Cable-stiffened foldable elastica for movable structures Valentina Beatini a , Gianni Royer-Carfagni b,⇑ a Department of Architecture, Izmir Institute of Technology, Turkey b Department of Industrial Engineering, University of Parma, Italy article info Article history: Received 6 September 2012 Revised 4 February 2013 Accepted 7 February 2013 Available online 28 May 2013 Keywords: Movable structures Euler elastica Cable-stiffened structure Foldable structures Free-form design Transformable architecture abstract A structural element is proposed, made of a row of rigid voussoirs joined by a passing through cable. When the cable is tensioned, the ensemble acquires stiffness and, for appropriate contact profiles of the voussoirs, the response of the element under applied loads is governed by the same equations of Euler’s elastica or, in equivalent terms, of a non-linear spline. Releasing the connecting cable, the struc- ture is loosened and can be closely packed. With this system one can reproduce, at least in principle, any desired profile in the ‘‘stiff’’ configuration, and construct free form foldable surfaces of any shape. Ó 2013 Published by Elsevier Ltd. 1. Introduction The proposed structural system belongs to the class of kinetic structures, and in particular aims at meeting the recent demands for mobile solutions capable of achieving different-in-type equilib- rium configurations. The demand for transforming spaces is pushing researchers to revise and improve the kinematic structures built in the past. The category wider constructed are line-supported structures (large roofs of stadiums and theaters [1]), that use a minor number of joints and members, but usually they are very heavy and do not allow to obtain complex surfaces. Between these Hoberman, inside the Adaptive Building Initiative, suggested rigid panels, variously connected through hinges or sliding joints [2]. The proposed solu- tions, however, are a juxtaposition of modules, each of which oper- ates isolated, they are not self-supporting and must therefore be used in fixed façades or roofs. Because of this, they are actually un- able to architectonically transform a whole space. Major of research regards point-supported structures made of bars. Calatrava designed simple kinematic chain structures [3] and conceived symmetrical solids made of bars [4]. Others are modular scissor-like structures as those introduced by Pinero [5], deployable tensegrity grids researched by Motro [6] and symmet- rical assemblies of over constrained mechanisms [7,8]. To stabilize the structure in the final configuration, further rods can be temporarily added. Kokawa used cables that run along spe- cific lines of the articulated mechanism: once tensioned, they block the mechanism itself, eliminating the degrees of freedom [9]. Re- cently, the opportunity to use a snap-trough effect (first introduced by Zeigler [10]) has been studied: some modules have incompati- bilities with the length of the structural components, so that, while moving, the bars instantaneously pass from an equilibrium config- uration to another, significantly distant from the former [11]. In or- der to obtain a better control of the structural movement, especially for space applications, Pellegrino creatively uses or com- bines complaints, cables, pistons, achieving systems which can be controlled by one motor [12]. Although of great aesthetic value, all the aforementioned appli- cations have only two configurations, typically open/closed or ex- tended/compact. They have to be designed to achieve the specific shape of a planar surface, barrel vault, a platonic solid, or to approximate a sphere. When instead a few topical configurations are desired, more degrees of freedom have to be controlled. For example, Inoue and his colleagues have developed a flexible truss, called Variable Geometry Truss – VGT. This is a beam with some piston actuators and hinges: by controlling the lengths of extend- able rods, it is possible to partially fold the beam’s pieces and thus obtain curved shapes [13]. Within this class, the most studied examples are the scissors-like structures. These are formed essen- tially by a basic modulus composed of two cross bars connected by an intermediate pivot. Assembled together in 2D or 3D arrange- ments, they can form various shapes by maintaining a single de- gree of freedom. If one or more hinges are added, the number of 0141-0296/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.engstruct.2013.02.009 ⇑ Corresponding author. E-mail addresses: v.beatini@pec.it (V. Beatini), gianni.royer@unipr.it (G. Royer- Carfagni). Engineering Structures 56 (2013) 126–136 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct