Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.     

Bending and extension of a multilayered plate with body forces and thermal loading

A multilayered plate composed of thin layers of isotropic materials is analyzed. The problem for the multilayered plate with body forces is formulated by using the lamination theory in which displacement fields are expressed in terms of in-plane displacements on a main plane and transverse displacement. Placing the main plane at an appropriate distance from the lower surface of the plate, a set of equilibrium equations is shown to be written in uncoupled forms, which are identical to those for an uncoupled plate such as a single layer plate. It is proved that the complete solutions of the multilayered plates subject to the specified in-plane resultant tractions or in-plane displacements on its whole boundary can be obtained from the sum of solutions for uncoupled plates. Closed form solutions are obtained for a circular laminate clamped or simply supported on its the boundary as well as for a rotating disk with a constant angular velocity. The calculations of thermoelastic stresses and displacements in multilayered plates are also discussed. Closed form solutions are obtained for a circular laminate with distributed temperature varying in the radial direction and through the thickness.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

Free vibration of laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates

This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed.     

Vibration analysis of piezoelectric FGM sensors using an accurate method

In this study, free vibration analysis of moderately thick smart FG annular/circular plates with different boundary conditions is presented on the basis of the Mindlin plate theory. This structure comprised a host FG plate and two bonded piezoelectric layers. Piezoelectric layers are open circuit therefore this plate can be used as a sensor. According to power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate while Poisson's ratio is set to be constant. Using Hamilton's principle and Maxwell electrostatic equation yields six complex coupled equations which are solved via an exact closed-form method. The accuracy of the frequencies is verified by the available literature, finite element method (FEM) and the Kirchhoff theory. The effects of plate parameters like boundary condition and gradient index are investigated and significance of coupling between in-plane and transverse displacements on the resonant frequency is proved.     

Comparative analysis of two impact-sensitive sensors for improved estimation of time difference of arrivals

In this study, an intensive comparative analysis of two impact-sensitive cantilever sensors for improved estimation of time difference of arrivals on plates is presented. Precise locating of the impact point requires exact bending wave propagation velocity due to the impact and time differences between each sensor. However since the bending wave velocity is dispersive on a plate, it is very difficult to decide the velocity when a vibration sensor such as an accelerometer is used. The new sensor philosophy is to provide a more exact bending wave velocity by designing the sensor as a cantilever type with a piezo transducer. Thus in this paper, two different cantilever sensors are considered: a rectangular type with a piezoelectric magnesium niobate ?? lead titanate single crystal patch and a circular type with a piezoelectric zirconate titanate disc. Their responses against impacts are compared with the response of a commercial accelerometer. The two sensors were manufactured and tested on a glass plate and an MDF plate separately. The impact test results showed that the two sensors give narrow band responses against impacts on the both plates and can provide a constant bending wave propagation velocity.     

Deflections and free vibrations of laminated plates—Levy-type solutions

This study is concerned with the static deflections and natural frequencies of isotropic, orthotropic/laminated composite plates using a Levy-type solution. Mindlin plate theory is applied in conjunction with the state-space concept to find such solutions. A state-space formulation of such plates is composed of variables having physical meanings, such as moments, shear forces, displacements and rotations. The influences of aspect ratio, ratio, fiber orientation angle, laminate-layer arrangement and ratio of moduli have been investigated. Some numerical results from the present analyses are compared with published results and good agreement is found.     

Analysis of stiffened corrugated plates based on the FSDT via the mesh-free method

The elastic bending of unstiffened and stiffened corrugated plates is studied in this paper, and a mesh-free Galerkin method is presented for the analyses. A corrugated plate is treated as an orthotropic plate that has different flexure properties in two perpendicular directions. The equivalent flexure properties are estimated by applying constant curvature conditions to the corrugated sheet. The stiffened corrugated plate is considered as a composite structure of an orthotropic plate with beams. By superimposing the strain energy of the orthotropic plate and the beams, and imposing the displacement compatibility conditions between the plate and the beams, the stiffness matrix of the structure is obtained. Because no mesh is needed in the proposed method, there is no limitation to the position of the stiffeners (beams). Changes in the positions of the stiffeners do not require the re-meshing of the plate. Several numerical examples are employed to show the accuracy and convergence of the proposed method. The computation results demonstrate good agreement with the solutions given by ANSYS, and different profiles of corrugated plates are considered.     

Stochastic analysis of compositionally graded plates with system randomness under static loading

This paper presents an investigation of the stochastic bending response of moderately thick, compositionally graded plates with uncertainties of low variability and subjected to lateral load and uniform temperature change. System parameters such as the thermal and mechanical material properties of each constituent material, volume fraction index, and load intensity are taken as independent random variables. The basic formulations are based on Reddy's higher-order shear deformation plate theory and a semi-analytical method. A first-order perturbation technique is employed to obtain the second-order response statistics-mean and variance of the flexural deflection of plates with various boundary conditions. Typical results are presented for two types of plates containing functionally graded materials made of metallic phase Ni and ceramic phase Al₂O₃. It is found that the response sensitivity of the plate is very much dependent on the material composition. Variations in Young's modulus and lateral load have dominant effects on the stochastic characteristics compared to other random parameters. The deflection dispersion of compositionally graded plates shows the so-called “non-intermediate” characteristic even when thermal loading is absent.     

