, Volume 44, Issue 3, pp 255–281 | Cite as

Free vibration analysis of functionally graded panels and shells of revolution

  • Francesco Tornabene
  • Erasmo Viola


The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one.

The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning eight types of shell structures illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behaviour of parabolic and circular shell structures.


Functionally graded materials Doubly curved shells FSD theory Free vibrations Generalized differential quadrature 


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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.DISTART - Department, Faculty of EngineeringUniversity of BolognaBolognaItaly

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