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Geometry of Structural Form
Abstract
This paper describes a precise geometric method for the inscription of structural constraints into architectural form. Based
on techniques from graphic statics, the force distribution in building structures is visualized using geometric diagrams.
This diagrammatic representation allows a formal description that shows the relationship between the force flow and the structural
form. The formal character of this description enables the direct implementation of a parametric truss geometry that maintains
major structural behavioral characteristics under deformation. An interactive model of a structural freeform roof is developed
through this link between a parametric truss definition and a design-driving NURBS surface. This allows for an intuitive exploration
of the constrained design space in real time. Formal explorations and the comparison with built examples demonstrate the effectiveness
of this approach.
KeywordsArchitectural Freeform Surface–Structural Geometry–Interactive Design–Reciprocal Diagrams–Parametric Modeling–Graphic Statics
... In the presence of bending moments, passive stiffening systems, such as trusses, can be designed to follow the bending moment diagram to improve structural efficiency, dividing moments into compression and tension elements. Lachauer (Lachauer and Kotnik 2010) showed how trussed elements can be added to an arbitrary two-dimensional curved geometry to improve structural efficiency in an integrated and interactive framework. In contrast, active systems are defined as systems in which it is possible to introduce external forces to the main structure through a system of stressed tension cables and compressive/tension struts to change internal force distributions. ...
... However, new computational tools have recently been developed to implement graphic statics in a very intuitive and user-friendly way (Block Research Group 2014;Fivet and Zastavni 2014;Greenwold and Allen 2014). As mentioned previously, an interactive tool for finding the geometry of passive external systems using graphic statics has already been developed (Lachauer and Kotnik 2010). The new tool presented in this paper for active external posttensioning systems is also interactive, in that a change of the starting shape results automatically in a new posttensioning system configuration. ...
Funicular geometries, which follow the idealized shapes of hanging chains under a given loading, are recognized as materially efficient structural solutions because they exhibit no bending under design loading, usually self-weight. However, there are circumstances in which nonstructural conditions make a funicular geometry difficult or impossible. This paper presents a new design philosophy, based on graphic statics, that shows how bending moments in a nonfunicular two-dimensional curved geometry can be eliminated by adding forces through an external posttensioning system. An interactive parametric tool is introduced for finding the layout of a posttensioning tendon for any structural geometry. The effectiveness of this approach is shown with several new design proposals.
... Furthermore, as seen in Figure 2, once such a construction has been set up in an interactive drawing environment as described in Section 2, an entire range of freeform constant-force trusses can be designed and evaluated in an intuitive and exploratory manner using simple controls, such as b-spline control points or shape superposition sliders (Fig. 2) [4]. Moreover, such explorations can lead to new insights, such as the realisation that the formally very different structures shown in Figure 3 could be designed using the same structural concept, i.e. with the same geometric constraints applied to their force diagrams [5]. ...
... Although the highly indeterminate 3D networks and the force equilibrium in space are significantly more complex than in twodimensional problems, the inherent simplicity and intuitive readability of the graphical representation Fig. 3: Interactive, parameter-driven explorations of the relation between form and forces demonstrate how different structures can be based on the same structural principles. [5] of form and forces through geometrically linked diagrams results in a deeper understanding of the structural form, especially in comparison to analytical and numerical approaches. Moreover, this approach allows the flexible generation of freeformlike compression-only networks with complex geometry [8]. ...
The development of structures with complex, curved geometry typically consists of a series of iterative steps in which formal and structural considerations are addressed separately. Since such a sequential process often results in solutions that fail to realise the aesthetic and structural intentions of the original design, there is clearly a need for an integrated approach providing bidirectional control over both form and forces at every stage of the design exploration. This paper therefore describes and advocates an approach based on elements of graphic statics, in which the design of structures is based on geometrical rather than analytical or numerical representations of the relation between form and forces. The presented approach adopts the key principles of graphic statics and extends them with the latest research on form finding, structural design and optimisation techniques.
... Structural design optimisation is commonly limited to a postrationalisation of free-form architectural shapes. Focusing only on minor shape adjustments in the late-designing phase [51]. While modern possibilities of using multivariate optimisation allow already in the preliminary design phase, construct a statement, parameters for further modifications based on structural logic and generate multiple solutions to consider by the architect [52]. ...
