Classical Plate Theory Research Articles

Three-dimensional analysis of composite FGM rectangular plates with in-plane heterogeneity

A three-dimensional elasticity solution is presented for unidirectionally reinforced composite plates with in-plane uniaxial grading using a series approach. Convergent results useful for future benchmarking studies are tabulated for two configurations. Displacement and stress variations are critically examined. Studies of thick and thin plates show that the benefit of grading depends significantly on the fiber orientation as well as the thickness ratio, and that this benefit cannot be correctly gauged using classical plate theory. Further, comparison of different configurations with the same total fiber content shows that a significant increase in stiffness may be achieved by a proper choice of fiber orientation and the degree of grading.

Implementation of the Nodal Discontinuous Galerkin Method for the Plate Vibration Problem using Linear Elasticity Equations

This work presents a numerical solution of the forced plate vibration problem using the nodal discontinuous Galerkin (DG) method. The plate is modelled as a three-dimensional domain, and its vibration is governed by the linear elasticity equations. The nodal DG method discretises the spatial domain and computes the spatial derivatives of the equations element-wise, while the time integration is conducted using the Runge-Kutta method. This method is in particular of interest as it is very favourable to carry out the computation by a parallel implementation. Several aspects regarding the numerical implementation such as the plate boundary conditions, the point force excitation, and the upwind numerical flux are presented. The numerical results are validated for rectangular concrete plates with different sets of boundary conditions and thicknesses, by a comparison with the exact mobilities that are derived from the classical plate theory (CPT) and the first order shear deformation theory (FSDT). The plate thickness is varied to understand its effect regarding the comparison with the CPT. An excellent agreement between the numerical solution and the FSDT was found. The agreement with the CPT occurs only at the first couple of resonance frequencies, and as the plate is getting thinner. Furthermore, the numerical example is extended to an L-shaped concrete plate. The mobility is then compared with the mobilities obtained by the CPT, FSDT, and linear elasticity equations.

Experimental Investigation of Nonlinear Vibration of a Thin Rectangular Plate

In this paper, the geometrical nonlinear vibrations of a rectangular plate have been investigated experimentally and numerically. The experiment was conducted on a thin rectangular plate. The plate was excited close to the first fundamental natural frequency. The time history of velocities of the central point has been measured by using a laser vibrometer. While the numerical investigation has been carried using the Finite Element Method (FEM), the numerical results are validated by analytical and experimental results. In order to develop and test the extraction procedure of the bifurcation plot of a dynamical system, a chaotic pendulum has been analyzed. Then, the same successful code has been used again for the experimental dynamics of the investigated plate. The plate has been forced with a sinusoidal input at a gradually stepped and increased amplitude. For every step, the phase portrait is determined, and then processed to extract the bifurcation map. The resulted map has shown successfully the linear range where the classical plate theory is adequate, and the boundary at which the transition to nonlinearity has occurred. The bifurcation has occurred when the lateral amplitude has reached 50% of the plate thickness.

Low-Stress State of Simply-Supported Anisotropic Composite Plates

The stress state in the neighborhood of the common edge of a simply-supported composite wedge-shaped plate under transverse load has been studied. The materials of the constituent plates are cylindrically orthotropic. The problem is solved using the classical theory of anisotropic plates. Equations are derived for a hypersurface which, in the space of physical and geometric parameters, is the boundary of the region of parameters for which the neighborhood of the common edge of the composite plate is in a low-stress state (the stresses at the points of the common edge of the composite plate are limited).

