Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

摘要

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton's principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.