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Steel and Composite Structures
  Volume 11, Number 6, December 2011 , pages 489-504

Mathematical solution for free vibration of sigmoidfunctionally graded beams with varying cross-section
Hassen Ait Atmane, Abdelouahed Tounsi, Noureddine Ziane, and Ismail Mechab

    This paper presents a theoretical investigation in free vibration of sigmoid functionally graded beams with variable cross-section by using Bernoulli-Euler beam theory. The mechanical properties are assumed to vary continuously through the thickness of the beam, and obey a two power law of the volume fraction of the constituents. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the S-FGM beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Results show that, all other parameters remaining the same, the natural frequencies of S-FGM beams are always proportional to those of homogeneous isotropic beams. Therefore, one can predict the behaviour of S-FGM beams knowing that of similar homogeneous beams.
Key Words
    functionally graded materials; beams; variable cross-section; free vibration
Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes, BP 89 Cite Ben M

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