Vibration, Buckling and Deflection Analysis of Cracked Thin Magneto Electro Elastic Plate Under Thermal Environment

The Magneto-Electro-Elastic (MEE) material exhibits pyroelectric and pyromagnetic effects under thermal environment. The effects of such pyroelectric and pyromagnetic behavior on vibration, buckling and deflection analysis of partially cracked thin MEE plate is presented and discussed in this paper. The aim of the study is to develop an analytical model for the vibration and geometrically linear thermal buckling analysis of cracked MEE plate based on the classical plate theory (CPT). The line spring model (LSM) is modified for the crack terms to accommodate the effect of electric and magnetic field rigidities, whereas the effect of thermal environment is accommodated in the form of thermal moment and in-plane forces. A classical relation for thermal buckling phenomenon of cracked MEE plate is also proposed. The governing equation for cracked MEE plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack and temperature rise. The results evaluated for natural frequencies as affected by crack length, plate aspect ratio and critical buckling temperature are presented for first four modes of vibration. The obtained results reveal that the fundamental frequency of the cracked plate decreases with increase in temperature and crack length. Furthermore the variation of the critical buckling temperature with plate aspect ratio and crack length is also established for different modes of vibration.

An analytical method for predicting the net (thermal plus intrinsic) residual stresses in multilayered yttrium barium copper oxide coated conductors

In this paper, the net (thermal plus intrinsic) residual stresses in the films of American Superconductor (AMSC 344®) tape was analyzed via a displacement- energy model (DEM) solution approach. The DEM solution approach was developed to predict both the thermal mismatch and intrinsic film stresses in the coated conductor (CC) films thereby avoid the costly experiments usually required to capture the intrinsic film’s stress part in CC films. The analysis was based on the force and moment balances method, classical plate theory, and Hamaker method. On this model, the authors studied the generation of misfit strains in the CC films due to the thermal and free-vibration mismatch between the layers, and further the role of these actions in the residual stresses evolution in the CC films. The results showed that the thermal film stress in the yttrium barium copper oxide (YBCO) films calculated by the DEM solution approach was 460.80 MPa, whereas that calculated by the closed-form solution method was 455.30 MPa. Moreover, for the non-textured yttrium-stabilized zirconium (YSZ) films deposited on the Hastelloy C substrate, the calculated net residual stress by the DEM solution approach was 673 MPa, whereas the measured net residual stress by the x-ray Diffraction ( Method) was 598 ± 75 MPa. Therefore, the DEM solution approach successfully predicted the net residual stresses in the CC films analyzed because the calculated results by the DEM solution approach agree well with those by the closed-form solution and x-ray Diffraction ( methods).

Stability and dynamic analysis of partially cracked thin orthotropic microplates under thermal environment: an analytical approach

An analytical model is proposed to analyze the vibration and buckling problem of partially cracked thin orthotropic microplate in the presence of thermal environment. The differential governing equation for the cracked plate is derived using the classical plate theory in conjunction with the strain gradient theory of elasticity. The crack is modeled using appropriate crack compliance coefficients based on the simplified line spring model. The influence of thermal environment is incorporated in governing equation in form thermal moments and in-plane compressive forces. The governing equation for cracked plate has been solved analytically to get fundamental frequency and central deflection of plate. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The stability and dynamic characteristics of the cracked plate are studied considering various parameters such as crack length, plate thickness, change in temperature, and internal length scale of microstructure. It has been concluded that the frequency and deflection are affected by crack length, temperature, and internal length scale of microstructure. Furthermore, to study the buckling behavior of cracked plate, the classical relations for critical buckling load and critical buckling temperature is also proposed considering the effect of crack length, temperature, and internal length scale of microstructure.

Hydroelastic stability of partially filled coaxial cylindrical shells

The work investigates numerically the dynamic behavior of the system, which consists of horizontally oriented, elastic coaxial shells and a flowing compressible fluid partially or completely filling the annular gap between the shells. The solution of the problem is carried out in a three-dimensional formulation using the finite element method. The motion of the compressible non-viscous liquid is described by the wave equation, which together with the impermeability condition and the corresponding boundary conditions is transformed by the Bubnov–Galerkin method. The mathematical formulation of the problem of the dynamics of thin-walled constructions is based on the variational principle of virtual displacements. The simulation of the shell behavior is performed under the assumption that the curvilinear surface is accurately approximated by a set of flat segments, in which the strains are determined using the relations of the classical plate theory. The stability estimate is based on the results of computation and analysis of the complex eigenvalues of the coupled system of equations. The influence of the filling level and the size of the annular gap on the boundaries of hydroelastic stability of rigidly fixed coaxial shells is analyzed. It is shown that a decrease in the filling level leads to an increase in the stability boundaries.

