Nonlinear Dynamics

, Volume 93, Issue 1, pp 189–204 | Cite as

R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells

  • Tetyana Shmatko
  • Atul Bhaskar
Original Paper


Nonlinear free vibration of functionally graded shallow shells with complex planform is investigated using the R-functions method and variational Ritz method. The proposed method is developed in the framework of the first-order shear deformation shallow shell theory. Effect of transverse shear strains and rotary inertia is taken into account. The properties of functionally graded materials are assumed to be varying continuously through the thickness according to a power law distribution. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. Admissible functions are constructed by the R-functions theory. To implement the proposed approach, the corresponding software has been developed. Comprehensive numerical results for three types of shallow shells with positive, zero and negative curvature with complex planform are presented in tabular and graphical forms. The convergence of the natural frequencies with increasing number of admissible functions has been checked out. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural and nonlinear frequencies is brought out. For simply supported rectangular FG shallow shells, the results obtained are compared with those available in the literature. Comparison demonstrates a good accuracy of the approach proposed.


Functionally graded shallow shells The R-functions theory Method by Ritz Geometrically nonlinear vibrations 



This work was carried out as a part of the Erasmus Mundus postdoctoral exchange research program ACTIVE.


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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Higher MathematicsNational Technical University “Kharkiv Polytechnic Institute”KharkivUkraine
  2. 2.Faculty of Engineering and the EnvironmentUniversity of SouthamptonSouthamptonUK

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