Contemporary interdisciplinary design requires architects' knowledge and cooperation with such fields as construction, material engineering, fabrication methods, and knowledge in optimisation of the design process, production, and minimisation of used materials and energy. Following the example of other disciplines, contemporary architecture seeks inspiration from Nature on various levels. The development of modern tools and materials opens unprecedented opportunities for designers to shape free forms with precision, following sustainable development guidelines. The article presents the influence of biomimicry inspiration on shaping spatial structures of 20th and 21st-century architecture. The primary conclusion of the review indicates the need for further implementing bio-logic strategies into interdisciplinary, holistic building design.
... Today new computational tools have been developed to implement graphic statics in a very intuitive and user-friendly way [36,98]. In addition, an interactive tool for finding the geometry of trusses using graphic statics has already been developed [59]. ...
Curved structures are characterized by the critical relationship between their geometry and structural behaviour, and selecting an appropriate shape in the conceptual design of such structures is important for achieving material efficiency. However, the set of bending-free geometries are limited and, often, non-structural design criteria (e.g., usability, architectural needs, aesthetics) prohibit the selection of purely funicular or antifunicular shapes.
In response to this issue, this thesis studies the possibility of achieving an axial-only behaviour even if the geometry departs from the ideally bending-free shape.
This dissertation presents a new design approach, based on graphic statics that shows how bending moments in a two-dimensional geometry can be eliminated by adding forces through an external post-tensioning system. his results in bending-free structures that provide innovative answers to combined demands on versatility and material optimization.
The graphical procedure has been implemented in a free-downloadable design-driven software (EXOEQUILIBRIUM) where structural performance evaluations and geometric variation are embedded within an interactive and parametric working environment. This provides greater versatility in finding new efficient structural configurations during the first design stages, bridging the gap between architectural shaping and structural analysis.
The thesis includes the application of the developed graphical procedure to shapes with random curvature and distribution of loads. Furthermore, the effect of different design criteria on the internal force distribution has been analyzed.
Finally, the construction of reduced- and large-scale models provides further physical validation of the method and insights about the structural behaviour of these structures.
In summary, this work strongly expands the range of possible forms that exhibit a bending-free behaviour and, de facto, opens up new possibilities for designs that combine high-performing solutions with architectural freedom.
Free download at: http://oa.upm.es/39733/1/Leonardo_Todisco.pdf
... Tight integration in the contemporary design workflow, visualization and interactive feedback in real time will lead to an intuitive understanding of structural correlations and thereby advance creativity in the design process. The conceptual framework for the integration of structural constraints in a parametric model has been described earlier [16]. The presented tools now offer a systematic approach for the integration of structural constraints into parametric systems in general. ...
This paper presents a concept for a novel typology of computational design tools which are able to bridge the gap between the intuitive "structural sketch" and sophisticated analysis software. Based on equilibrium solutions and graphic statics, a set of interactive parametric tools have been implemented as a prototype, within an existing computer-aided design system. These tools allow for the powerful synthesis between structural design approaches based on funicular forms and associative geometric modelling techniques.
... Tight integration in the contemporary design workflow, visualization and interactive feedback in real time will lead to an intuitive understanding of structural correlations and thereby advance creativity in the design process. The conceptual framework for the integration of structural constraints in a parametric model has been described earlier [16]. The presented tools now offer a systematic approach for the integration of structural constraints into parametric systems in general. ...
This paper presents a concept for a novel typology of computational design tools which are able to bridge the gap between the intuitive "structural sketch" and sophisticated analysis software. Based on equilibrium solutions and graphic statics, a set of interactive parametric tools have been implemented as a prototype, within an existing computer-aided design system. These tools allow for the powerful synthesis between structural design approaches based on funicular forms and associative geometric modelling techniques.
This book publishes the peer-reviewed proceeding of the third Design Modeling Symposium Berlin . The conference constitutes a platform for dialogue on experimental practice and research within the field of computationally informed architectural design. More than 60 leading experts the computational processes within the field of computationally informed architectural design to develop a broader and less exotic building practice that bears more subtle but powerful traces of the complex tool set and approaches we have developed and studied over recent years. The outcome are new strategies for a reasonable and innovative implementation of digital potential in truly innovative and radical design guided by both responsibility towards processes and the consequences they initiate.