Ritz method analysis of rectilinear orthotropic composite circular plates undergoing in-plane bending and torsional moments

In this article, the elastic problem of composite circular plates with rectilinear orthotropic characteristics, in case of in-plane bending and torsional moments applied, is resolved. According to classical lamination plate theory, the analysis makes use of the Ritz method along with the principle of virtual displacements, originally combined in order to obtain displacements components. The procedure represents an important step for the setting-up of a spot joint element able to simulate in FE analyses the structural behavior of point junctions applied to composite components. The accuracy of the procedure has been demonstrated comparing analytical results to FE reference analyses.

Thermal effect on radially symmetric vibrations of temperature-dependent FGM circular plates with nonlinear thickness variation

Thermo-elastic analysis for the free axisymmetric vibration of functionally graded circular plates with quadratic thickness variation along radial direction has been presented on the basis of classical theory of plates. Following a power law model, the plate is assumed to be graded in thickness direction and mechanical properties of the material are temperature-dependent. The plate is subjected to a non-linear temperature distribution in thickness direction while the thermal environment over the top and bottom surfaces is uniform. The equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model have been derived using Hamilton’s principle. Employing differential quadrature method, the numerical values of thermal displacements and frequencies for clamped and simply supported plates have been computed. The influence of thickness parameters, material graded index and temperature difference on the vibration characteristics for the first three modes of vibration has been analyzed. The benchmark results for uniform and linear temperature distributions have been computed. A study with the plate material having temperature-independent mechanical properties has also been performed. Because, functionally graded materials inherently withstand high temperature gradients due to low thermal conductivity, core ductility, low thermal expansion coefficient, etc, the performance of results can help for optimal designing of non-uniform circular plates by controlling one or both the taper parameters. In special cases, the results have been compared with those obtained by other methods.

Free vibration analysis of polyethylene/CNT plates

In this work, we analyzed the free vibration of single-walled carbon nanotubes (SWCNTs)-reinforced composite plates with carbon nanotubes (CNTs) embedded in amorphous polyethylene. Here, the governing differential equations of simply supported and clamped boundary conditions were found using the generalized differential quadrature (GDQ) method. We used the rules of mixture according to different plate models including first-order shear deformation theory (FSDT), classical plate theory (CLPT), and higher-order shear deformation theory (HSDT) to find the fundamental frequencies of nanocomposite plates. The properties of the materials used in the fabrication of nanocomposite plates were investigated using the Multiscale Finite Element Method (FEM) simulation for both short (10, 10) and long (10, 10) SWCNTs composites. The results of FEM simulations were fitted using those of the rule of mixture to obtain optimum values of CNT efficiency parameters. A few selected numerical results have been provided to investigate the effects of the volume fractions of CNTs and the types of edge supports on the value of fundamental frequency of long- and short-CNTs reinforced composite plates.

Computing Buckling Characteristics of Double-Layered Graphene Film Embedded in an Elastic Foundation

Given the unique and extremely valuable properties, research has significantly focussed on graphene sheets (GSs). To premeditate the small-scale effect, the present work applies the nonlocal theory to study the buckling behavior of a double-layered GS (DLGS) embedded in an elastic foundation. To derive the equation, classical plate theory is adopted. For the elastic foundation, Pasternak-type model is used. In terms of buckling response, a meshless method is utilized to compute simulation results. Accordingly, we examine the effects of aspect ratio, geometry, boundary conditions and nonlocal parameters on the buckling responses of DLGSs.

