Geometrically nonlinear multi-patch isogeometric analysis of spatial Euler–Bernoulli beam structures

Abstract

This study presents a novel isogeometric Euler–Bernoulli beam formulation for geometrically nonlinear analysis of multi-patch beam structures. The proposed formulation is derived from the three-dimensional continuum theory where the beam axis and the director vectors of cross-sections are used to characterize beam configurations. The translational displacements of the beam axis and the axial cross-sectional rotation along the beam axis are considered as unknown kinematics. The orthogonality between the cross-sections and the beam axis is satisfied by using the smallest rotation mapping for the description of finite cross-sectional rotations. The use of the smallest rotation mapping reduces the nonlinearity of the employed strain measurements with respect to the unknown kinematics and offers highly efficient linearization. Furthermore, a penalty-free approach is introduced to deal with rigid connections in multi-patch beam structures in the context of geometrically nonlinear analysis. A novel nonlinear transformation between the total cross-sectional rotation and the unknown kinematics is derived, which facilitates the use of the total cross-sectional rotations at the ends of patches as discrete unknowns. This approach also allows straightforward enforcement of rotational boundary conditions. The proposed formulation is investigated by several well-established examples and great accuracy and efficiency are observed.