  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. Incompatible linear in-plane strain components for the
cross-section have been added to alleviate Poisson locking by using an enhanced
assumed strain (EAS) method. 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.
