• Advances in nano research
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• v.5 no.4
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• pp.313-336
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• 2017

This article investigates vibration behavior of magneto-electro-elastic functionally graded (MEE-FG) nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of MEE-FG nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen's nonlocal elasticity theory which captures the small size effects and using the Hamilton's principle, the nonlocal governing equations of motions are derived and then solved analytically. Then the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index and slenderness ratio on the frequencies of the embedded MEE-FG nanobeams are studied.

• Earthquakes and Structures
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• v.17 no.5
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• pp.447-462
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• 2019

This present paper concerned with the analytic modelling for vibration of the functionally graded (FG) plates resting on non-variable and variable two parameter elastic foundation, based on two-dimensional elasticity using higher shear deformation theory. Our present theory has four unknown, which mean that have less than other higher order and lower theory, and we denote do not require the factor of correction like the first shear deformation theory. The indeterminate integral are introduced in the fields of displacement, it is allowed to reduce the number from five unknown to only four variables. The elastic foundations are assumed a classical model of Winkler-Pasternak with uniform distribution stiffness of the Winkler coefficient (kw), or it is with variables distribution coefficient (kw). The variable's stiffness of elastic foundation is supposed linear, parabolic and trigonometry along the length of functionally plate. The properties of the FG plates vary according to the thickness, following a simple distribution of the power law in terms of volume fractions of the constituents of the material. The equations of motions for natural frequency of the functionally graded plates resting on variables elastic foundation are derived using Hamilton principal. The government equations are resolved, with respect boundary condition for simply supported FG plate, employing Navier series solution. The extensive validation with other works found in the literature and our results are present in this work to demonstrate the efficient and accuracy of this analytic model to predict free vibration of FG plates, with and without the effect of variables elastic foundations.

• Structural Engineering and Mechanics
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• v.66 no.6
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• pp.737-748
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• 2018

The current study presents a new technique in the framework of the nonlocal elasticity theory for a comprehensive buckling analysis of Euler-Bernoulli nano-beams made up of bidirectional functionally graded material (BDFGM). The mechanical properties are considered by exponential and arbitrary variations for axial and transverse directions, respectively. The various circumstances including tapering, resting on two-parameter elastic foundation, step-wise or continuous variations of axial loading, various shapes of sections with various distribution laws of mechanical properties and various boundary conditions like the multi-span beams are taken into account. As far as we know, for the first time in the current work, the buckling analyses of BDFGM nano-beams are carried out under mentioned circumstances. The critical buckling loads and mode shapes are calculated by using energy method and a new technique based on calculus of variations and collocation method. Fast convergence and excellent agreement with the known data in literature, wherever possible, presents the efficiency of proposed technique. The effects of boundary conditions, material and taper constants, foundation moduli, variable axial compression and small-scale of nano-beam on the buckling loads and mode shapes are investigated. Moreover the analytical solutions, for the simpler cases are provided in appendices.

• Steel and Composite Structures
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• v.39 no.1
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• pp.95-107
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• 2021

In this work, the dynamic response of functionally graded beams on variable elastic foundations is studied using a novel higher-order shear deformation theory (HSDT). Unlike the conventional HSDT, the present one has a new displacement field which introduces undetermined integral variables. The FG beams were assumed to be supported on Winkler-Pasternak type foundations in which the Winkler modulus is supposed to be variable in the length of the beam. The variable rigidity of the elastic foundation is assumed to be linear, parabolic and sinusoidal along the length of the beam. The material properties of the FG porous beam vary according to a power law distribution in terms of the volume fraction of the constituents. The equations of motion are determined using the virtual working principle. For the analytical solution, Navier method is used to solve the governing equations for simply supported porous FG beams. Numerical results of the present theory for the free vibration of FG beams resting on elastic foundations are presented and compared to existing solutions in the literature. A parametric study will be detailed to investigate the effects of several parameters such as gradient index, thickness ratio, porosity factor and foundation parameters on the frequency response of porous FG beams.

• Structural Engineering and Mechanics
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• v.15 no.2
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• pp.249-267
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• 2003

In this study, circular plates subjected to general type of loads and supported on a two-parameter elastic foundation are analysed. The stiffness, elastic bedding and soil shear effect matrices of a fully compatible ring sector plate element, developed by Saygun (1974), are obtained numerically assuming variable thickness of the element. Ring sector soil finite element is also defined to determine the deflection of the soil surface outside the domain of the plate in order to establish the interaction between the plate and the soil. According to Vallabhan and Das (1991) the elastic bedding (C) and shear parameters ($C_T$) of the foundation are expressed depending on the elastic constants ($E_s$, $V_s$) and the thickness of compressible soil layer ($H_s$) and they are calculated with a suitable iterative procedure. Using ring sector elements presented in this paper, permits the generalization of the loading and the boundary conditions of the soil outside the plate.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.171-180
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• 2017

