Free vibrations of continuous rectangular plates on oblique supports

The Finite Strip Method has been employed in the vibration problem of continuous rectangular plates on oblique supports. The structure has been divided into quadrilateral finite strips. The various properties of a quadrilateral finite strip have been derived using the displacement approach. The results are obtained for a two span rectangular plate with an oblique support at various angles and compared with other solutions. The results obtained due to various layouts of finite strips in the structure have been compared.     

Buckling studies for simply supported symmetrically laminated rectangular plates

Extensive and accurate numerical results are presented for the critical buckling loads of simply supported, rectangular, laminated composite plates subjected to five types of loading conditions: (1) uniaxial, (2) hydrostatic biaxial, (3) compression-tension biaxial, (4) positive shear and (5) negative shear. Considerably different results are found for the two types of shear loading for angle-ply composites. The Ritz method, along with displacements assumed in the form of a double sine series, is used to solve the problems. Convergence studies are presented to demonstrate the accuracy of the results. Contour plots of the buckled mode shapes are shown for some of the more interesting plate and loading configurations.     

Analysis of rectangular laminated composite plates via FSDT meshless method

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.     

Buckling of symmetrically laminated rectangular plates under parabolic edge compressions

Buckling analysis of symmetrically laminated rectangular plates with parabolic distributed in-plane compressive loadings along two opposite edges is performed using the Rayleigh-Ritz method. Classical laminated plate theory is adopted. Stress functions satisfying all stress boundary conditions are constructed based on the Chebyshev polynomials. Displacement functions for buckling analysis are constructed by Chebyshev polynomials multiplying with functions that satisfy either simply supported or clamped boundary condition along four edges. Methodology and procedures are worked out in detail. Buckling loads for symmetrically laminated plates with four combinations of boundary conditions are obtained. The proposed method is verified by comparing results to data obtained by the differential quadrature method (DQM) and the finite element method (FEM). Numerical example also shows that the double sine series displacement for simply supported symmetrically laminated plates having bending-twisting coupling may overestimate the stiffness, thus providing higher buckling loads.     

Generic nonlinear behavior of antisymmetric angle-ply laminated plates

A generic presentation of the large deflection, postbuckling and nonlinear flexural vibration behavior of antisymmetric angle-ply laminated plates is investigated. A transformation between on-axis and off-axis stiffnesses is presented such that all the stiffnesses, including bending-stretching coupling, in the nonlinear governing equations of angle-ply laminates can be expressed in terms of bounded global constants in orthotropic plates. The results presented are applicable to a large variety of composite materials and show that a higher generalized rigidity ratio will lead to smaller deflection of plates, higher buckling loads and lower postbuckling deflection, and higher relative large-amplitude natural frequencies. The information presented should be helpful to anyone seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

Exact solutions for the free in-plane vibrations of rectangular plates

All classical boundary conditions including two distinct types of simple support boundary conditions are formulated by using the Rayleigh quotient variational principle for rectangular plates undergoing in-plane free vibrations. The direct separation of variables is employed to obtain the exact solutions for all possible cases. It is shown that the exact solutions of natural frequencies and mode shapes can be obtained when at least two opposite plate edges have either type of the simply-supported conditions, and some of the exact solutions were not available before. The present results agree well with FEM results, which show that the present solutions are correct and the direct separation of variables is practical. The exact solutions can be taken as the benchmarks for the validation of approximate methods.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates

The geometrically nonlinear free vibrations of polar orthotropic circular plates are analyzed by the axisymmetric finite element. The formulation is based on the three-dimensional elasticity with all the nonlinear terms in the strains included and is considered to be more general and precise theoretically compared to the plate-theory-based nonlinear approaches. Numerical results with various geometries, orthotropies, amplitudes and boundary conditions by the present formulation are also shown and compared to those by other approaches.     

FEM buckling analysis of quasi-isotropic symmetrically laminated rectangular composite plates with a square/rectangular cutout

