Conference Paper

Transforming Nature's Forest into Manmade Forest: Fractal‐Based Computational Morphogenesis Approach for a Dendriform Pavilion Design

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This paper aims to explore the scope of applying the concept of fractal geometry in the field of architecture and construction. There are mainly two different types of fractals – self-similar fractal and random fractal. In this paper, both types of fractals are used to design a nature-inspired architectural structure with the strategy of exploring the potency of fractal geometry as a geometric framework that can offer new structural forms. Based on the mathematical formulations of self-similar fractal shape and random fractal shape, tree-inspired branching supports and natural terrain inspired unsmooth crinkled roof are modeled using the algorithms of Iterated Function System and Midpoint Displacement (Diamond Square Algorithm) method respectively. Fractal dimensions are calculated to assess the visual complexity of the roof surface and branching supports. Finite element analysis is performed to assess the structural strength of the model with respect to changing of fractal dimensions.
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The paper presents a detailed study of the columns in Pier Luigi Nervi's architecture. Geometrical analyses highlight the columns' complex formal gestures, such as the recurring use of ruled surfaces, as key in meeting the essential principles of structural architecture. The study links geometry articulations to functional, static, technological and economic considerations in six projects. Nervi's possible relationship with contemporary technological advancements is also reflected upon to enhance understanding of his design approach.
Conference Paper
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The paper illustrates the results of an interdisciplinary work flow concerning the design and construction of a prefabricated structure using poplar plywood. The study, promoted by Politecnico and University of Torino, is related to the development of a local wood resource focussing on the most traditional and important panel in Italy which, from a consolidated position in the furniture market, is now searching potential and innovative applications especially in the building sector, where the use of modern wooden materials is growing quickly. The Monalisa Wood Pavillon investigates the wood design issue from three main points of view. First, the structure is part of the promotion path of poplar plywood, conceptualizing its manufacturing process in an architectural project; then it highlights the role of technology by using an engineered veneer-based product with interesting properties; finally it explores the role of digital design in exploiting the potentialities of the material, in particular through a parametric approach which permits the optimization of design and construction. The project was carried out thanks to the joined efforts of complementary sectors and to the coordinated work of architects, engineers and wood experts. The resulting architecture was built on the occasion of the 2012 Made Expo event, with the patronage of the Italian woodworking associations Federlegno, Assopannelli and Conlegno, together with the technical contribution of several industrial companies, which have sustained this research with the aim of developing a manifesto for a better knowledge of poplar plywood.
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The work of Gaudí embraces all the facets of architectural design. The present paper studies the analysis and design of masonry arches, vaults and buildings. It is well known that Gaudí used hanging models and graphical methods as design tools. These methods can be traced back to the end of the 17th Century. In addition, it was not original the use of equilibrated, catenarian forms. What was completely original was the idea of basing all the structural design in considerations of equilibrium. Gaudí also employed unusual geometrical forms for some of his vaults and ruled surfaces, showing a deep structural insight. Finally, he designed tree-forms of equilibrium for the supports of the vaults in the Sagrada Familia. In the present paper Gaudí's equilibrium methods are studied with some detail, stressing their validity within the frame of Limit Analysis.
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Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the café where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
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Examining botanical trees, Leonardo da Vinci noted that the total cross section of branches is conserved across branching nodes. In this Letter, it is proposed that this rule is a consequence of the tree skeleton having a self-similar structure and the branch diameters being adjusted to resist wind-induced loads.
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Tree stability in windstorms and tree failure are important issues in urban areas where there can be risks of damage to people and property and in forests where wind damage causes economic loss. Current methods of managing trees, including pruning and assessment of mechanical strength, are mainly based on visual assessment or the experience of people such as trained arborists. Only limited data are available to assess tree strength and stability in winds, and most recent methods have used a static approach to estimate loads. Recent research on the measurement of dynamic wind loads and the effect on tree stability is giving a better understanding of how different trees cope with winds. Dynamic loads have been measured on trees with different canopy shapes and branch structures including a palm (Washingtonia robusta), a slender Italian cypress (Cupressus sempervirens) and trees with many branches and broad canopies including hoop pine (Araucaria cunninghamii) and two species of eucalypt (Eucalyptus grandis, E. teretecornus). Results indicate that sway is not a harmonic, but is very complex due to the dynamic interaction of branches. A new dynamic model of a tree is described, incorporating the dynamic structural properties of the trunk and branches. The branch mass contributes a dynamic damping, termed mass damping, which acts to reduce dangerous harmonic sway motion of the trunk and so minimizes loads and increases the mechanical stability of the tree. The results from 12 months of monitoring sway motion and wind loading forces are presented and discussed.
