Transplaced by Porosity V1

transplaced by porosity geometry, structure, and space:

kadim alasady

volume 1 | part 1 - part 3

Thesis Student Kadim Alasady Masters of Architecture Track I Advisor Bruce Johnson Assistant Professor University of Kansas School of Architecture, Design and Planning Department of Architecture terra@ku.edu 102 Marvin Studios Advisor Genevieve Baudoin Assistant Professor University of Kansas School of Architecture, Design and Planning Department of Architecture gbaudoin@ku.edu 102 Marvin Studios

A Independent Study Proposal Presented to the Graduate School University of Kansas

In Partial Fulfilment Of the Requirements of the Degree Master of Architecture Track I

May 2013

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THESIS FRAMEWORK section 1.0 | Thesis Components section 1.1 | Reference Data

PRELIMINARY DESIGN section 2.0 | Structural Column Decoding section 2.1 | Site Context Analysis section 2.2 | Space Frame Analysis

SCHEMATIC DESIGN section 3.0 | Structural Column section 3.1 | Space Frame

DESIGN DEVELOPMENT section 4.0 | Exhibition Program section 4.1 | Pavilion Program section 4.2 | Pavilion Design

FABRICATION section 5.0 | Fabrication Drawings

section 5.1 | Fabrication Photography

EXHIBITION section 6.0 | Exhibit Design

section 6.1 | Exhibition Photography

Part O Thesis Frame Sectio

One s ework on 01 Part One

Thesis Framework

In this part you will find the underlying framework guiding the process of the thesis and referential data to be used within the research.

Table of Contents:

Section 1.0 | Thesis Components Section 1.1 | Reference Data

Thesis Framework

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1.0 | Thesis Components

Section 1.0

Table of Contents

Section 1.0.1 | Preface Section 1.0.2 | Problem, Impediments, Case Study Section 1.0.3 | Annotated Bibliography Section 1.0.4 | Methodology Framework

Description

Thesis Framework

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1.0 | Thesis Components

Section 1.0 — Thesis Components

Section 1.0 will list the four components which structure the thesis and the research. The first component, preface, is a brief outlining the definitions of material, structure and geometry and the case study (the Sagrada Familia Structural Column). The second component will explore the problem, impediments, and the case study of the research, the third components gives an annotated bibliography and the fourth drafts a framework for the methodology.

Preface

Thesis Framework

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1.0.1 | Preface

Section 1.0.1 —

The proposed thesis is cultivated out of three parts: an exploration of the contents of materials, structure, and geometry as they relate to an understanding of architecture; a study of Antonio Gaudi’s Expiatory Church of the Sagrada Familia in Barcelona as a venue towards understanding the principles of part one; and a design project with an idea derived from the analysis of part two. All of these parts serve to chisel away a personal framework to solidify my ideas of design.

Part 1: Materials, Structure, and Geometry Mark Burry says “considering the built architecture in its possible fulfilments may work backwards to influence the ideation at a point where the physical manifestation of an idea is still hardly defined.” Burry illustrates a reciprocal of the common design process; he is advocating the process of fabrication to ideation.1 The common design process I speak of was made clear to me during my undergraduate studies as the process of ideation to fabrication, which sprung from the ‘architectural equation.’

The ‘architectural equation’ was introduced to me during my second year of undergraduate studio and it stated five variables: form, concept, program, function, and technics.2 Through design projects I explored the parts of the architectural equation and used them to expand my understanding of it. Although the equation specifies variables on architecture it does not state a thesis for design. A thesis for design should provide a process that enforces criticality and rigor. However, the process of ideation to fabrication that emerged during my undergraduate studios lacked criticality and rigor because it neglected the tangible; the tangible being the materiality of architecture. Instead, the process began with an idea derived from the analysis of a program, site, context, and/or precedent.

1. Mark, Burry. “Homo Faber.” Architectural design 75.4 (2005): 30-37. Print.

2. Eisenman, Peter.“Eisenman inside Out : Selected Writings, 1963-1988.” Theoretical Perspectives in Architectural History and Criticism. New Haven: Yale University Press, 2004. xv, 247 p. Print.

Preface

Preface

Thesis Framework 1.0.1 | Preface

Realizing that this process left the substance of my projects unfulfilled and my thoughts fragmented, I revisited all previous studio projects and critically analysed them. The analysis extracted three consistent parts: geometry, structure, and materials.

Geometry revealed itself as syntax for design.3,4 The Oxford dictionary defines syntax as “the arrangement of words and phrases to create well-formed sentences in a language.”5 In other words, syntax is a set of rules and within my projects this provided two sets of operational transformations: conceptual and formal. The conceptual tools deal with the mapping, decoding, and in depth understanding of existing conditions. The formal tools give hierarchy and organization through the transformation of geometry.6 These operational transformations dictate a discipline within the systems of architecture, which include program organization, circulation, structure, and form (interiority and exteriority).

The application of a syntax is executed by many architects. For instance, Antonio Gaudi observed and analysed nature to give him a syntax, which disciplined the geometries of his architecture. Two of the most complex geometries applied at his Sagrada Familia Church in Barcelona are ruled double curve surfaces and double twisted columns. Ruled double curve surfaces are, “as their name suggests, surfaces that contain straight lines because they are generated by the movement of one straight line that follows a particular route.” The double twisted columns are columns that “begin at the base with a regular or starred polygon with straight or parabolic sides or with a combination of polygons which, as the column rises, are transformed into different sections with an increasing number of vertices, until they reach the circle at the top.”7 In the way that syntax allowed Gaudi to create such geometries, syntax also guided my own work by providing a framework of geometric discipline. However, my process was reciprocal to that of Gaudi in two aspects: I, first, had a preconceived geometry and found a syntax that best described it versus extracting geometry from analysis and secondly the material selection and structural type were ostensibly imposed onto the geometry versus them unfolding from process.

3. Oxford dictionary defines geometry as the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. The different parts of geometry are studied by different fields; the field which best explains Gaudi’s geometry is algebraic geometry. Algebraic geometry studies plane algebraic curves such as lines, circles, parabolas, ellipses, hyperbolas, and cassini curves.

4. Syntax as a set of rules, in other words, a modus operandi. It is a set of rules prescribed via a diagram mapping a method of operation to discipline my designs.

5. “Oxford Dictionaries Online.” Oxford Dictionaries Online. N.p., n.d. Web. 19 July 2012. <http://oxforddictionaries. com/>.

6. Eisenman, Peter. Diagram Diaries. New York, NY: Universe Pub., 1999. Print.

7. Burry, Mark. Expiatory Church of the Sagrada Família : Antoni Gaudí. Architecture in Detail. 1 vols. London: Phaidon Press, 1993. Print.

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Preface

Thesis Framework 1.0.1 | Preface

Clearly, geometry dictated the greater parts of my previous designs, although this left them lacking in meaningful selections of structure and materials.8 When a geometric principle is first laid down for a design, the structure tends to follow that syntax, leaving the material selection to be dictated by structure. This works counter to architecture because architecture manifests itself in materials; in my designs, the material was the last thing considered. I believe this was the result of following the process of ideation to fabrication; my idea for each design began with a preconceived geometry for which a syntax was later created to control its development.

