Isogeometric analysis of space rods: considerations on stress Locking

"the paper deals with the isogeometric analysis via B-splines of spaces rods under Kirchhoff-Love hypotheses. The approach was used by Gontier and Vollmer [8] for devoloping a plane curve element within the framework of the Timoshenko rod model, but they adopted only one patch to represent entirely the geometry of the rod; furthermore the authors developed their theory only plane elements. . In this work we develop a multi-patch isogeometric approach for the numeric analysis of the 3D Kirchhoff-Love rod theory. We use Bezier and B-Splines interpolations and we show that they are able to attain very good accuracy for rod structures, particularly for developing a 3D exact curve element with geometric torsion. The pathces in general present a Cn-continuity in the interior and are joined with C0-continuity, so that the global tangent stiffness operator in general is singular. In order to avoid the singularity in the stiffness operator several continuity conditions at the joints of the patches are required. Either parametric continuity (C2 or C3) or geometric continuity (G1 or G2) conditions can be imposed.. The geometric continuity conditions are weaker than the parametric conditions. The continuity conditions in the CAD-literature are known as the beta-constraint and represent constraint condition for the displacements of the control points where the scalar beta-quantity represents additional unknowns. In this work, we don't impose the continuity conditions via beta-constraints but directly by means of the Lagrange's multipliers methods."