Three‐Dimensional Finite Element Formulation for Thin‐Walled Beam Elements with Seven Degrees of Freedom per Node

This paper presents the complete theoretical framework for a three-dimensional thin-walled beam finite element with seven degrees of freedom per node: axial displacement u, transverse displacements v and w, torsional rotation θx, bending rotations θy and θz, and warping rotation φ. The formulation rigorously incorporates flexural–torsional coupling, shear flexibility in bending and torsion, bimoment effects, and geometric non-linearity through an exact geometric stiffness matrix. The variational (weak) form is derived in full from the principle of virtual work, yielding all generalized elastic and geometric stress resultants. Equilibrium equations in strong form and natural boundary conditions follow as the Euler–Lagrange equations of the variational statement. The resulting element enables analysis of lateral-torsional buckling, second-order effects, and general 3D instability of open and closed thin-walled members.