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 Earthquakes and Structures   Volume 17, Number 5, November 2019 , pages 447-462 DOI: https://doi.org/10.12989/eas.2019.17.5.447 Effect of nonlinear elastic foundations on dynamic behavior of FG plates using four-unknown plate theory Mokhtar Nebab, Hassen Ait Atmane, Riadh Bennai and Benabdallah Tahar Abstract This present paper concerned with the analytic modelling for vibration of the functionally graded (FG) plates resting on non-variable and variable two parameter elastic foundation, based on two-dimensional elasticity using higher shear deformation theory. Our present theory has four unknown, which mean that have less than other higher order and lower theory, and we denote do not require the factor of correction like the first shear deformation theory. The indeterminate integral are introduced in the fields of displacement, it is allowed to reduce the number from five unknown to only four variables. The elastic foundations are assumed a classical model of Winkler-Pasternak with uniform distribution stiffness of the Winkler coefficient (kw), or it is with variables distribution coefficient (kw). The variable\'s stiffness of elastic foundation is supposed linear, parabolic and trigonometry along the length of functionally plate. The properties of the FG plates vary according to the thickness, following a simple distribution of the power law in terms of volume fractions of the constituents of the material. The equations of motions for natural frequency of the functionally graded plates resting on variables elastic foundation are derived using Hamilton principal. The government equations are resolved, with respect boundary condition for simply supported FG plate, employing Navier series solution. The extensive validation with other works found in the literature and our results are present in this work to demonstrate the efficient and accuracy of this analytic model to predict free vibration of FG plates, with and without the effect of variables elastic foundations. Key Words free vibration; variables elastic foundations; Functionally graded plate; higher shear deformation theory Address Mokhtar Nebab: Faculty Civil Engineering and Architecture, Civil Engineering Department, University Hassiba Benbouali of Chlef, Algeria Hassen Ait Atmane: Faculty Civil Engineering and Architecture, Civil Engineering Department, University Hassiba Benbouali of Chlef, Algeria; Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Algeria Riadh Bennai: Faculty Civil Engineering and Architecture, Civil Engineering Department, University Hassiba Benbouali of Chlef, Algeria Benabdallah Tahar: Deanship of Quality and Academic Accreditation, King Abdualziz University, Jeddah, Saudi Arabia

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