An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam using Euler parameters

An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross-section and stretch and shear strains. The rotation is parametrized by Euler parameters, which can naturally avoid the singularity problem that Euler angles or the rotation vector can have. While stretch and shear strains are interpolated using polynomial functions, Euler parameters are interpolated using a C¹-continuous interpolation function to guarantee that curvatures of the centroid line of the beam are continuous at nodes of its elements. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several examples are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear locking and Poisson locking. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement, which verifies accuracy of the current formulation. Results from the current formulation for some other cases are also in excellent agreement with those from commercial software ABAQUS and ADAMS.