At that point, the remaining variables of the architectural equation (concept, program, function, and technics) were later imposed onto the design as the result of a geometric (formal) syntax.

Fabrication to ideation, on the other hand, is a potentially better method for the integration of material, structure, and geometry. It begins with the more tangible aspect of architecture; materials.

One principle sources that reinforce the successful use of this process is Antonio Gaudi and his modus operandi. Gaudi was interested in the observation of nature to yield an in-depth understanding of specific geometries which were inherently structural.9 That allowed Gaudi to approach his thinking on design through material and structural lenses. Therefore, the resulting geometries of his design were all derivatives of a rigorous analysis of nature which led to the testing of his finding via models and full scale prototypes.

Part 2: Antonio Gaudi’s La Sagrada Familia in Barcelona To acquire the contents of material, structure, and geometry Antonio Gaudi’s La Sagrada Familia Church in Barcelona will be used as a case study. Given the scale of the Sagrada Familia Church, only a primary column at the crossing will be used for the project (right). The project will be conducted through three parts: decoding the geometries of the column and digitally modelling it for production, using digital fabrication technology to reproduce the column at full scale, and provide a critical analysis.10 Part 3: Projected Outcomes The projected outcomes of this thesis are two-fold: the first being a final exhibition of scaled models, a digital screening, and an explanation of the Sagrada Familia and the second, a demonstration of my design syntax drafting the genealogy and evolution of my architecture.

The exhibition’s primary focus is a structural member (a primary column of the Sagrada

8. The selection of material is dependent on the fabrication methods. This thesis will be exploring 3-axis CNC milling, laser cutter, and welding. Respectively, plainer (sheet) material such as polymers and woods and dimensional material such as steel members with specified cross sectional profiles.

9. Burry, Mark.“Paramorph: Anti-Accident Methodologies.” Architectural design 69.9-10 (1999): 78-83. Print.

10. The methodology describe is furthered explained in section 1.2.1

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Preface

Thesis Framework 1.0.1 | Preface

Famiia Church – 65 feet in length) fabricated at full-scale, for which the material will be determined by fabrication methods (CNC milling, laser, and welding) and budget. The structural member surpasses the commonly recognized purpose; it is a structural element that transmits, through compression, the loads of structure from above to other elements below.

For Antonio Gaudi, a structural element (a primary column) is the component in space that is expressed through skin, structure, and geometry. Gaudi recognized the column as a segment containing the genetics of his architecture and expressed that meaning versus its contemporary treatment as clad object. Like Gaudi, other architects such as Mies Van Der Rohe recognized a similar meaning for the structural element (a primary column) and created either an atectonic or tectonic application.

The structural element (a primary column) will include the substance of material, structure, and geometry and simultaneously being skin and surface all in a single component, with the structure being a space-frame requiring the capability to support the column. Thus, the design will primarily focus on the parametric mapping of the internalization of the column to derive the space-frame, which will work in unison with the existing geometries. An example of such geometrically measurable architecture is found in the stone walls of Cusco, where the Incas, using dry stone construction, built a walled complex with each stone being either a derivative or an integral of the adjacent stone(s). The intent is to understand Gaudi’s geometrical method by fabricating and supporting a skin and surface through a primary column.

Alongside the full-scale structural member (the primary column) contextual models will show the primary column in its original context. Further process models will exhibit samples of parametric mapping while a digitally published book will exhibit the results of a year-long process and study. The greater part of its content will be a ‘Codex of Explicit Geometries,’ including geometric shape and construction, equations, and structural definitions. The Codex will visually display the parameters of the geometries. In addition to this Codex there will be another titled ‘Material Codex,’ expressing material type, behaviour, and stress analysis (via Robot Structural Analysis).

The final element of the exhibition involves a detailed explanation of the Sagrada Familia based on the studies conducted at the beginning of the research. The studies will be composed in a manner that is appropriate for the explanation of the Sagrada Familia Church to person(s) both familiar and unfamiliar (i.e. laypersons and architects).

Lastly and in addition to the final exhibition, I anticipate that the research and conclusions involved in the thesis will result in a defined personal design intent. I am in search of a design syntax; a set of rules or a system by which to discipline my design. The design syntax will be expressed partly through writing and partly through diagramming which will describe the genealogy and evolution of my architecture to date and potentially in the future.

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Problem, Impediments

Thesis Framework

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1.0.2 | Problem, Impediments, Case Study

Section 1.0.2 — Problem, Impediments, Case Study

Problem: Among the multiplicity of contentions in design brought about in the past few decades by increasing computer technology (CAD) and fabrication efficiency (CAM), this thesis aims to focus on just a few. First of all, the study of Antonio Gaudi’s Sagrada Familia through the use of CAD software demonstrates a contrast between traditional architectural methods and digital architectural methods. While neither claims to be right or wrong, they can be used in conjunction for a better understanding of the whole and can produce variability in process. The second topic the research addresses is the current lack of criticality in design, or, how one goes about being critical in design. The intimate study of Gaudi’s Sagrada Familia reveals a rigor and process that has slipped through the cracks in design today. The third topic the research aims to address is form-making without intent, a problem primarily associated with the use of computer software as a method for design rather than a tool for the design process. Compared with Gaudi’s unprecedented forms and his methods for creating them, this becomes a highly valid and controversial issue of design. The last idea this research will attempt to test is the potential of the process of fabrication-ideation versus ideation-fabrication.

Impediments: Due to the particular nature of this research topic and its methodology both literary and physical research barriers exist

which could limit its full potential. A substantial limitation to the scope of the research is the lack of available literature on Antonio Gaudi’s Sagrada Familia; Mark Burry is currently the leading expert on the cathedral and has the greatest number of publications on it. While other sources do exist, Burry’s perspective on the topic could lead to biased understandings of Gaudi’s methods or intentions. Practical limitations to the research include lack of space allocation for provision by the University of Kansas School of Architecture, Design, and Planning as well as limited fabrication equipment. While the school does have laser cutters, a CNC mill, and wood shops, it lacks a 3D printer and plasma cutter which could greatly contribute to a larger material palette for the fabrication phase of the research. Similarly, the school has computer labs with several CAD software, but the performance of these computers is limiting when the complexity of the CAD model reaches a certain level.

Case Study

Thesis Framework

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1.0.2 | Problem, Impediments, Case Study

Mark Burry on Antoni Guadi and The Expiatory Church. ______________________________ ANTONI GAUDI:

every year and many more study its architectural and religious content.

It has always been an expiatory church, which means that since the outset, 130 years ago now, it has been built from “Antoni Gaudí is one of the outstanding figures of Catalan cul- donations. Gaudí himself said: “The expiatory church of La ture and international architecture. He was born in Baix Camp Sagrada Família is made by the people and is mirrored in (Reus, Riudoms), but it was in Barcelona that he studied, them. It is a work that is in the hands of God and the will of worked and lived with his family. It is also in the city that we the people.” The building is still going on and could be finfind most of his work. He was first and foremost an architect, ished some time in the first third of the 21st century.” but he also designed furniture and objects and worked in town planning and landscaping, amongst other disciplines. In all those fields he developed a highly expressive language of his own and created a body of work that speaks directly to the senses.”

