Variable Refined Plate Theory Research Articles

Buckling analysis of plates using an efficient sinusoidal shear deformation theory

Mechanical buckling response of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain-displacement of Von Karman relations are also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results are presented for the present efficient sinusoidal shear deformation theory, demonstrating its importance and accuracy in comparison to other theories.

A Two Variable Refined Plate Theory for Isogeometric Vibration Analysis of The Functionally Graded Piezoelectric Microplates with Porosities

In this article, a free vibration analysis of the functionally graded porous piezoelectric (FGPP) microplates is firstly solved by using a combination of two variable refined plate theory (RPT), modified strain gradient theory (MSGT) and isogeometric analysis (IGA). The FGPP microplate is composed of piezoelectric material with pores, which are distributed across the plate thickness in uniform and non-uniform distributions. The modified strain gradient theory is used to capture the size effect on the natural frequency of the FGPP microplates. According to the variational principle of RPT with two variables, the governing equations are derived and solved by the IGA. The influence of the length scale parameters (LSPs), external electric voltage, power law index, length-to-thickness ratio, aspect ratio and boundary conditions (BCs) on the natural frequency of the FGPP microplates is studied. The numerical results show that a rise in the porosity coefficient makes a decrease in the microplate’s stiffness, while an increase in LSPs leads to a rise in the microplate’s stiffness.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Nonlinear free vibration of rotating FG trapezoidal microplates in thermal environment

The size-dependent nonlinear free vibration characteristics of rotating functionally graded (FG) trapezoidal microplates are investigated based on a four variable refined plate theory (FVRPT) coupled with the modified strain gradient theory (MSGT). The geometric nonlinear strain–displacement relations are derived by considering the von Kármán geometric nonlinearity hypothesis. The material composition is assumed to be graded in the thickness direction according to the power law function. The properties of the constituent materials are assumed temperature-dependent and the effective material properties are determined by employing Mori–Tanaka micromechanical homogenization technique. By applying the Chebyshev–Ritz method, the system of nonlinear equations governing the nonlinear free vibration characteristics of rotating trapezoidal FG microplates is derived. The nonlinear frequencies are determined through a direct iterative process by considering both positive and negative deflection cycles. Through the numerical investigations, the effects of different geometric and material parameters on the nonlinear to linear frequency ratio are studied and discussed. The results show that the rectangular-shaped microblades have greater frequencies than the trapezoidal ones. Also, it is found that the temperature rise has hardening effect, meanwhile the length scale parameter has softening effect on the variations of the frequency ratio versus amplitude ratio.

Acoustic response behavior of porous 3D graphene foam plate

Sound radiation and sound transmission loss (STL) behavior of porous 3D graphene (3D-GrF) foam plate are presented. Two variable refined plate theory which includes both transverse bending and shear stresses is used to model the plate and Navier’s solution is used to calculate the vibration responses while Rayleigh integral is used to analyze the acoustic response. Variation in free vibration frequencies with the nature of porosity distribution is significant for the 3D-GrF plates having higher porosity co-efficient. The natural frequency of the 3D-GrF plate with more porosity around the center and less porosity at the outer surfaces is high. However, resonant amplitudes of the responses and STL of the plates are controlled by both the nature of the porosity distribution pattern and porosity co-efficient. In general, STL of the plate with less porosity around the center and high porosity at the extreme surfaces is high compared to the other cases.

Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory

The thermo-mechanical bending behavior of the antisymmetric cross-ply laminates is examined using a new simple four variable trigonometric plate theory. The proposed theory utilizes a novel displacement field which introduces undetermined integral terms and needs only four variables. The validity of the present model is proved by comparison with solutions available in the literature.

Free vibration of orthotropic Levy-type solution plates by using SEM

Orthotropic Levy-type solution plates considering shear deformation were solved by using the Spectral Element Method (SEM). In SEM, the exact solution of structural dynamic related problems can be solved by using the frequency-dependent shape functions. The solutions of SEM are accurate in decreasing the amount of Degree Of Freedoms (DOF) to discard the cost and computational disadvantages in the Finite Element Method (FEM). This study investigates the free vibration problems of orthotropic Levy-type solution plates solutions based on the two variable Refined Plate Theory (RPT). SEM for orthotropic Levy-type solution plates in the frequency domain was formulated to solve free vibration problems. The differential governing equations of the orthotropic plate element in SEM are formulated into the form of transcendental stiffness matrices. The boundary conditions of the spectral element model consist of four edge-type DOFs at each side of the orthotropic Levy-type solution plates. The natural frequencies of the plate are calculated by means of the Wittrick-Williams procedure. The numerical investigations are presented to show the accuracy, efficiency, and effectiveness of SEM without any discretization scheme.

