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Shock and Vibration
Volume 17 (2010), Issue 1, Pages 21-37

Non-linear Vibrations of Deep Cylindrical Shells by the p-Version Finite Element Method

Pedro Ribeiro,1 Bruno Cochelin,2 and Sergio Bellizzi2

1IDMEC/DEMEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
2Laboratoire de Mécanique et d'Acoustique, CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France

Received 23 May 2008; Revised 24 October 2008

Copyright © 2010 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A p-version shell finite element based on the so-called shallow shell theory is for the first time employed to study vibrations of deep cylindrical shells. The finite element formulation for deep shells is presented and the linear natural frequencies of different shells, with various boundary conditions, are computed. These linear natural frequencies are compared with published results and with results obtained using a commercial software finite element package; good agreement is found. External forces are applied and the displacements in the geometrically non-linear regime computed with the p-model are found to be close to the ones computed using a commercial FE package. In all numerical tests the p-FE model requires far fewer degrees of freedom than the regular FE models. A numerical study on the dynamic behaviour of deep shells is finally carried out.