Asymptotic Behavior of Stable Structures Made of Beams

In this paper, we study the asymptotic behavior of an e-periodic 3D stable structure made of beams of circular cross-section of radius r when the periodicity parameter e and the ratio r/e simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.