Acta Mechanica Sinica

, Volume 32, Issue 6, pp 1088–1103 | Cite as

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

  • T. F. Xu
  • Y. F. XingEmail author
Research Paper


This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton’s principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson’s ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.


Functionally graded material Free vibration Rectangular plate Close form solutions Neutral surface 



The project was supported by the National Natural Science Foundation of China (Grants 11172028, 1372021), Research Fund for the Doctoral Program of Higher Education of China (Grant 20131102110039), and the Innovation Foundation of Beihang University for PhD graduates.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Solid MechanicsBeihang UniversityBeijingChina
  2. 2.China Academy of Launch Vehicle Technology R&D CenterBeijingChina

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