This paper presents a novel and rigorous approach for the design of efficient spatial support structures for flat slabs. Based on the given dead load of the slab and the topology of the supporting structure, this method allows form-finding of support structures that are in equilibrium with axial compression forces only. During the form-finding process, the resulting horizontal forces from the compression support structure are balanced through translation and rotation of the slab, and subsequently resolved in the plane of the slab using funicular or trussed strut-and-tie systems. The presented method can be applied for the design of a variety of different typologies of supporting structures, such as inclined columns, curved walls, branching structures and shells. A computational prototype of the approach is implemented as CAD modeling tool, and the potential of the method is shown through two formal design explorations.
An essential step for the realization of free-form surface structures is to create an efficient structural gird that satisfies not only the architectural aesthetics, but also the structural performance. Employing the main stress trajectories as the representation of force flows on a free-form surface, an automatic grid generation approach is proposed for the architectural design. The algorithm automatically plots the main stress trajectories on a 3D free-form surface, and adopts a modified advancing front meshing technique to generate the structural grid. Based on the proposed algorithm, an automatic grid generator named “St-Surmesh” is developed for the practical architectural design of free-form surface structure. The surface geometry of one of the Sun Valleys in Expo Axis for the Expo Shanghai 2010 is selected as a numerical example for validating the proposed approach. Comparative studies are performed to demonstrate how different structural grids affect the design of a free-form surface structure.
The quality of a quad-mesh depends on the shape of the individual quadrilaterals. The ideal shape from an architectural point of view is the planar square or rectangles with fixed aspect ratio. A parameterization that divides a surface into such shapes is called isothermic, i.e., angle-preserving and curvature-aligned. Such a parameterization exists only for the special class of isothermic surfaces. We extend this notion and introduce quasiisothermic parameterizations for arbitrary triangulated surfaces.
We describe an algorithm that creates quasiisothermic meshes. Interestingly many surfaces appearing in architecture are close to isothermic surfaces, namely those coming from form finding methods and physical simulation. For those surfaces our method works particularly well and gives a high quality and robust mesh layout. We show how to optimize such meshes further to obtain disk packing representations. The quadrilaterals of these meshes are planar and possess touching incircles.
This paper presents a novel approach for the interactive design of linear structures in space. A method is introduced, that
provides a maximum of formal freedom in the design of funicular, hence efficient, structures. For a given deck geometry, defined
by a design driving NURBS curve via parametric modeling techniques, a tailored relaxation routine allows for controlled, real-time
form-finding of the spatial funicular. Subsequently, the equilibrium of the deck is constructed using techniques from graphic
statics, combined with a least-square optimization technique. Finally, the method is applied to a design example.
The paper traces the development of a digital hanging chain modeler in Java inspired by Antonio Gaudi?s physical hanging chain models. More importantly, it demonstrates how fabrication schemas for physical mockups of the digitally simulated hanging chain can be linked to the real time form finding simulation. Fabrication output is an integral part of the iterative process and not a post-design process. The current implementation is still limited and currently requires programming for reconfiguration. The paper proposes the link of form-finding and fabrication finding and lays out several examples and first steps of how to do so.
Le thème de la structure constitue depuis toujours un aspect fondamental de la construction, intéressant aussi bien les ingénieurs que les architectes. Cet ouvrage consacré aux structures en architecture s'est donné pour objectif de contribuer à faciliter le dialogue entre ces deux domaines professionnels. «L'art des structures» offre un panorama complet sur les structures portantes et leur fonctionnement, en décrivant la manière dont les charges sont reprises et transmises jusqu'au sol. A cet effet, une approche intuitive est privilégiée: les bases de l'équilibre sont notamment expliquées en visualisant les efforts à l'intérieur d'ouvrages d'art historiques et modernes, à l'aide de simples outils graphiques.
This book argues for novel strategies to integrate engineering design procedures and structural analysis data into architectural design. Algorithmic procedures that recently migrated into the architectural practice are utilized to improve the interface of both disciplines. Architectural design is predominately conducted as a negotiation process of various factors but often lacks rigor and data structures to link it to quantitative procedures. Numerical structural design on the other hand could act as a role model for handling data and robust optimization but it often lacks the complexity of architectural design. The goal of this research is to bring together robust
methods from structural design and complex dependency networks from architectural design
processes. The book presents three case studies of tools and methods that are developed to exemplify, analyze and evaluate a collaborative work flow.
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