Inextensibility and Its Effect on the Number of Equilibria of Shallow Buckled Beams

A buckled beam with shallow rise under lateral constraint is considered. The initial rise results from a prescribed end displacement. The beam is modeled as inextensible, and analytical solutions of the equilibria are obtained from a constrained energy minimization problem. For simplicity, the results are derived for the archetypal beam with pinned ends. It is found that there are an infinite number of zero lateral-load equilibria, each corresponding to an Euler buckling mode. A numerical model is used to verify the accuracy of the model and also to explore the effects of extensibility.

Deformation Property and Suppression of Ultra-Thin-Walled Rectangular Tube in Rotary Draw Bending

Recently, miniaturization and weight reduction have become important issues in various industries such as automobile and aerospace. To achieve weight reduction, it is effective to reduce the material thickness. Generally, a secondary forming process such as bending is performed on the tube, and it is applied as a structural member for various products and a member for transmitting electromagnetic waves and fluids. If the wall thickness of this tube can be thinned and the bending technology can be established, it will contribute to further weight reduction. Therefore, in this study, we fabricated an aluminum alloy rectangular tube with a height H0 = 20 mm, width W0 = 10 mm, wall thickness t0 = 0.5 mm (H0/t0 = 40) and investigated the deformation properties in the rotary draw bending. As a result, the deformation in the height direction of the tube was suppressed applying the laminated mandrel. In contrast, it was found that the pear-shaped deformation peculiar to the ultra-thin wall tube occurs. In addition, axial tension and lateral constraint were applied. Furthermore, the widthwise clearance of the mandrel was adjusted to be bumpy. As a result, the pear-shaped deformation was suppressed, and a more accurate cross-section was obtained.