René Motro

Multiparametered Formfinding Method: Application to Tensegrity Systems:

A method allowing a multiparametered formfinding for prestressed and selfstressed reticulated systems with tensile and compressive members is presented. Known methods, based on geometric analysis and dynamic (dynamic relaxation) considerations have been developed for these systems but they allow generally the evolution of only one parameter. But, in case of shape finding of non-regular new forms or when the sought-after form is subject to a set of geometrical constraints, it becomes obligatory to elaborate a multiparametered form-finding process. The proposed numerical method, which is described in this paper, exploits the force density method, already used for form finding of pure tensile structures. However, equilibrium matrix of pure tensile structures as cable nets systems, admits always an inverse, which might be false when tensile and compressive members coexist in the system. In this paper, different processes allowing to define prestressed (or selfstressed) equilibrium geometry are described. Except for the relational structure which is considered as known at the beginning of the process, two sets of form-finding parameters can be identified for this method: prestress (or selfstress) coefficients of members and coordinates or redundant nodes. The proposed method does not yield a unique geometry but it is very convenient for a multiparametered formfinding, and has produced very interesting results, especially for Tensegrity Systems. Application of this method of multiparametered formfinding to Tensegrity Systems, provides the designer with an efficient way to achieve interesting new selfstressed geometries, such as the generation of double-layer grids by agglomeration of Tensegrity modules.

Designing tensegrity systems: the case of a double layer grid

Abstract Tensegrity systems are innovative strut and cable systems used in Civil Engineering. Their lightness and the impression of transparency they convey represent new sources of inspiration for architects. Nevertheless, their conception and their design are not easy insofar as these systems are reticulate, spatial and self-stressed. In this article we set out to present the different stages of the conception and the design of tensegrity systems. The study of the selfstress, the choice of its level, the design of the elements and the study of the sensitivity to manufacturing element errors are the different subjects described. We will then present the concrete case of a double layer grid of 81 m 2 area.

The surface stress density method as a form-finding tool for tensile membranes

Form-finding for membrane tension structures is a delicate operation which must ensure both the absence of compressive areas and interactive control of the forms generated. Until now, methods have generally been based on large displacements and strain analysis that provide non-linear formulations; resolution and computation are, therefore, too complex and cumbersome. This paper describes a new method of form-finding which reflects a wish to provide architects with a simple, effective and reliable investigation suited to their needs. The surface stress density method uses surface triangular elements with an isotropic stress tensor and leads to an iterative procedure which converges on configurations that satisfy the laws of static equilibrium. Mathematical considerations ensure the convergence conditions. This method enables the designer to generate a broad range of structures (pneumatic membranes or cable-reinforced membranes), while conserving control of shape and internal stress distribution. Moreover, the formulation draws a parallel with the force density method and a combined approach may be specified, particularly for the monitoring of reinforcing cable.

Determination of mechanism’s order for kinematically and statically indetermined systems

Abstract This paper deals with the mechanisms in kinematically and statically indeterminate reticulated systems. Knowledge of length variation amplitude for members in association with an assigned mechanism allows determination of mechanism order. This is a fundamental characteristic of these systems, mainly for stability considerations. We submit on one part, simple tests allowing distinction between “order one mechanisms” and mechanisms of higher order, and on the other part an algorithm giving access to order exact value for all mechanisms associated with a given reticulated system. With this algorithm, order one mechanisms and higher order mechanisms are identified. Simple examples are given in the text and illustrate these aspects. In conclusion, we submit a stop criterion for the algorithm which gives access to the finite mechanisms for most of constructive reticulated systems.

Design Aspects of a Deployable Tensegrity-Hollow-rope Footbridge

Tensegrity structures are composed of cables and struts in a prestressed self-equilibrium. Although tensegrity first appeared in the 1950s, it is seldom used in civil engineering. This paper focuses on the design aspects of a deployable tensegrity-hollow-rope footbridge. Deployment is usually not a critical design case for traditional deployable structures. However, for tensegrity systems deployment may be critical due to the actuation required. In this paper, deployment is investigated in a general design framework. The influence of clustered (continuous) cables and spring elements in statics and dynamics is studied. Finally, actuation schemes are explored to identify cases where deployment becomes a critical design case. For this configuration, deployment is a critical design case when the structure has spring elements and continuous cables.

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

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.

Selfstress States Identification and Localization in Modular Tensegrity Grids

The design of a modular tensegrity grid requires the determination of its selfstress states, before choosing an appropriate combination defining the systems initial stresses. However, the computation of the vectorial basis associated with selfstress states generally produces results that are difficult to exploit. We therefore propose two different strategies to identify and localize selfstress states in a modular tensegrity grid more pertinently. The first is based on a heuristic approach that exploits the systems structural composition of modularity and regularity. The second is numerical and aims at redefining the vectors of the basis in a more convenient and useful way. Two methods based on transformations of the vectorial basis of selfstress states have been developed for a minimal number of involved components. Finally, we suggest a selfstress state classification based on the number of components and their localization as well as on their mechanical behavior.

Cutting Pattern of Fabric Membranes with the Stress Composition Method

Study of fabric membranes emphasises on numerous difficulties which have to be overcome according to the willingness of reducing errors. We meet hence the first step of form-finding which must be achieved such as it ensures the determination of tensile shapes characterised by the lack of compressive areas. Therefore, it implies the knowledge of both the whole geometry and the prestress tensor at each point of the surface. Nevertheless, these considerations may be regarded as theoretical ones since the designer must then focus on the realisation of the membrane, in other words, on the determination the plane fabric strips which once assembled together and placed into position in space with anchorage points define the required configuration. This paper thus describes a new method devoted to the calculation of the strips with the aim of overcoming the drawbacks of the traditional processes. The used mechanical approaches allow to reduce distortions and errors by taking into account the whole geometrical characteristics of the strip, its stress distribution and the material constitutive laws. The theoretical formulation associates the operations of development and reduction and is based upon an iterative numerical procedure which converges to appropriate plane strips where distortions are minimised by least square methods. Several illustrative applications point out the efficiency of the method.

Visual spatial learning of complex object morphologies through the interaction with virtual and real-world data

Conceptual design relies on extensive manipulation of the morphological properties of real or virtual objects. This study aims to investigate the nature of the perceptual information that could be retrieved in different representation modalities to learn a complex structure. An abstract and complex object was presented to two study populations, experts and non-experts, in three different representation modalities: 2D view; digital 3D model; real object. After viewing, observers had to draw some parts of the structure into a 2D reference frame. The results reveal a considerable performance advantage of digital 3D compared with real 3D, especially in the expert population. The results are discussed in terms of the nature of the morphological cues made available in the different representation modalities.

Investigation of minimal forms with conjugate gradient method

Abstract Amongst the numerous calculation methods available to investigate minimal forms, an approach based on the mechanical consideration of uniform tension in the domain leads to the writing of innovative relationships. Indeed, by establishing the equivalence between the vector of nodal internal forces due to the prestressed domain and the gradient of the function to be minimized, this study proposes to use the conjugate gradient method as a minimal area shape form-finding tool. The determination of descent directions may refer to Fletcher–Reeves suggestion or to an optimized value based on the Polak–Ribiere formula. Moreover, the steplength calculation is envisaged in accordance with the More and Thuentes line search algorithm or with a modified approach especially adapted to the studied configurations. Numerical experiments illustrate the use of the conjugate gradient method and thus point out the efficiency of the suggested modifications related to required computation times. In a second time, the conjugate gradient method is compared with density methods and a mixed formulation is therefore put forward. Numerical tests enable the comparison between these different approaches.