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This paper proposes a new general framework of higher-order shear deformation theory (HSDT) and solves the structural responses of the functionally graded (FG) plates using novel exponential shape functions for the Ritz method. Based on the fundamental equations of the elasticity theory, the displacement field is expanded in a unified form which can recover to many different shear deformation plate theories such as zeroth-order shear deformation plate theory, third-order shear deformation plate theory, various HSDTs and refined four-unknown HSDTs. The characteristic equations of motion are derived from Lagrange's equations. Ritz-type solutions are developed for bending, free vibration and thermal buckling analysis of the FG plates with various boundary conditions. Three types of temperature variation through the thickness are considered. Numerical results are compared with those from previous studies to verify the accuracy and validity of the present theory. In addition, a parametric study is also performed to investigate the effects of the material parameters, side-to-thickness ratio, temperature rise and boundary conditions on the structural responses of the FG plates. [ABSTRACT FROM AUTHOR]

In this paper, the Ritz-based solutions are developed for the bending, buckling and vibration behaviors of laminated composite beams with arbitrary lay-ups. A quasi-3D theory, which accounts for a higher-order variation of both the axial and transverse displacements, is used to capture the effects of both shear and normal deformations on the behaviors of composite beams. Numerical results for various boundary conditions are presented and compared with existing ones available in the literature. Besides, the effects of fiber angle, span-to-height ratio, material anisotropy and Poisson's ratio on the displacements, stresses, natural frequencies and buckling loads of the composite beams are investigated. [ABSTRACT FROM AUTHOR]

This paper presents a finite element model for free vibration and buckling analyses of functionally graded (FG) sandwich beams by using a quasi-3D theory in which both shear deformation and thickness stretching effects are included. Sandwich beams with FG skins-homogeneous core and homogeneous skins-FG core are considered. By using the Hamilton’s principle, governing equations of motion for coupled axial–shear–flexural–stretching response are derived. The resulting coupling is referred to as fourfold coupled vibration and buckling. Numerical examples are carried out to investigate the thickness stretching effect on natural frequencies and critical buckling loads as well as mode shapes of sandwich beams for various power-law indexes, skin–core–skin thickness ratios and boundary conditions. [ABSTRACT FROM AUTHOR]

A new four-node quadrilateral plate that accounts for shear deformation effect and all couplings from the material anisotropy is developed for laminated composite plates. Lagrangian linear interpolation functions are used to describe the primary variables corresponding to the in-plane displacements, while Hermitian cubic interpolation functions are considered for the transverse displacement. Since the present element is derived based on a refined plate theory that has strong similarity with the classical plate theory, it is capable of modeling both thin and very thick plates without shear locking. The accuracy of the present formulation is verified by comparing the results obtained with those available in the open literature. Numerical results are presented to investigate the effects of thickness ratio, lamination angle and lay-up on the shear deformation and response of laminates. [ABSTRACT FROM PUBLISHER]

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small-scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated. [ABSTRACT FROM PUBLISHER]

An improved transverse shear stiffness for vibration and buckling analysis of functionally graded sandwich plates based on the first-order shear deformation theory is proposed in this paper. The transverse shear stress obtained from the in-plane stress and equilibrium equation allows to analytically derive an improved transverse shear stiffness and associated shear correction factor of the functionally graded sandwich plate. Sandwich plates with functionally graded faces and both homogeneous hardcore and softcore are considered. The material property is assumed to be isotropic at each point and vary through the plate thickness according to a power-law distribution of the volume fraction of the constituents. Equations of motion and boundary conditions are derived from Hamilton’s principle. The Navier-type solutions are obtained for simply supported boundary conditions, and exact formulae are proposed and compared with the existing solutions to verify the validity of the developed model. Numerical results are obtained for simply supported functionally graded sandwich plates made of three sets of material combinations of metal and ceramic, Al/Al2O3, Al/SiC and Al/WC to investigate the effects of the power-law index, thickness ratio of layer, material contrast on the shear correction factors, natural frequencies and critical buckling loads as well as load–frequency curves. [ABSTRACT FROM PUBLISHER]

Abstract: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory is presented. The core of sandwich beam is fully metal or ceramic and skins are composed of a functionally graded material across the depth. Governing equations of motion and boundary conditions are derived from the Hamilton’s principle. Effects of power-law index, span-to-height ratio, core thickness and boundary conditions on the natural frequencies, critical buckling loads and load–frequency curves of sandwich beams are discussed. Numerical results show that the above-mentioned effects play very important role on the vibration and buckling analysis of functionally graded sandwich beams. [Copyright &y& Elsevier]

Abstract: This paper presents a zeroth-order shear deformation theory for bending and vibration analyses of functionally graded plates resting on elastic foundation. In the present theory, the shear deformation effect is incorporated in the in-plane displacements through the use of shear forces instead of rotational displacements as in existing shear deformation theories. Equations of motion and boundary conditions are derived from Hamilton's principle. Analytical solutions of simply supported plates are presented, and the obtained results are compared with available solutions to verify the accuracy of the present theory. Numerical results show that the present theory gives a very good prediction of bending and vibration responses of functionally graded plates resting on elastic foundation. [Copyright &y& Elsevier]

