A non-linear symmetric G1-conforming Bézier finite element formulation for the analysis of Kirchhoff beam assemblies

We present a formulation for slender space curved rods and rod assemblies that implicitly accounts for the Kirchhoff constraints and for the G1-continuity conditions (i.e. continuity of the geometric tangent) between elements. The whole formulation is developed in tensorial coordinate free form, apt to any numerical interpolation to be implemented. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the internal energy functional, for which the Levi-Civita connection of the configurations manifold of the rod is needed. The G1-continuity conditions are fulfilled by means of a change of basis, from the original configuration parameters (position of the beam axis and rotation around the axis tangent) to a new set of configuration parameters, whose relation to the original set is non-linear. For this reason an additional geometric term, specific for the G1-formulation, appears in the tangent stiffness matrix. The robustness and accuracy of the obtained Kirchhoff model is demonstrated with numerical examples that employ Bézier interpolation for the position and for the rotation angle.