On the locking free isogeometric formulations for 3-D curved Timoshenko beams

Locking-free isogeometric formulations of 3-D curved Timoshenko beams are studied. In particular, the global B¯ projection method and a mixed formulation are examined and compared, showing equivalently optimal convergence with regard to displacement solutions. These two formulations are proved to be equal when the beam cross-sectional properties are uniform. For beams with non-uniform cross-sectional properties, the mixed formulation is superior in terms of stress recovery. In addition to these two methods, an alternative locking-free formulation, a C0 NURBS element with selectively reduced integration is suggested in this study, making use of the traditional selective reduced integration (SRI) rule designed for Lagrangian elements. This SRI C0 NURBS element is simple to implement, preserves the sparsity of the global stiffness matrix and requires fewer quadrature point evaluations. Most importantly, due to the use of NURBS basis functions, the exact curve geometry is preserved with all three locking-free isogeometric elements. Benchmark problems and illustrative examples with complex curved geometries are examined for a detailed investigation of the considered locking-free elements.