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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 838924, 22 pages
http://dx.doi.org/10.1155/2011/838924
Research Article

Exact Solutions of Nonlinear Equation of Rod Deflections Involving the Lauricella Hypergeometric Functions

1Via Negroli, 6, 20136 Milan, Italy
2Dipartimento di Matematica per le Scienze Economiche e Sociali, Viale Filopanti, 5, 40126 Bologna, Italy

Received 4 December 2010; Revised 29 March 2011; Accepted 16 June 2011

Academic Editor: Kenneth Berenhaut

Copyright © 2011 Giovanni Mingari Scarpello and Daniele Ritelli. 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.

Abstract

The stress induced in a loaded beam will not exceed some threshold, but also its maximum deflection, as for all the elastic systems, will be controlled. Nevertheless, the linear beam theory fails to describe the large deflections; highly flexible linear elements, namely, rods, typically found in aerospace or oil applications, may experience large displacements—but small strains, for not leaving the field of linear elasticity—so that geometric nonlinearities become significant. In this article, we provide analytical solutions to large deflections problem of a straight, cantilevered rod under different coplanar loadings. Our researches are led by means of the elliptic integrals, but the main achievement concerns the Lauricella 𝐹 𝐷 ( 3 ) hypergeometric functions use for solving elasticity problems. Each of our analytic solutions has been individually validated by comparison with other tools, so that it can be used in turn as a benchmark, that is, for testing other methods based on the finite elements approximation.