A new beam element with transversal and warping eigenmodes

• A new beam element with an enriched kinematics is derived.
• A full description for the method to determine warping and distortion modes is given.
• An exact solution of the equilibrium equation is performed.

In this work, we present a new formulation of a 3D beam element, with a new method to describe the transversal deformation of the beam cross section and its warping. With this new method we use an enriched kinematics, allowing us to overcome the classical assumptions in beam theory, which states that the plane section remains plane after deformation and the cross section is infinitely rigid in its own plane. The transversal deformation modes are determined by decomposing the cross section into 1D elements for thin walled profiles and triangular elements for arbitrary sections, and assembling its rigidity matrix from which we extracts the Eigen-pairs. For each transversal deformation mode, we determine the corresponding warping modes by using an iterative equilibrium scheme. The additional degree of freedom in the enriched kinematics will give rise to new equilibrium equations, these have the same form as for a gyroscopic system in an unstable state, these equations will be solved exactly, leading to the formulation of a mesh free element. The results obtained from this new beam finite element are compared with the ones obtained with a shell model of the beam.