EXPIATORY CHURCH: “The expiatory church of La Sagrada Família is a work on a grand scale which was begun on 19 March 1882 from a project by the diocesan architect Francisco de Paula del Villar (1828-1901). At the end of 1883 Gaudí was commissioned to carry on the works, a task which he did not abandon until his death in 1926. Since then different architects have continued the work after his original idea.

The building is in the centre of Barcelona, and over the years it has become one of the most universal signs of identity of the city and the country. It is visited by millions of people

Cross Section

Thesis Framework 1.0.2 | Problem, Impediments, Case Study

The cross section on the left is drawing through the south east entrance (Glory Facade), the nave, the apse, and the sacarsite. The section is cutting through the major tower (tower of Jesus Christ.

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Annotated Bibliography

Thesis Framework

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1.0.3 | Annotated Bibliography

Section 1.0.3 —

The literature review below is in the form of an annotated bibliography. The goal is to expand the sources into a literature review throughout the duration of the thesis. ______________________________ Burry, Mark. Expiatory Church of the Sagrada Familia : Antoni Gaudi. Architecture in Detail. 1 vols. London: Phaidon Press, 1993. Print.

“The Expiatory Church of the Sagrada Familia is Gaudi’s undoubted masterpiece. It charts the styles the architect evolved during his career. At the crypt level a Gothic design is used, but as the building climbs towards the sky the structure passes through an Art Nouveau stage before becoming more surreal and fanciful, finishing in four intricately carved, open-work cone-shaped spires. For all its apparent fantasy, however, the Sagrada Familia is rooted in structural principles (based on the parabolic arch) and an elaborate personal symbolism. Unfinished upon the architect’s accidental death in 1926, the church is currently being completed following Gaudi’s plans.”

---. “Paramorph: Anti-Accident Methodologies.” Architectural design 69.9-10 (1999): 78-83. Print.

This article focuses on the difference between “art through accident” and “art through design” and the use of parametric modelling through associative geometry as an approach to design. Mark Burry describes “art through accident” by con-

Annotated Bibliography

trasting Jackson Pollock paintings with architectural surfaces.

A Jackson Pollock painting’s value lies in its randomness and unrepeated splashes, while an architectural surface, if created using random programming, chaos theory, or serendipity, lacks substance. The problematic begins when the architectural surface is taken a step further to manufacturing or building, which requires an understanding of the operations taken to create it (process). An accidental approach to surface generation lacks process. On the other hand, “art through design” requires the understanding of process consisting of decision-making in sequential order. The definition of a paramorph is therefore stated as “unstable form but stable characteristics (parameters).” The concept of a paramorph was applied in reconstructing the high triforium columns of Gaudi’s Sagrada Familia Church in Barcelona; this was done with all of the characteristics as parameters so that they could be changed to give an exact replica of Gaudi’s 1:10 gypsum model fragments.

---. “Between Surface and Substance.” Architectural design 73.2 (2003): 8-19. Print.

Mark Burry investigates in this article the sector lying between surface and substance. Surface being the ostensible form subjected to questions of style, and substance being the tectonic trueness of material, as in the case of Louis Kahn’s expression asking a brick what it wants to be. In quest to locate insights for “new formal engagements,” Burry argues that there is a tendency to classify the questions of form, technics,

Annotated Bibliography

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and material as contemporary, but that is only because of the deception that digital applications allowed for the realization of the Guggenheim in Bilbao. However, preceding digital applications there was Gaudi’s catenary curve hanging model for the Colonia Guell Church, Le Corbusier’s Chapel at Ronchamp, and Eero Saarinen’s TWA Terminal. To establish “new formal engagements,” Burry used Eliel Saarinen’s The Search for Form: a Fundamental Approach to Art and Architecture to first support an understanding of form, and secondly to use mathematics as a concise and consistent approach to form.

In order to support the use of mathematics as a possible approach Burry employs four projects: Euclidian 1: Gaudi and the Nave Roof for the Sagrada Familia Church, Euclidian 2: Paramorphs, Non-Euclidian 1: Perplications, and Non-Euclidian 2: Hyposurface.

---. “Virtually Gaudi.” Digital Tectonics. Eds. Leach, Neil, David Turnbull and Chris Williams. Chichester, West Sussex, U.K. ; Hoboken, NJ: Wiley-Academy, 2004. 152 p.

Print.

This chapter’s main focus is to describe the geometric constructs executed by Antoni Gaudi at the Sagrada Familia Church in Barcelona. Mark Burry argues that Gaudi’s understanding and application of “wrapped surfaces” (second-order geometry) more than three-quarters of a century ago could have only been realized through the aid of computer software in the past twenty years. In describing the geometries Burry prescribes four methods: mathematics, physical modelling, analytical drawing, and digital modelling. Mathematics being only an abstraction of patterns via symbols, it cannot alone aid a designer in understanding geometric construction, and physical modelling and analytical drawing pursued separately are not fully effective means to decode geometry. However, digital modelling has the capability to express both analytical mathematics and iterative modelling, which leads Burry to conclude his argument by stating that digital modelling has thus far been the only efficient method applied to decode the geometries of Gaudi

Print.

Mark Burry drafts in this article a plot by which the pre-digital and the post-digital modus operandi could be combined as a possible ground for practicing and teaching architecture.

Although today the creation process begins with the digital skill set, Homo Faber (man the maker) still retains a requirement to see his or her creative endeavours at full scale and interact with it physically as a possible design precursor.

Burry argues that we need to “apply the brakes” and critically assess the handicraft skill set and the digital skill set. When software and hardware reach an affordable price, and the arguments surrounding the agendas of theory and practice reach a conclusion the design studio environment may come to an end. Graduating students being trained adequately with software coupled with a generation that lacks the digital skill set has created “an unbridgeable gulf.” A common ground has to be achieved so that there is not an end to the handicraft skill set. To create a mediating line between the pre-digital and post-digital mediums Burry experiments with three projects in different design studio environments: Student Project: 1:1 – pre-digital, Sagrada Familia: pre- and post digital design development, and Student Project: 10:1 – post digital.

Chaszar, Andrea. “Blurring the Lines: An Exploration of Current Cad/Cam Techniques.” Architectural design 73.1 (2003): 111-17. Print.

“Computer-aided design and manufacturing are relatively well-established practices in the aeronautics/aerospace, automotive and shipbuilding industries. But although these practices have elicited fields of architecture and building engineering, their application here is as yet in a nascent state.

At present there is (fortunately) no widespread agreement as to how, or indeed even why, CAD/CAM should be employed in designing and fabricating the built environment. This is the first of a series of ‘Engineering Exegesis’ articles, edited by Andre Chaszar, in this volume of AD that will aim to survey some of the areas of current development underlying them.

Though ‘blurring the lines’ may seem a strange phrase to ---. “Homo Faber.” Architectural design 75.4 (2005): 30-37. associate with digital tools the underlying premise of which is

Annotated Bibliography

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precision, we will see that the communicative and cognitive opportunities they offer break down many of the distinctions now entrenched in building.”

De Landa, Manuel. “Material Complexity.” Digital Tectonics. Eds. Leach, Neil, David Turnbull and Chris Williams.

