• Steel and Composite Structures
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• v.36 no.6
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• pp.711-727
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• 2020

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton's principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.161-169
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• 2017

In this paper, using consistent couple stress theory and Hamilton's principle, the free vibration analysis of Euler-Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

• Steel and Composite Structures
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• v.26 no.6
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• pp.663-672
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• 2018

This paper contains a consistent couple-stress theory to capture size effects in Euler-Bernoulli nano-beams made of three-directional functionally graded materials (TDFGMs). These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in all three axial, thickness and width directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of minimum potential energy. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of TDFG nano-beam. At the end, some numerical results are performed to investigate some effective parameter on buckling load. In this theory the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor.

• Advances in nano research
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• v.8 no.2
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• pp.115-134
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• 2020

In this paper, the free vibrations analysis of the nanoplates made of three-directional functionally graded material (TDFGM) with small scale effects is presented. To study the small-scale effects on natural frequency, modified strain gradient theory (MSGT) has been used. Material properties of the nanoplate follow an arbitrary function that changes in three directions along the length, width and thickness of the plate. The equilibrium equations and boundary conditions of nanoplate are obtained using the Hamilton's principle. The generalized differential quadrature method (GDQM) is used to solve the governing equations and different boundary conditions for obtaining the natural frequency of nanoplate made of three-directional functionally graded material. The present model can be transformed into a couple stress plate model or a classic plate model if two or all parameters of the length scales set to zero. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants and the aspect ratio with different boundary conditions on the free vibrations of nanoplates. To the best of the researchers' knowledge, in the literature, there is no study carried out into MSGT for free vibration analysis of FGM nanoplate with arbitrary functions.

• Steel and Composite Structures
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• v.24 no.6
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• pp.727-740
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• 2017

The present paper studies the stability analysis of the continuously graded CNT-Reinforced Composite (CNTRC) panel stiffened by rings and stringers. The Stiffened Panel (SP) subjected to axial and lateral loads is reinforced by agglomerated CNTs smoothly graded through the thickness. A two-parameter Eshelby-Mori-Tanaka (EMT) model is adopted to derive the effective material moduli of the CNTRC. The stability equations of the CNRTC SP are obtained by means of the adjacent equilibrium criterion. Notwithstanding most available literature in which the stiffener effects were smeared out over the respective stiffener spacing, in the present work, the stiffeners are modeled as Euler-Bernoulli beams. The Generalized Differential Quadrature Method (GDQM) is employed to discretize the stability equations. A numerical study is performed to investigate the influences of different types of parameters involved on the critical buckling of the SP reinforced by agglomerated CNTs. The results achieved reveal that continuously distributing of CNTs adjacent to the inner and outer panel's surface results in improving the stiffness of the SP and, as a consequence, inclining the critical buckling load. Furthermore, it has been concluded that the decline rate of buckling load intensity factor owing to the increase of the panel angle is significantly more sensible for the smaller values of panel angle.

• Structural Engineering and Mechanics
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• v.71 no.5
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• pp.525-544
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• 2019

In the present study, buckling and free vibration analyses of annular thin sector plate made of functionally graded materials (FGMs) resting on visco-elastic Pasternak foundation, subjected to external radial, circumferential and shear in-plane loads is investigated. Material properties are assumed to vary along the thickness according to an power law with Poisson's ratio held constant. First, based on the classical plate theory (CPT), the governing equation of motion is derived using Hamilton's principle and then is solved using the generalized differential quadrature method (GDQM). Numerical results are compared to those available in the literature to validate the convergence and accuracy of the present approach. Finally, the effects of power-law exponent, ratio of radii, thickness of the plate, sector angle, and coefficients of foundation on the fundamental and higher natural frequencies of transverse vibration and critical buckling loads are considered for various boundary conditions. Also, vibration and buckling mode shapes of functionally graded (FG) sector plate have been shown in this research. One of the important obtained results from this work show that ratio of the frequency of FG annular sector plate to the corresponding values of homogeneous plate are independent from boundary conditions and frequency number.

