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Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model

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Abstract

A free vibration analysis of shallow and deep curved functionally graded (FG) nanobeam is presented. Differential equations and boundary conditions are obtained using Hamilton’s principle, and then, nonlocal theory is employed to derive differential equations in small scale. Properties of the material are FG in radial direction. In order to investigate the effects of deep curved beam, extensional stiffness, bending–extension coupling stiffness, and bending stiffness are calculated in the deep case, analytically. By employing Navier method, an analytical solution is presented. Results are compared and validated with available studies, and a good agreement is seen. The influences of effective parameters such as geometrical deep term, nonlocal parameter, opening angle, aspect ratio, mode number, and gradient index are discussed in detail. It is found that the frequency of deep curved nanobeam is higher than that of shallow one, and the aspect ratio significantly affects this difference to decrease. Also, it is concluded that the opening angle, nonlocal parameter, and power gradient index can notably influence the amount of frequency.

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Authors and Affiliations

  1. Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran

    S. A. H. Hosseini & O. Rahmani

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  1. S. A. H. Hosseini
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  2. O. Rahmani
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Correspondence to O. Rahmani.

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Hosseini, S.A.H., Rahmani, O. Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Appl. Phys. A 122, 169 (2016). https://doi.org/10.1007/s00339-016-9696-4

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