Isogeometric analysis of 3D beams for arbitrarily large rotations: Locking-free and path-independent solution without displacement DOFs inside the patch

Abstract

In large rotation analysis of shear deformable 3D beams, some features characterize the ideal discretization method: locking-free solution, small number of unknowns, accurate description of curved geometries, no singularity for large rotations, objectivity, additivity and robustness in the iterative solution, path independence and symmetric matrix for conservative loads. This work shows how to achieve all these features using a novel isogeometric weak formulation. Incremental rotation vectors are associated to the control points of quadratic NURBS describing the geometry. The control point vectors are used to define a novel path-independent, objective and singularity-free approximation of the rotation matrix used for the evaluation of the exact strain measure, based on an element-wise corotational system. The inter-element NURBS regularity is preserved also for the rotation. Additivity and symmetry are assured by the vector-like parameterization. Imposing constant spatial forces over each patch results in a locking-free discretization without displacement DOFs within the patch. Boundary conditions can be imposed directly without any ad hoc procedure. Accurate results are obtained using coarse meshes with reduced number of unknowns compared to displacement-based isogeometric analyses and locking-free finite elements.