Chichester, West Sussex, U.K. ; Hoboken, NJ: WileyAcademy, 2004. 152 p. Print.

De Landa Manuel, currently a Gilles Deleuze Chair of Contemporary Philosophy and Science at the European Graduate School in Saas-Fee, Switzerland, states that the study of materials and material behiavor today has reached its maturity in the sciences and academic institution. The beginnings of the study of material finds itself surrounding empirically oriented craftsman and engineers not scientist or philosophers. De Landa argues that metallurgists for example have an essential understanding of metal even before the scientific understanding of metals breached the surface. Metallurgists have always understood that forming metal is a process emerging out of metal itself. One cannot force the material to behave differently than its natural propensity. During the high period of Greek philosophy, the understanding of material behaviour was lost and the philosophy of matter took place. Metaphysical explantation gave meaning to the four elements and that could have only been expanded symbolically. The original understanding of actual material behaviour was lost until its revival by sixteenth century alchemists. De Landa further examines examples of materials and their properties and concludes by stating “we may now be in a postilion to think about the origin of form and structure, not as something imposed form the outside on an inert matter, not as a hierarchical command form above as in assembly line, but as something that may come from within the materials, a form that we tease out of those materials as we allow them to have their say in the structures we create.”

Frampton, Kenneth. “Modernizaton and the New Monumentality, 1944-1972.” Studies in Tectonic Culture : The Poetics of Construction in Nineteenth and Twentieth Century Architecture. Ed. Cava, John. Cambridge, Mass.: MIT Press, 1995, p. 225.

“Kahn’s preoccupation with the idea of hollow structure -“now we can build with hollow stone” -- was to unify his thinking at both an architectural and an urban level. And where the former announced itself, in various ways, in the triagrid floors

of the Yale Art Gallery, in the Vierendeel trusses projected for the Trenton Jewish Community Centre, and finally in the hollow floor beams of the Richards Medical Research Building of 1957, the latter was to manifest itself as a viaduct architecture of his last urban proposal for Philadelphia, dating from 1962.”

Louis Kahn’s idea of ‘hollow stone’ surpasses the concept of structural and system integration. “Hollow stone’ as a modus operandi, was used to execute all variables of the architectural equation at all scales. Architecture is an entity carved away, creating ‘hollow stones.’ And the ‘hollow stones’ used to harmoniously bind the parts of architecture.

Leach, Neil. “Digital Morphogenesis.” Architectural design 79.1 (2009): [32]-37. Print.

Taking its inspiration from biology, digital morphogenesis operates through a logic of optimization. Departing from the notion of architecture primarily as form-finding that privileges appearance, Neil Leach describes how morphogenesis places emphasis on ‘material performance’ and ‘processes over representation’.

Saarinen, Eliel. The Search for Form in Art and Architecture. New York: Dover Publications, 1985. Print.

“In this important volume, the father of the “American Bauhaus” has set down the principles that guided his work, making them comprehensible to the general reader and student.

Emphasising design on an organic level and an interrelated study of all forms of art, Saarinen rejects the slavish mimicry of old, exhausted styles (e’g’ Classicism and romanticism). Instead, he promotes the development of fresh modes of artistic expression, appropriate to each new generation.”

Williams, Chris. “Design by Algorithm.” Digital Tectonics. Eds. Leach, Neil, David Turnbull and Chris Williams.

Chichester, West Sussex, U.K. ; Hoboken, NJ: WileyAcademy, 2004. 152 p. Print.

Chris Williams, a professor of structural engineering at the University of BATH, argues the use of an algorithm is inevitable. The Oxford English Dictionary states that an algorithm is a process, or set of rules, usually one expressed in algebraic notation, now used especially in computing, machine translation and linguistics.” Chris argues that when designing an object the designer “consciously or subconsciously adopt a set of rules. These set of rules, however, can only be used to

Annotated Bibliography

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design a part of the complex whole (object). For instance, the wing of an aeroplane or the window of a building. Chris states as examples two methods for constructing an algorithm: conformal mapping and differential equations. Through these two methods he calls examples such as Gaudi’s hanging model for structural analysis, D’Arcy Thompson geometric mapping of living forms, and the roof British Museum by Foster and Partners.

Kenneth Frampton: Studies in Tectonic Culture Chapter 1: Introduction- Reflections on the Scope of the Tectonic; Tectonic/Atectonic Kenneth Frampton describes the tectonic in architecture as having a resultant expression or character, which stems from structure and construction to inform one another directly. In an attempt to describe the atectonic Frampton uses a deviated definition of tectonics prescribed by Eduard Sekler which states “tectonics is a certain expressivity arising from the statical resistance of constructional form in such a way that the resultant expression could not be accounted for in terms of structure and construction alone.” Therefore the atectonic is an expressivity where structure and construction oppose one another by neglecting or obscuring the loading and support of structure. An elemental case study of the atectonic is the Stoclet House whose walls appear to be made of large sheets of thin material joined at the corners with metal band while they are load bearing walls.

Kenneth Frampton: Studies in Tectonic Culture Chapter 1: Introduction- Reflections on the Scope of the Tectonic; Etymology Kenneth Frampton, in order to clarify the use of the word “tectonic” in the realm of architecture, breaks down its origin and establishes its incidental meaning in order to project. The word’s essential origin comes from the Greek tekton, meaning carpenter or builder. As the act of making and building transcends throughout history into the poetic, tectonic comes to mean the “art of joining,” such as assembling and weaving.

Frampton refers to Karl Botticher’s analysis of the Greek temple and his understanding of the tectonic “as signifying a complete system binding all the parts… into a single whole” to project the significance of joining parts to compose a whole.

Frampton then turns to Gottfried Semper, who identifies a built space as a whole being composed of four fundamental elements: the hearth, the mound (earthwork), the roof (frame),

and the enclosure (lightweight membrane encompassing a spatial matrix). Frampton further distinguishes these elements into two fundamental procedures: the tectonic, made of the frame and the lightweight membrane, having a relationship with the sky, and the stereotomic, made of earthwork and the hearth, having a relationship with the earth. Throughout the historic development of building techniques and material capacities, people engineered structures in which the tectonic and the stereotomic began to extend and merge to become multiple elements simultaneously.

Gottfried Semper: The Four Elements of Architecture Through the early phases of society humankind began constructing architecture with what Gottfried Semper identifies as the four elements: the hearth, the mound, the roof, and the enclosure. The hearth is central to all of these elements, serving to congregate the individuals of a group, while the other three elements serve to protect the hearth and the group from the environment around them. The enclosure in particular was significant to Semper as a “vertical space divider” that had its origins in woven carpets. As humans became more domestic, an artisan and a weaver were the resultants, specializing in the art of the wall filter. The modus operandi included weaving of branches that were joined by entwining and knotting, perhaps the oldest form of ornamentation.

This woven carpet wall began as a spatial division between the exteriority and the interiority of architecture, later moving internally to articulate further spatial divisions and externally becoming articulated in different characters through masonry construction and mosaics.