Abstract: In this paper, finite element formulation of various four unknown shear deformation theories is presented for the bending and vibration analyses of functionally graded plates. The present theories have strong similarity with the classical plate theory and accounts for shear deformation effects without using any shear correction factors. A four-node quadrilateral finite element is developed using Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement, respectively. Material properties are assumed to be graded in the thickness direction according to a power-law distribution in terms of volume fractions of the constituents. Convergence test and comparison studies are performed for thin and very thick plates to demonstrate the accuracy of the present formulation. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: A simple refined shear deformation theory is proposed for bending, buckling, and vibration of thick plates resting on elastic foundation. The theory accounts for parabolic 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. The number of unknowns of present theory is two as against three in the case of other shear deformation theories. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived from Hamilton's principle. Analytical solutions 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 can be concluded that the proposed theory is accurate and efficient in predicting the bending, buckling, and vibration responses of thick plates resting on elastic foundation. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: A size-dependent model for bending and free vibration of functionally graded plate is developed based on the modified couple stress theory and sinusoidal shear deformation theory. In the former theory, the small scale effect is taken into consideration, while the effect of shear deformation is accounted for in the latter theory. The equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for the bending and vibration problems of simply supported plates are obtained. Numerical examples are presented to illustrate the influences of small scale on the responses of functionally graded microplates. The results indicate that the inclusion of small scale effects results in an increase in plate stiffness, and consequently, leads to a reduction of deflection and an increase in frequency. Such small scale effects are significant when the plate thickness is small, but become negligible with increasing plate thickness. [Copyright &y& Elsevier]

Abstract: Size-dependent models for bending, buckling, and vibration of functionally graded Kirchhoff and Mindlin plates are developed using a modified couple stress theory. The present models contain one material length scale parameter and can capture the size effect, geometric nonlinearity, and two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton’s principle based on a modified couple stress theory, the von Karman nonlinear strains, and the power law variation of the material through the thickness of the plate. Analytical solutions for deflection, buckling load, and frequency of a simply supported plate are presented to bring out the effect of the material length scale parameter on the bending, buckling, and vibration responses of microplates. [Copyright &y& Elsevier]

Abstract: In this paper, various higher-order shear deformation beam theories for bending and free vibration of functionally graded beams are developed. The developed theories account for higher-order variation of transverse shear strain through the depth of the beam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. A shear correction factor, therefore, is not required. In addition, these theories have strong similarities with Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion and boundary conditions are derived from Hamilton''s principle. Analytical solutions are presented, and the obtained results are compared with the existing solutions to verify the validity of the developed theories. Finally, the influences of power law index and shear deformation on the bending and free vibration responses of functionally graded beams are investigated. [Copyright &y& Elsevier]

This paper proposes novel Ritz functions for the size-dependent analysis of micro laminated composite beams with arbitrary lay-ups. Displacement field is based on a higher-order deformation beam theory and size effect is captured by the modified couple stress theory. Lagrange’s equations are used to obtain the governing equations of motion. The present beam model, which can recover the classical one by neglecting the material length scale parameter, is used to predict the size-dependent responses of micro composite beams. The results indicate that the present study is efficient for bending, vibration and buckling problems of micro composite beams. Some new results are given to serve as benchmarks for future studies. [ABSTRACT FROM AUTHOR]

An analytical method for vibration and buckling behaviours of Functionally Graded (FG) beams with various boundary conditions under mechanical and thermal loads is presented. Based on linear strain-displacement relations, equations of motion and essential boundary conditions are derived from Hamilton’s principle. In order to account for thermal effects, three cases of the temperature rise through the thickness, which are uniform, linear and nonlinear, are considered. The exact solutions are derived using the state space approach. Numerical results are presented to investigate the effects of boundary conditions, temperature distributions, material parameters and slenderness ratios on the critical temperatures, critical buckling loads, and natural frequencies as well as load-frequencies curves, temperature-frequencies curves of FG beams under thermal/mechanical loads. The accuracy and effectiveness of proposed model are verified by comparison with previous research. [ABSTRACT FROM AUTHOR]

The hygro-thermal effects on vibration and buckling analysis of functionally graded beams are presented in this paper. The present work is based on a higher-order shear deformation theory which accounts for a hyperbolic distribution of transverse shear stress and higher-order variation of in-plane and out-of-plane displacements. Equations of motion are obtained from Lagrange’s equations. Ritz solution method is used to solve problems with different boundary conditions. Numerical results for natural frequencies and critical buckling temperatures of functionally graded beams are compared with those obtained from previous works. Effects of power-law index, span-to-depth ratio, transverse normal strain, temperature and moisture changes on the results are discussed. [ABSTRACT FROM AUTHOR]

A new analytical solution based on a higher-order beam theory for static, buckling and vibration of laminated composite beams is proposed in this paper. The governing equations of motion are derived from Lagrange’s equations. An analytical solution based on trigonometric series, which satisfies various boundary conditions, is developed to solve the problem. Numerical results are obtained to compare with previous studies and to investigate the effects of length-to-depth ratio, fibre angles and material anisotropy on the deflections, stresses, natural frequencies and critical buckling loads of composite beams with various configurations. [ABSTRACT FROM AUTHOR]

Based on fundamental equations of the elasticity theory, a novel unified beam model is developed for laminated composite beams. In this model, the displacement field is selected in a unified form which can be recovered to that of existing shear deformation beam theories available in the literature. Based on Lagrange's equations, the governing equations of the present theory are derived. They are then solved for deflections, stresses, natural frequencies and critical buckling loads of composite beams under different boundary conditions and lay-ups by using the Ritz approach with novel hybrid trigonometric functions. Various examples are also presented to verify the accuracy and generalization of the present theory, as well as investigate the influences of fibre angle on the behaviour of composite beams under different boundary conditions and lay-ups. [ABSTRACT FROM AUTHOR]