Thermodynamic effect on the bending response of elastic foundation FG plate by using a novel four variable refined plate theory

ABSTRACTThis article deals with the study of the thermodynamic behavior of functionally graded material plates resting on two-parameter elastic foundation. An analytical solution based on a new shear refined deformation theory is presented. The displacement field used in the present refined theory contains undetermined integral forms and involves only four unknowns to derive. The plate is assumed simply supported and subjected to two different temperatures fields across its thickness. The mechanical characteristics of the plate are assumed to be varied across the thickness according to a simple exponential law distribution. The governing equations and boundary conditions are derived using the principle of virtual displacements and Navier solution technique is adopted to derive analytical solutions. A detailed numerical study of the present new refined theory is carried out to examine the influence of the time’s parameter, foundation’s parameters and deflection on the bending response of the FG plate.

Free vibration and buckling analysis of orthotropic plates using a new two variable refined plate theory

The present work presents a free vibration and buckling analysis of orthotropic plates by proposing a novel two variable refined plate theory. Contrary to the conventional higher order shear deformation theories (HSDT) and the first shear deformation theory (FSDT), the proposed theory utilizes a novel displacement field which incorporates undetermined integral terms and involves only two unknowns. The governing equations are obtained from the dynamic version of principle of virtual works. The analytical solution of a simply supported orthotropic plate has been determined by using the Navier method. Numerical investigations are performed by employing the proposed model and the obtained results are compared with the existing HSDTs.

A novel refined plate theory for stability analysis of hybrid and symmetric S-FGM plates

In this paper, buckling analysis of hybrid functionally graded plates using a novel four variable refined plate theory is presented. In this theory the distribution of transverse shear deformation is parabolic across the thickness of the plate by satisfying the surface conditions. Therefore, it is unnecessary to use a shear correction factor. The variations of properties of the plate through the thickness are according to a symmetric sigmoid law (symmetric S-FGM). The principle virtual works is used herein to extract equilibrium equations. The analytical solution is determined using the Navier method for a simply supported rectangular plate subjected to axial forces. The precision of this theory is verified by comparing it with the various solutions available in the literature.

A novel four variable refined plate theory for wave propagation in functionally graded material plates

In This work an analysis of the propagation of waves of functionally graduated plates is presented by using a high order hyperbolic (HSDT) shear deformation theory. This theory has only four variables, which is less than the theory of first order shear deformation (FSDT). Therefore, a shear correction coefficient is not required. Unlike other conventional shear deformation theories, the present work includes a new field of displacement which introduces indeterminate integral variables. The properties of materials are supposed classified in the direction of the thickness according to two simple distributions of a power law in terms of volume fractions of constituents. The governing equations of the wave propagation in the functionally graded plate are derived by employing the Hamilton's principle. The analytical dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. The convergence and the validation of the proposed theoretical numerical model are performed to demonstrate the efficacy of the model.

Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory

ABSTRACTIn this work, analytical solutions are presented for the wave propagation in functionally graded (FG) nanoplates using a nonlocal strain gradient theory and four-variable refined plate theory considering the magnetic field. The size effects are included using nonlocal strain gradient theory that has two length scale parameters, and the nanoplate is modeled as a plate using four-variable refined plate theory. From the knowledge of authors, it is the first time that the influences of magnetic field on the wave propagation in FG nanoplates are investigated based on present methodology.

Levy solution for bending analysis of functionally graded sandwich plates based on four variable plate theory

In the present study, static analysis of a functionally graded sandwich plate with two opposite simply supported edges is studied using a four variable refined plate theory. The core of sandwich plate is assumed isotropic and material properties of face layers are assumed graded through thickness direction by the power law. Displacement fields are defined by using the four variable plate theory. Equilibrium equations of the sandwich plate are obtained using virtual work principle. Levy type solution with state space concept is used to derive governing equations. The results of deflection, normal and shear stress of uniformly loaded functionally graded square sandwich plates with two opposite simply supported edges are presented. The accuracy of the solution is verified by comparing it with available results in the literature.

The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular piezoelectric nanoplate based on the two variable-refined plate theory

The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular and skew piezoelectric nanoplate were investigated based on the two variable-refined plate theory (TVRPT). The governing equations were derived using the principle of virtual work and then it was solved using the finite difference method. Finite difference solutions were validated by Navier’s method and journal references. A parametric study was conducted to investigate the influences of the external electric voltage, nonlocal parameter, different boundary condition, nanoplate thickness, and aspect ratio on the surface effect of uniaxial and biaxial buckling. Numerical results obtained showed that the influence of surface stress on the buckling of the rectangular piezoelectric nanoplate was enhanced by augmenting the nonlocal parameter, external electric voltage, and aspect ratio while it was decreased in the thicker nanoplate. The present work could be very useful in designing micro and nano-devices using smart nanocomposites.

