Normal Deformable Plate Theory Research Articles

Buckling analysis of thick functionally graded piezoelectric plates based on the higher-order shear and normal deformable theory

In this paper, buckling analysis of thick functionally graded piezoelectric rectangular plates is investigated based on the higher-order shear and normal deformable plate theory. Two cases consisting of closed and open–closed circuits are considered as electrical conditions. Using the principle of minimum total potential energy and utilizing the variational approach, nonlinear governing equations for buckling analysis of thick functionally graded rectangular plates are derived. Applying the adjacent equilibrium criterion, the linear form of the governing stability equations is determined. The electric potential function is assumed to be quadratic in terms of the thickness variable. Also, it is supposed that material properties of the functionally graded plates vary through the thickness according to the power law function. Finally, the Maxwell and stability equations are solved analytically for a simply supported thick plate to obtain the critical buckling loads. Consequently, the effects of loading conditions, aspect ratio, thickness and material properties on the critical buckling loads are investigated in detail.

Simplified Theory for Hygrothermal Response of Angle-Ply Composite Plates

A simplified four-unknown shear and normal deformation plate theory is proposed. It is used to study the response of multilayered angle-ply composite plates due to a variation in temperature and moisture concentrations. The effects of temperature and moisture concentrations on the material properties and the hygrothermal response of multilayered angle-ply composite plates are studied. A number of examples are solved to illustrate the numerical results concerning bending response of multilayered angle-ply composite plates subjected to hygrothermal effects. It is suggested that temperature-dependent and/or moisture-dependent material properties ought to be used in the analysis of laminated plates subjected to hygrothermal loads. Additional comparisons between the present theory and the classical, uniform, and parabolic shear deformation plate theories are made to show how the effect of transverse normal strain is highly significant.

Vibration analysis of functionally graded rectangular plates resting on elastic foundation using higher-order shear and normal deformable plate theory

The free vibration of simply supported functionally graded rectangular plates resting on two-parameter elastic foundation is studied using the higher-order shear and normal deformable plate theory of Batra and Vidoli by an analytical approach. All three displacement components are expanded in the thickness direction using the Legendre polynomials. The effects of transverse shear and normal deformations are considered and the equations of motion are derived using the principle of virtual work. A power law distribution is used to explain the variation of mechanical and physical properties through the thickness of the functionally graded plate. Governing equations are then derived and the natural frequencies and the corresponding mode shapes are obtained up to the fifth-order expansion. The numerical results are given in detail and compared with the existing works. It is shown that when fifth-order expansion is used, the results of this theory for natural frequencies of thick functionally graded rectangular plates are very close to those obtained from three dimensional elasticity theory.

Analysis of thick plates by local radial basis functions-finite differences method

Although global collocation with radial basis functions proved to be a very accurate means of solving interpolation and partial differential equations problems, ill-conditioned matrices are produced, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces better conditioned matrices. For scattered points, a combination of finite differences and radial basis functions avoids the limitation of finite differences to be used on special grids. In this paper, we use a higher-order shear and normal deformation plate theory and a radial basis function—finite difference technique for predicting the static behavior of thick plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.

Vibration of an incompressible isotropic linear elastic rectangular plate with a higher-order shear and normal deformable theory

We use a mixed higher-order shear and normal deformable plate theory (HOSNDPT) of Batra and Vidoli with Poisson's ratio equal to 0.49 and the finite element method to analyze vibrations of a homogeneous isotropic rectangular plate made of an incompressible linear elastic material. Through-the-thickness integrals are evaluated exactly, and those over an element in the midplane of the plate are evaluated by using the 2 × 2 Gauss quadrature rule. The plate theory equations are used to ascertain frequencies of a clamped–clamped and a clamped–free square/rectangular plate of different aspect ratios. Through-the-thickness modes of vibration valid for compressible and incompressible materials and missed by previous investigators are also identified. Computed frequencies are found to match well with those deduced from the analytical solution.

Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method

The meshless local Petrov–Galerkin (MLPG) method with radial basis functions (RBFs), and the higher order shear and normal deformable plate theory (HOSNDPT) are used to analyze static infinitesimal deformations of thick laminated composite elastic plates under different boundary conditions. Two types of RBFs, namely, multiquadrics (MQ) and thin plate splines (TPS), are employed for constructing trial functions while a fourth order spline function is used as the test function. Computed results for different lamination schemes are found to match well with those obtained by other researchers. A benefit of using RBFs over those generated by the moving least squares approximation is that no special treatment is needed to impose essential boundary conditions, which substantially reduces the computational cost. Furthermore, the MLPG method does not require nodal connectivity which reduces the time required to prepare the input data.

Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a homogeneous and isotropic thick elastic plate have been analyzed by using a meshless local Petrov–Galerkin (MLPG) method and a higher-order shear and normal deformable plate theory (HONSDPT). Radial basis functions (RBF) are employed for constructing trial solutions, while a spline function is used as the weight function over a local subdomain. The present method uses a number of randomly distributed nodes in the domain and is truly meshless. Two types of RBFs, i.e. multiquadrics (MQ) and thin plate splines (TPS), are employed and effects of their shape parameters on the quality of the computed solution are examined for deformations of thick plates under different boundary conditions. It is found that the present MLPG formulations give results very close to those obtained by other researchers. A benefit of using RBFs is that no special treatment is needed to impose the essential boundary conditions, which substantially reduces the computational cost.

Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a thick functionally graded elastic plate have been analyzed by using the meshless local Petrov-Galerkin (MLPG) method and the higher-order shear and normal deformable plate theory (HOSNDPT). Radial basis functions (RBF) are employed for constructing trial functions, while a spline function is used as the weighting function over a local subdomain. The present method employs a number of randomly located nodes in the domain. It does not require a mesh for either interpolation of the field variables or integration of the weak form and hence is truly meshless. Two types of RBFs, i.e. Multiquadrics (MQ) and Thin Plate Splines (TPS), are used. Effective material moduli of the plate, made of two isotropic constituents with volume contents varying only in the thickness direction, are computed using the Mori-Tanaka homogenization technique. Computed results for a simply supported plate are found to agree well with their analytical solutions. The through-the-thickness distributions of the deflection and stresses in the plate are also presented for three types of boundary conditions: simply supported, clamped and free.

A further study on nonlinear vibration of initially stressed plates

Numerical solutions of an arbitrary initially stressed plate based on various displacement fields under nonlinear vibration are presented. Nonlinear partial differential equations of plate vibration are formulated from Lo’s high order transverse shear and transverse normal deformation plate theory. The higher order terms for Lo’s displacement field are neglected to obtain various simpler forms of equations such as the first order theory and other higher order theories for isotropic plate. These nonlinear partial equations of different forms are then transformed by the Galerkin method to ordinary nonlinear differential equations. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequencies of vibration. By using these equations, the nonlinear vibration of simply supported initially stress plates with various plate theories are investigated. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, Poisson ratio, initial stress and plate theory. The study concludes that the surprising discrepancies exist among the various plate theories, which indicate the transverse normal strain, nonuniform shear stress and initial stress have great effect on the vibration behavior of plate under nonlinear vibration.

Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories

We use a higher order shear and normal deformable plate theory of Batra and Vidoli and the finite element method to analyze free vibrations and stress distribution in a thick isotropic and homogeneous plate. The transverse shear and the transverse normal stresses and strains in the plate are considered and traction boundary conditions on the top and the bottom surfaces of the plate are exactly satisfied. All components of the stress tensor are computed from equations of the plate theory. Equations governing deformations of the plate involve second-order spatial derivatives of generalized displacements with respect to in-plane coordinates. Thus triangular or quadrilateral elements with Lagrange basis functions can be employed to find their numerical solution. Results have been computed for rectangular plates of aspect ratios varying from 4 to 20 and with all edges either simply supported or clamped, or two opposite edges clamped and the other two free. Computed frequencies, mode shapes, and through the thickness distribution of stresses for a simply supported plate are found to match very well with the corresponding analytical solutions. Advantages of the present approach include the use of Lagrange shape functions, satisfaction of traction boundary conditions on the top and the bottom surfaces and the use of the plate theory equations for accurate determination of transverse stresses. The order of the plate theory to be used depends upon several factors including the aspect ratio of the plate.

TRANSIENT THERMOELASTIC DEFORMATIONS OF A THICK FUNCTIONALLY GRADED PLATE

We study transient thermoelastic deformations of a thick functionally graded plate with edges held at a uniform temperature and either simply supported or clamped. Either the temperature or the heat flux is prescribed on the top surface of the plate with the bottom surface of the plate kept at either a uniform temperature or thermally insulated. Stresses and deformations induced due to the simultaneous application of the transient thermal and mechanical loads are also computed. The problem is solved by using a higher order shear and normal deformable plate theory and a meshless local Petrov–Galerkin method. Only nodal coordinates are needed, and neither nodal connectivity nor a background mesh is employed. The validity of the method and of the computer code is established by comparing computed results with the analytical solution of the three-dimensional thermoelasticity equations for a simply supported plate. Results are then computed for clamped plates. It is found that the centroidal deflection and the axial stress induced at the centroid of the top surface of the plate are significantly influenced by boundary conditions at the plate edges.

Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method

Static deformations, and free and forced vibrations of a thick rectangular functionally graded elastic plate are analyzed by using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless local Petrov–Galerkin (MLPG) method. All components of the stress tensor are computed from equations of the plate theory. The plate material, made of two isotropic constituents, is assumed to be macroscopically isotropic with material properties varying in the thickness direction only. Effective material moduli are computed by using the Mori–Tanaka homogenization technique. Computed results for static and free vibration problems are found to agree well with their analytical solutions. The response of the plate to impulse loads is also computed for different volume fractions of the two constituents. Contributions of the work include the use of the HOSNDPT and the MLPG method for the analysis of functionally graded plates.

Free and Forced Vibrations of Thick Rectangular Plates using Higher-Order Shear and Normal Deformable Plate Theory and Meshless Petrov-Galerkin (MLPG) Method

We use a meshless local Petrov-Galerkin Liu (2001) have used the MLPG method to find natu- (MLPG) method to analyze three-dimensional infinitesi- ral frequencies and forced plane ,strain deformations.of mal elastodynamic deformations of a homogeneous rect- a cantilever beam. Batra and Chmg .(2002~ have de!m- angular plate subjected to different edge conditions. We eated the time evolution of the stress-mtenslty factor m a employ a higher-order plate theory in which both trans- double edge-cracked plate. verse shear and transverse normal deformations are con- Atluri and Shen (2002a,b) have compared the perfor- sidered. Natural frequencies and the transient response manceof six' variants of the MLPG method for solving to external loads have been computed for isotropic Poisson's equation. Qian et al. (2002) used two of these and orthotropic plates. Computed results are found to formulations to study elastostatic deformations of a thick agree with those obtained from the analysis of the 3- rectangular plate with a compatible higher-order shear dimensional problem either analytically or by the finite and normal deformable plate theory (HOSNDPT) pro- element method.

Elastostatic Deformations of a Thick Plate by using a Higher-Order Shear and Normal Deformable Plate Theory and two Meshless Local Petrov-Galerkin (MLPG) Methods

We use two meshless local Petrov-Galerkin formulations, namely, the MLPG1 and the MLPG5, to analyze infinitesimal deformations of a homogeneous and isotropic thick elastic plate with a higher-order shear and normal deformable plate theory. It is found that the two MLPG formulations give results very close to those obtained by other researchers and also by the three- dimensional analysis of the problem by the finite element method.

PLANE WAVE SOLUTIONS AND MODAL ANALYSIS IN HIGHER ORDER SHEAR AND NORMAL DEFORMABLE PLATE THEORIES

We use the three-dimensional Hellinger–Reissner mixed variational principle to derive a K th order (K = 0,1,2,...) shear and normal deformable plate theory. The balance laws, the constitutive relations and the boundary conditions for the plate theory are deduced. The constitutive relations incorporate the shear and the normal tractions applied on the top and the bottom surfaces of the plate. For a K th order plate theory with displacements expressed as a power series in the thickness co-ordinate z with terms up to z K , the transverse shear and the transverse normal stresses involve terms upto z K +2 while in-plane stress components have terms up to z K . The equations for the plate theory are expressed in a compact form by taking Legendre polynomials as the basis functions. The plate theory is used to study plane travelling waves and in particular the lengths of decay of the displacement components; this allows for a rigorous ordering of the importance of the displacement descriptors in terms of decaying properties.Finally, we study the free vibrations of a simply supported rectangular orthotropic thick plate; results from the present theory are compared with an exact three-dimensional solution and with other plate theories. To this end, a K th order compatible plate theory is also deduced; the term “compatible” alludes to the fact that the reduction map for the stress fields is induced by the kinematical reduction map, whilst in the “mixed” models it is postulated independently. It is found that the frequencies up to the fifth mode of vibration computed with the fifth order theory and without introducing any shear correction factors match very well with the corresponding analytical solution. Also, through-the-thickness distribution of all of the stress components is found to agree well with the three-dimensional elasticitysolution, while the stress distribution obtained from the compatible plate theories deviatesconsiderably, especially for the higher modes.