• Advances in nano research
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• 제14권5호
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• pp.421-433
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• 2023

In the present paper, the influences of the variation of exponent of volume fraction of carbon nanotubes (CNTs) on the natural frequencies (NFs) of the carbon nanotube-reinforced composite (CNTRC) beams under four different boundary conditions (BCs) are investigated. The single-walled carbon nanotubes (SWCNTs) are assumed to be aligned and dispersed in a polymeric matrix with various reinforcing patterns, according to the variation of exponent of volume fraction of CNTs for functionally graded (FG) reinforcements. Besides, uniform distribution (UD) of reinforcement is also considered to analyze the influence of the non-linear (NL) variation of the reinforcement of CNTs. Using Hamilton's principle and third-order shear deformation theory (TSDT), the equations of motion of the CNTRC beam are derived. Under four different BCs, the resulting equations are solved analytically. To verify the present formulation, comparison investigations are conducted. To examine the impacts of several factors on the NFs of the CNTRC beams, numerical examples and some benchmark results are presented.

• Geomechanics and Engineering
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• 제14권6호
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• pp.519-532
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• 2018

In this work, two dimensional (2D) and quasi three-dimensional (quasi-3D) HSDTs are proposed for bending and free vibration investigation of functionally graded (FG) plates using hyperbolic shape function. Unlike the existing HSDT, the proposed theories have a novel displacement field which include undetermined integral terms and contains fewer unknowns. The material properties of the plate is inhomogeneous and are considered to vary continuously in the thickness direction by three different distributions; power-law, exponential and Mori-Tanaka model, in terms of the volume fractions of the constituents. The governing equations which consider the effects of both transverse shear and thickness stretching are determined through the Hamilton's principle. The closed form solutions are deduced by employing Navier method and then fundamental frequencies are obtained by solving the results of eigenvalue problems. In-plane stress components have been determined by the constitutive equations of composite plates. The transverse stress components have been determined by integrating the 3D stress equilibrium equations in the thickness direction of the FG plate. The accuracy of the present formulation is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature.

• Wind and Structures
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• 제22권4호
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• pp.429-454
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• 2016

In this paper, a shear deformation plate theory based on neutral surface position is developed for free vibration analysis of functionally graded material (FGM) plates. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. During manufacture, defects such as porosities can appear. It is therefore necessary to consider the vibration behavior of FG plates having porosities in this investigation. The proposed theory is based on 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 neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The equation of motion for FG rectangular plates is obtained through Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Numerical results are presented and the influences of the volume fraction index and porosity volume fraction on frequencies of FGM plates are clearly discussed.

• Structural Engineering and Mechanics
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• 제65권1호
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• pp.19-31
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• 2018

In this paper, a new quasi-3D sinusoidal shear deformation theory for functionally graded (FG) plates is proposed. The theory considers both shear deformation and thickness-stretching influences by a trigonometric distribution of all displacements within the thickness, and respects the stress-free boundary conditions on the upper and lower faces of the plate without employing any shear correction coefficient. The advantage of the proposed model is that it posses a smaller number of variables and governing equations than the existing quasi-3D models, but its results compare well with those of 3D and quasi-3D theories. This benefit is due to the use of undetermined integral unknowns in the displacement field of the present theory. By employing the Hamilton principle, equations of motion are obtained in the present formulation. Closed-form solutions for bending and free vibration problems are determined for simply supported plates. Numerical examples are proposed to check the accuracy of the developed theory.

• Earthquakes and Structures
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• 제14권2호
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• pp.117-128
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• 2018

In this research work, free vibrations of simply supported functionally graded plate resting on a Winkler-Pasternak elastic foundation are investigated by a new shear deformation theory. The influence of alternative micromechanical models on the macroscopic behavior of a functionally graded plate based on shear-deformation plate theories is examined. Several micromechanical models are tested to obtain the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which continuously varies through the thickness of a functionally graded plate. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The energy functional of the system is obtained using Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Finally, the numerical results are provided to reveal the effect of explicit micromechanical models on natural fundamental frequencies.

• Steel and Composite Structures
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• 제28권1호
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• pp.13-24
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• 2018

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (${\varepsilon}_z{\neq}0$) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

• Steel and Composite Structures
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• 제38권1호
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• pp.1-15
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• 2021

In this paper, a higher order shear deformation theory for bending analysis of functionally graded plates resting on Pasternak foundation and under various boundary conditions is exposed. The proposed theory is based on the assumption that porosities can be produced within functionally graded plate which may lead to decline in strength of materials. In this research a novel distribution of porosity according to the thickness of FG plate are supposing. Governing equations of the present theory are derived by employing the virtual work principle, and the closed-form solutions of functionally graded plates have been obtained using Navier solution. Numerical results for deflections and stresses of several types of boundary conditions are presented. The exactitude of the present study is confirmed by comparing the obtained results with those available in the literature. The effects of porosity parameter, slenderness ratio, foundation parameters, power law index and boundary condition types on the deflections and stresses are presented.

• Steel and Composite Structures
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• 제43권5호
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• pp.581-601
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• 2022

The main objective of this research work is to investigate the free vibration behavior of annular sandwich plates resting on the Kerr foundation at thermal conditions. This sandwich configuration is composed of two FGM face sheets as coating layer and a porous GPLRC (GPL reinforced composite) core. It is supposed that the GPL nanofillers and the porosity coefficient vary continuously along the core thickness direction. To model closed-cell FG porous material reinforced with GPLs, Halpin-Tsai micromechanical modeling in conjunction with Gaussian-Random field scheme is used, while the Poisson's ratio and density are computed by the rule of mixtures. Besides, the material properties of two FGM face sheets change continuously through the thickness according to the power-law distribution. To capture fundamental frequencies of the annular sandwich plate resting on the Kerr foundation in a thermal environment, the analysis procedure is with the aid of Reddy's shear-deformation plate theory based high-order shear deformation plate theory (HSDT) to derive and solve the equations of motion and boundary conditions. The governing equations together with related boundary conditions are discretized using the generalized differential quadrature (GDQ) method in the spatial domain. Numerical results are compared with those published in the literature to examine the accuracy and validity of the present approach. A parametric solution for temperature variation across the thickness of the sandwich plate is employed taking into account the thermal conductivity, the inhomogeneity parameter, and the sandwich schemes. The numerical results indicate the influence of volume fraction index, GPLs volume fraction, porosity coefficient, three independent coefficients of Kerr elastic foundation, and temperature difference on the free vibration behavior of annular sandwich plate. This study provides essential information to engineers seeking innovative ways to promote composite structures in a practical way.