A new simple shear deformation plate theory

This paper proposes a new simple shear deformation theory for isotropic plates. The present theory involves one unknown and one governing equation as in the classical plate theory, but it is capable of accurately capturing shear deformation effects. The displacement field of the present theory was based on a two variable refined plate theory in which the transverse displacement is partitioned into the bending and shear parts. Based on the equilibrium equations of three-dimensional (3D) elasticity theory, the relationship between the bending and shear parts was established. Therefore, the number of unknowns of the present theory was reduced from two to one. Closed-form solutions were presented for both Navier- and Levy-type plates. Numerical results indicate that the obtained predictions are comparable with those generated by ABAQUS and available results predicted by 3D elasticity theory, first-order and third-order shear deformation theories.

A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation

This paper presents a free vibration analysis of plates made of functionally graded materials and resting on twolayer elastic foundations by proposing a non-polynomial four variable refined plate theory. Undetermined integral terms are introduced in the proposed displacement field and unlike the conventional higher shear deformation theory (HSDT), the present one contains only four unknowns. Equations of motion are derived via the Hamilton\'s principles and solved using Navier\'s procedure. Accuracy of the present theory is demonstrated by comparing the results of numerical examples with the ones available in literature.

Free vibrations of laminated composite plates using a novel four variable refined plate theory

In this research, the free vibration response of laminated composite plates is investigated using a novel and simple higher order shear deformation plate theory. The model considers a non-linear distribution of the transverse shear strains, and verifies the zero traction boundary conditions on the surfaces of the plate without introducing shear correction coefficient. The developed kinematic uses undetermined integral terms with only four unknowns. Equations of motion are obtained from the Hamilton's principle and the Navier method is used to determine the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical examples studied using the present formulation is compared with three-dimensional elasticity solutions and those calculated using the first-order and the other higher-order theories. It can be concluded that the present model is not only accurate but also efficient and simple in studying the free vibration response of laminated composite plates.

An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates

In this article, an efficient and simple refined theory is proposed for buckling analysis of functionally graded plates by using a new displacement field which includes undetermined integral variables. This theory contains only four unknowns, with is even less than the first shear deformation theory (FSDT). Governing equations are obtained from the principle of virtual works. The closed-form solutions of rectangular plates are determined. Comparison studies are carried out to check the validity of obtained results. The influences of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of functionally graded plates are examined and discussed.

A novel four variable refined plate theory for laminated composite plates

A novel four variable refined plate theory is proposed in this work for laminated composite plates. The theory considers a parabolic distribution of the transverse shear strains, and respects the zero traction boundary conditions on the surfaces of the plate without employing shear correction coefficient. The displacement field is based on a novel kinematic in which the undetermined integral terms are used, and only four unknowns are involved. The analytical solutions of antisymmetric cross-ply and angle-ply laminates are determined via Navier technique. The obtained results from the present model are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories reported in the literature. It can be concluded that the developed theory is accurate and simple in investigating the bending and buckling responses of laminated composite plates.

Static bending behaviors of piezoelectric nanoplate considering thermal and mechanical loadings based on the surface elasticity and two variable refined plate theories

This study zoomed on the maximum static bending and stress behaviors of piezoelectric nanoplate based on surface effect and nonlocal elasticity theories. The piezoelectric nanoplate was subjected to thermal and mechanical loads, the governing equations are derived via two variable refined plate theory, using Hamilton’s principle. The equations of motion were solved through Navier’s method. By ensuring the accuracy of the results, the effects of some important parameters such as the nonlocal parameter, external electric voltage, thermal and mechanical loading, nanoplate thickness and aspect ratio on the surface on bending were studied. Numerical results showed that the effect of surface energy on the bending of piezoelectric nanoplate was increased by diminishing the nonlocal parameter, external electric voltage and nanoplate length, and raising temperature change, mechanical loading and nanoplate thickness. In contrast, the influence of surface layer on the normalized stress of piezoelectric nanoplate was augmented by increasing the in-plane loading, nanoplate length and thickness, and diminishing temperature change.

A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates

This work presents a bending, buckling, and vibration analysis of functionally graded plates by employing a novel higher-order shear deformation theory (HSDT). This theory has only four unknowns, which is even less than the first shear deformation theory (FSDT). A shear correction coefficient is, thus, not needed. Unlike the conventional HSDT, the present one has a new displacement field which introduces undetermined integral variables. Equations of motion are obtained by utilizing the Hamilton's principles and solved via Navier's procedure. The convergence and the validation of the proposed theoretical numerical model are performed to demonstrate the efficacy of the model.