• Steel and Composite Structures
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• v.26 no.5
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• pp.649-662
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• 2018

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of sandwich sectorial plates with multiwalled carbon nanotube-(MWCNT)-reinforced composite core are considered. Modified Halpin-Tsai equation is used to evaluate the Young's modulus of the MWCNT/epoxy composite samples by the incorporation of an orientation as well as an exponential shape factor in the equation. The exponential shape factor modifies the Halpin-Tsai equation from expressing a straight line to a nonlinear one in the MWCNTs wt% range considered. In this paper, free vibration of thick functionally graded sandwich annular sectorial plates with simply supported radial edges and different circular edge conditions including simply supported-clamped, clamped-clamped, and free-clamped is investigated. A semi-analytical approach composed of two-dimensional differential quadrature method and series solution are adopted to solve the equations of motion. The material properties change continuously through the core thickness of the plate, which can vary according to a power-law, exponentially, or any other formulations in this direction. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated sectorial plates.

• Structural Engineering and Mechanics
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• v.65 no.1
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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.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.16 no.10 s.115
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• pp.1024-1035
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• 2006

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components $u_r,\;u_\theta\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be periodic in ${\theta}$ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and ones obtained from thin and thick ring theories, experiments, and another 3-D method.

• Geomechanics and Engineering
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• v.12 no.1
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• pp.9-34
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• 2017

In this work, an efficient and simple quasi-3D hyperbolic shear deformation theory is developed for bending and vibration analyses of functionally graded (FG) plates resting on two-parameter elastic foundation. The significant feature of this theory is that, in addition to including the thickness stretching effect, it deals with only 5 unknowns as the first order shear deformation theory (FSDT). The foundation is described by the Pasternak (two-parameter) model. The material properties of the plate are assumed to vary continuously in the thickness direction by a simple power law distribution in terms of the volume fractions of the constituents. Equations of motion for thick FG plates are obtained within the Hamilton's principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The numerical results are given in detail and compared with the existing works such as 3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates resting on elastic foundation.

• Structural Engineering and Mechanics
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• v.81 no.3
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• pp.259-266
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• 2022

Based upon differential quadrature method (DQM) and nonlocal strain gradient theory (NSGT), an investigation on the free vibrations of 2D plate systems with nano-dimensions has been provided taking into account the effects of different mechanical loadings. In order to capture different mechanical loadings, a general form of variable compressive load applied in the axial direction of the plate system has been introduced. The studied plate has been constructed from two types of particles which results in graded material properties and nanoscale pores. The established formulation for the plate is in the context of a novel shear deformable model and the equations have been solved via a semi-numerical trend. Presented results indicate the prominence of material composition, nonlocal coefficient, strain gradient coefficient and boundary conditions on vibrational frequencies of nano-size plate.

• Steel and Composite Structures
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• v.20 no.3
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• pp.623-649
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• 2016

Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.

• Structural Engineering and Mechanics
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• v.52 no.4
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• pp.663-686
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• 2014

This paper deals with free vibration analysis of bidirectional functionally graded annular plates resting on a two-parameter elastic foundation. The formulations are based on the three-dimensional elasticity theory. This study presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D functionally graded materials that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated by using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D functionally graded materials.

• Steel and Composite Structures
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• v.19 no.3
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• pp.521-546
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• 2015

A new refined hyperbolic shear and normal deformation beam theory is developed to study the free vibration and buckling of functionally graded (FG) sandwich beams under various boundary conditions. The effects of transverse shear strains as well as the transverse normal strain are taken into account. Material properties of the sandwich beam faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. Equations of motion are derived from Hamilton's principle. Analytical solutions for the bending, free vibration and buckling analyses are obtained for simply supported sandwich beams. Illustrative examples are given to show the effects of varying gradients, thickness stretching, boundary conditions, and thickness to length ratios on the bending, free vibration and buckling of functionally graded sandwich beams.

• Structural Engineering and Mechanics
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• v.85 no.6
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• pp.717-728
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• 2023

The effect of temperature dependent material properties on the free vibration of FG porous beams is investigated in the present paper. A quasi-3D shear deformation solution is used involves only three unknown function. The mechanical properties which are considered to be temperature-dependent as well as the porosity distributions are assumed to gradually change along the thickness direction according to defined law. The beam is supposed to be simply supported and lying on variable elastic foundation. The differential equation system governing the free vibration behavior of porous beams is derived based on the Hamilton principle. Navier's method for simply supported systems is then used to determine and compute the frequencies of FG porous beam. The results of the present formulation are validated by comparing with those available literatures. Finally, the effects of several parameters such as porosity distribution and the parameters of variable elastic foundation on the free vibration behavior of temperature-dependent FG beams are presented and discussed in detail.

• Smart Structures and Systems
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• v.20 no.4
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• pp.509-524
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• 2017

In this article, a free vibration analysis of functionally graded (FG) plates resting on elastic foundations is presented using a quasi-3D hybrid-type higher order shear deformation theory. Undetermined integral terms are employed in the proposed displacement field and modeled based on a hybrid-type (sinusoidal and parabolic) quasi-3D HSDT with five unknowns in which the stretching effect is taken into account. Thus, it can be said that the significant feature of this theory is that it deals with only 5 unknowns as the first order shear deformation theory (FSDT). The elastic foundation parameters are introduced in the present formulation by following the Pasternak (two-parameter) mathematical model. Equations of motion are obtained via the Hamilton's principles and solved using Navier's method. Accuracy of the proposed theory is confirmed by comparing the results of numerical examples with the ones available in literature.