Showing papers on "Plate theory published in 2011"

A non-classical Mindlin plate model based on a modified couple stress theory

Abstract: A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.

A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient

Abstract: The novelty of this paper is the use of four variable refined plate theory for free vibration analysis of plates made of functionally graded materials with an arbitrary gradient. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. 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. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents with an arbitrary gradient. The equation of motion for FG rectangular plates is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. In the case of FG clamped plates, the free vibration frequencies are obtained by applying the Ritz method where the four displacement components are assumed as the series of simple algebraic polynomials. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the free vibration behavior of FG plates. Illustrative examples are given also to show the effects of varying gradients, aspect ratios, and thickness to length ratios on the free vibration of the FG plates.

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

Abstract: A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.

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

Abstract: 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.

Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method

Abstract: Nonlinear dynamic analysis of a cantilever functionally graded materials (FGM) rectangular plate subjected to the transversal excitation in thermal environment is presented for the first time in this paper. Material properties are assumed to be temperature dependent. The nonlinear governing equations of motion for the FGM plate are derived based on Reddy’s third-order plate theory and Hamilton’s principle. The first two vibration mode functions satisfying the boundary conditions of the cantilever FGM rectangular plates are chosen to be the admissible displacement functions. Galerkin’s method is utilized to convert the governing partial differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under combined external excitations. The present study focuses on resonance case with 1:1 internal resonance and subharmonic resonance of order 1/2. The asymptotic perturbation method is employed to obtain four nonlinear averaged equations which are then solved by using Runge–Kutta method to find the nonlinear dynamic responses of the plate. It is found that chaotic, periodic and quasi-periodic motions of the plate exist under certain conditions and the forcing excitations can change the form of motions for the FGM rectangular plate.

An efficient meshfree method for vibration analysis of laminated composite plates

Abstract: A detailed analysis of natural frequencies of laminated composite plates using the meshfree moving Kriging interpolation method is presented. The present formulation is based on the classical plate theory while the moving Kriging interpolation satisfying the delta property is employed to construct the shape functions. Since the advantage of the interpolation functions, the method is more convenient and no special techniques are needed in enforcing the essential boundary conditions. Numerical examples with different shapes of plates are presented and the achieved results are compared with reference solutions available in the literature. Several aspects of the model involving relevant parameters, fiber orientations, lay-up number, length-to-length, stiffness ratios, etc. affected on frequency are analyzed numerically in details. The convergence of the method on the natural frequency is also given. As a consequence, the applicability and the effectiveness of the present method for accurately computing natural frequencies of generally shaped laminates are demonstrated.

An Analytical Model for Free Vibration Analysis of Smart Annular FGM Plates Integrated with Piezoelectric Layers

Abstract: In this paper, a nonlinear free vibration analysis of thin annular functionally graded (FG) plate integrated with two uniformly distributed actuator layers made of piezoelectric (PZT4) material on the top and bottom surfaces of the annular FG plate is presented based on Kirchhoff plate theory. The material properties of the FGM core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents and the distribution of electric potential field along the thickness direction of piezoelectric layers is simulated by a sinusoidal function such that the Maxwell static electricity equation is satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. The analytical solutions are derived and validated by comparing the obtained resonant frequencies of the piezoelectric coupled FG annular plate with those of an isotropic core plate. In numerical study, the emphasis is placed on investigating the effect of varying the gradient index of FG plate on free vibration characteristics of the structure. In addition, good agreement between the results of this paper and those of finite element (FE) analyses validated the present approach. The analytical solutions and findings contribute towards a simplified model for the parametric study and understanding of vibration of piezoelectric-coupled FGM annular plate.

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

Abstract: This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton's principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-order theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.

Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method

Abstract: Buckling study of plates subjected to uniformly uniaxial, biaxial in-plane compression and pure shear loads using an efficient novel meshfree method is presented in this paper. The moving Kriging (MK) interpolation technique satisfying the Kronecker delta function property is employed to construct the shape functions. To allow for the effect of transverse shear deformation on thick plates, the first-order Reissner–Mindlin plate theory (FSDT) is adopted. The new formulation enables us to eliminate shear-locking demonstrated by various numerical examples involving both thin and moderately thick plates. It is found that the results achieved by the present approach match well with those obtained by other existing numerical approaches and analytical solutions, which illustrates the applicability, the effectiveness and the accuracy of the method.

An exact solution for free vibration of thin functionally graded rectangular plates

Abstract: The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been...

Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory

Abstract: The present study proposes an analytical solution for the buckling analysis of rectangular nanoplates. In order to extract characteristic equations of the micro/nanoscale plate under in-plane load, the analysis procedure is based on the nonlocal Mindlin plate theory considering the small scale effects. The nonlocal Mindlin plate theory allows for small scale effects. The results show that buckling loads of biaxially compressed micro/nanoscale plate depend on the nonlocal parameter. In addition, the effects of small length scale on buckling loads are graphically presented for different geometrical parameters such as aspect ratios and loading factors. This study might be useful for the design of nanoelectronic devices such as atomic dust detectors and biological sensors.