Simulation of three-dimensional deformation in rolling by the finite-element method

A method for simulating three-dimensional deformation in plate rolling and edge rolling is established on the basis of the rigid-plastic finite-element method. For the simulation with small number of elements, simplified elements which represent three-dimensional deformation in rolling with grooveless rolls are developed. Solutions for some problems associated with simulation of steady-state rolling of work-hardening plates are given. The three-dimensional behaviour of deforming plate in single-pass plate rolling and multi-pass edge rolling is simulated by assuming a constant coefficient of friction between the rolls and the plate. The computed shapes of the plates and loads are found to be in good agreement with the experimental ones carried out with aluminium plates.     

Mixed-type finite element formulation of higher order shear deformation theory for the linear and nonlinear analyses of a laminated composite plate

For the linear and nonlinear analyses of a laminated composite plate structure, the mixed type finite element program is developed on the basis of higher order shear deformation theory of laminated plates. The accuracy of this program is checked by means of comparing with the existing results for laminated rectangular plates and is found to agree well with them. Deformations and interlaminar stresses of laminated plates are calculated according to the variation of layer numbers, fiber orientations, and plate thicknesses, so that the shear and nonlinear effects on their behaviors are studied. It is found that plate deformations are reduced by means of arranging the fiber direction into the angle-ply and increasing layer numbers.     

A four variable refined plate theory for nonlinear cylindrical bending analysis of functionally graded plates under thermomechanical loadings

In this paper we present a new application for a four variable refined plate theory to analyse the nonlinear cylindrical bending behavior of functionally graded plates subjected to thermomechanical loadings. This recent theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The non-linear strain-displacement relations in the von Karman sense are used to study the effect of geometric non-linearity. The solutions are achieved by minimizing the total potential energy and the results are compared to the classical and the first-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the nonlinear cylindrical bending behavior of functionally graded plates.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-free Galerkin method

A mesh-free Galerkin method for the free vibration analysis of unstiffened and stiffened corrugated plates is introduced in this paper, in which the corrugated plates are simulated with an equivalent orthotropic plate model. To obtain the corresponding equivalent elastic properties for the model, a constant curvature state is applied to the corrugated sheet. The stiffened corrugated plates are treated as composite structures of equivalent orthotropic plates and beams, and the strain energies of the plates and beams are added up by the imposition of displacement compatible conditions between the plate and the beams. The stiffness matrix of the whole structure is then derived. The proposed method is superior to the finite element methods (FEMs) because no mesh is needed, and thus stiffeners (beams) do not need to be placed along the mesh lines and the necessity of remeshing when the positions of the stiffeners change is avoided. To demonstrate the accuracy and convergence of the proposed method, several numerical examples are analyzed both with the proposed method and the finite element commercial software ANSYS. Examples from other research are also employed. A good agreement between the results for the proposed method, the results of the ANSYS analysis, and the results from other research is observed. Both sinusoidally and trapezoidally corrugated plates are studied.     

Free vibration of isosceles triangular mindlin plates

The free flexural vibration of thick isosceles triangular plates based on the Mindlin shear deformation theory is investigated. The pb-2 Rayleigh-Ritz method, proposed earlier by the authors, is employed for solution. New sets of vibration frequency parameters for isosceles triangular Mindlin plates with various apex angles, thickness-to-width ratios and different combinations of free, simply supported and clamped edges are presented. These results are valuable to designers and may serve as the benchmark values for future numerical techniques and software packages in thick plate analyses. Since no results are available to date for such thick triangular plates, comparisons of results can only be made in the special case of small thickness-to-width ratios with existing thin plate results. In all cases, they are found to be in very close agreement. The influences of shear deformation and rotatory inertia on the frequency parameters are examined.     

Vibrational characteristics of annular plates and rings of radially varying thinkness

In this paper, annular plates having thickness variation are studied by deriving the equations of motion on the basis of the Mindlin plate theory. The Chebyshev collocation method is employed to solve the differential equation governing the transverse motion of such plates. The dimensionless frequencies are evaluated for different values of taper constant (α), thickness ratio (h _u). radii ratio (ε) and power (n). The results of an experimental investigation are also presented, and the agreement between these findings and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and transverse shear deformation reduce the natural frequencies for all boundary conditions and for all values ofn. h _o, ∈, a ands (mode number). This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (n?1)th power function for positive values of α, and vice versa for negative values ofe for all three boundary conditions. Moreover, there is a proof that the natural frequencies of annular plates tend to be higher as the taper constant decrease and/or as the radii ratio increase for all three boundary conditions and for all values ofn, s andh _o.     

Buckling of laminated sandwich plates subjected to partial edge compression

The buckling characteristics of sandwich plates having laminated stiff layers are studied for different types of partial edge loadings using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piecewise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. It is quite interesting to note that this plate model having all these refined features requires unknown parameters only at the reference plane. To have a generality in the present analysis, finite element technique is adopted and it is carried out with newly developed triangular element, as existing finite elements cannot accommodate this plate model. So far, no solution exists in the literature for the problem of sandwich plate subjected to partial edge loading. The present analysis is first validated for the case of an isotropic plate subjected to partial edge compression and then it is extended to analyze sandwich plates. Few results are presented.