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This study explored the disparities in bending, buckling, and vibration results of ideal and non-ideal functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) beams. The smooth and continuous profiles of material distributions of ideal FG-GPLRC beams were modified for making the controlling tracks to produce two different forms of non-ideal FG-GPLRC beams which had in-and out-stepwise distributions of material constituents across the beam’s thickness. The Halpin–Tsai model and the rule of mixture were used to predict the effective material properties of the nanocomposite beams. The closed-form solution possessing less time of computation was provided for predicting the mechanical behavior of the beams, and it was validated for accuracy by comparing with the results of the Ritz method. The study’s results suggest that non-ideal beams with an out-stepwise distribution of material constituents have a better dispersion of reinforcing nanomaterials than in-stepwise distribution. Therefore, the results of the beams with an out-stepwise distribution are closer to those of ideal beams than with in-stepwise distribution.

This study examined the transient or dynamic response of sandwich plates with a functionally graded porous core under the action of time-dependent loads. The plates had two isotropic faces at the top and bottom layers, and the middle layer was made of an open-cell material with functionally graded internal pores. By using the first-order shear deformation theory, the equations of motion used to describe the dynamic behavior of the plates were applied to generate accurate results with less computational effort. To solve the equations of motion, the Ritz method based on the Jacobi polynomials for the admissible displacements, cooperating with the time integration of Newmark, was used to find out the dynamic response of the plates. The results of the numerical experiments revealed that the plates carrying a larger number of internal pores at the middle zone of the core had a great improvement in flexural stiffness, providing less deflection under dynamic loads. The observed results of the plates’ dynamic behavior related to the effects of the porosity coefficient, plate’s geometrical ratio, dynamic loading types, porous distributions of the core, etc. are shown in the form of graphs and tables, which can be used as a benchmark for future research.

The objective of this study is to apply Gram-Schmidt orthogonalization procedure for generating displacement functions. This procedure allows us to obtain numerically stable functions to be used in...

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

In this paper, vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory. The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs. The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams. The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton’s principle, which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions. Based on the numerical experiments, it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Karman. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique . The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio , loading types, porous distributions of the core, etc., are presented in graphical and tabular forms , serving as a benchmark solution for future studies.

This paper investigates flexural vibration of functionally graded sandwich microbeams supported by different axially immovable boundary conditions. The governing equations of free vibration problem are based on Timoshenko beam theory and the modified couple stress theory which are taking into account the important effects of shear deformation, rotary inertia and material length scale parameter. To solve the governing equations presented in the forms of coupled differential equations for vibration analysis of the beams with various boundary conditions, an effective tool, namely Chebyshev collocation method, is employed to find out accurate solutions with many important parametric studies. The effects of material volume fraction index, layer thickness ratio, slenderness ratio, boundary condition, temperature rise, etc. on natural frequencies of the beams are taken into account and discussed in details. The numerical results of the beams in ambient temperature and high thermal environment are presented in several tables and figures that can serve as benchmarks for further investigations in the field of FG sandwich microbeam analysis.

This paper aims to present accurate solutions for flexural vibration of functionally graded sandwich plates resting on two-parameter elastic foundation with any combined boundary conditions. The governing equations of free vibration problem are derived from the first-order shear deformation theory that covers the important effects of shear deformation and rotary inertia. To solve the coupled differential equations governing vibration behavior of the plates with various boundary conditions, an effective tool, namely Chebyshev collocation method, is implemented to obtain the accurate solutions with several parametric studies. The influences of material volume fraction index, layer thickness ratio, side-to-height ratio, boundary conditions, etc., on natural frequencies of the plates are taken into investigation and discussed in details. Our numerical experiments reveal that the proposed method can offer the accurate frequency results of the plates as compared to those available in the literature. Additionally, the spring constants of elastic foundation have a significant impact on frequency changes of the plates. Increasing the values of spring constants leads to considerable increases of the frequencies.

