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Research Papers

Design and Analysis of a Foldable/Unfoldable Corrugated Architectural Curved Envelop

[+] Author and Article Information
Francesco Gioia

 Dipartimento di Architettura, Costruzioni e Strutture (DACS),Università Politecnica delle Marche, I-60121 Ancona, Italyfrancescogioia@email.it

David Dureisseix1

 Contact and Structural Mechanics Laboratory (LaMCoS),INSA de Lyon/CNRS UMR 5259, F-69621 Villeurbanne, FranceDavid.Dureisseix@insa-lyon.fr

René Motro

 Mechanics and Civil Engineering Laboratory (LMGC),University Montpellier 2/CNRS UMR 5508, F-34095 Montpellier, FranceRene.Motro@univ-montp2.fr

Bernard Maurin

 Mechanics and Civil Engineering Laboratory (LMGC),University Montpellier 2/CNRS UMR 5508, F-34095 Montpellier, FranceBernard.Maurin@univ-montp2.fr

1

Corresponding author.

J. Mech. Des 134(3), 031003 (Feb 18, 2012) (11 pages) doi:10.1115/1.4005601 History: Received February 04, 2011; Revised November 10, 2011; Published February 17, 2012; Online February 18, 2012

Origami and paperfolding techniques may inspire the design of structures that have the ability to be folded and unfolded: their geometry can be changed from an extended, servicing state to a compact one, and back-forth. In traditional origami, folds are introduced in a sheet of paper (a developable surface) for transforming its shape, with artistic, or decorative intent; in recent times the ideas behind origami techniques were transferred in various design disciplines to build developable foldable/unfoldable structures, mostly in aerospace industry (Miura, 1985, “Method of Packaging and Deployment of Large Membranes in Space,” Inst. Space Astronaut. Sci. Rep., 618 , pp. 1–9; Ikema , 2009, “Deformation Analysis of a Joint Structure Designed for Space Suit With the Aid of an Origami Technology,” 27th International Symposium on Space Technology and Science (ISTS)). The geometrical arrangement of folds allows a folding mechanism of great efficiency and is often derived from the buckling patterns of simple geometries, like a plane or a cylinder (e.g., Miura-ori and Yoshimura folding pattern) (Wu , 2007, “Optimization of Crush Characteristics of the Cylindrical Origami Structure,” Int. J. Veh. Des., 43 , pp. 66–81; Hunt and Ario, 2005, “Twist Buckling and the Foldable Cylinder: An Exercise in Origami,” Int. J. Non-Linear Mech., 40 (6), pp. 833–843). Here, we interest ourselves to the conception of foldable/unfoldable structures for civil engineering and architecture. In those disciplines, the need for folding efficiency comes along with the need for structural efficiency (stiffness); for this purpose, we will explore nondevelopable foldable/unfoldable structures: those structures exhibit potential stiffness because, when unfolded, they cannot be flattened to a plane (nondevelopability). In this paper, we propose a classification for foldable/unfoldable surfaces that comprehend non fully developable (and also non fully foldable) surfaces and a method for the description of folding motion. Then, we propose innovative geometrical configurations for those structures by generalizing the Miura-ori folding pattern to nondevelopable surfaces that, once unfolded, exhibit curvature.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Using folds in architecture. Top left: kinematic aspect of a foldable gallery (Tachi [10]); top right: morphogenetic aspect of the Art Tower Mito (A. Isozaki, image by Korall, Creative Commons 2006); bottom: static aspect for the Saint Loup Chaptel (Buri and Weinand [11])

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Figure 2

Definition of a configuration and orientation of its surface

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Figure 3

Mountain fold (upper left, denoted with straight lines) and valley fold (upper right, denotes with dotted lines)

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Figure 4

Possible evolutions of folds during folding movement, and local folding angle

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Figure 5

Illustration of local flat foldability, for the flat unfolded state (left), and two nonflat states (center and right)

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Figure 6

Global descriptor GΩ as the parameter of a unique folding path (GΩ'≤GΩ≤GΩ")

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Figure 7

Classification and examples of foldable corrugated meshes; (a) developable and flat-foldable, (b) developable and nonflat-foldable, (c) nondevelopable and flat-foldable, (d) nondevelopable and nonflat-foldable, and (e) unfoldable (rigid)

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Figure 8

Controlability of folding path: typical evolution of two degrees of freedom versus GΩ

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Figure 9

Examples of Miura-ori generalizations; top: genuine Miura-ori fold; middle: curved pleated sheet structure [8]; bottom: generalization of quadrilateral mesh origami [9]

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Figure 10

Foldability properties of the Miura-ori tessellation (straight lines are mountain folds, dotted lines are valley folds)

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Figure 11

Characteristics of the Miura-ori basic unit: expansion coefficients along x and y directions: ex and ey

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Figure 12

Folding path for the Miura-ori basic unit

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Figure 13

Modification of the Miura-ori basic unit to a nondevelopable corrugated unit (straight lines are mountain folds, dotted lines are valley folds); (a) original Miura-ori unit, (b) modified unit, (c) another assembly process of the same modified unit, (d) dedicated coordinate basis for the analysis

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Figure 14

Folding path for the nondevelopable unit

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Figure 15

Expansion coefficients for different design parameters

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Figure 16

Thickness of the unfolded state of the nondevelopable unit (left) and of the Miura-ori unit (right, with zero thickness)

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Figure 17

Evolution of the thickness with design parameters (left, lighter color: not admissible states), folded state for the maximal thickness (right)

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Figure 18

Product of the thickness with the covered area of the unfolded state

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Figure 19

Folding path of a three-by-three unit assembly. Two arrays of orthogonal planes are conserved along the path

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Figure 20

Modification of the Miura-ori basic unit to a nondevelopable corrugated units with single curvature (straight lines are mountain folds, dotted lines are valley folds); (a) original Miura-ori unit, (b) modified unit of first type (x-curved surface), (c) modified unit of second type (y-curved surface)

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Figure 21

Four different three-by-three unit assemblies to get single-curved flat-foldable corrugated meshes. From left to right and top to bottom: concave x-curvature, convex x-curvature, concave y-curvature, convex y-curvature. Two arrays of orthogonal planes are conserved along the path; one being a radial array.

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Figure 22

Design process for modifying the generalized unit to engender a curved global surface

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Figure 23

Assembly of two modified units for the second type of curved surface (x-curved, within symmetry plane)

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Figure 24

Assembly of two modified units for the first type of curved surface (y-curved, within the plane orthogonal to CA )

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Figure 25

Convex y-curvature folding path

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Figure 26

Concave y-curvature folding path

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Figure 27

Concave x-curvature folding path with a zoom when an interpenetration occurs

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Figure 28

Physical model of a single curvature design (unfolded state, and the almost flat-folded state at the same scale)

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Figure 29

Free-form single-curved foldable corrugated surface

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