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The purpose of this research is to study the free vibration response of composite plates made up of three-phase polymer–glass fiber (FG)–carbon nanotubes (CNTs). Polymer matrix has an important property, i.e. the elastic modulus and Poisson’s ratio decrease over time. In this study, the waviness of the CNTs is also taken into consideration. A three-step hierarchical approach is adopted to estimate the elastic properties of the composite media. The governing equations are derived by quasi-3D plate theory and Hamilton’s principle. The adopted quasi-3D model takes into account the nonuniform shear and normal strain distribution and also satisfies the condition of traction-free on top and bottom surfaces. The natural frequencies of this composite are found using the Navier solution method for simply supported boundary conditions. This work investigates how the dimensionless frequency is affected by time, plate geometry, fiber mass fraction, and CNT mass fraction, all of which have a substantial impact. For example, it is highlighted that for a specific plate, after 60s, the frequency drops by about 14% if one takes the time parameter into account. In addition, increasing the mass fraction of CNTs or FGs results in higher frequencies.

This paper presents the uniaxial and biaxial buckling analysis of rectangular plates based on new trigonometric shear and normal deformation theory. The theory accounts for the cosine distribution of the transverse shear strain through the plate thickness and on the free boundary conditions on the plate surfaces without using the shear correction factor. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The Navier type solutions for the buckling analysis of simply supported isotropic, transversely isotropic, orthotropic and symmetric cross-ply laminated composite rectangular plates subjected to uniaxial and biaxial compressions are presented. The effects of variations in the degree of orthotropy of the individual layers, side-to-thickness ratio and aspect ratio of the plate are examined on the buckling characteristics of composite plates. The present results are compared with those of the classical plate theory (CPT), first order shear deformation theory (FSDT) and exact three-dimensional (3D) elasticity theory wherever applicable. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory.

The transverse vibration control of a clamped, rectangular, isotropic plate by a vibratory flap subjected to harmonic excitation has been investigated by Finite Element Analysis (FEA) and experimental technique. The vibratory flap is a new plate-type dynamic vibration absorber, which can vibrate on the plate when attached as a cantilever plate. The study has been focused specifically on the influence of aspect ratio of vibratory flap on the dynamic response of the plate at constant mass ratio and constant tuning frequency ratio. The study has revealed that the dynamic response of the plate varies with respect to the aspect ratio for aforementioned conditions. An optimum aspect ratio has also been obtained by minimizing the mass ratio with maximum attenuation in the first and second target frequencies. The results have shown that the optimized flap can trim down the plate vibrations by up to 90–95% in the fundamental mode. Moreover, the dynamic response of the plate can be improved to a great extent due to the adoption of an optimal aspect ratio of the flap. Finally, the experimental outcomes have shown fairly good agreement with the results obtained from the finite element analysis.

In engineering practice, a massive machine may be placed on a plate supported by beams considered as elastic boundary conditions. The vibration of the plate due to the periodic excitation of the massive machine will cause noises or damages to the building in which the machine is housed. An analytical approach for the vibration analysis of a rectangular plate carrying a massive machine with uniform elastic supports is presented. The machine is simplified as a distributed mass. The transverse plate displacement is determined by the superposition of a two-dimensional (2D) Fourier cosine series and several supplementary functions. All the unknown Fourier coefficients are calculated directly from the Rayleigh–Ritz formulation. To validate the present approach, several numerical examples with classical boundary conditions are presented. The results reveal good agreement between the analytical results and those based on the finite element analysis (ANSYS). The effects of the plate size, location of the machine, and support stiffness on the modal, and transient response of the plate are investigated. From the results it is found that the transient displacement amplitude of the plate decreases almost linearly as the thickness increases, it increases nonlinearly along with the increase in the support stiffness, and that the optimal position for deploying the transformer is the center of the plate.

This paper is concerned with the development of the Hencky bar-net model (HBM) for free vibration analyses of rectangular plates with mixed boundary conditions and point supports. The HBM is a two-dimensional discrete net system composed of rigid segments connected by frictionless hinges and rotational springs. In the model, bending is accommodated by rotational springs at each joint while the twisting by a diagonal spring system in each grid cell. The total mass of the plate is distributed as lumped mass at each joint and the continuous boundary stiffness of plate is simulated by springs located at the edge joints. Owing to the discrete property of HBM, it is able to readily handle any boundary conditions of plates including mixed boundary conditions and point supports. The HBM is herein used to solve some vibration problems of rectangular plates with mixed boundary conditions and point supports to demonstrate its accuracy and convenience for plate analyses.