Effect of Fluid–Structure Interaction on Vibration and Deflection Analysis of Generally Orthotropic Submerged Micro-plate with Crack Under Thermal Environment: An Analytical Approach

To develop a new analytical model for vibration analysis of cracked-submerged orthotropic micro-plate affected by fibre orientation and thermal environment. The proposed analytical model is based on Kirchhoff’s classical thin plate theory and the size effect is introduced using the modified couple stress theory. Effect of crack is deduced using appropriate crack compliance coefficients based on line spring model while the effect of thermal environment is introduced in terms of thermal in-plane moments and forces. The coupling of shear and normal stresses for fibre orientation is represented using the coefficient of mutual influence. The fluid forces associated with its inertial effects are added in the governing differential equation to incorporate the fluid–structure interaction effect. The results are presented for frequency response as affected by different fibre orientation, crack length, crack location, level of submergence, temperature variation and material length-scale parameter for simply supported boundary condition. Furthermore, to study the phenomenon of shifting of primary resonance in a cracked micro-plate, the classical relations for central deflection of plate is also proposed. The results show that the fundamental frequency of micro-plate decreases by the presence of crack and thermal environment and this decrease in frequency is further intensified by the presence of surrounding fluid medium in present study. Another important conclusion is that with increase in temperature variation the reduction in frequency at 45° of fibre orientation is less when compared to 0 and 90° for both intact and cracked orthotropic plates.

On radially symmetric vibrations of functionally graded non-uniform circular plate including non-linear temperature rise

In this study, the free axisymmetric vibrations of functionally graded circular plate of linearly varying thickness controlled by a taper parameter in radial direction and non-linear temperature rise along the thickness have been investigated on the basis of classical plate theory. The plate material is graded in transverse direction and its mechanical properties are temperature-dependent. The temperature over the top and bottom surfaces is assumed to be uniformly distributed. Hamilton's principle has been used to derive the governing equations for thermo-elastic equilibrium and axisymmetric motion of such a plate model. The generalized differential quadrature method has been employed to obtain the thermal displacements and characteristic equations, for clamped and simply supported plates. The lowest three roots of these equations have been computed and reported as the values of frequency parameter for the first three modes of vibration. Effect of thickness parameter, volume fraction index and temperature difference has been analyzed on the vibration characteristics of the plate. A study with temperature-independent material properties has also been performed. Results have been compared with the published work.

Vibration analysis of multi-scale hybrid nanocomposite plates based on a Halpin-Tsai homogenization model

An analytical answer to the natural vibration problem of a composite plate consisted of multi-scale hybrid nanocomposites, is presented here for the first time. In this paper, the constituent material of the structure is made of an epoxy matrix which is reinforced by both macro- and nano-size reinforcements, namely carbon fiber (CF) and carbon nanotube (CNT). The effective material properties like Young's modulus or density are derived utilizing a micromechanical scheme incorporated with the Halpin-Tsai model. To present a more realistic problem, the plate is placed on a two-parameter elastic substrate. Then, on the basis of an energy-based Hamiltonian approach, the equations of motion are derived using the classical theory of plates. Finally, the governing equations will be solved analytically to obtain the natural frequency of the system's oscillation. Afterward, the normalized form of the results will be presented to put emphasis on the impact of each parameter on the dimensionless frequency of nanocomposite plates. It is worth mentioning that the effects of various boundary conditions on the frequency of the plate are covered, too. To show the efficiency of presented modeling, the results of this article are compared to those of former attempts. Numerical results declare that plates fabricated from the hybrid nanocomposites can endure higher frequencies compared with those consisted of conventional composites.

Size-dependent nonlinear forced vibration and dynamic stability of electrically actuated micro-plates

A size-dependent structural dynamic model that incorporates the effect of geometric nonlinearity is developed in this paper for the forced vibration and dynamic stability of thin rectangular micro-plates. The equations of motion for micro-plates are derived within the framework of classical plate theory, modified couple stress theory (MCST), and von Kármán geometric nonlinearity using Hamilton's principle. Galerkin method is used to convert the governing partial differential equations to a nonlinear second-order ordinary differential equation, which is solved by a Runge-Kutta method. The static instability analysis of the micro-plate is performed to determine the critical electrostatic voltages, and to avoid the pull-in instability. By tracking the static behavior of the microplate, and determining the electrostatic pull-in voltage, the frequency response curves are plotted. In dynamic response, primary, superharmonic, and subharmonic resonance are studied, and the frequency response equation is obtained for each case by the method of multiple scales. Further efforts are made to investigate the influence of size effect, electrical loading (DC and AC voltages), and excitation frequency on the static, and dynamic responses, critical AC voltages, and dynamic stability of micro-plates. It is found that the critical dynamic voltage is a function of the frequency of excitation force. It is shown that the stiffness of micro-plate decreases by increasing the constant DC voltage; however, the increase in the alternating AC voltage does not considerably affect the stiffness of the micro-plate.

