the underlying material. In this paper we present a set of stress measurements from different ohmic contact materials deposited on short- and long-wavelength HgCdTe films grown by liquid-phase epitaxy (LPE). Using a new experimental technique we remove the substrate and measure the stress induced on single- and multilayered HgCdTe cantilevers. To interpret our results, we develop a theoretical model that describes the physics of elastic deformation in HgCdTe layers. Our model is based on classical thin-plate bending theory and explicitly takes into account the realistic boundary conditions that are present in the experimental setup by using a variational approach.

Free vibration and forced harmonic response of an electrorheological fluid-filled sandwichplate

A dynamic model for the electric field-dependent steady-state vibrational responseof a rectangular sandwich plate with a tunable electrorheological fluid (ERF)interlayer, subjected to a general harmonic transverse excitation, is developed.Hamilton’s principle and the classical thin plate theory are applied to derivea set of fully coupled dynamic equations of motion along with the associatedgeneral boundary conditions. Assuming simply-supported edge conditions, thedisplacement components of the ERF-based sandwich plate are postulated by meansof generalized double Fourier series with frequency-dependent coefficients. Thenatural frequencies and modal loss factors are subsequently determined by solving acomplex eigenvalue problem. Analytical solutions are also obtained for the forcedvibration characteristics of the adaptive structure under different external transverseexcitations of varying frequency (0–300 Hz) and applied electric field strength (0–3.5 kV mm−1). Primary attention is focused on the effects of electric field magnitude, geometric aspectratio, loading type, and ER core layer thickness on the dynamic characteristics of thesandwich plate. In addition, an effort is made to find the optimal electric field which yieldsminimized vibration amplitude for each excitation frequency. Limiting cases are consideredand good agreements with the numerical solutions available in the literature areobtained.

Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies

A general, global theory is developed for nano-scaled functionally graded films considering surface effects. In addition to the Kirchhoff hypothesis of the classical thin plate theory, the surface layers of the film are modeled by the continuum theory of surface elasticity. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Unlike the classical plate theory, the bulk transverse normal stress is preserved here. By incorporating the surface energies into the principle of minimum potential energy, a series of non-classical governing differential equations which include intrinsic length scales are derived. To illustrate application of the theory, a simply supported nano-scaled film in cylindrical bending is investigated. Numerical examples are presented to clarify the effects of surface energies on the bending behavior of FGM films, whose effective elastic moduli are predicted using the Mori–Tanaka method. Finally, the nature of intrinsic length scales, and the effects of gradient index and aspect ratio on the displacements are discussed.

Multipoles in an elastic multilayer structure in bending and their applications (continual approximation)

We consider the equilibrium of a multilayer structure in bending. This structure is assumed to be formed by plane homogeneous and isotropic layers. The strain of each of the layers is described by the classical small bending theory of thin plates with regard to the fact that the layer is acted upon by the neighboring layers, which contact it either directly or through sufficiently compliant interlayers whose flexural rigidity can be neglected. The characteristic scales of variation in all the fields under study treated as functions of coordinates are assumed to exceed the layer thicknesses so much as to justify the description of the structure deformation in the continual approximation. We assume that the layer deflections are completely determined by the transverse loads and are not affected by tangential stresses. We consider multipoles in this structure. Attention is mainly paid to the multipole asymptotics and their applications to the study of solutions far from the regions where the action exerted on structure are mainly concentrated. We present the results of field computations with the use of these asymptotics and of the exact solutions of the corresponding problems for the plane and spatial cases. These results are used to discuss the possibilities of applying the multipole asymptotics for solving the problems approximately.

A theoretical analysis of FGM thin plates based on physical neutral surface

This paper presents a theoretical analysis to the FGM (functionally graded materials) thin plates based on the physical neutral surface. Under the assumption of changeable parameters such as the Young’s modulus E and the mass density ρ along the thickness of the plate, the physical neutral surface of a FGM plate is determined by the theory of thin plate when the Poisson’s ratio ν is assumed to be constant in whole plate. After that, the displacements in-surface (u0 and v0) can be neglected in small deflection problems, there is no stretching–bending coupling effect in constitutive equations of physical neutral surface thin plate theory in both small and large deflection problems, and governing equations and boundary conditions of physical neutral surface thin plate theory have the simple forms as those of classical thin plate theory about homogeneous isotropic materials, so the solution procedure of physical neutral surface thin plate theory is as easy as homogeneous isotropic plate. The validity of FGM thin theory based on physical neutral surface is verified by comparison with related literatures. Physical neutral surface thin plate theory has more merits in the engineering application, because it is easier and simpler than classical laminated plate theory based on geometric middle surface.

