Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 575131, 32 pages
Research Article

Unconstrained Finite Element for Geometrical Nonlinear Dynamics of Shells

Departamento de Engenharia de Estruturas, Universidade de São Paulo, Av Trabalhador Sãocarlense 400, 13570-960 São Carlos, SP, Brazil

Received 27 June 2008; Revised 3 February 2009; Accepted 25 March 2009

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2009 Humberto Breves Coda and Rodrigo Ribeiro Paccola. 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.


This paper presents a positional FEM formulation to deal with geometrical nonlinear dynamics of shells. The main objective is to develop a new FEM methodology based on the minimum potential energy theorem written regarding nodal positions and generalized unconstrained vectors not displacements and rotations. These characteristics are the novelty of the present work and avoid the use of large rotation approximations. A nondimensional auxiliary coordinate system is created, and the change of configuration function is written following two independent mappings from which the strain energy function is derived. This methodology is called positional and, as far as the authors' knowledge goes, is a new procedure to approximated geometrical nonlinear structures. In this paper a proof for the linear and angular momentum conservation property of the Newmark β algorithm is provided for total Lagrangian description. The proposed shell element is locking free for elastic stress-strain relations due to the presence of linear strain variation along the shell thickness. The curved, high-order element together with an implicit procedure to solve nonlinear equations guarantees precision in calculations. The momentum conserving, the locking free behavior, and the frame invariance of the adopted mapping are numerically confirmed by examples.