An isogeometric finite element formulation for geometrically exact Timoshenko beams with extensible directors

An isogeometric finite element formulation for geometrically and materially nonlinear Timoshenko beams is presented, which incorporates in-plane deformation of the cross-section described by two extensible director vectors. Since those directors belong to the space ${\Bbb R}^3$, a configuration can be additively updated. The developed formulation allows direct application of nonlinear three-dimensional constitutive equations without zero stress conditions. Especially, the significance of considering correct surface loads rather than applying an equivalent load directly on the central axis is investigated. In various numerical examples exhibiting large deformations, the accuracy and efficiency of the presented beam formulation is assessed in comparison to brick elements. We particularly use hyperelastic materials of the St.$\,$Venant-Kirchhoff and compressible Neo-Hookean types.