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Branching structures are based on geometric systems that expand through bifurcation without returning to form closed cells. In this sense, branching structures resemble the structure of trees that branch continually outward. In architectural engineering, these forms can be used either as tension or compression systems. Numerous built examples have been produced since the initial inspiring studies made by Frei Otto in the early 1960's. Form finding techniques based on models have been used in the past to study these forms. Although thread models can be effective in the study of force paths, they cannot distinguish between tension and compression and have no way to take member buckling into account. But buckling does have an influence on appropriate geometry of a compression system. Also, minimal paths (or pseudo minimal paths based on surface tension thread models) have been used to explore possible geometries for branching structures. In this paper, both surface tension thread models dipped in water, and weighted string models are shown in comparison with ideal tension and compression forms found with a computational method based on Genetic Algorithms. The same computational model is used to find geometries with minimal overall member length. Both 2D and 3D geometries are derived. Peer Reviewed
Pier Luigi Nervi's first series of RC airplane hangars, built at Orvieto after 1935, are known for their cast in situ ribbed vault with crisscrossing elements. The paper aims to point out how a series of parallel experiments in steel construction and in airplane engineering also partly dealt with Nervi's objectives; and these experiments, alongside the developments in RC thin shell construction, need to be brought to attention in discussing Nervi's achievements.
Through low cyclic loadings of single Dougong, two Dougongs and four Dougongs, the degradation of the lateral stiffness of Dougong was studied. According to the test, the hysteresis curve and skeleton curve of force-displacement were obtained and the formula of resilience model and mechanical model were established. From the experiment, it is by all appearances that slippage is the main deformation of Dougong. Energy dissipation due to frictional slippage is an important reason for the excellent seismic behavior of ancient wooden structures. Hysteretic loops of Dougong give the appearance of parallelogram. Plump area of the hysteretic loop accounts for its excellent ability of energy dissipation performance. The mechanics model of Dougong belongs to linear strengthening elastic-plastic model. Both the fitting equation of force-displacement and resilience model reflect the changes in the stiffness of Dougong, and can be used to the research on static load delivery as well as dynamic structural analysis and calculation of similar wooden structures.
An experimental study of interactions between a high Reynolds number fluid flow and multi-scale, fractal, objects is performed. Studying such interactions is required to improve our current understanding of wind or ocean current effects on vegetation elements, which often display fractal-like branching geometries. The main objectives of the study are to investigate the effects of the range of scales (generation numbers) of the fractal object and of the incoming flow condition on the drag force and drag coefficient, and to observe flow features in the near wake region resulting from the interaction. In this study, Sierpinski carpets and triangles with the scale ratios of 1/3 and 1/2, respectively, are employed. The fractal dimensions of the Sierpinski carpet and triangle are D = 1.893 and 1.585, respectively. Each pre-fractal object is mounted on a load cell at the centerline in a wind tunnel. Two types of inflow conditions are considered: laminar flow and high-turbulence level, active-grid-generated, flow. As a first approximation, we find the drag coefficients are approximately constant of order unity, and do not depend upon generation number of the pre-fractal when defined using the actual frontal area that varies as function of generation number. Still, the drag coefficient of the Sierpinski carpet increases weakly with number of generations indicating that the drag force decreases less than the cross-sectional area. For the Sierpinski triangle a similar trend is observed at large scales. However, the drag coefficient displays a peak at the third generation and then shows a decreasing trend as smaller scales are included for higher generation cases. The drag coefficient for the turbulent flow is larger than that for the laminar flow for all the fractal generations observed. Flow features (mean velocity, mean vorticity, and turbulence root-mean-square distributions) are measured by using stereoscopic Particle Image Velocimetry to observe various scales of the motion in the near wake of the pre-fractal objects. Strong shear layers are formed behind the fractal objects depending on the hole locations of different generations, which results in the formation of various length scales of the dominant turbulence structures. The smaller scale wakes are found to merge behind the Sierpinski carpet, whereas they are merely damped behind the Sierpinski triangle.
