Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

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.     

FSDPT based study for vibration analysis of piezoelectric coupled annular FGM plate

The vibration behavior of a piezoelectrically actuated thick functionally graded (FG) annular plate is studied based on first order shear deformation plate theory (FSDPT). A consistent formulation that satisfies the Maxwell static electricity equation is presented so that the full coupling effect of the piezoelectric layer on the dynamic characteristics of the annular FG plate can be estimated based on the free vibration results. 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 composite plate with those of an isotropic core plate. As a special case, assuming that the material composition of core plate varies continuously in the direction of the thickness according to a power law distribution, a comprehensive study is conducted to show the influence of functionally graded index on the vibration behavior of smart structure. Also, the good agreement between the results of this paper and those of the finite element (FE) analyses validates the presented approach. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Transient bending of a piezoelectric circular plate

Thin, piezoelectric circular plates are frequently used as active components in transducer and smart materials applications. This paper reports on the exact, explicit solution for the transient motion of a piezoelectric circular plate, built-in or simply supported on the edge and electrically grounded over the entire surface. Expressed by elementary Bessel functions and obtained via exact inverse Laplace transforms, the solution enables the efficient calculation of accurate system parameters.     

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.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint

In this study, three-dimensional free vibration and stress analyses of an adhesively bonded functionally graded single lap joint were carried. The effects of the adhesive material properties, such as modulus of elasticity, Poisson's ratio and density were found to be negligible on the first ten natural frequencies and mode shapes of the adhesive joint. Both the finite element method and the back-propagation artificial neural network (ANN) method were used to investigate the effects of the geometrical parameters, such as overlap length, plate thickness and adhesive thickness; and the material composition variation through the plate thickness on the natural frequencies, mode shapes and modal strain energy of the adhesive joint. The suitable ANN models were trained successfully using a series of free vibration and stress analyses for various random geometrical parameters and compositional gradient exponents. The ANN models showed that the support length, the plate thickness and the compositional gradient exponent played important role on the natural frequencies, mode shapes and modal strain energies of the adhesive joint whereas the adhesive thickness had a minor effect. In addition, the optimal joint dimensions and compositional gradient exponent were determined using genetic algorithm and ANN models so that the maximum natural frequency and the minimum modal strain energy conditions are satisfied for each natural frequency of the adhesively bonded functionally graded single lap joint.     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.     

Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a functionally graded rectangular plate due to nonuniform heat supply. The thermal and thermoelastic constants of the rectangular plate are assumed to vary exponentially in the thickness direction. The transient three-dimensional temperature is analyzed by the methods of Laplace and finite cosine transformations. We obtain the three-dimensional solution for the simple supported rectangular plate. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the nonhomogeneity of the material is investigated.     

Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.     

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.     

Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation

A theoretical model for geometrically nonlinear vibration analysis of piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s-Love hypothesis with von-Karman type geometrical large nonlinear deformations. To determine the initial stress state and pre-vibration deformations of the smart plate a nonlinear static problem is solved followed by adding an incremental dynamic state to the pre-vibration state. The derived governing equations of the structure are solved by exact series expansion method combined with perturbation approach. 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. Control of the FGM plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixture of ceramic and metal are presented in dimensionless forms. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FGM plate on vibration characteristics of the smart structure. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.     

Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams

An improved third order shear deformation theory is employed to investigate thermal buckling and vibration of the functionally graded beams. A power law distribution is used to describe the variation of volume fraction of material compositions. The functionally graded material properties are assumed to vary smoothly and continuously across the thickness of the beams. The Ritz method is adopted to solve the eigenvalue problems that are associated with thermal buckling and vibration in various types of immovable boundary conditions. The parametric study covered in this paper includes the effects of material composition, temperature-dependent material properties, and slenderness ratio.     

Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity

This paper presents a new approach for analyzing transverse bending and vibration of circular cylindrical beams with radial nonhomogeneity. The radial nonhomogeneity may be continuous or piecewise-constant, corresponding a functionally graded circular cylinder or a multi-layered circular cylinder, respectively. Different from the Euler-Bernoulli and Timoshenko theories of beams, our analysis considers shear deformation, but does not need to introduce a shear correction factor. Using the shear-stress-free condition at the surface of the cylinder, coupled governing equations for deflection and rotation angle are derived, and then converted to a single governing equation. The influences of gradient index on the deflection and stress distribution for cantilever and simply-supported beams are studied. Natural frequencies of free vibration of a cylindrical beam with circular cross-section are calculated for different power-law gradients. In particular, a circular cylindrical shell may be taken as a special case of a bi-layered cylinder where the material properties of the inmost cylinder vanish. For this case, the natural frequencies for simply-supported and clamped-clamped cylindrical shells are evaluated and compared with those using three-dimensional theory. Our results coincide well with the previous.     

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.