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Bending and vibration analyses of coupled axially functionally graded tapered beams

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Abstract

The nonlinear bending and vibrations of tapered beams made of axially functionally graded (AFG) material are analysed numerically. For a clamped–clamped boundary conditions, Hamilton’s principle is employed so as to balance the potential and kinetic energies, the virtual work done by the damping, and that done by external distributed load. The nonlinear strain–displacement relations are employed to address the geometric nonlinearities originating from large deflections and induced nonlinear tension. Exponential distributions along the length are assumed for the mass density, moduli of elasticity, Poisson’s ratio, and cross-sectional area of the AFG tapered beam; the non-uniform mechanical properties and geometry of the beam along the length make the system asymmetric with respect to the axial coordinate. This non-uniform continuous system is discretised via the Galerkin modal decomposition approach, taking into account a large number of symmetric and asymmetric modes. The linear results are compared and validated with the published results in the literature. The nonlinear results are computed for both static and dynamic cases. The effect of different tapered ratios as well as the gradient index is investigated; the numerical results highlight the importance of employing a high-dimensional discretised model in the analysis of AFG tapered beams.

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Acknowledgements

The financial support to this research by the start-up grant of the University of Adelaide is gratefully acknowledged.

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Adelaide, Adelaide, SA, 5005, Australia

    Mergen H. Ghayesh

  2. Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada

    Hamed Farokhi

Authors
  1. Mergen H. Ghayesh
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  2. Hamed Farokhi
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Correspondence to Mergen H. Ghayesh.

Appendix A: Validation

Appendix A: Validation

In order to validate the performance of the present study, the numerical results are compared with the work of Ref. [29] and are demonstrated in Table 1. In particular, the first dimensionless frequency parameter for the transverse motion of the AFG beams with different ratio of modules of elasticity, \(E_{\mathrm{ratio}}= E_{\mathrm{left}} /E_{\mathrm{right}}\), is obtained versus the material gradient index and length-to-thickness ratio. As seen in this table, the first dimensionless frequency parameters for different values of n are in excellent agreement with those of Ref. [29].

Table 1 The first dimensionless frequency parameter \((\lambda = \omega ^{0.5})\) for the transverse motion of a pinned–pinned AFG beam for different material distribution (\(E_{{\text {ratio}}}=E_{\mathrm{left}}/E_{\mathrm{right}})\)
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Ghayesh, M.H., Farokhi, H. Bending and vibration analyses of coupled axially functionally graded tapered beams. Nonlinear Dyn 91, 17–28 (2018). https://doi.org/10.1007/s11071-017-3783-8

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