A Timoshenko–Ehrenfest planar beam element for geometrically nonlinear analysis using rational Bézier representations with varying weights

This study presents an enhanced Timoshenko–Ehrenfest beam element designed for geometrically nonlinear analysis of planar straight beams subjected to small strains. The methodology adopts the isogeometric analysis (IGA) approach, utilizing rational cubic Bernstein basis functions to construct the interpolations of kinematic unknowns. However, in contrast to the traditional IGA concept, the isoparameterization between the reference geometry and kinematic unknowns is relaxed by treating specific weights in the basis functions as degrees of freedom. This approach significantly improves the accuracy of interpolations, providing superior descriptions of complex deformed configurations with only a minimal increase in element degrees of freedom. Notably, the proposed beam element is shown to be free from the locking phenomena, i.e., membrane and shear locking, without the need for additional treatments. Through rigorous numerical experiments, the study demonstrates the exceptional efficiency and robustness of the proposed beam element, confirming its superior capabilities.