This paper investigates the dynamic behavior of functionally graded sandwich plates under multiple moving loads. The first-order shear deformation theory of plates is adopted with the effects of shear deformation and rotary inertia included. By using Lagrange’s equations, the equations of motion for the dynamic behavior of the plate are derived. Then they are solved by the Ritz and Newmark time integration methods for the free and forced vibrations of the plates with different boundary conditions. To guarantee that all terms in the admissible functions can cope with the essential boundary conditions, the Gram–Schmidt procedure is used to generate the shape functions for the Ritz method. The influences of several factors on the dynamic response of the plates, such as layer thickness ratio, boundary condition, velocity, excitation frequency, phase angle, etc., are examined and discussed in detail. The numerical study indicates that the dynamic deflection has initial fluctuated growth in the low range of moving load velocity before reaching the peak at the critical velocity, which is followed by the considerable decrease in magnitude. Besides, the gaps or distances between the moving loads also play an important role in predicting the dynamic deflections of the plate when subjected to more than one moving loads.

Practical applications such as airplane wings are usually subjected to combined thermal and mechanical loads, and they hence are prone to buckling failure. Preceding works on the buckling of advanced materials, e.g., functionally graded materials, under combined thermal and mechanical loads are rather rare in literature. In this paper, we report new numerical results of thermal-mechanical buckling of functionally graded rectangular and skew plates (FGPs) under combined thermal and mechanical loads. The numerical responses of buckling are computed using isogeometric analysis (IGA) based on the first-order shear deformation plate theory (FSDT) without shear-locking effect. We present formulations and then provide validation of numerical results computed by the proposed formulation against reference existing solutions. Parametric study is also performed to explore insight into the effects of various numerical aspect ratios such as gradient index, plate aspect ratio, loading type, skew angle, and boundary condition, etc. on mechanical response of FGPs. The stability diagrams are also presented.

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail a...

In this present study, Chebyshev collocation method is utilized to solve buckling and vibration problems of functionally graded (FG) sandwich beams resting on two-parameter elastic foundation including Winkler and shear layer springs. The faces of FG sandwich beam are assumed to be made by functionally graded materials (FGMs) composing of ceramic and metal phases and the core of the beam is made from homogenous material. Timoshenko beam theory is employed to construct the governing equations of motion in order to cover the significant effects of shear deformation and rotary inertia. The beams with various boundary conditions are considered to find out their critical loadings and natural frequencies. An accuracy of the present solutions is confirmed by comparing with some available results in the literature. Moreover, many important parametric studies of layer and beam thickness ratios, material volume fraction index, spring constants, etc. are taken into investigation. According to numerical exercises, it is revealed that the spring constants of elastic foundation have significant impact on buckling and vibration results of such beams.

This paper investigates static and dynamic behavior of carbon nanotube-reinforced composite plates resting on the Pasternak elastic foundation including shear layer and Winkler springs. The plates are reinforced by single-walled carbon nanotubes with four types of distributions of uni-axially aligned reinforcement material. Exact solutions obtained from closed-form formulation based on generalized shear deformation plat theory which can be adapted to various plate theories for bending, buckling and vibration analyses of such plates are presented. An accuracy of the present solutions is validated numerically by comparisons with some available results in the literature. Various significant parameters of carbon nanotube volume fraction, spring constant factors, plate thickness and aspect ratios, etc. are taken into investigation. According to the numerical results, it is revealed that the deflection of the plates is found to decrease as the increase of spring constant factors; while, the buckling load and natural frequency increase as the increment of the factors for every type of plate.

The main objective of this paper is to apply an Adomian modified decomposition method for solving large amplitude vibration analysis of stepped beams with various general and elastic boundary conditions. Damaged or imperfect supports of beams can be modeled by using elastic boundary conditions composing of translational and rotational springs. For the beams subjected to dynamic severe loading, it is important to include the nonlinear term of axial stretching force developed by the large vibration amplitude in the governing equation for more accurate design. By using the method, the convergence studies for linear and nonlinear vibration analyses of stepped beams are shown for determining an appropriate number of terms in the solutions. The accuracy of the present results is validated numerically by comparing with some available results in the literature. New results of nonlinear frequency ratios of stepped beams with different boundary conditions are presented and discussed in detail. Aspects of st...

Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution.