• Steel and Composite Structures
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• v.21 no.1
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• pp.1-36
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• 2016

In the present study, the nonlinear magneto-electro-mechanical free vibration behavior of rectangular double-bonded sandwich microbeams based on the modified strain gradient theory (MSGT) is investigated. It is noted that the top and bottom sandwich microbeams are considered with boron nitride nanotube reinforced composite face sheets (BNNTRC-SB) with electrical properties and carbon nanotube reinforced composite face sheets (CNTRC-SB) with magnetic fields, respectively, and also the homogenous core is used for both sandwich beams. The connections of every sandwich beam with its surrounding medium and also between them have been carried out by considering Pasternak foundations. To take size effect into account, the MSGT is introduced into the classical Timoshenko beam theory (CT) to develop a size-dependent beam model containing three additional material length scale parameters. For the CNTRC and BNNTRC face sheets of sandwich microbeams, uniform distribution (UD) and functionally graded (FG) distribution patterns of CNTs or BNNTs in four cases FG-X, FG-O, FG-A, and FG-V are employed. It is assumed that the material properties of face sheets for both sandwich beams are varied in the thickness direction and estimated through the extended rule of mixture. On the basis of the Hamilton's principle, the size-dependent nonlinear governing differential equations of motion and associated boundary conditions are derived and then discretized by using generalized differential quadrature method (GDQM). A detailed parametric study is presented to indicate the influences of electric and magnetic fields, slenderness ratio, thickness ratio of both sandwich microbeams, thickness ratio of every sandwich microbeam, dimensionless three material length scale parameters, Winkler spring modulus and various distribution types of face sheets on the first two natural frequencies of double-bonded sandwich microbeams. Furthermore, a comparison between the various beam models on the basis of the CT, modified couple stress theory (MCST), and MSGT is performed. It is illustrated that the thickness ratio of sandwich microbeams plays an important role in the vibrational behavior of the double-bonded sandwich microstructures. Meanwhile, it is concluded that by increasing H/lm, the values of first two natural frequencies tend to decrease for all amounts of the Winkler spring modulus.

• Advances in nano research
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• v.8 no.1
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• pp.13-24
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• 2020

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

• Steel and Composite Structures
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• v.33 no.2
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• pp.307-318
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• 2019

This is the first attempt to consider the nonlinear bending analysis of porous functionally graded (FG) thick annular and circular nanoplates resting on Kerr foundation. The size effects are captured based on modified couple stress theory (MCST). The material properties of the porous FG nanostructure are assumed to vary smoothly through the thickness according to a power law distribution of the volume fraction of the constituent materials. The elastic medium is modeled by Kerr elastic foundation which consists of two spring layers and one shear layer. The governing equations are extracted based on Hamilton's principle and two variables refined plate theory. Utilizing generalized differential quadrature method (GDQM), the nonlinear static behavior of the nanostructure is obtained under different boundary conditions. The effects of various parameters such as material length scale parameter, boundary conditions, and geometrical parameters of the nanoplate, elastic medium constants, porosity and FG index are shown on the nonlinear deflection of the annular and circular nanoplates. The results indicate that with increasing the material length scale parameter, the nonlinear deflection is decreased. In addition, the dimensionless nonlinear deflection of the porous annular nanoplate is diminished with the increase of porosity parameter. It is hoped that the present work may provide a benchmark in the study of nonlinear static behavior of porous nanoplates.

• Steel and Composite Structures
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• v.43 no.5
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• pp.603-623
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• 2022

This article is dedicated to predict the natural frequencies of joined conical shell structures made of Functionally Graded Material (FGM). The structure includes two conical segments. The equivalent material properties are found by using the rule of mixture based on Voigt model. In addition, three well-known patterns are employed for distribution of material properties throughout the thickness of the structure. The main objective of the present research is to propose a novel exponential pattern and obtain the related equivalent material properties. Furthermore, the Donnell type shell theory is used to obtain the governing equations of motion. Note that these equations are obtained by employing First-order Shear Deformation Theory (FSDT). In order to discretize the governing system of differential equations, well-known and efficient semi-analytical scheme, namely Generalized Differential Quadrature Method (GDQM), is utilized. Different boundary conditions are considered for various types of single and joined conical shell structures. Moreover, an applicable modification is considered for the continuity conditions at intersection position. In the first step, the proposed formulation is verified by solving some well-known benchmark problems. Besides, some new numerical examples are analyzed to show the accuracy and high capability of the suggested technique. Additionally, several geometric and material parameters are studied numerically.