Free Vibration Analysis of Functionally Graded Material Plates Resting on Elastic Foundation Using Dynamic Stiffness Method

Abstract

In this present work, the dynamic stiffness method (DSM) is developed for free vibration analysis of functionally graded material (FGM) plates resting on Winkler and Pasternak elastic foundations. The material properties of the FGM plate vary continuously with according to different volume fraction laws such as power-law (P-FGM), sigmoid-law (S-FGM), and exponential-law (E-FGM) through the thickness direction of the FGM plate. A well-known, Classical plate theory (CPT) or Kirchhoff s plate theory is implemented along with the effect of physical neutral surface (PNS) to develop the kinematic variables or displacement components of the FGM plate. Hamilton s principle is applied to formulate the standard governing differential equation of motion for FGM plate resting on elastic foundation. A Levy-type solution technique is used to develop the dynamic stiffness matrix along with the displacement and force boundary conditions. The application of the Wittrick and Williams (W-W) algorithm is implemented to explain the complex behaviour of the obtained dynamic stiffness matrix and compute the accurate natural frequencies and mode shapes of the FGM plate resting on the elastic foundation. The obtained DSM natural frequencies results are compared with the available reported results and noticed that the DSM results are highly accurate with the existing results and can be further referred as a standard solution to compare the accuracy of various other analytical methods applied for analysing the free vibration response of FGM plate resting on elastic foundation. The effects of material and geometric property parameters (material gradient index, Young s modulus ratio, density ratio, aspect ratio, boundary conditions, and elastic modulus of foundations) on natural frequency are also examined. It is concluded that the present DSM method is efficient, reliable and accurate for examining the free vibration response of FGM plate resting on Winkler and Pasternak elastic foundations.