Anne Griswold Tyng: Geometric Extensions of Consciousness Anne Tyng, as an elemental explanation of the transformation of man’s consciousness of space through time as a cyclic process of simplicity to complexity, describes four fundamental geometric principles: the bilateral, the rotational, the helical, and the spiral, significant in that respective order.

The bilateral identifies a synthesis in which there is an understanding of symmetry (in relation to a node, the human body).

The rotational identifies an understanding of the continuity of space beyond the physical body. The helical identifies time, in which duration is perceived as part of continuous space.

The spiral identifies space-time, a hierarchical flux of the helix as it moves through time marking a beginning and an end.

Annotated Bibliography

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1.0.3 | Annotated Bibliography

Tyng provides multiple analogies within nature to prove the continuous nature of this cycle; one of which is hemoglobin in its most elemental construction. Its bilateral nature begins with carbon bonds that transcend to a rotational nature as the bonds begin clustering. The clusters develop to a helical nature in which they form alpha and beta chains and gravitate to a spiral nature as they form irregular spirals. At this point, the hemoglobin is once again in a basic bilateral state; the process continues to build infinitely into larger and more complex forms.

Shea, Kristina, and Jonathan Cagan. “Languages and Semantics of Grammatical Discrete Structures.” Artificial Intelligence for Enineering Design, Anaylsis and Manufacturing (1999): 241-51. Web. “Applying grammatical formalism to engineering problems requires consideration of spatial, functional, and behavioral deisgn attributes. This paper explores structural deisgn languages and semantics for the geneartrion of feasible and purposeful discrete structures. In an applicaiton of shape annealing, a combination of grammatical design generation and search, to the generation of discrete structures, rule syntax, and semantics are used to model desired relations between structural form and function as well as control deisgn generation. Explicit domain knowledege is placed withing the grammer through rule and syntax formulaiton, resulting in th egenration of only forms that make funcitonal sense and adhere to preferred visual styles. Design interpretations, or semantics, is then used to select forms that meet funcitonal and visual goals. The distincition between syntax used in grammer rules to explicitly drive geometric deisgn and semantics used in deisgn interpretation to implicitly guide geometric form is shown.

Overall, the desings presented show the validity of applying a grammatical fromalism to an engineering deisgn problem and illustrated a range of possibilities for modeling functional and visual criteria.”

Cardoso, Daniel. “The Melnikov Grammer.” Thesis. Massachusetts Institute of Technology, 2008. Print.

“The Melnikov Grammer is presented. A computer program that playfully re-interprets an iconic work of architecture by Konstantin Melnikov, implementing a rule-based system that semi-autonomously computes uexpected “Melnikove” designs in a nondeterministic manner while satisfying certain architectural constraints.”

Evans, Robin. Translations from Drawing to Building and Other Essays. London: Architectural Association, 1997.

Print.

“Robin Evans delivers a seminal and stunningly beautiful argument in ‘Figures, Doors and Passages’ that is based on comparing paintings and architectural plans to gain insights into the relation between spatial organisation and social arrangements and formations. He distinguished between the Italian medieval matrix of connected spaces and a social context based on closeness, carnality and accidental social encounter and the British corridor and cellular room model and a social context based on privacy, distance and segregation. He completes his accomplished discourse with the question why the corridor model is today still the predominant spatial organisation and questions its relevance vis-à-vis today’s prevailing social pattern.”

Tyng, Anne Griswold, and Srdjan J. Weiss. “Inhabitation Is Unusual and Important.” Anne Tyng: Inhabiting Geometry. Philadelphia: Institute of Contemporary Art, 2011. N. pag. Print.

“The exhibition Anne Tyng: Inhabiting Geometry proposes to investigate and reorganize the priorities of architecture by way of a word, the history and use of which are still partial.

Understood primarily through the history of applications, this word can be sued to reconnect diverse camps within theory, practice, and education in architecture today -- to reconceptualize qualities between contemporary humanity and construction. The word is “inhabitation.”

Tyng, Anne Griswold, and Alicia Imperiale. “Dynamic Symmetries.” Anne Tyng: Inhabiting Geometry. Philadelphia: Institute of Contemporary Art, 2011. N. pag. Print.

“In conjunction with the Graham’s exhibition, Anne Tyng: Inhabiting Geometry, Alicia Imperiale will discuss Tyng’s seminal essay Geometric Extensions of Consciousness, published in the Italian architectural journal Zodiac 19 in 1969. In the article, Tyng proposes geometries that would open up architectural form by following natural laws of growth in plants and organisms. She writes, “I have found a geometric progression from simplicity to complexity of symmetric forms linked by asymmetric process, ”* and goes on to demonstrate the power of geometry as the invisible driving force in natural forms intrinsic to her research, astonishing drawings, and

Annotated Bibliography

Thesis Framework 1.0.3 | Annotated Bibliography

architectural projects.

Zodiac was published in Milan from 1959-1973. In a number of issues, editor Maria Bottero assembled an extraordinary group of international architects around the themes of geometric studies and the influence of natural systems on architectural design. Imperiale will discuss Tyng’s work in relation to her contemporaries, such as Buckminster Fuller, Zvi Hecker and Alfred Neumann, Keith Critchlow, Robert LeRicolais, Moshe Safdie, Rinaldo Semino, Michael Burt, Renzo Piano, and others, also published in Zodiac.”

Tyng, Anne Griswold, and Jenny E. Sabin. “Geometry in Transformation: Computing Mind and Matter.” Anne Tyng: Inhabiting Geometry. Philadelphia: Institute of Contemporary Art, 2011. N. pag. Print.

“Anne Tyng inhabits geometry not as figure or material, but as formation of mind and immanent matter. Is geometry a practice of defining shapes through mathematics, or is it one of projecting and then inhabiting shapes in the ether of mathematical space?”

25

Methodology

Thesis Framework

26

1.0.4 | Methodology Framework

Section 1.0.4 —

“Considering the built architecture in its possible fulfilments may work backwards to influence the ideation at a point where the physical manifestation of an idea is still hardly defined.”

-- Mark Burry The widely applied common procedure of design is ascribed by first having an idea, secondly through a process developing the idea, and lastly the fabrication of the idea. Since architecture concerns itself with tangible substance, I find that Burry’s suggestion to reverse the common procedure is applicable. Instead of creating an idea from studies and analysis, an idea should emerge from fabrication.

The framework on the right illustrates how the concept of reversing the procedure will work in relation to my research and its methods.

This analysis of Antonio Gaudi’s La Sagrada Familia falls under the research paradigm of interpretive/constructivist, in which studies lead to a qualitative conclusion. The focus of the research is centred around the interpretation of existing conditions. The analysis of these existing conditions targets values and symbols in relationship to context.

To conduct research under the interpretive/constructivist paradigm is to do qualitative research which answers the question of reason and method versus quantitative characteristics. In such cases, the research produces information on one ex-

Methodology Framework

ample which could lead to the binding of larger ideas. Thus, an in-depth analysis of an individual entity leading to a broad conclusion, results Through qualitative research a case study can provide the in-depth analysis of one subject, which in this case is the arch of La Sagrada Familia Church in Barcelona. Such analysis will explore the causes of the underlying principles executed

by Gaudi.