A numerical study was conducted using the finite element method to determine the effects of square and rectangular cutouts on the buckling behavior of a 16-ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate. The square/ rectangular cutouts were subjected to uniaxial compression loading. This study addresses the effects of the size of the square/rectangular cutout, orientation of the square/rectangular cutout, plate aspect ratio (a/b), and plate length/thickness ratio (a/t) on the buckling behavior of the symmetrically laminated rectangular composite plate under uniaxial compression loading. Buckling loads were computed for seven different quasi-isotropic laminate configurations [0°/+45°/?45°/90°]_2s, [15°/+60°/?30°/?75°]_2s, [30°/+75°/?15°/?60°]_2s, [45°/+90°/0°/?45°]_2s, [60°/?75°/+15°/?30°]_2s, [75°/?60°/+30°/?15°]_2s, [90°/?45°/+45/°0°]_2s. Results showed that the magnitudes of the buckling loads decrease with increasing cutout positioned angle as well as c/b and d/b ratios for plates with a rectangular cutout. The symmetrically laminated quasi-isotropic [0°/+45°/?45°/90°]_2s composite plate is stronger than all other symmetrically analyzed laminated quasi-isotropic composite plates. The magnitudes of the buckling loads of a rectangular composite plate with square/rectangular cutout decrease with increasing plate aspect ratio (a/b) and plate length/thickness (a/t) ratio.     

Buckling and post-buckling behavior of laminated plates using the generalized nonlinear formulation

In this paper the buckling and post-buckling analysis of symmetrically laminated cross-ply plates has been considered using the generalized formulation of which the von Karman type formulation is a special case. Unlike the von Karman type formulation, the governing equations of the present formulation are nonlinear in all the displacement parameters of the plate. This obviates the use of hitherto known solution techniques for nonlinear plate analysis. To overcome this difficulty the perturbation technique has been followed in this paper. Numerical results indicate that the two formulations differ by about 2.5% for thicker perfect plates and the difference ceases to exist as the imperfection value of the plate increases.     

Analytical solutions for bending analysis of rectangular laminated plates with arbitrary lamination and boundary conditions

The intent of the present study is to employ the extended Kantorovich method for semi-analytical solutions of laminated composite plates with arbitrary lamination and boundary conditions subjected to transverse loads. The method based on separation of spatial variables of displacement field components. Within the displacement field of a first-order shear deformation theory, a laminated plate theory is developed. Using the principle of minimum total potential energy, two systems of coupled ordinary differential equations with constant coefficients are obtained. The equations are solved analytically by using the state-space approach. The results obtained are compared with the Levy-type solutions of cross-ply and antisymmetric angle-ply laminates with various admissible boundary conditions to verify the validity and accuracy of the present theory. Also, for other laminations and boundary conditions that there exist no Levy-type solutions the present results are compared with those obtained by other investigators and finite element method. It is found that the present results have very good agreements with those obtained by other methods. This paper was recommended for publication in revised form by Associate Editor Heoung Jae Chun Ali Mohammad Naserian Nik received his M.S. in Mechanical Engineering from Ferdowsi University of Mashhad, Iran, in 2006. Currently, he is a doctoral student at the Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran. His research interests are in the area of computational mechanics and nanobiotechnology. Masoud Tahani is currently an Associate Professor at the Department of Mechanical Engineering at Ferdowsi University of Mashhad, Iran. He received his B.S. in Mechanical Engineering from Ferdowsi University of Mashhad, Iran, in 1995. He then received his M.S. and Ph.D. degrees from Sharif University of Technology, Iran, in 1997 and 2003, respectively. Dr. Tahani’s research interests include design of structures using advanced composites, mechanics of anisotropic materials, smart materials and structures, mechanics of plates and shells and biomechanics.     

Free vibration analysis of moderately thick antisymmetric cross-ply laminated rectangular plates with elastic edge constraints

This study presents a simple formulation for studying the free vibration of shear-deformable antisymmetric cross-ply laminated rectangular plates having translational as well as rotational edge constraints. The aim is to fill the void in the available literature with respect to the free vibration results of antisymmetric cross-ply laminated rectangular plates. The spatial discretization of the resulting mathematical model in five field variables is carried out using the two-dimensional Differential Quadrature Method (DQM). Several combinations of translational and rotational elastic edge constraints are considered. Convergence study with respect to the number of nodes has been carried out and the results are compared with those from past investigations available only for simpler problems. Effects of stiffness parameters, geometrical features, moduli ratio and lamination schemes on the natural frequencies are studied.     

A semi-analytical solution for forced vibrations response of rectangular orthotropic plates with various boundary conditions

Orthotropic plates form an essential part of many marine, aerospace, and automobile structures. The recent increase in the use of composite materials for plate-type structures intensified the need for solutions utilizing orthotropic plates. These structural components, in many instances, are subjected to vibration. The main purpose of this paper is to present a semi-analytical analysis of the response of orthotropic plates to forced vibrations with different combinations of boundary conditions subjected to a general distributed excitation. The solution of the partial differential equation was reduced to an iterative sequential solution of ordinary differential equations using extended Kantorovich method. The efficiency and accuracy of the proposed method were examined through comparison with available literature and was in good agreement with them.     

Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods

In this paper, the buckling analysis of laminated composite plates reinforced by single-walled carbon nanotubes (SWCNTs) is carried out using an analytical approach as well as the finite element method. The developed model is based on the classical laminated plate theory (CLPT) and the third-order shear deformation theory for moderately thick laminated plates. The critical buckling loads for the symmetrical layup are determined for different support edges. The Mori-Tanaka method is employed to calculate the effective elastic modulus of composites having aligned oriented straight nanotubes. The effect of the agglomeration of the randomly oriented straight nanotubes on the critical buckling load is also analyzed. The results of analytical solution are compared and verified with the FEM calculations The critical buckling loads obtained by the finite element and the analytical methods for different layup and boundary conditions are in good agreement with each other. In this article, the effects of the carbon nanotubes (CNTs) orientation angle, the edge conditions, and the aspect ratio on the critical buckling load are also demonstrated using both the analytical and finite element methods.     

Thermal buckling analysis of cross-ply laminated rectangular plates under nonuniform temperature distribution: A differential quadrature approach

Differential quadrature method (DQM) is implemented for analyzing the thermal buckling behavior of the symmetric cross-ply laminated rectangular thin plates subjected to uniform and/or non-uniform temperature fields. The approach includes two steps: (1) solving the problem of in-plane thermo-elasticity to obtain the in-plane force resultants and (2) solving the buckling problem under the force distribution obtained in the previous step. Solution procedures are numerically performed by discretizing the governing differential equations and boundary conditions using DQM method. Applying the developed DQ formulation, the buckling loads are obtained for several sample plates. The numerical results compared well with those available in the literature as well as those obtained by ABAQUS. Parametric studies are conducted to investigate the influence of some important parameters including the plate aspect ratio, cross-ply ratio, and stiffness ratio on the critical temperature and mode shape of buckling.     

Nonlinear vibrations of a pre-stressed laminated thin plate

In this communication, we present the numerical analysis of nonlinear vibrations of a pre-stressed, laminated, thin, square plate. The laminate is modeled as an isotropic nonlinear plate with effective elastic properties and the model is validated experimentally. The geometrically nonlinear vibrations are analyzed using the finite difference method, and the time integration technique based on the Newmark method combined with the fixed-point iterations to satisfy compatibility. The system exhibits behavior similar, but not identical, to the standard Duffing oscillator. Superharmonics are examined using phase space representation, Fourier transforms, deflection history, and centerline deflection shapes. The deflection shapes show significant presence of higher order spatial harmonics.     

Postbuckling analysis of rectangular orthotropic plates

The postbuckling analysis of orthotropic simply supported rectangular plates of symmetric cross-section is studied and curves of load-shortening and effective widths are presented for various cases of orthotropic plates having different elastic properties. The results obtained are compared with results already published.     

Nonlinear analysis of rectangular orthotropic plates

The present study deals with the large deflections and nonlinear flexural vibrations of orthotropic rectangular plates. In this paper, generalized double Fourier series in terms of generalized beam eigenfunctions for deflection and force functions are presented. The influences of aspect ratio, material properties, transverse load, rotational edge restraint coefficients, and two types of in-plane conditions are studied. The results of deflection and frequency ratio for special cases are compared with those values given in the literature and good agreement is obtained.     

Unsteady flow between rectangular plates and between circular plates

An analysis is presented of the unsteady flow of a viscous incompressible fluid between two parallel infinitely long rectangular plates and between two parallel circular plates. The lower plate is fixed and the upper plate moves towards the lower plate. Full Navier-Stokes equations are used to obtain the pressure distribution as a function of the film thickness h(t) and the velocity $\text{h}\text{?}\text{(t)}$ of the upper plate. The inertia is taken into account and, for a given load on the upper plate, the sinkage relation between h and t is determined for various values of the Reynolds number. The departure from the classical inertialess solution is exhibited for various values of the two dimensionless parameters involved, one characterizing the load and the other gravity.     

Nonlinear flexural vibration of rectangular moderately thick plates and sandwich plates

Nonlinear flexural vibration is investigated for rectangular Reissner moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Highly accurate solutions of series form with many different movable and immovable boundary conditions, especially with unsymmetrical boundary conditions, are obtained by means of the method of harmonic balance and by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated in numerical computations. Solutions of the nonlinear fundamental frequency of rectangular plates are obtained by iteration. The multimode approach includes not only the influences of transverse shearing deformation and rotatory inertia, but also the coupling effect of vibrating modes on the nonlinear fundamental frequency. The present solutions are satisfactory in comparison with other available results.