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Finite Element Vibration Analysis of Pre-Stressed Functionally Graded Plates

Venkataramana Naik1 , G Prasanthi2 , D Sudhakara1, and M Jayapal Reddy1
1.Department of Mechanical Engineering, Vignanan Bharathi Institute of Technology, Proddatur 516361, YSR Kadapa (Dist.), Andhra Pradesh, India.
2.Department of Mechanical, Jawaharlal Nehru Technological University College of Engineering, Anantapur 516361, Anantapur (Dist.), Andra Pradesh, India.

Abstract—The present theory is based on a Higher-Order displacement model and the three-dimensional Hooke’s laws for plate material. The theory represents a more realistic quadratic variation of the transverse shearing and normal strains through the thickness of the plate. Nine-node Lagrangian elements have been used for the purpose of discretization using a refined Higher Order Shear deformation Theory (HOST12) that includes the effects of transverse shear deformations, transverse normal deformation and rotary inertia. A C0 isoperimetric finite element formulation is presented to calculate the required number of lowest natural frequencies of Functionally Graded Plates (Functionally graded plates) subjected to in-plane pre-stress. The material properties of the functionally graded plates are assumed to vary continuously from one surface to another, according to a simple power law distribution in terms of the constituent volume fractions. The formulation is applicable to thin as well as thick plates. The plate structure is idealized into an assemblage of nine-noded iso-parametric quadrilateral elements with Twelve degrees of freedom per node. Poisson’s ratio has been assumed to be constant throughout the thickness and the material is assumed to be isotropic at a point. Hamilton’s principle is used for the formulation. The effect of in-plane pre-stress is taken care by calculating the geometric stiffness matrix. The same shape functions are used to calculate the elastic stiffness matrix, geometric stiffness matrix and the element mass matrix. The consistent mass matrix is diagonalized by a special mass lumping scheme which conserves the total mass of the element and includes the effects due to rotary inertia terms. Subspace Iteration technique is applied to extract the natural frequencies. Numerical results for first seven natural frequencies are presented for rectangular and square plates under various boundary conditions. A parametric study has been carried out. Variations of natural frequencies with the constituent volume fractions are presented. The results show good agreement with three-dimensional analytical formulation.

Index Terms—Vibration analysis-functional graded plate

Cite: Venkataramana Naik, G Prasanthi, D Sudhakara, and M Jayapal Reddy, "Finite Element Vibration Analysis of Pre-Stressed Functionally Graded Plates," International Journal of Mechanical Engineering and Robotics Research, Vol.1, No.3, pp. 457-466, October 2012.