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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]

This paper presents a robust numerical model, which takes into account both size-dependent and shear deformation effects, for the bending, buckling and free vibration analyses of functionally graded microplates. The size-dependent effect is captured by using the modified strain gradient elasticity theory with three length scale parameters, whilst the shear deformation effect is accounted by using the third-order shear deformation theory. The rule of mixture is employed to describe the distributions of material phrases through the plate thickness. By using Hamilton’s principle, the governing equations are derived and then discretized by employing an Isogeometric Analysis (IGA) approach, where the Non-Uniform Rational B-Splines (NURBS) basis functions are adopted to meet the C 2 -continuity requirement. Physical mesh convergence and verification studies are performed to prove the accuracy and reliability of the present model. In addition, parametric studies are also carried out to investigate the size effect in conjunction with the influences of gradient index, shear deformation effect and boundary conditions on the responses of microplates. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive review on the development of higher-order continuum models for capturing size effects in small-scale structures. The review mainly focus on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple stress theory and strain gradient theory due to their common use in predicting the global behaviour of small-scale structures. In each higher-order continuum theory, various size-dependent models based on the classical theory, first-order shear deformation theory and higher-order shear deformation theory were reviewed and discussed. In addition, the development of finite element solutions for size-dependent analysis of beams and plates was also highlighted. Finally a summary and recommendations for future research are presented. It is hoped that this review paper will provide current knowledge on the development of higher-order continuum models and inspire further applications of these models in predicting the behaviour of micro- and nano-structures. [ABSTRACT FROM AUTHOR]

This paper presents flexural analysis of composite and sandwich beams using a quasi-3D theory, which considers simultaneously three effects such as normal and shear deformation as well as anisotropy coupling. The axial and transverse displacements are assumed to be cubic and parabolic variation through the beam depth. In order to solve problem, two-node C 1 beam elements with six degrees of freedom per node are developed. Numerical examples are carried out and the results are compared with those available in literature to validate the accuracy of the present theory. The effects of fibre angle, lay-up and span-to-height ratio on displacements and stresses are studied. Some new results, which can be useful for future references, are also given. [ABSTRACT FROM AUTHOR]

This paper proposes a new simple shear deformation theory for isotropic plates. The present theory involves one unknown and one governing equation as in the classical plate theory, but it is capable of accurately capturing shear deformation effects. The displacement field of the present theory was based on a two variable refined plate theory in which the transverse displacement is partitioned into the bending and shear parts. Based on the equilibrium equations of three-dimensional (3D) elasticity theory, the relationship between the bending and shear parts was established. Therefore, the number of unknowns of the present theory was reduced from two to one. Closed-form solutions were presented for both Navier- and Levy-type plates. Numerical results indicate that the obtained predictions are comparable with those generated by ABAQUS and available results predicted by 3D elasticity theory, first-order and third-order shear deformation theories. [ABSTRACT FROM AUTHOR]

This paper aims to develop a detailed three-dimensional finite element model to study the behaviour of blind bolted endplate composite joints that connect composite beams to a concrete filled steel tubular column. All connection components such as steel beam, profiled sheeting, shear stud, endplate, blind bolt, concrete infill, concrete slab, hollow steel column and reinforcing bars were separately modelled using shell, solid and truss elements. The obtained predictions of the moment–rotation curve and failure mode were compared with experimental results and a very good agreement was established. In addition, an analytical model for predicting the moment–rotation response of composite joints up to failure was also presented based on the component method. Finally, extensive parametric studies were carried out to investigate the influences of shear studs and reinforcement ratio, blind bolt types and diameters, column sections and thickness ratios on the joint performance. [ABSTRACT FROM AUTHOR]

This paper presents static behaviour of functionally graded (FG) sandwich beams by using a quasi-3D theory, which includes both shear deformation and thickness stretching effects. Various symmetric and non-symmetric sandwich beams with FG material in the core or skins under the uniformly distributed load are considered. Finite element model (FEM) and Navier solutions are developed to determine the displacement and stresses of FG sandwich beams for various power-law index, skin-core-skin thickness ratios and boundary conditions. Numerical results are compared with those predicted by other theories to show the effects of shear deformation and thickness stretching on displacement and stresses. [ABSTRACT FROM AUTHOR]

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: In this paper, the refined pate theory is improved to account for the effect of thickness stretching in functionally graded plates. The refined plate theory has fewer number of unknowns and equations of motion than the first-order shear deformation theory, but accounts for the transverse shear deformation effects without requiring shear correction factors. The displacement field of the refined plate theory is modified by assuming a parabolic variation of the transverse displacement through the thickness, and consequently, the thickness stretching effect is taken into consideration. Closed-form solutions of simply supported rectangular plates are presented, and the obtained results are compared with 3D solutions and those predicted by higher-order shear deformation theories. Verification studies show that the proposed theory is not only more accurate than the refined plate theory, but also comparable with the higher-order shear deformation theories which contain more number of unknowns. [Copyright &y& Elsevier]

Abstract: In this paper, a simple first-order shear deformation theory is presented for laminated composite plates. Unlike the existing first-order shear deformation theory, the present one contains only four 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 and boundary conditions are derived from Hamilton’s principle. Analytical solutions of simply supported antisymmetric cross-ply and angle-ply laminates are obtained and the results are compared with the exact three-dimensional (3D) solutions and those predicted by existing theories. Comparison studies show that this new first-order shear deformation theory can achieve the same accuracy of the existing first-order shear deformation theory which has more number of unknowns. [Copyright &y& Elsevier]

