Constitutive models for strongly curved beams in the frame of isogeometric analysis

The current development of the isogeometric approach in various fields of mechanics is explained by the high-accuracy results which can be achieved at a reduced computational cost by codes based on non-uniform rational B-splines (NURBS). In the case of strongly curved beams the simple diagonal de Saint-Venant's constitutive model can lead to significant errors as it has been reported in the classic literature. Other models such as Winkler's have been proposed and seem more suitable for these kinds of structures. Unfortunately several numerical codes are based on a diagonal constitutive model which neglects the coupling effect of elongation and curvature even if a highly refined geometry description can be developed by means of NURBS. The results obtained by means of numerical codes based on isogeometrical analysis for curved beams are here reported and basic choices, computational costs and numerical accuracy of the above-mentioned constitutive models are discussed, from a qualitative and quantitative point of view. This comparison, in the authors' opinion, is necessary to avoid an excessive gap between the computational efficiency of NURBS, which are capable of very accurate geometry description, and a simplistic representation of the constitutive relations that is efficient for straight beams but not so much for curved beams whose curvature is large. The results of some selected tests are presented and discussed to highlight differences between the two approaches, showing that the small increase of computational cost of Winkler's model is well compensated by the accuracy gain.