Flexure and torsion locking phenomena in out-of-plane deformation of Timoshenko curved beam element

The various locking phenomena in a linear Timoshenko curved beam element are identified in the context of out-of-plane deformations. A study on performance sensitivity of straight and curved Timoshenko beam finite elements to flexure-to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios are carried out. The use of consistent field interpolation for shear strain is shown to eliminate shear locking effects, which depends on the ratio EI/GA, in both straight and curved beams. However, in the case of curved beam, a curvature related additional spurious stiffening which depends on EI/GJ ratio is observed. This additional stiffening effect is attributed to inconsistencies present in the out-of-plane flexure and torsion strain definitions, which are critically examined to characterize their adverse effects on the solution convergence. The results reveal the existence of two additional locking phenomena which we introduce here as flexure locking and torsion locking. The field consistency requirements to eliminate these locking effects have been identified. In the numerical examples, the field consistency in flexure and torsion strains is applied, independently and in combination, to understand the hidden perspectives of these locking phenomena. The regimes where these locking effects become significant are identified based on the relative magnitudes of flexural and torsional stiffnesses. The convergence characteristics in various locking regimes are studied for different models and their performances are discussed in detail.