Framework

Thesis Framework

27

1.0.4 | Methodology Framework

program site context precedent

Existing Process

1. Ideation program organization structural logic form iteration system of assembly

2. Process

3. Fabrication

digital analog

Design Project analog

Speculative Process

1. Fabrication

mapping

model full/half scale

associative geometry modeling (parametric)

digital

2. Ideation

3. Process geometry structure material

analysis

revit rhino 3d grasshopper 3d autocad

Design Project

Identifies the underlying principles of application executed by Gaudi.

Decoding

Realizing Gaudiâ&#x20AC;&#x2122;s geometries at full scale to influence the design idea.

Fabrication

Unpacking the idea from decoding and fabrication.

Critical Analysis

partitioning variables

coding analytical

factoring relationships between variables

conceptual coherence

explanatory

workshop polymer

analog

wood CNC

material

digital metal

laser

CAD Digital Modeling

Mathematical Analysis

associative geometry modelling Differential Geometry rhino 3d grasshopper 3d

mathematica

differential calculus Integral calculus linear algebra multilinear algebra

Framework

Thesis Framework 1.0.4 | Methodology Framework

Research methods applied for the analysis of La Sagrada Familia include decoding, fabrication, and critical analysis. These methods inform, respectively, the instruments of research: mathematics software, CAD and associative geometry modelling (Rhino and Grasshopper), and CNC and laser fabrication. The wire frame on the right illustrates the relationships between the types of analysis and their respective instruments.

As stated in section 1.0.2 (Research Impediments) a studio or work space is paramount to the success of the research as there will be production of several largescale models. Space to store and assemble materials is necessary for the construction of these models. A properly equipped computer is also necessary for this space as the vast majority of the analysis would be conducted through software.

29

Thesis Framework

30

1.1 | Reference Data

Section 1.1

Table of Contents

Section 1.1.1 | Sagrada Familia Geometries Section 1.1.2 | Mark Burry’s Geometric Mapping Section 1.1.3 | Peter Eisenman’s Geometric Toolset

Description

Thesis Framework

31

1.1 | Reference Data

Section 1.1 â&#x20AC;&#x201D;

Section 1.1 will display a collection of background reference data which will be used for the analysis and design of the thesis. First, the Sagrada Familia geometries will be surveyed and defined. Second, Mark Buryâ&#x20AC;&#x2122;s mapping of the Sagrada Familia structural column will explain the complexity of the double-twisted column. And third, Peter Eisenman toolset which lists formal and conceptual tools.

Reference Data

Geometric Definitions

Thesis Framework

32

1.1.1 | Sagrada Familia Geometries

Section 1.1.1 — Sagrada Familia Geometries

1. RULED SURFACE: “Ruled double curve surfaces are, as their name suggests, surfaces that contain straight lines because they are generated by the movement of one straight line that follows a particular route.

With the use of twisted ruled surfaces (hyperboloids, paraboloids, helicoids and conoids), Gaudí planned a naturalistic architecture formed only of geometrical surfaces with hyperbolic and parabolic sections, of fine structural, acoustic and light distribution qualities. The fact of being generated by straight lines makes the construction easier.”

1. HYPERBOLOID: “The hyperboloid is a surface generated by a hyperbola which revolves around a circle or ellipse. They can be solid or empty, solid to pass from the column to the vaults, empty where the light enters the interior of the church. The hyperboloid contains two sheaves of inclined straight lines tangent to the circle or the ellipse. On the vaults and windows, the hyperboloid is bounded by star shapes created by the straight lines. The vaults and the windows are intersections between hyperboloids, linked with paraboloids thanks to straight lines that are common to two surfaces.” Figure 3.

The conoid is a surface formed by a straight line which is displaced above another straight line and above a curve, for example a sinusoid.”

ELLLIPSOID: “The ellipsoid is a solid in which all the flat sections are ellipses. Because of its elliptical shape it was chosen by Gaudí for the knots or capitals that subdivide the lower columns into branches. The different knots are the result of adding and subtracting ellipsoids from one another.”

3. HYPERBOLIC PARABOLOID: “The paraboloid is a twisted surface of parabolic sections which is the result of a displacement of a straight line above two other lines that cross in the space. It is generally bounded by four straight lines.”

PROPORTIONS: “A single system of proportions, based on the twelfth parts of the largest dimension, orders in series the general measurements of the church (width, length and height of each part), the diameters of the columns and the diameters of the openings of windows and vaults.”

2. HELICOID AND CONOID: “The helicoid is a ruled surface generated by a straight line that revolves according to a spiral around a vertical axis.

Figure 2.

Definitions are stated on the Geometry Section of (http://www.sagradafamilia.cat)

Geometry

Thesis Framework

33

1.1.1 | Sagrada Familia Geometries

1

1-3. Temple Expiatory Sagrada Familia

2

3

3

Mark Burry

Thesis Framework

34

1.1.2 | Mark Burry’s Geometric Mapping

Section 1.1.2 — Mark Burry’s Geometric Mapping

MARK BURRY MAPPING: Mark Burry is Executive Architect and Researcher at the Temple Sagrada Familia. The geometries on the right, created

by Mark Burry and his colleagues attempt to reverse engineer Gaudi’s plaster models to aid in the construction process.

They are all modelled digitally through associative geometry modelling software. This method of modelling allows the digital model to be adjusted parametrically to agree with the existing conditions. Double ruled surfaces constructs the ceiling at the Crossing, 1. “The double twisted column begins at the base with a regular or starred polygon with straight or parabolic sides or with a combination of polygons which, as the column rises, are transformed into different sections with an increasing number of vertices, until they reach the circle at the top, 2, 3, 4.”

1

Mark Burry

Thesis Framework

Geometry

1.1.2 | Mark Burryâ&#x20AC;&#x2122;s Geometric Mapping

2

35

3

4

Peter Eisenman Geometric Toolset

Thesis Framework

36

1.1.3 | Peter Eisenman Geometric Toolset

Section 1.1.3 â&#x20AC;&#x201D; Peter Eisenman Geometric Toolset

EISENMANâ&#x20AC;&#x2122;S TOOLSET: The toolset on the right is more than a list to manipulate form, it is a list which displays disciplines for diagramming to provide a new set of work. The act of diagramming separates itself from drawing as it deals with performative divisions versus the representational (the virtual versus the real).

The toolset is group in two categories: Formal tools and conceptual tools. The formal tools are used to manipulate, alter, or transform existing or speculative conditions. For example, they can potentially be used to organize spaces, transform geometries, and integrate with context.

The conceptual, however, provide a basis to decode existing or speculative conditions for analytical purposes. A way of diagramming, or having a process to diagram for analysis could potentially lead to a higher level of coherence. As the study is concerned with the reverse engineering of existing conditions, the conceptual tools will be used as a starting basis to discipline the diagrams.

FORMAL TOOLS CONCEPTUAL TOOLS extrusion twisting extension interweaving displacement disassembling shear morphing interference projection torquing distortion superposition nesting warping repetition shifting scaling imprinting slippage transformation rotation doubling

inversion mapping artificial excavation folding grafting tracing marking layering montage voiding decomposition blurring striation gridding laminar flow

Peter Eisenman Geometric Toolset

Thesis Framework 1.1.3 | Peter Eisenman Geometric Toolset

37

Part Tw Prelim Design Sectio

wo minary n on 012 Part Two

Preliminary Design

In this part you will find the research and analysis for the point-star structural column, site and context of Barcelona and Catalonia, and space frame research, mapping, and geometric analysis.

