Analytical solution for free vibration analysis of GPL-RP beam integrated with piezoelectric layers

Author affiliations


  • Tran Quang Hung The University of Da Nang, University of Science and Technology, Da Nang, Vietnam
  • Tran Minh Tu Hanoi University of Civil Engineering, Hanoi, Vietnam
  • Do Minh Duc The University of Da Nang, University of Science and Technology, Da Nang, Vietnam



piezoelectric material, porous beam, graphene platelets, Navier's solution


This report presents an analytical approach to the natural frequency analysis of a porous beam consisting of a host porous layer reinforced with graphene platelets (GPLs), namely GPL-reinforced porous core, and two piezoelectric outer layers. In the modelling, symmetric distributions of both porosity and GPLs in the core are supposed. The effective mechanical properties of the GPL-reinforced porous core are estimated by Halpin–Tsai model and the rule of mixture. The electric potential in each piezoelectric layer is assumed to vary linearly across its thickness. Two types of electrical boundary conditions, which are open- and closed-circuits, are considered for the free surfaces of the piezoelectric layers. Parabolic shear deformation beam theory associated with Hamilton’s principle is adopted to derive the governing equations of the free vibration. Afterwards these equations are solved analytically by Navier’s solution. Comparative and comprehensive studies are carried out to examine the accuracy and effects of parameters and conditions, such as GPL weight fraction, porosity coefficient, and electrical boundary conditions on the natural frequencies of the beam.


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