Free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports on the diagonals

This computational study reports the free vibration frequencies (corresponding to the first three symmetrical and antisymmetrical modes), and the minimum buckling load (in case of in-plane uniform pressure along the periphery) of point-supported super-elliptical plates of uniform thickness. The plate perimeter was defined by a super-elliptic function with a power corresponding to the shape ranging from an ellipse to a rectangle. The analysis was based on the classical theory of thin plates and the computations were carried out by the Ritz method. The geometrical boundary conditions were satisfied by the Lagrange multipliers. The results were compared with those of rectangular plates and good agreement was obtained.

Three-dimensional modeling of curved structures containing and/or submerged in fluid

The dynamic behavior of a 3D thin flexible structure in inviscid incompressible stationary fluid is studied numerically. A finite element is developed using a combination of classical thin plate theory and finite element analysis, in which the finite elements are rectangular four-noded flat shell with five degrees of freedom per node. The displacement functions are derived from Sanders’ thin shell equations. The velocity potential function and Bernoulli's equation for liquid yield an expression for fluid pressure as a function of nodal displacement of the element and inertial force of the quiescent fluid. An analytical integration of the fluid pressure over the element produces the virtual added-mass matrix of fluid. Calculations are presented to illustrate the dynamic behavior of a rectangular reservoir containing fluid as well as a completely submerged blade.

NONLINEAR ANALYSIS OF AXIALLY MOVING PLATES USING FEM

In the present study, a nonlinear finite element formulation is developed for analysis of axially moving two-dimensional materials, based on the classical thin plate theory. Using Green's strain definition, the membrane stresses variation due to transverse displacements is considered. Hamilton's principle is employed to obtain the secant stiffness matrix, the in-plane and out-of-plane gyroscopic matrices and the dynamic stability matrix due to centripetal acceleration for a traveling thin plate. In order to extract the numerical results, a p-version finite element is adopted by selecting only an quadrilateral super element with Lagrangian interpolation functions. With a few test cases, the reliability of the formulation and the solution procedure is shown.

Calculation of eigenfrequencies for rectangular free orthotropic plates – An overview

AbstractIn this paper, the free transverse vibrations of rectangular orthotropic plates with completely free boundary conditions are considered. Some methodologies for calculating the eigenfrequencies and eigenmodes, based on the classical thin plate theory are presented in a comparative overview. The advantages and disadvantages of these methodologies are discussed for an orthotropic plate application with specific material and geometric properties.

Vertical Vibration of a Flexible Plate with Rigid Core on Saturated Ground

In this paper, the vertical vibration of a flexible plate with rigid core resting on a semi-infinite saturated soil is studied analytically. The behavior of the soil is assumed to follow Biot's poroelastodynamic theory with compressible soil skeleton and pore water, and the response of the time-harmonic excited plate is governed by the classical thin-plate theory. By virtue of the Hankel transform technique, the fundamental solutions of the skeleton displacements, stresses, and pore pressure are derived, and a set of dual integral equations associated with the relaxed boundary and completely drained condition at the soil-foundation contact interface are also developed. These governing integral equations are further reduced to the standard Fredholm integral equations of the second kind and solved by numerical procedures. Comparison with existing solutions for a rigid permeable plate on saturated soil confirms the accuracy of the present solution. Selected numerical results are presented to show the influence of the permeability, the size of the rigid core, and the plate flexibility on the dynamic interaction between the elastic plate with rigid core and the underlying saturated soil.

Applicability of the Large-deflection Theory on Thick Plates

In recent years, composite materials including synthetic resins are widely used as structural materials for plates. For such structures, large-deflection is often allowed, so that large deformation analysis becomes necessary. To such an end, von Kármán large-deflection theory of plate bending has been mainly used. This theory is, however, based on essentially the same assumptions as those of classical Lagrange's thin plate theory for infinitesimal deformation, except that the magnitude of the deflection is permitted up to the order of the plate thickness. Thus the applicability of such a theory may be doubtful for thick plate. In this paper, we examine the applicability of the above theory to uniformly loaded clamped circular plates with various different thickness by comparing with appropriate numerical solutions. The numerical method enployed here is a three-dimensional finite element method based on the incremental finite-deformation theory formulated in convected coordinates. From the obtained numerical results, the applicability of the considered theory is confirmed if the magnitude of maximum deflection is within 4 times the plate thickness.

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.