Non-prismatic Timoshenko-like beam model: Numerical solution via isogeometric collocation

The present paper combines an effective beam theorywith a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross-section geometry, that in general deeply influences the stress distribution and, therefore, leads to non-trivial constitutive relations. Accounting for these peculiar aspects, the beam theory is described by a mixed formulation of the problem represented by six linear Ordinary Differential Equations (ODEs) with non-constant coefficients depending on both the cross-section displacements and the internal forces. Due to theODEs’ complexity, the solution can be typically computed only numerically also for relatively simple geometries, loads, and boundary conditions; however, the use of classical numerical tools for this problem, like a (six-field) mixed finite element approach, might entail several issues (e.g., shear locking, ill-conditioned matrices, etc.). Conversely, the recently proposed isogeometric collocation method, consisting of the direct discretization of the ODEs in strong form and using the higher-continuity properties typical of spline shape functions, allows an equal order approximation of all unknown fields, without affecting the stability of the solution. This makes such an approach simple, robust, efficient, andparticularly suitable for solving the systemofODEs governing the nonprismatic beam problem. Several numerical experiments confirm that the proposedmixed isogeometric collocationmethod is actually cost-effective and able to attain high accuracy. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).