A 4-degree-of-freedom Kirchhoff beam model for the modeling of bending–torsion couplings in active-bending structures

Abstract

Gridshells are lightweight structures made of interconnected slender beams. Due to large displacements, high interaction between the beams, and bending–torsion coupling, modeling gridshells requires specific non-linear numerical tools to reach convergence within a reasonable time. In this article, the development of such a tool is presented. It is based on the Kirchhoff beam theory and uses the dynamic relaxation method. First, from Kirchhoff’s equations, the internal forces and moments acting on a beam are obtained. Once this mathematical work is done, the dynamic relaxation method is used in order to get the static equilibrium configuration of the beam. This new approach is tested on several examples and validated for slender beams with arbitrary rest-state configuration and cross sections. In particular, results for ribbons with high bending–torsion coupling are presented. Finally, this process enables the fast and precise modeling of gridshells including bending–torsion coupling.