This paper presents an investigation of free vibration of stiffened doubly curved shallow shells made of functionally graded materials under thermal environment. Two types of temperature rise throughout the shell thickness; namely linear and nonlinear temperature rises are considered in the present investigation. The power law distribution and Mori–Tanaka homogenization scheme are used to describe the material graduation throughout the shell thickness. In order to take into account the significant effects of shear deformation and rotatory inertia of the shell skin and its stiffeners, the first-order shear deformation theory is employed to derive the governing equations used for determining natural frequencies of the stiffened shells. The governing equations can be solved analytically to obtain exact solutions for this problem. The stiffened shells can be specialized into different forms of spherical, cylindrical and hyperbolic shells by setting components of curvature. Several parameters of material volume fraction index, geometrical ratio, temperature change, number of stiffeners, etc. that affect vibration results of the shells are investigated and discussed in detail. Based on the numerical results, it is revealed that increasing number of stiffeners leads to considerable changes in natural frequencies of the stiffened shells.

This paper investigates the dynamic response of Timoshenko beams made of functionally graded materials (FGMs). The beams are supported by various classical and non-classical boundary conditions. By using Timoshenko beam theory to establish the governing equations of motion for describing the vibration behavior of the beams, the significant effects of shear deformation and rotary inertia are taken into account. Different mathematical models that is the power law, exponential and Mori–Tanaka models are used to describe material composition across the beam thickness. To predict accurate vibration behavior of the beams, the Chebyshev collocation method (CCM) is applied to solve the vibration problem of such beams. According to the numerical results, it is revealed that the proposed modeling and analysis method can provide accurate frequency results of the beams as compared to some cases in the literature. New frequency results of the beams with different material compositions and boundary conditions are also presented for future comparison and development.

The present paper is given to investigate free vibration analysis of stepped beams produced from functionally graded materials (FGMs). The differential transformation method is employed to solve the governing differential equations of the beams to obtain their natural frequencies and mode shapes. The power law distribution is used and modified for describing material compositions across the thickness of the stepped beams made of FGM. Two main types of the stepped FGM beams in which their material compositions can be described by the modified power law distribution are selected to investigate the free vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions and material volume fraction are also covered in this paper.

The objective of this paper is to apply the differential transformation method (DTM) to solve linear and nonlinear vibration problems of elastically end-restrained beams. The method demonstrates many advantages such as rapid convergence, high accuracy, and computational stability to determine linear and nonlinear natural frequencies as well as mode shapes of such beams. The mathematical models provided in this paper can be solved easily using symbolic tools in available software packages such as Maple and Matlab. An accuracy of the present solutions is confirmed by comparing with some published results in the open literature. New numerical results of nonlinear frequency ratio of beams supported by various types of elastic boundary conditions are presented and discussed in detail. The significant effects of translational and rotational springs including vibration amplitudes on linear and nonlinear vibration results are also taken into investigation. Based on the numerical exercises, it is revealed that the...

In this paper, free vibration response of stepped beams made from functionally graded materials (FGMs) is investigated. The beams are supported by various types of elastically end constraints. The differential transformation method (DTM) is employed to solve the governing differential equations of such beams in order to obtain natural frequencies and mode shapes. The power law distribution is used and modified to describe material compositions across the thickness of the beams made of FGMs. Two main types of the stepped FGM beams in which their material compositions can be described by using the modified power law distribution are selected to investigate their vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions, spring constants and material volume fraction are taken into investigation.

Linear and nonlinear vibration problems of elastically end restrained beams made of functionally graded materials (FGMs) are investigated in this present paper. Due to porosities, possibly occurring inside FGMs during fabrication, it is therefore necessary to consider the vibration behavior of beams having porosities in this investigation. The rule of mixture is modified to describe and approximate material properties of the functionally graded beams with porosity phases. The differential transformation method (DTM) is employed to solve linear and nonlinear vibration responses of FGM beams with different kinds of elastic supports. The effects of material property distribution, spring constants and porosity volume fraction on linear and nonlinear frequencies of FGM beams are also presented and discussed in detail.

An improved third-order shear deformation theory is employed to investigate free and forced vibration responses of functionally graded plates. A power law distribution is used to describe the variation of material compositions across the plate thickness. The governing equations for vibration analysis obtained using an energy approach are then solved using the Ritz method. Two types of solutions, temperature independent and dependent material properties, are considered. Many effects of the volume fraction index, temperature, material pairs, thickness, plate aspect ratio, etc., which have significant impact on dynamic behaviour of the plates, are considered in the numerical illustrations of free and forced vibration results. At high temperatures, it is observed that the maximum deflections of the functionally graded plates subjected to the dynamic loading increase with the increase of frequency ratio and temperature.

