• Advances in aircraft and spacecraft science
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• v.5 no.1
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• pp.107-128
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• 2018

Thermo-mechanical vibration of sandwich beams with a stiff core and face sheets made of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) is investigated within the framework of Timoshenko beam theory. The material properties of FG-CNTRC are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture and are considered to be temperature dependent. The governing equations and boundary conditions are derived by using Hamilton's principle and are solved using an efficient semi-analytical technique of the differential transform method (DTM). Comparison between the results of the present work and those available in literature shows the accuracy of this method. A parametric study is conducted to study the effects of carbon nanotube volume fraction, slenderness ratio, core-to-face sheet thickness ratio, and various boundary conditions on free vibration behavior of sandwich beams with FG-CNTRC face sheets. It is explicitly shown that the vibration characteristics of the curved nanosize beams are significantly influenced by the surface density effects.

• Steel and Composite Structures
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• v.39 no.1
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• pp.1-20
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• 2021

An exact solution based on refined third-order theory (TOT) has been presented for functionally graded porous curved beams having deep curvature. The displacement field of the refined TOT is derived by imposing the shear free conditions at the outer and inner surfaces of curved beams. The properties of the two phase composite are tailored according the power law rule and the effective properties are computed using Mori-Tanaka homogenization scheme. The equations of motion as well as consistent boundary conditions are derived using the Hamilton's principle. The curved beam stiffness coefficients (A, B, D) are obtained numerically using six-point Gauss integration scheme without compromising the accuracy due to deepness (1 + z/R) terms. The porosity has been modeled assuming symmetric (even) as well as asymmetric (uneven) distributions across the cross section of curved beam. The programming has been performed in MATLAB and is validated with the results available in the literature as well as 2D finite element model developed in ABAQUS. The effect of inclusion of 1 + z/R terms is studied for deflection, stresses and natural frequencies for FG curved beams of different radii of curvature. Results presented in this work will be useful for comparison of future studies.

• Steel and Composite Structures
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• v.49 no.6
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• pp.603-614
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• 2023

In this paper, frequency analysis of curved functionally graded (FG) nanobeam by consideration of deepness effect has been studied. Differential transform method (DTM) has been used to obtain frequency responses. The nonlocal theory of Eringen has been applied to consider nanoscales. Material properties are supposed to vary in radial direction according to power-law distribution. Differential equations and related boundary conditions have been derived using Hamilton's principle. Finally, by consideration of nonlocal theory, the governing equations have been derived. Natural frequencies have been obtained using semi analytical method (DTM) for different boundary conditions. In order to study the effect of deepness, the deepness term is considered in strain field. The effects of the gradient index, radius of curvature, the aspect ratio, the nonlocal parameter and interaction of aforementioned parameters on frequency value for different boundary conditions such as clamped-clamped (C-C), clamped-hinged (C-H), and clamped-free (C-F) have been investigated. In addition, the obtained results are compared with the results in previous literature in order to validate present study, a good agreement was observed in the present results.

• Smart Structures and Systems
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• v.20 no.3
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• pp.351-368
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• 2017

To peruse the free vibration of curved functionally graded piezoelectric (FGP) nanosize beam in thermal environment, nonlocal elasticity theory is applied for modeling the nano scale effect. The governing equations are obtained via the energy method. Analytically Navier solution is employed to solve the governing equations for simply supported boundary conditions. Solving these equations enables us to estimate the natural frequency for curved FGP nanobeam under the effect of a uniform temperature change and external electric voltage. The results determined are verified by comparing the results by available ones in literature. The effects of various parameters such as nonlocality, uniform temperature changes, external electric voltage, gradient index, opening angle and aspect ratio of curved FGP nanobeam on the natural frequency are successfully discussed. The results revealed that the natural frequency of curved FGP nanobeam is significantly influenced by these effects.

• Advances in aircraft and spacecraft science
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• v.10 no.1
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• pp.51-65
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• 2023

In this paper, the effect of deepness on in-plane free vibration behavior of a curved functionally graded (FG) nanobeam based on nonlocal elasticity theory has been investigated. Differential equations and boundary conditions have been developed based on Hamilton's principle. In order to figure out the size effect, nonlocal theory has been adopted. Properties of material vary in radial direction. By using Navier solution technique, the amount of natural frequencies has been obtained. Also, to take into account the deepness effect on vibrations, thickness to radius ratio has been considered. Differences percentage between results of cases in which deepness effect is included and excluded are obtained and influences of power-law exponent, nonlocal parameter and arc angle on these differences percentage are studied. Results show that arc angle and power law exponent parameters have the most influences on the amount of the differences percentage due to deepness effect. It has been observed that the inclusion of geometrical deep term and material distribution results in an increase in sensitivity of dimensionless natural frequency about variation of aforementioned parameters and a change in variation range of natural frequency. Finally, several numerical results of deep and shallow curved functionally graded nanobeams with different geometry dimensions are presented, which may serve as benchmark solutions for the future research in this field.

• Advances in nano research
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• v.13 no.6
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• pp.599-617
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• 2022

In the present research, for the first time, the vibrational as well as buckling characteristics of a three-layered curved nanobeam including a core made of functionally graded (FG) material and two layers of smart material-piezo-magneto-electric-resting on a Winkler Pasternak elastic foundation are examined. The displacement field for the nanobeam is chosen via Timoshenko beam theory. Also, the size dependency is taken into account by using nonlocal strain gradient theory, aka NSGT. Then, by employing Hamilton's principle, energy procedure, the governing equations together with the boundary conditions are achieved. The solution procedure is a numerical solution called generalized differential quadrature method, or GDQM. The accuracy and reliability of the formulation alongside solution method is examined by using other published articles. Lastly, the parameter which can alter and affect the buckling or vocational behavior of the curved nanobeam is investigated in details.