The Folding of Triangulated Cylinders, Part I: Geometric Considerations

Abstract

This study was inspired by a model of a triangulated cylindrical shell made by C. R. Calladine during an investigation of the mechanics of biological structures. The model consisted of identical triangular panels on a helical strip and had a small-displacement internal inextensional mechanism. It is shown that many triangulated cylinders broadly similar to Calladine’s model can be folded down to a compact stack of plates: only small strains, whose magnitude can be made arbitrarily small by the choice of suitable design parameters, are imposed during folding. A general geometric formulation of the problem is presented and then, assuming that the folding process is uniform, the folding properties of any triangulated cylinder of this generic type are discussed.

Geometry and design of origami bellows with tunable response

“…After the crease bend emerges, it does not remain in the same location. By traversing the face, the vertex of the crease bend introduces a mobile crease to the fold pattern, thereby bypassing the preconditions required for the bellows theorem to apply [27].…”

The geometry of unfolding tree leaves

“…A series of studies using a cylindrical shell model based on a biological structure (Calladine 1978) was carried out by Guest & Pellegrino (1994a,b, 1996. The force required to fold the cylinder was examined theoretically and experimentally using a model consisting of identical triangular panels along a helical strip.…”

Review of Inflatable Booms for Deployable Space Structures: Packing and Rigidization

“…It is important to note that while the folded buckling states are inextensional, they are an isolated configuration and cannot fold or unfold without material strains: using these patterns for deployable booms will therefore necessitate some stretching of the material. This feature may also be used to advantage by designing multi-stable booms, which are undeformed in the stowed, a partly-deployed and fully inflated configuration [31,37,38].…”

“…For example, You and Kuribayashi [30] use a distortion factor: a simple geometric incompatibility between adjoining folding unit cells is taken as a measure of the deployment strain. Guest and Pellegrino [31] constructed 'triangulated' folded cylinders where the fold 8 lines were described by three helical patterns. They assumed that only one of the helices changes length during deployment, and its strain was taken as a measure of the total deformation.…”