The objective of the present paper is to investigate the bending, buckling and vibration behaviors of carbon nanotube-reinforced composite (CNTRC) beams. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are estimated by using the rule of mixture. Various shear deformation theories are employed to deal with the problems. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of bending, buckling and vibration analyses of CNTRC beams based on several higher-order shear deformation theories are presented and discussed in details. Several aspects of beam types, spring constant factors, carbon nanotube volume fraction, etc., are taken into investigation.

The purpose of the present paper is to apply the differential transformation method (DTM) to deal with the vibration problem of stepped beams with general end supports and elastically constrained ends. The method demonstrates several advantages, such as rapid convergence, high accuracy, and computational stability. Unlike some approximate approaches which require correct assumed admissible function, the differential transformation method gives all natural frequencies and their mode shapes without any frequency missing. By using the DTM algorithms fully provided in this paper with general mathematical software packets, natural frequencies and mode shapes of these beams can be obtained easily for every boundary condition. Aspects such as boundary conditions, spring constant values, stepped beam types, step ratio and step location, which have a significant impact on frequencies and mode shapes, are taken into investigation in this paper.

An improved third order shear deformation theory is employed to formulate a governing equation for predicting free vibration of layered functionally graded beams. The Ritz method is adopted to solve the governing equation for various types of boundary conditions and the frequency results are validated by some available and experimental results. A multi-step sequential infiltration technique is used to fabricate the layered functionally graded beams for vibration testing. For the first time, a simple mathematical model, based on a power law distribution, is introduced to approximate material volume fraction of the layered beams. The details of layered beam fabrication according to the infiltration technique, microstructure and volume fraction analysis as well as vibration experimental set up are included and described in this investigation. Aspects which affect natural frequencies, such as material compositions, thickness ratio, and boundary conditions, are then taken into consideration. The impact on frequency of added mass is presented and discussed.

The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.

An improved third order shear deformation theory is employed to investigate thermal buckling and vibration of the functionally graded beams. A power law distribution is used to describe the variation of volume fraction of material compositions. The functionally graded material properties are assumed to vary smoothly and continuously across the thickness of the beams. The Ritz method is adopted to solve the eigenvalue problems that are associated with thermal buckling and vibration in various types of immovable boundary conditions. The parametric study covered in this paper includes the effects of material composition, temperature-dependent material properties, and slenderness ratio.

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

Developed herein are the scaling laws for physical modeling of anti-symmetric cross-ply laminated circular cylindrical shells for buckling and free vibration experiments. In the absence of experimental data, the validity of the scaling laws is verified by numerical experiments. This is accomplished by calculating theoretically the buckling loads and fundamental frequencies of the model and substituting into the scaling laws to obtain the corresponding values of the prototype. The predicted values of the prototype from the scaling laws are then compared with existing closed-form solutions. Examples for the complete similitude cases with various stacking sequences, number of plies, and length-to-radius ratios show exact agreement. The derived relationships between the model and prototype will greatly facilitate and reduce the need for costly experiments. In reality, either due to the complexity of the scaling laws or to economize experimental cost and time, it may not be feasible to construct the model to fulfil the scaling laws completely. Thus, several possible models of partial similitude are investigated numerically. These include models with distortion in laminated material properties, stacking sequences and number of plies. Model with distortion in material properties yields a high percentage of discrepancy and is not recommended.

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends.

This paper applies the Chebyshev collocation method to finding accurate solutions of natural frequencies for circular cylindrical shells. The shells with different boundary conditions are considered in the parametric study. By using the method to solve the coupled differential equations of motion governing the vibration of the shell, numerical results are obtained from the algebraic eigenvalue equation using the Chebyshev differentiation matrices. And the results satisfy both the geometric and force boundary conditions. Based on the numerical examples, the proposed method shows its capacity and reliability in predicting accurate frequency results for circular cylindrical shells with various boundary conditions as compared to some exact solutions available in the literature.