Using modified Halpin-Tsai approach for vibrational analysis of thick functionally graded multi-walled carbon nanotube plates

  • Tahouneh, Vahid (Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University)
  • Received : 2016.10.27
  • Accepted : 2017.02.19
  • Published : 2017.04.30


In the most of previous studies, researchers have restricted their own studies to consider the effect of single walled carbon nanotubes as a reinforcement on the vibrational behavior of structures. In the present work, free vibration characteristics of functionally graded annular plates reinforced by multi-walled carbon nanotubes resting on Pasternak foundation are presented. The response of the elastic medium is formulated by the Winkler/Pasternak model. Modified Halpin-Tsai equation was used to evaluate the Young's modulus of the multi-walled carbon nanotube/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 multi-walled carbon nanotubes wt% range considered. The 2-D generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The effects of two-parameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the plates are investigated. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of annular plates.


multi-walled carbon nanotubes;vibration;thick plates;modified Halpin-Tsai equation


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