Table of Contents:

Section 2.0 | Structural Column Analysis Section 2.1 | Site and Context Analysis Section 2.2 | Space Frame Analysis

Preliminary Design

40

Section 2.0 | Structural Column Analysis

Section 2.0

Table of Contents

Section 2.0.1 | Construction Analysis Section 2.0.2 | Floor Plan Analysis Section 2.0.3 | Geometric Decoding Section 2.0.4 | Parametric Mapping Section 2.0.5 | (Column) Geometric Value System

Description

Preliminary Design

41

Section 2.0 | Structural Column Analysis

Section 2.0 â&#x20AC;&#x201D; Structural Column Analysis

Section 2.0 will explore the structural column at the Sagrada Familia Church. The structural column (the double twisted column) will be studied through a series of analytical analysis and geometric mapping. Section one will explain the actual construction of the column and how the geometry informs the construction process, section two will give a analysis of the floor plan as it relates to the structural column placement, section three will geometrically decode the column and section four will parametrically map the geometries.

Construction

Preliminary Design

42

2.0.1 | Construction Method Analysis

Section 2.0.1 â&#x20AC;&#x201D; Construction Method Analysis

1

The structural columns at the Sagrada Familia Church are finished with stone (type of stone being a dependent on the number of stars used to construct each column) and the core is primarily reinforced concrete, 3, 4. They are constructed by first building the framework and placing the natural stone tambours in the specified sections. Once the tambours are placed, the void is filled with concrete, 1, 2.

2

1. Temple Expiatory Sagrada Familia 2. Temple Expiatory Sagrada Familia

Construction

Preliminary Design

43

2.0.1 | Construction Method Analysis

4

3

3. Temple Expiatory Sagrada Familia 4. Screen shot of Virtual Visit (La Sagrada Familia)

Mold and Tambours

Preliminary Design

44

2.0.1 | Construction Method Analysis

1,2

The cross sections of the tambours are designed from a compound curve derived from the 12-point star. The compound curve is used to produce the natural stone tambours at each section, 5, 6, 7. The geometric figures were used to construct wood templates to begin forming plaster prototypes, 1, 2, 3, 4. The process illustrates in the images was the process that Guadi used to demonstrated the technics of the column.

3,4

1-4. Mark Burry, Expiatory Church of Sagrada Familia

Geometry to Section

Preliminary Design

45

2.0.1 | Construction Method Analysis

5

6

7

Floor Plan

Preliminary Design

46

2.0.2 | Floor Plan Analysis

4

9

The floor plan of the church is composed of 5 naves and crossing with 3, intersecting together to form the traditional Latin cross.

The internal measurements are: Main Nave 45m x 30m, Crossing 60m x 30m, and the Apse 45m.

9

8

7

6

1

2

5

5

The 3 naves at the Crossing connect Nativity and Passion Facade. Nativity Facade is dedicated to the birth of Christ and Passion Facade is dedicated to the passion, death, and resurrection of Christ.

1. Crossing 2. Main Nave 3. Glory Facade 4. Chapels

4

4

5. Cloister 6. Nativity Facade 7. Passion Facade 8. Apse 9. Sacristies

3

Floor Plan Geometric Decoding and Column Placements

Preliminary Design

47

2.0.2 | Floor Plan Analysis

Section 2.0.2 â&#x20AC;&#x201D; Floor Plan Analysis

The structural columns are hierarchically placed based on the load bearing capacity. There is a total of 4 types used throughout the floor plan, each type yields its own color, diameter, length, and stone.

Crossing: The four central columns at the crossing use a 12-point star and they carry the load of the Central Tower of Jesus Christ which rises to 170 meters and part of the weight of the evangelists which surround the tower. They use red porphyry stone for their tambours Evangelists: The eight columns at the Passion and Nativity Facades placed around the central four will bear the primary weight of the evangelists. They are 10-point star and use basalt stone for the tambours.

Apse: The columns at the apse are placed on two concentric tetradecagon. The outer tetradecagon uses 6-point star and inner tetradecagon uses 8-point star. These columns bear the weight of the tower of Our Lady, the vaults of the ambulatory, and the 45m triforium. 6 point star uses sedimentary sandstone and the 8 point star uses granite for the tambours.

Central Nave: The columns at the central naves are 8-point star columns and the carry the weight of the vaults and part of the side nave.

Side Nave: The side nave columns bear the loads of the choir which is the least amount of load. They use a 6-point star.

Column Mathematics

Preliminary Design

49

2.0.2 | Floor Plan Analysis

The cross section on the right is cut through the main nave.

The two central columns carrying the weight of the main vault are 8-point star, and the side nave columns are 6-point star.

ch

The number of points of the star (x) determines the total height (th); th = x(2m). Each column is constructed from 3 stretches made of cross sectional profiles (CS), the length of each stretch is based on the total height; th/2^(CS); therefore, the shaft height (sh) is the sum of all the stretches; sh = th/2 + th/4 + th/8. The height of the base (bh) is the number of points of the star multiplied by 10cm; bh = x(10cm), while the capital height (ch) is a function of the base, and shaft and total height; ch = th - (sh + bh). Lastly, the number of points of the star determines the internal diameter (it) of the column; it = sh(10)/100.

Therefore, the number of points of the star is the most important value in constructing the column. It determines not only the structural placement of the column and materiality, it also determines the physical arrangements of parts within each column.

x(8)

CS.4

x(4)

CS.3

x(2)

CS.2

x(1)

CS.1

th/8

th/4

th

sh

th/2

bh

Right: Cross Section. Mark Burry, Expiatory Church of Sagrada Familia

Plan, Cross Section, and Lofts

Preliminary Design

50

2.0.3 | Geometric Decoding

Section 2.0.2 â&#x20AC;&#x201D; Geometric Decoding

The figures on the right illustrate the proportion and relative scale of each column system. Beginning with the 12-point star system (far left) to the 6-point star system (far right).

Above is the cross sectional profile at the base and a legend display the placement of the column system in the plan.

6 Point Star Cross Section 6.PS.CS

1

Preliminary Design

52

2.0.3 | Geometric Decoding

4

2

5

3

6

The following drawings display the six-part process to construct the stars used at the Sagrada Familia Church.

The process begins by inscribing a six-point star in a circle, 1, 2. All axes of symmetry are then constructed by connecting the vertices of the star, 3.

Three circles are tangentially constructed within the external and internal edges of the star; two of those circlesâ&#x20AC;&#x2122; centroids are identified by the intersections of the axes of symmetry and the edges of the star, 4, 5.

By finding the intersection of those circles at the tangent points an arc is derived.

Finally, by joining all the arcs a compound curve is created, 6.

6.PS.CS.00

6.PS.CS.01

6.PS.CS.02

6.PS.CS.03

8 Point Star Cross Section 8.PS.CS

1

Preliminary Design

54

2.0.3 | Geometric Decoding

4

2

5

3

6

The following drawings display the six-part process to construct the stars used at the Sagrada Familia Church.

