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Gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1992‐1995.

A new simple parametric shear deformation theory applicable to isotropic and functionally graded plates is developed. This new theory has five degrees of freedom, provides parabolic transverse shear strains across the thickness direction and hence, it does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the plate are satisfied rigorously. The paper aims to discuss these issues.

Material properties are temperature-dependent and vary continuously through the thickness according to a power law distribution. The plate is assumed to be initially stressed by a temperature rise through the thickness. The energy functional of the system is obtained using Hamilton’s principle. Free vibration frequencies are then calculated using a set of characteristic orthogonal polynomials and by applying Ritz method for different boundary conditions.

In the light of good performance of the present theory for all boundary conditions considered, it can be considered as an excellent alternative to some two-dimensional (2D) theories for approximating the tedious and time consuming three-dimensional plate problems.

To the best of the authors’ knowledge and according to literature survey, almost all published higher order shear deformation theories have been limited to simply supported boundary conditions and without taking into account the thermal stresses effects. The existing 2D shear deformation theories of Reddy, Karama and Touratier can be easily recovered. Furthermore, this feature can be highly appreciated in an iterative design process where a large number of derived plate models can be tested by selecting only two parameters in a simple polynomial function which is computationally efficient. Finally, new results are presented to show the effect of material variation, and temperature rise on natural frequencies of the FG plate for different boundary conditions.

The purpose of this paper is to investigate an analytical modeling for the thermoelastic buckling behavior of functionally graded (FG) rectangular plates (FGM) under thermal loadings. The material properties of FGM are assumed to vary continuously through the thickness of the plate, according to the simple power-law distribution. Derivations of equations are based on novel refined theory using a new hyperbolic shear deformation theory. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. In addition, numerical results for a variety of FG plates with simply supported edge are presented and compared with those available in the literature. Moreover, the effects of geometrical parameters of dimension the length to width aspect ratio (a/b), the plate width to thickness ratio (b/h), and material properties index (k) on the FGM buckling temperature difference are determined and discussed.

In the current paper, the application of the refined theory proposed by Shimpi is based on the assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it 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. It is extended to the analysis of buckling behavior of ceramic-metal FG plates subjected to the three types of thermal loadings, namely; uniform temperature rise, linear temperature change across the thickness, and nonlinear temperature change across the thickness. The material properties of the FG plates are assumed to vary continuously through the thickness of the plate, according to the simple power-law distribution. Numerical results for a variety of FG plates with simply supported edges are given and compared with the available results, wherever possible. Additionally, the effects of geometrical parameters and material properties on the buckling temperature difference of FGM plates are determined and discussed.

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.

To the best of the authors’ knowledge, there are no research works for thermal buckling analysis of FG rectangular plates based on new four-variable refined plate theory (RPT). The novelty of this paper is extended the use of the above-mentioned RPT with the addition of a new function proposed by Shimpi for thermal buckling analysis of plates made of FG materials. Unlike any other theory, the number of unknown functions involved is only four, as against five in the case of other shear deformation theories. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. The plates subjected to the two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. Numerical results for a variety of FG plates with simply supported edges are given and compared with the available results.

The purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO₂) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.

In this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.

The bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.

For the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

The purpose of this article is to carry out the thermal buckling analysis of power and sigmoid functionally graded material Sandwich plate (P-FGM and S-FGM) under uniform, linear, nonlinear and sinusoidal temperature rise.

Thermal buckling of FGM Sandwich plates namely, FGM face with ceramic core (Type-A) and homogeneous face layers with FGM core (Type-B), incorporated with nonpolynomial shear deformation theories are considered for an analytical solution in this investigation. Effective material properties and thermal expansion coefficients of FGM Sandwich plates are evaluated based on Voigt's micromechanical model considering power and sigmoid law. The governing equilibrium and stability equations for the thermal buckling analysis are derived based on sinusoidal shear deformation theory (SSDT) and inverse trigonometric shear deformation theory (ITSDT) along with Von Karman nonlinearity. Analytical solutions for thermal buckling are carried out using the principle of minimum potential energy and Navier's solution technique.

Critical buckling temperature of P-FGM and S-FGM Sandwich plates Type-A and B under uniform, linear, non-linear, and sinusoidal temperature rise are obtained and analyzed based on SSDT and ITSDT. Influence of power law, sigmoid law, span to thickness ratio, aspect ratio, volume fraction index, different types of thermal loadings and Sandwich plate types over critical buckling temperature are investigated. An analytical method of solution for thermal buckling of power and sigmoid FGM Sandwich plates with efficient shear deformation theories has been successfully analyzed and validated.

The temperature distribution across FGM plate under a high thermal environment may be uniform, linear, nonlinear, etc. In practice, temperature variation is an unpredictable phenomenon; therefore, it is essential to have a temperature distribution model which can address a sinusoidal temperature variation too. In the present work, a new sinusoidal temperature rise is proposed to describe the effect of sinusoidal temperature variation over critical buckling temperature for P-FGM and S-FGM Sandwich plates. For the first time, the FGM Sandwich plate is modeled using the sigmoid function to investigate the thermal buckling behavior under the uniform, linear, nonlinear and sinusoidal temperature rise. Nonpolynomial shear deformation theories are utilized to obtain the equilibrium and stability equations for thermal buckling analysis of P-FGM and S-FGM Sandwich plates.

