Rotation-free isogeometric analysis of an arbitrarily curved plane Bernoulli-Euler beam

The present study elucidates linear static analysis for plane beam structures using the isogeometric approach. A novel methodology for rotation-free analysis of an arbitrarily curved Bernoulli-Euler beam in the convective frame of reference is derived in detail. The full degeneration of a 3D continuum beam to a 1D line has been presented and a fully applicable isogeometric finite element has been obtained. The driving force behind developing the present research has been the derivation of the NURBS-based isogeometric analysis which will enable an elegant formulation of the plane Bernoulli-Euler beams, being a function only of the global rectangular Cartesian coordinates. The verification and accuracy of the research are obtained via a thorough comparison between theory, finite element analyses and relevant examples from literature. An excellent agreement of results is achieved and usefulness for academic and practical purposes alike are proved. The effects of the hpk-refinements are illuminated and it is observed that the convergences for the most variables and refinement techniques are not monotonic. A special attention is paid to the influence of the product of maximum curvature and thickness of beam on the accuracy of the solution. The limits of applicability of the present approach are defined for a few specific types of analyses. The derived formulation is geometrically exact and appropriate for the analysis of strongly curved Bernoulli-Euler beams.