Pore pressure and porosity effects on bending and thermal postbuckling behavior of FG saturated porous circular plates

Postbuckling and nonlinear bending of a geometrically perfect circular plate arisen by the temperature of one side of a axisymmetric circular plate made from a poroelastic solid whose matrix pores have been saturated by fluid have been numerically analyzed. The plate porosity varies continuously through the plate thickness according to some given specific functions. The postbuckling and nonlinear bending configurations of respectively clamped and simply-supported plates, as well as the critical postbuckling temperature, have been obtained under the influence of the plate one side surface temperature, thickness, type of pore distribution, porosity and compressibility of fluid confined by the pores. Equilibrium equations of the plate have been derived on the basis of the classical plate theory, Love–Kirchhoff hypothesis and the Sander’s nonlinear strain–displacement relationship. They have been discretized via differential quadrature method. The numerical results appropriately coincide with the relevant literature, namely the results of saturated porous and mathematically equivalent plates made from functionally grated materials.

On the High-Temperature Free Vibration Analysis of Elastically Supported Functionally Graded Material Plates Under Mechanical In-PlaneForce Via GDQR

In this article, the effect of Pasternak foundation on free axisymmetric vibration of functionally graded circular plates subjected to mechanical in-plane force and a nonlinear temperature distribution (NTD) along the thickness direction has been investigated on the basis of classical plate theory. The plate material is graded in thickness direction according to a power-law distribution and its mechanical properties are assumed to be temperature-dependent (TD). At first, the equation for thermo-elastic equilibrium and then equation of motion for such a plate model have been derived by Hamilton's principle. Employing generalized differential quadrature rule (GDQR), the numerical values of thermal displacements and frequencies for clamped and simply supported plates vibrating in the first three modes have been computed. Values of in-plane force parameter for which the plate ceases to vibrate have been reported as critical buckling loads. The effect of temperature difference, material graded index, in-plane force, and foundation parameters on the frequencies has been analyzed. The benchmark results for uniform and linear temperature distributions (LTDs) have been computed. A study for plates made with the material having temperature-independent (TI) mechanical properties has also been performed as a special case. Comparison of results with the published work has been presented.

Nonlinear aeroelastic stability analysis of three-phase nano-composite plates

The main aim of this study is to investigate the geometrical nonlinear flutter of three-phase nano-composite plates reinforced with carbon nano-tubes (CNTs). The effective material properties are obtained using Halpin-Tsai relation and rule of mixture. The classical plate theory and von Karman geometrical nonlinearities assumptions are employed to obtain the governing equation of the system. The homotopy perturbation method (HPM) is utilized to conduct a numerical investigation to study the aeroelastic behavior of multiscale composite plates. The effects of different parameters such as CNTs weight percentage, aspect ratio, and elastic modulus ratio on flutter pressure of nano-composite plates are considered.Communicated by S. Velinsky.

Functionally Graded Sandwich Circular Plate of Non-Uniform Varying Thickness with Homogenous Core Resting on Elastic Foundation: Investigation on Bending via Differential Quadrature Method

This paper illustrates the effectiveness of two functionally graded (FG) layers on a homogenous circular plate for bending analysis. The classical plate theory (CPT) serves as the basis of the analysis. Differential quadrature method (DQM) as semi-analytical method is employed to solve the governing equations. The material properties is varied to obey power-law in terms of the plate thickness direction. The plate is subjected to uniform transverse loading and resting on Winkler elastic foundation. In this study, the effect of the different profile of the plate thickness, elastic foundation coefficient, the volume fraction FG index, and effect of the boundary conditions, namely, simply supported and clamped edge on static response are demonstrated. The results are compared with finite element method and published literature that observed to be in accordance with each other.