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.     

Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko's beam theory

The static, dynamic, and buckling behavior of partial interaction composite members is investigated in this paper by taking into account for the influences of rotary inertia and shear deformations. The governing differential equations obtained are very comprehensive, covering and extending the current models for the problems that are based on Euler–Bernoulli beam theory. The analytical solutions of the deflection are then found for the beam with uniformly distributing load under common boundary conditions. The free vibration and buckling behavior are also studied and the analytical expressions of the frequencies of the simply supported beam are obtained explicitly, as are the buckling loads. For other boundary conditions, the eigen-equations are transcendental and thus some numerical examples are presented to demonstrate the effects of the shear deformation and rotary inertia on the resonant frequencies and buckling loads.     

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.     

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.     

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.     

Thermally induced buckling of functionally graded hybrid composite plates

In this paper, thermal buckling analysis is performed on hybrid functionally graded plates (FGPs) with an arbitrary initial stress. The governing equations are derived using the average stress method, including the effect of transverse shear deformation. Then, an eigenvalue problem is formed to evaluate thermal buckling temperatures for simple supported initially stressed ceramic-FGM-metal plates. The effects of functionally graded material (FGM) layer thickness, volume fraction index, layer thickness ratio, thickness ratio, aspect ratio and initial stress on the thermal buckling temperature of hybrid FGPs are investigated. The results reveal that the volume fraction index, initial stresses and FGM layer thickness have significant influence on the thermal buckling of hybrid FGPs.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

Nonlinear stability analysis of infinitely long laminated cylindrical shallow shells including shear deformation under lateral pressure

This paper presents a theoretical analysis for the various kinds of buckling behaviour of infinitely long laminated cylindrical shallow shells subjected to lateral uniform pressure. The exact solutions of the nonlinear equilibrium equations, in which first-order shear deformation is included, are obtained and the buckling criteria corresponding to different kinds of buckling are constructed taking into account the effects of the transverse shear deformation.     

Vibrations of an initially stressed transversely isotropic circular thick plate

Equations of motion for a transversely isotropic circular thick plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The natural frequencies of axisymmetric clamped plates and simply supported plates subjected to initial stresses are investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are solved by the Galerkin method. The effects of various parameters on the natural frequencies are studied.     

Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness

The free vibration analysis of laminated conical shells with variable stiffness is presented using the method of differential quadrature (DQ). The stiffness coefficients are assumed to be functions of the circumferential coordinate that may be more close to the realistic applications. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations. In the DQ method, the governing equations and the corresponding boundary conditions are replaced by a system of simultaneously algebraic equations in terms of the function values of all the sampling points in the whole domain. These equations constitute a well-posed eigenvalue problem where the total number of equations is identical to that of unknowns and they can be solved readily. By vanishing the semivertex angle (α) of the conical shell, we can reduce the formulation of laminated conical shells to that of laminated cylindrical shells of which stiffness coefficients are the constants. Besides, the present formulation is also applicable to the analysis of annular plates by letting α=π/2. Illustrative examples are given to demonstrate the performance of the present DQ method for the analysis of various structures (annular plates, cylindrical shells and conical shells). The discrepancies between the analyses of laminated conical shells considering the constant stiffness and the variable stiffness are mainly concerned.     

Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses

Natural frequencies and buckling stresses of angle-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Numerical results are compared with those of the published existing theories. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. The present global higher-order approximate theories can predict the natural frequencies, buckling stresses and modal stresses of thick multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers.     

Free vibration of antisymmetric angle-ply-laminated plates including transverse shear deformation: Spline method

A free vibration study of antisymmetric angle-ply composite plates including shear deformation and rotatory inertia using the point collocation method and applying spline function approximations is presented. The equations of motion for the plate are derived using the theory of Yang, Norris and Stavsky. The solution is assumed in a separable form to obtain a system of coupled differential equations in displacement and rotational functions and these functions were approximated by Bickley-type splines of order three. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two- and four-layered plates, made up of several types of layer materials and subjected to two types of boundary conditions are considered. Parametric studies were made of the variation of frequency parameters with respect to the aspect ratio, side-to-thickness ratio and ply angle. The numerical results are presented through diagrams and, in some cases, are compared with results obtained by FEM.     

Hygrothermal effects on the postbuckling of shear deformable laminated plates

The influence of hygrothermal effects on the postbuckling of shear deformable laminated plates subjected to a uniaxial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected by the variation of temperature and moisture, and are based on a micro-mechanical model of a laminate. The governing equations of a laminated plate are based on Reddy's higher-order shear deformation plate theory that includes hygrothermal effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, fiber volume fraction and initial geometric imperfections are studied.     

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.     

Nonlinear cylindrical bending analysis of shear deformable functionally graded plates under different loadings using analytical methods

An exact solution is presented for the nonlinear cylindrical bending and postbuckling of shear deformable functionally graded plates in this paper. A simple power law function and the Mori–Tanaka scheme are used to model the through-the-thickness continuous gradual variation of the material properties. The von Karman nonlinear strains are used and then the nonlinear equilibrium equations and the relevant boundary conditions are obtained using Hamilton's principle. The Navier equations are reduced to a linear ordinary differential equation for transverse deflection with nonlinear boundary conditions, which can be solved by exact methods. Finally, by solving some numeral examples for simply supported plates, the effects of volume fraction index and length-to-thickness ratio are studied. It is shown that there is no bifurcation point for simply supported functionally graded plates under compression. The behavior of near-boundary areas predicted by the shear deformation theory and the classical theory is remarkably different.     

Nonlinear flexural vibration of rectangular moderately thick plates and sandwich plates

Nonlinear flexural vibration is investigated for rectangular Reissner moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Highly accurate solutions of series form with many different movable and immovable boundary conditions, especially with unsymmetrical boundary conditions, are obtained by means of the method of harmonic balance and by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated in numerical computations. Solutions of the nonlinear fundamental frequency of rectangular plates are obtained by iteration. The multimode approach includes not only the influences of transverse shearing deformation and rotatory inertia, but also the coupling effect of vibrating modes on the nonlinear fundamental frequency. The present solutions are satisfactory in comparison with other available results.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

Buckling and bending behaviour of initially stressed specially orthotropic thick plates

The virtual work theorem is employed to derive the governing equations of specially orthotropic plates in a general state of non-uniform initial stress, where the effects of transverse shear are included. The equations were adjusted to a generic form by using appropriate transformations. Considering the transformations, introducing generalized parameters and employing a similarity parameter, comprehensive solutions are found to this problem; hence, the curves presented in the text are generic rather than specific. The thick plate equations are solved for a simply-supported rectangular specially orthotropic plate in a state of combined uniform uniaxial compression and bending. The results of the reduced set of equations agree with those obtained by previous investigators.