A study of the behaviour of shear deformable plate finite elements is carried out to determine why and under what conditions these elements lock, or become overly stiff. A new analytical technique is developed to derive the exact form of the shear constraints which are imposed on an element when its side‐to‐thickness ratio is large. The constraints are expressed in terms of the nodal degrees of freedom, and are interpreted as being either the proper Kirchhoff constraints or spurious locking constraints. To gain a better understanding of locking phenomena, the constraints which arise under full and reduced integration are derived for various plate elements. These include bilinear, biquadratic, eight‐node serendipity and heterosis elements. These analytical findings are compared with numerical results of isotropic and laminated composite plates, verifying the role that shear constraints play in determining the behaviour of thin shear deformable elements. The results of the present study lead to definitive conclusions regarding the origin of locking phenomena and the effect of reduced integration.

The purpose of this paper is to study the static buckling and free vibration of continuously graded ceramic-metal beams by employing a refined higher-order shear deformation, which is also the primary goal of this paper.

The proposed model is able to catch both the microstructural and shear deformation impacts without employing any shear correction factors, due to the realistic distribution of transverse shear stresses. The material properties are supposed to vary across the thickness direction in a graded form and are estimated by a power-law model. The equations of motion and related boundary conditions are extracted using Hamilton’s principle and then resolved by analytical solutions for calculating the critical buckling loads and natural frequencies.

The obtained results are checked and compared with those of other theories that exist in the literature. At last, a parametric study is provided to exhibit the influence of different parameters such as the power-law index, beam geometrical parameters, modulus ratio and axial load on the dynamic and buckling characteristics of FG beams.

Searching in the literature and to the best of the authors’ knowledge, there are limited works that consider the coupled effect between the vibration and the axial load of FG beams based on new four-variable refined beam theory. In comparison with a beam model, the number of unknown variables resulting is only four in the general cases, as against five in the case of other shear deformation theories. The actual model represents a real distribution of transverse shear effects besides a parabolic arrangement of the transverse shear strains over the thickness of the beam, so it is needless to use of any shear correction factors.

The purpose of this paper is to carry out free vibration and buckling analysis of functionally graded material (FGM) plate.

Equilibrium and stability equations of FGM rectangular plate under different boundary conditions are derived using finite element method-based inverse trigonometric shear deformation theory (ITSDT). Eight-noded rectangular plate element with seven degrees of freedom at each node is used for the present analysis. The power-law distribution method has been considered for the continuously graded variation in composition of the ceramic and metal phases across the thickness of a functionally graded plate.

The finite element formulation incorporated with ITSDT and provisions of the constitutive model of FGM plate has been implemented in a numerical code to obtain the natural frequency and critical buckling load under uniaxial and biaxial compressive load. The influence of material gradation, volume fraction index, span to thickness ratio and boundary constraints over free vibration and buckling response has been studied.

Development and validation of finite element methodology using ITSDT to predict the structural response of the FGM plates under different loading, geometric and boundary conditions.

The purpose of this paper is to assess different five variables shear deformation plate theories for the buckling analysis of FGM plates.

Governing differential equations (GDEs) of the theories are derived by employing the Hamilton Principle. A polynomial radial basis function (RBF)-based Meshless method is used to discretize the GDEs, and a MATLAB code is developed to solve these discretize equations.

Numerical results are obtained for buckling loads. The results are compared with other available results for validation purpose. The effect of the span-to-thickness ratio and grading index is observed. It is observed that some theories underpredict the deflection for thick plates, while at the same time they seem to be in good agreement with other theories for thin plates.

This paper assesses the different theories with the same method to determine their applicability.

The purpose of this paper is to propose an efficient and accurate numerical model that employs isogeometric analysis (IGA) for the geometrically nonlinear analysis of functionally graded plates (FGPs). This model is utilized to investigate the effects of boundary conditions, gradient index, and geometric shape on the nonlinear responses of FGPs.

A geometrically nonlinear analysis of thin and moderately thick functionally graded ceramic-metal plates based on IGA in conjunction with first-order shear deformation theory and von Kármán strains is presented. The displacement fields and geometric description are approximated with nonuniform rational B-splines (NURBS) basis functions. The Newton-Raphson iterative scheme is employed to solve the nonlinear equation system. Material properties are assumed to vary along the thickness direction with a power law distribution of the volume fraction of the constituents.

The present model for analysis of the geometrically nonlinear behavior of thin and moderately thick FGPs exhibited high accuracy. The shear locking phenomenon is avoided without extra numerical efforts when cubic or high-order NURBS basis functions are utilized.

This paper shows that IGA is particularly well suited for the geometrically nonlinear analysis of plates because of its exact geometrical modelling and high-order continuity.