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Mathematical Problems in Engineering
Volume 2018, Article ID 1040973, 16 pages
Research Article

Alternative Solutions of the Geodesic Differential Equations Applied to the Mechanical Analysis of the Tensile Armors of Flexible Pipes under Bending

COPPE/UFRJ, Department of Civil Engineering, Centro de Tecnologia, Bloco I2000, Sala I116, Cidade Universitária, 21945-970 Ilha do Fundão, RJ, Brazil

Correspondence should be addressed to Gabriel M. Gonzalez; moc.liamg@gsottamg

Received 8 December 2017; Accepted 23 January 2018; Published 26 February 2018

Academic Editor: Francesco Marotti de Sciarra

Copyright © 2018 Gabriel M. Gonzalez et al. 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.


In some mechanical models, the tensile armors of bent flexible pipes are treated as geodesics on a torus and, based on this hypothesis, the curvatures of these curves are calculated to obtain the acting stresses. However, a closed-form solution of the geodesic differential equations is not possible, which imposes difficulties on determining these curvatures. This work, therefore, proposes two alternative solutions to the nonlinear geodesic differential equations. The first relies on an artificial neural network (ANN) and the second is obtained by symbolic regression (SR). Both employ data from the numerical solution of the geodesic differential equations and showed good correlation with the complete dataset. Nevertheless, when tested against new data, the SR equations led to results almost equal to those obtained with the numerical solution of the differential equations and to null geodesic curvature. Despite also agreeing well with the numerical solution, the ANN indicates nonnull geodesic curvatures. Moreover, when compared to equations often employed in the design of flexible pipes, the SR equations may indicate different results, which can impact, for example, the fatigue or the instability analysis of the tensile armors of these pipes.