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton's principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns. [ABSTRACT FROM AUTHOR]

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: Closed-form solution for buckling analysis of thick functionally graded plate resting on elastic foundation is presented using the third-order shear deformation theory. It is assumed that the plate rests on Pasternak foundation and its material properties vary through the plate thickness as a power function. By decoupling the governing equations, the neutral surface position for such plate is determined, and the third-order shear deformation theory based on exact neutral surface position is employed to derive the governing equations. Comparing with the middle surface based formulations, the neutral surface based formulations do not exhibit the stretching–bending coupling; hence, the values of buckling load can be obtained by eigenvalue analysis. Closed-form solutions are obtained for rectangular with different boundary conditions. Numerical results are presented and discussed for a wide range of plate and foundation parameters. [Copyright &y& Elsevier]

Abstract: This paper presents a simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Unlike the conventional first-order shear deformation theory, the present first-order shear deformation theory contains only four unknowns and has strong similarities with the classical plate theory in many aspects such as governing equations of motion, boundary conditions, and stress resultant expressions. Equations of motion and boundary conditions are derived from Hamilton’s principle. Closed-form solutions of simply supported plates are obtained and the results are compared with the exact 3D and quasi-3D solutions and those predicted by other plate theories. Comparison studies show that the present theory can achieve the same accuracy of the conventional first-order shear deformation theory which has more number of unknowns. [Copyright &y& Elsevier]

Abstract: This paper presents a simple quasi-3D theory for the bending analysis of functionally graded plates. This theory accounts for both shear deformation and thickness stretching effects by a sinusoidal variation of all displacements through the thickness. By dividing the transverse displacement into bending and shear parts, the number of unknowns and governing equations of the present theory is reduced, and hence, makes it simple to use. The governing equations and boundary conditions are derived using the principle of virtual displacements. Analytical solutions are obtained for simply supported plates. The accuracy of the present theory is verified by comparing the obtained results with 3D and quasi-3D solutions and those predicted by higher-order shear deformation theories. The comparison studies show that the obtained results are not only more accurate than those obtained by higher-order shear deformation theories, but also comparable with those predicted by quasi-3D theories with a greater number of unknowns. [Copyright &y& Elsevier]

A finite element model based on sinusoidal shear deformation theory is developed to study vibration and buckling analysis of composite beams with arbitrary lay-ups. This theory satisfies the zero traction boundary conditions on the top and bottom surfaces of beam without using shear correction factors. Besides, it has strong similarity with Euler-Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. By using Hamilton's principle, governing equations of motion are derived. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results for cross-ply and angle-ply composite beams are obtained as special cases and are compared with other solutions available in the literature. A variety of parametric studies are conducted to demonstrate the effect of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads, and load-frequency curves as well as corresponding mode shapes of composite beams. [ABSTRACT FROM AUTHOR]

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: In this paper, a new higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates is developed. The present theory has only four unknowns, but it accounts for a parabolic variation of transverse shear strains through the thickness of the plate. A shear correction factor is, therefore, not required. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3D and quasi-3D solutions and those predicted by other plate theories. Results show that the present theory can achieve the same accuracy of the existing higher-order shear deformation theories which have more number of unknowns, but its accuracy is not comparable with those of 3D and quasi-3D models which include the thickness stretching effect. [Copyright &y& Elsevier]

This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton's principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates. [ABSTRACT FROM AUTHOR]

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]

Abstract: Vibration and buckling analysis of composite beams with arbitrary lay-ups using refined shear deformation theory is presented. The theory accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from Hamilton''s principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are obtained for composite beams to investigate effects of fibre orientation and modulus ratio on the natural frequencies, critical buckling loads and corresponding mode shapes. [Copyright &y& Elsevier]

Abstract: Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. 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. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed. [Copyright &y& Elsevier]

Abstract: A refined plate theory for functionally graded plates resting on elastic foundation is developed in this paper. The theory 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 number of independent unknowns of present theory is four, as against five in other shear deformation theories. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived using Hamilton’s principle. The closed-form solutions of rectangular plates are obtained. Numerical results are presented to verify the accuracy of present theory. [Copyright &y& Elsevier]

Abstract: Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton''s principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates. [Copyright &y& Elsevier]

Abstract: A two variable refined plate theory of laminated composite plates is developed in this paper. The theory accounts for parabolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of antisymmetric cross-ply and angle-ply laminates are obtained using Navier solution. Numerical results of present theory are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static bending and buckling behaviors of laminated composite plates. [Copyright &y& Elsevier]

Abstract: The first-order shear deformation beam theory for static and free vibration of axially loaded rectangular functionally graded beams is developed. In this theory, the improved transverse shear stiffness is derived from the in-plane stress and equilibrium equation and thus, associated shear correction factor is then obtained analytically. Equations of motion are derived from the Hamilton’s principle. Analytical solutions are presented for simply-supported functionally graded beams. The obtained results are compared with the existing solutions to verify the validity of the developed theory. Effects of the power-law index, material contrast and Poisson’s ratio on the displacements, natural frequencies, buckling loads and load–frequency curves as well as corresponding mode shapes are investigated. [Copyright &y& Elsevier]

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed. [ABSTRACT FROM AUTHOR]