This paper first discusses the fundamental notion of shape morphing and morphing techniques. Then it sets out to introduce early applications of the notion in architecture. Antoni Gaudi’s Sagrada Familia Cathedral in Spain and Le Corbusier’s Firminy Chapel in France are examined with regard to shape morphing. The examination seeks to cast new light on the significance of the two designs whose morphing method has become the legacy of an innovative characteristic of considerable contemporary architecture.
Models predicting forest stand wind-firmness are usually based on the calculation of a critical wind speed above which the mean tree of a stand is broken or uprooted. This approach is well adapted to regular stands, but in heterogeneous stands, not all the trees are necessarily damaged at the same time. Models used to analyse the distribution of damage within a population of trees can be a good alternative. In this perspective we developed FOREOLE, an individual-based mechanical model of tree response to wind. FOREOLE is based on a numerical description of tree structure allowing both wind and self-weight loads to be calculated at every level of the stem, as well as the bending moment at the tree base and mechanical stresses along the stem. We use a static approach to model wind forces in which the turbulent aspect of wind is taken into account through a gust factor. Stem breakage or uprooting is then predicted from comparisons to failure criteria, i.e. critical bending moment and critical compressive stress, respectively. Implemented in the software called CAPSIS, FOREOLE is compatible with a model of coniferous forest stand dynamics and allows wind-firmness to be simulated both in measured and virtual populations of trees.
"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
na broad sense Design Science is the grammar of a language of images Irather than of words. Modern communication techniques enable us to transmit and reconstitute images without needing to know a specific verbal sequence language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multi­ dimensional. Architectural renderings commonly show projections onto three mutual­ ly perpendicular planes, or consist of cross sections at different altitudes capa­ ble of being stacked and representing different floor plans. Such renderings make it difficult to imagine buildings comprising ramps and other features which disguise the separation between floors, and consequently limit the cre­ ative process of the architect. Analogously, we tend to analyze natural struc­ tures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures. Perception is a complex process. Our senses record; they are analogous to audio or video devices. We cannot, however, claim that such devices perceive.
Extending the analysis of branching patterns of the drainage net of rivers, originated by Horton, the relation of average numbers and lengths of tree branches to size of branch was investigated. Size of branch was defined by branch order, or its position in the hierarchy of tributaries. It was found that, as in river drainage nets, there is a definite logarithmic relation between branch order and lengths and numbers.This definite relation is quantitatively comparable, within limits, among river networks, tree branching systems, and several random-walk models in both two and three dimensions. Such a relation appears to be the most probable under the applicable constraints. Moreover the most probable arrangement appears to minimize the total length of all stems in the branching system within other constraints and so, to that extent, achieves a certain efficiency.
Form and Forces: Designing Efficient, Expressive Structures
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The geometry of organic architecture: the works of Eduardo Torroja, Felix Candela and Miguel Fisac
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Lower-bound analysis of masonry vaults
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Structure as Architecture
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The morphogenesis of shell structures: a conceptual, computational and constructional challenge
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Combined timber plate and branching column systems-variations and development of system interaction
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Félix Candela: Engineer, Builder, Structural Artist. Princeton University Art Museum Monographs
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Gaudí: Geometria, Estructura i Construcció
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Shape design methods based on the optimisation of the structure. Historical background and application to contemporary architecture
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Frank Lloyd Wright and the Johnson Wax Buildings
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Gaudi I la Sagrada Familia Comentada per all Mateix
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Metropol Parasol, Seville by Jürgen Mayer H -Review. Art and Design. The Observer. Guardian
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Art Nouveau & Gaudí: The way of nature
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Toyo Ito fuses structure and wrapper in a network of concrete trees at the new Tod's Omotesando Building in Tokyo
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Morphogenesis of Flux Structure
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Gaudí i els seus coŀlaboradors: artistes i industrials a l'entorn del 1900
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