An efficient and simple refined theory for buckling analysis of functionally graded plates

In this paper, an efficient and simple refined theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, 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 mechanical properties of functionally graded material are assumed to vary according to a power law distribution of the volume fraction of the constituents. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates are obtained. Comparison studies are performed to verify the validity of present results. The effects of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of functionally graded plates are investigated and discussed.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.     

Post-bucking of angle-ply laminated plates under thermal loading

Notionsa. b, h Plate dimensionsL', [-. [1- mid-plane displacement componentsu- v- Ic dboensionless mid-plane displacement componentsVy., ac'~ slOPeS in xo and gi plane, ropectivelyJll, N number of terms in Cheby-shev series in x and y directions, respectivelyCCCC all edges clampedSSSS all edges simply supportedCCCS three edges (x = fi and y = 1) clamped and one (y = --1) simply supportedCCSS two edges (x = 11) clamped and two (y = fi) simply supportedCSSS one edge (x = --1) clamped …     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations

The static response of simply supported functionally graded plates (FGP) subjected to a transverse uniform load (UL) or a sinusoidally distributed load (SL) and resting on an elastic foundation is examined by using a new hyperbolic displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the hyperbolic shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

Generalized shear deformation theory for bending analysis of functionally graded plates

In this study, the static response is presented for a simply supported functionally graded rectangular plate subjected to a transverse uniform load. The generalized shear deformation theory obtained by the author in other recent papers is used. This theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. 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. The equilibrium equations of a functionally graded plate are given based on a generalized shear deformation plate theory. The numerical illustrations concern bending response of functionally graded rectangular plates with two constituent materials. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, and volume fraction distributions are studied. The results are verified with the known results in the literature.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Mechanical and thermal postbuckling of shear-deformable FGM plates with temperature-dependent properties

An analytical approach to investigating the stability of simply supported rectangular functionally graded plates under in-plane compressive, thermal, and combined loads is presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents. The equilibrium and compatibility equations for the plates are derived by using the first-order shear deformation theory of plates, taking into account both the geometrical nonlinearity in the von Karman sense and initial geometrical imperfections. The resulting equations are solved by employing the Galerkin procedure to obtain expressions from which the postbuckling load–deflection curves can be traced by an iterative procedure. A stability analysis performed for geometrically midplane-symmetric FGM plates shows the effects of material and geometric parameters, in-plane boundary conditions, temperature-dependent material properties, and imperfections on the postbuckling behavior of the plates.     

A new sinusoidal shear deformation theory for bending,buckling, and vibration of functionally graded plates

A new sinusoidal shear deformation theory is developed for bending, buckling, and vibration of functionally graded plates. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional sinusoidal shear deformation theory, the proposed sinusoidal shear deformation theory contains only four unknowns and has strong similarities with classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of simply supported plates are obtained and the results are compared with those of first-order shear deformation theory and higher-order shear deformation theory. It can be concluded that the proposed theory is accurate and efficient in predicting the bending, buckling, and vibration responses of functionally graded plates.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions

This paper presents an analytical investigation on the nonlinear response of thick functionally graded doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, explicit relations of load-deflection curves for simply supported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the panels are analyzed and discussed. The novelty of this study results from accounting for higher order transverse shear deformation and panel-foundation interaction in analyzing nonlinear stability of thick functionally graded cylindrical and spherical panels.     

Natural frequencies of rectangular Mindlin plates coupled with stationary fluid

The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated.     

Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory

In the present paper, a non-classical model for functionally graded annular sector microplates under distributed transverse loading is developed based on the modified couple stress theory and the first-order shear deformation plate theory. The model contains a single material length scale parameter which can capture the size effect. The material properties are graded through the thickness of plates according to a power-law distribution of the volume fraction of the constituents. The equilibrium equations and boundary conditions are simultaneously derived from the principle of minimum total potential energy. The system of equilibrium equations is then solved using the generalized differential quadrature method. The effects of length scale parameter, power-law index and geometrical parameters on the bending response of annular sector plates subjected to distributed transverse loading are investigated.     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

An explicit exact analytical approach for free vibration of circular/annular functionally graded plates bonded to piezoelectric actuator/sensor layers based on Reddy’s plate theory

In the present study, a novel exact closed-form procedure based on the third order shear deformation plate theory is developed to analyze in-plane and out-of-plane frequency responses of circular/annular functionally graded material (FGM) plates embedded in piezoelectric layers for both close/open circuit electrical boundary conditions. Introducing a new analytical method, five governing partial deferential equations of motion beside Maxwell electrostatic equation are solved via an exact closed-form method. The high accuracy and reliability of the present approach is confirmed by comparing some of the present data with their counterparts reported in the literature. Finally, the effect of material properties, power law index and boundary conditions on the free vibration of the smart FGM plate are studied and discussed in detail.     

功能梯度板的柱面弯曲弹性力学解

在推广后的England-Spencer功能梯度板理论基础上,研究了功能梯度板在不同荷载作用下的柱面弯曲问题.采用该理论中的位移展开公式,并且材料参数沿板厚方向可以任意连续变化,并将材料由各向同性推广到正交各向异性.假设板在y方向无限长,最终建立了一个从弹性力学理论出发的正交各向异性功能梯度板在横向分布荷载作用下柱面弯曲问题的板理论.通过算例分析,讨论了边界条件、材料梯度及板厚跨比等因素对功能梯度板静力响应的影响.