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 ${\mathbb{R}}^3$, 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.

提出了几何和材料非线性 Timoshenko 梁的等几何有限元公式，该公式结合了由两个可扩展导向向量描述的横截面的平面变形。因为那些导演属于这个空间${\mathbb{电阻}}^3$，配置可以附加更新。开发的公式允许在没有零应力条件的情况下直接应用非线性三维本构方程。特别是，研究了考虑正确的表面载荷而不是直接在中心轴上施加等效载荷的意义。通过使用增强的假定应变 (EAS) 方法，为横截面添加了不兼容的线性平面应变分量以缓解泊松锁定。在显示大变形的各种数值示例中，与砖元素相比，评估了所提出梁公式的准确性和效率。我们特别使用了 St.  Venant-Kirchhoff 和可压缩的 Neo-Hookean 类型的超弹性材料。