On invariance of spatial isogeometric Timoshenko–Ehrenfest beam formulations for static analysis

Abstract

This study elaborates the invariance of spatial Timoshenko–Ehrenfest​ beam formulations in the context of isogeometric analysis. Such invariance confirms that zero strain measures are always generated by rigid transformations, i.e., rigid translations and rotations. The violation of this property can degrade the performance of the formulations in predicting structural responses. In the setting of linear analysis, the invariance of planar beam formulations has already been studied, but a similar investigation for spatial beam formulations is not yet carried out. Most of the spatial beam formulations are developed in the local coordinate frame, and components of unknown kinematics in this frame are discretized by using rational B-spline basis functions. Unfortunately, those formulations are found to be non-invariant under such a discretization scheme, and the degradation in their performance is demonstrated. On the other hand, the local coordinate frame is widely defined by the so-called natural Frenet–Serret frame. The sole utilization of this frame does not allow the consideration of beams having twisting configuration. In this study, these shortcomings are resolved.