In this paper, the nonlinear thermally induced vibration in rectangular plates made of porous functionally graded materials (FGMs) due to thermal shock is studied based on a new numerical approach. The shear deformation influences are taken into consideration via Mindlin’s plate theory. The properties of the materials are also assumed to be temperature- and position-dependent. Moreover, the influence of elastic foundation is captured by the Winkler–Pasternak model. For deriving the governing equations, Hamilton’s principle, transient 1D Fourier-type heat conduction equation and von Kármán hypothesis are utilized. The relations of paper are written in a new matricized format for computational aims. The generalized differential quadarature (GDQ) method and Newmark’s direct integration scheme are used for solving the heat equation. Furthermore, solving motion equations is done based on the variational differential quadrature (VDQ) formulation. The effects of important parameters including material porosity and elastic foundation on the thermal shock response of porous FG plates are investigated in the numerical results.

The nonlinear vibration response of rectangular plates made of functionally graded porous materials (FGPs) induced by hygrothermal loading is investigated in this article using a numerical approach. The effect of elastic foundation on the vibrations is taken into account according to the Winkler–Pasternak model. Hygroscopic stresses produced due to nonlinear rise in moisture concentration are also considered. The temperature-dependent material properties of plate are computed based on the modified Voigt’s rule of mixture and Touloukian experiments for even and uneven distribution patterns of porosity. Within the framework of the first-order shear deformation plate theory and von-Kármán nonlinearity, Hamilton’s principle is utilized in order to derive the equations of motions. To achieve the temporal evolution of maximum lateral deflection of hygrothermally-induced plates, the generalized differential quadrature (GDQ) and Newmark integration methods are employed. Selected numerical results are presented to study the influences of temperature distribution, porosity volume fraction, moisture concentration, geometrical parameters, elastic foundation parameters and FG index on the geometrically nonlinear vibrations of FG porous plates with various boundary conditions.

Two cases of a rectangular plate having moment singularities at the ends of a partial internal line support are analytically investigated. The bending of the plate by uniform loading is formulated in terms of dual-series equations. Application of the finite Hankel integral transform reduces the dual-series equations to a Fredholm integral equation of the second kind that can be solved by standard techniques. Numerical results are given for the deflections and bending moments along the line outside of an internal line support and the change in strain energy due to the presence of a partial support.

In this research, the nonlinear free vibrations analysis of functionally graded (FG) rectangular plate which simply supported all edges are investigated analytically using modified Lindstedt–Poincare (MLP) method for the first time. For this purpose, with the aid of von Karman nonlinearity strain-displacement relations, the partial differential equations of motion are developed based on first-order shear deformation theory (FSDT). Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations. The nonlinear equation of motion is then solved analytically by MLP method to determine the nonlinear frequencies of the FG rectangular plate. The material properties are assumed to be graded through the direction of plate thickness according to power law distribution. The effects of some system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear to linear frequency ratio are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

In this paper, transient response of functionally graded materials (FGM) rectangular sandwich plates subjected to thermal loading based on three-dimensional theory of elasticity is carried out. The facing layers are made of metal and ceramics whereas core layer is FGM with exponential function distribution. For the case of simply supported boundary conditions, obtained state space differential equations are solved analytically by using Fourier series expansions along the in-plane coordinates and Laplace transformation. For other edges, boundary condition semi-analytical method is used by applying differential quadrature method (DQM) along the in-plane direction and state space technique across the transverse coordinate. After carrying out the convergence of the present method, accuracy of this study is validated by comparing the numerical results with those reported in the literature. Finally, influences of the boundary conditions, length-to-thickness ratios, length-to-width ratios, time constant and the FGM graded index on elastic behavior of sandwich plate are examined.

This study aims to explore the buckling behavior of rectangular plate with angle-ply overlay under thermal and mechanical loadings using meshless method. The governing differential equations (GDEs) of the plate are based on a new higher-order shear deformation theory (HSDT) and are discretized using polynomial radial basis function (RBF) to cast a set of equations simultaneously. A code is developed for the solution in MATLAB (2019). Uniform temperature differences and mechanical loads are the cause of in-plane loads. Convergence study has been conducted to check the current formulation’s soundness. The accuracy of the contemporary theory and formulation is validated with the published results. A parametric investigation is conducted for buckling behavior under the span-to-thickness ratio, lamination scheme, effect of the number of layers and effect of length-to-width ratio.

The usual assumption of isotropy of the material is shown here to be one of the reasons for the observed disparity between the theoretical and experimental values of the critical stress for plastic buckling of plates and shells. When the effect of plastic anisotropy that commonly occurs in thin-walled structures is included in the theoretical framework, the predicted load at the point of bifurcation can be significantly lower than that corresponding to the isotropic material. The present paper is intended to demonstrate the influence of plastic anisotropy on the critical stress for the elastic/plastic buckling of rectangular plates under unidirectional compression. The results are presented graphically to indicate how the critical stress is lowered by the presence of plastic anisotropy that is characterized by an R- value less than unity.

Considering the effects of shear deformation and stress wave, the dynamic buckling governing equations of rectangular plates under axial step load are established. Based on the Rayleigh-Ritz method, the expression of the critical load is got. The relation curve between the critical load and critical length is described by using MATLAB software. In this paper, the influences of thickness, first-order shear deformation (FSD), and the number of modes are discussed.