Static and dynamic instability of a viscoelastic carbon nanotube (CNT) embedded on a thermo-elastic foundation are investigated, in this research. The CNT is modeled based on Euler-Bernoulli beam (EBB) and nonlocal small scale elasticity theory is utilized to analyze the structure. Governing equations of the system are derived using Hamilton's principle and differential quadrature (DQ) method is applied to solve the partial differential equations. The effects of variable axial load and diverse boundary conditions on static/vibration instability are studied. To verify the result of the DQ method, the Galerkin weighted residual approach is used for the instability analysis. It is observed appropriate agreement for results of two different solution methods and satisfactory accuracy with those obtained in prior studies. The results of this work could be useful for engineers and designers in order to produce and design nano/micro structures in thermo-elastic medium.

• Structural Engineering and Mechanics
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• v.52 no.5
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• pp.1033-1049
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• 2014

The free vibration of functionally graded material (FGM) beams on an elastic foundation and spring supports is investigated. Young's modulus, mass density and width of the beam are assumed to vary in thickness and axial directions respectively following the exponential law. The spring supports are also taken into account at both ends of the beam. An analytical formulation is suggested to obtain eigen solutions of the FGM beams. Numerical analyses, based on finite element method by using a beam finite element developed in this study, are performed in order to show the legitimacy of the analytical solutions. Some results for the natural frequencies of the FGM beams are given considering the effect of various structural parameters. It is also shown that the spring supports show the greatest effect on the natural frequencies of FGM beams.

• Geomechanics and Engineering
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• v.22 no.4
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• pp.361-374
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• 2020

This paper is concerned with the thermoelastic bending of FG beams resting on two-layer elastic foundations. One of these layers is Winkler springs with a variable modulus while the other is considered as a shear layer with a constant modulus. The beams are considered simply supported and subjected to thermo-mechanical loading. Temperature-dependent material properties are considered for the FG beams, which are assumed to be graded continuously across the panel thickness. The used theories contain undetermined integral terms which lead to a reduction of unknowns functions. Several micromechanical models are used to estimate the effective two-phase FG material properties as a function of the particles' volume fraction considering thermal effects. Analytical solutions for the thermo-mechanical bending analysis are obtained based on Navier's method that satisfies the boundary conditions. Finally, the numerical results are provided to reveal the effect of explicit micromechanical models, geometric parameters, temperature distribution and elastic foundation parameters on the thermoelastic response of FG beams.

• Computers and Concrete
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• v.25 no.3
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• pp.215-224
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• 2020

In the present study, according to the important of porosity in low specific weight in comparison of high stiffness of carbon nanotubes reinforced composite, buckling and free vibration analysis of sandwich composite beam in two configurations, of laminates using differential quadrature method (DQM) is studied. Also, the effects of porosity coefficient and three types of porosity distribution on critical buckling load and natural frequency are discussed. It is shown the buckling loads and natural frequencies of laminate 1 are significantly larger than the results of laminate 2. When configuration 2 (the core is made of FRC) and laminate 1 ([0/90/0/45/90]_s) are used, the first natural frequency rises noticeably. It is also demonstrated that the influence of the core height in the case of lower carbon volume fractions is negligible. Even though, when volume fraction of fiber increases, the critical buckling load enhances smoothly. It should be noticed the amount of decline has inverse relationship with the beam aspect ratio. Investigating three porosity patterns, beam with the distribution of porosity Type 2 has the maximum critical buckling load and first natural frequency. Among three elastic foundations (constant, linear and parabolic), buckling load and natural frequency in linear variation has the least amount. For all kind of elastic foundations, when the porosity coefficient increases, critical buckling load and natural frequency decline significantly.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.16 no.6 s.111
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• pp.609-618
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• 2006

This study is undertaken for the vibration analysis of tapered thick plate with concentrated mass on elastic foundation. The boundary condition of the plate is analyzed with the 4-sides simply supported and 4-fixed basis. This study find out the frequency following the change in size for each foundational variable on Pasternak foundation, one of the two-parameter elastic foundation parameter that considered the shear layer to the Winkler foundation parameter. The concentrated mass is applied with the consideration of mass of the entire plate, and the change of frequency is studies on each location with the consideration of reacting for the three locations for concentrated mass. And, in order to find out the change of frequency on the thickness of the plate, it considered tapered ratio that linearly changes depending on the length of the plate with the thickness of the plate in x-direction, and the tapered ratio has changes with 4 types ($\alpha$=0.25, 0, 5, 0.75, and 1.0). For the interpretation, the program using finite element method (F.E.M.) is used and the element coordination is used the 8-node serendipity element. Therefore, the purpose of this study is to find out the characteristics of plate vibration under the mechanica vibration or external vibration factor to facilitate as the basic data of the design to secure the stability.