The process begins by inscribing a eight-point star in a circle, 1, 2. All axes of symmetry are then constructed by connecting the vertices of the star, 3.

Three circles are tangentially constructed within the external and internal edges of the star; two of those circlesâ&#x20AC;&#x2122; centroids are identified by the intersections of the axes of symmetry and the edges of the star, 4, 5.

By finding the intersection of those circles at the tangent points an arc is derived.

Finally, by joining all the arcs a compound curve is created, 6.

8.PS.CS.00

8.PS.CS.01

8.PS.CS.02

8.PS.CS.03

10 Point Star Cross Section 10.PS.CS

1

Preliminary Design

56

2.0.3 | Geometric Decoding

4

2

5

3

6

The following drawings display the six-part process to construct the stars used at the Sagrada Familia Church.

The process begins by inscribing a ten-point star in a circle, 1, 2. All axes of symmetry are then constructed by connecting the vertices of the star, 3.

Three circles are tangentially constructed within the external and internal edges of the star; two of those circlesâ&#x20AC;&#x2122; centroids are identified by the intersections of the axes of symmetry and the edges of the star, 4, 5.

By finding the intersection of those circles at the tangent points an arc is derived.

Finally, by joining all the arcs a compound curve is created, 6.

10.PS.CS.00

10.PS.CS.01

10.PS.CS.02

10.PS.CS.03

12 Point Star Cross Section 12.PS.CS

1

Preliminary Design

58

2.0.3 | Geometric Decoding

4

2

5

3

6

The following drawings display the six-part process to construct the stars used at the Sagrada Familia Church.

The process begins by inscribing a twelve-point star in a circle, 1, 2. All axes of symmetry are then constructed by connecting the vertices of the star, 3.

Three circles are tangentially constructed within the external and internal edges of the star; two of those circlesâ&#x20AC;&#x2122; centroids are identified by the intersections of the axes of symmetry and the edges of the star, 4, 5.

By finding the intersection of those circles at the tangent points an arc is derived.

Finally, by joining all the arcs a compound curve is created, 6.

12.PS.CS.1

12.PS.CS.2

12.PS.CS.3

12.PS.CS.4

12 Point Star Model

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60

2.0.3 | Geometric Decoding

The materials used for the column are MDF, .25in acrylic, 2in PVC pipe, rope and cord. The four cross sectional profiles and the MDF base were fabricated using the laser cutter.

Then 1,968in of 1/8in white nylon cord was fed through the acrylic section to represent the rebar for the concrete core.

And lastly, 3,936in of 3/2 Perle cotton rope was fed through to represent the surface deformation of the column.

The acrylic sections have 48 1/16in holes to represent the geometric deformations and 2 sets of 12 5/32in holes to represent the rebar.

12 Point Star Model

Preliminary Design 2.0.3 | Geometric Decoding

61

Description

Preliminary Design

62

2.0.4 | Parametric Mapping

Section 2.0.4 â&#x20AC;&#x201D; Parametric Mapping

The variables, equations, and values on the right were used to inform the construction of the parametric model (page 62-63). All of variables were first plotted on a diagram to understand the parametric relationships. Then, using Rhino Grasshopper, a parametric model was constructed so that one slider (number of points of the star) controls effects all of the geometric operations.

Equations

Preliminary Design

63

2.0.4 | Parametric Mapping

Variables:

total height (th)

x = number of points of the star

th = x(2m)

CS = cross sectional profiles sh = shaft height st = cross sectional stretch length

CS.2 = x(2)

bh = height of the base ch = height of the capital

number of points at each cross section along the height (CS)

it = internal core diameter c = color s = type of stone

CS.3 = x(4)

CS.4 = x(8)

p = structural grid placement Equations: interior diameter (it)

th = x(2m)

it = [sh(10)] /100

st = th/2^(CS) sh = th/2^(1) + th/2^(2) + th/2^(3) sh = th/2 + th/4 + th/8 bh = x(10cm)

base height (bh)

ch = th - (sh + bh)

Values: 06PS = 60, 90, 30, 60;

rotate = [30, 15, 7.5]

08PS = 45, 45, 00, 45;

rotate = [22.5, 11.25, 5.625]

10PS = 36, 18, -18, 36;

rotate = [18, 11.25, 4.5]

12PS = 30, 00, -30, 90;

rotate = [15, 7.5, 3.75]

number of points on each star (x)

it = sh(10)/100 bh = x(10cm)

shaft height (sh)

sh = th/2 + th/4 + th/8

capital height (ch)

ch = th - (bh + sh)

Description

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66

2.0.5 | (Column) Geometric Value System

Section 2.0.5 â&#x20AC;&#x201D; (Column) Geometric Value System

The geometry of the column is valued based on the number of tessellations Antonio Gaudi takes to reach the fourth cross sectional profile. Beginning with a triangle base, the profile is tessellated until arriving at the 48, 64, 80, and 96 GONs, right.

The triangle is the simplest and most natural form and the 96 GON is the most complex form, on the scale that is identified as the triangle being negative and the 96 GON being positive. Therefore, the origin has to be a range meeting in the middle of the column. On the next two pages, this value system is applied for all 21 cross sectional profiles.

Value Scale

Preliminary Design 2.0.5 | (Column) Geometric Value System

67

Preliminary Design

70

Section 2.1 | Site and Context Analysis

Section 2.1

Table of Contents

Section 2.1.1 | Spain, Catalonia, and Barcelona Section 2.1.2 | Barcelona L’ Eixample Cerda’s Proposal Section 2.1.3 | L’Eixample District Present Decoding Section 2.1.4 | (Block) Geometric Value System Section 2.1.5 | Sagrada Familia Block Placement

Description

Preliminary Design

71

Section 2.1 | Site and Context Analysis

Section 2.1 — Site and Context Analysis

Section 2.1 will explore the site and context of Catalonia, Barcelona, L’Eixample, and the Sagrada Familia Church.

The analysis will be conducted through a geometric filter, meaning only the actual geometry that composes the city will be analyzed. The existing L’Eixample district plan and the original Iigegons Cerdà proposal will be decoded based on explicit formal and conceptual operations. And lastly, the Sagrada Familia Church will be analyzed based on the geometries of the L’Eixample Block.

Reference Maps

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2.1.1 | Spain, Catalonia, and Barcelona

Section 2.1.1 — Spain, Catalonia, and Barcelona

The following series of maps are to situate Sagrada Familia Church within the broader context of Catalonia and respectively Spain, 1, 2. The Province of Barcelona which is on the southern edge of Catalonia is subdivided into districts, 3, 4. The L’ Eixample district is subdivide further creating neighborhoods, 5.

1. Spain 2. Catalonia 1

2

3. Province of Barcelona 4. Barcelona 5. L’ Eixample

Reference Maps

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2.1.1 | Spain, Catalonia, and Barcelona

3

4

5

Block Typology

Preliminary Design

74

2.1.2 | Barcelona L’ Eixample Cerda’s Proposal

Section 2.1.2 — Barcelona L’Eixample Cerda’s Proposal a.00

b.00

a.01

b.01

a.02

b.02

a.03

b.03

a.04

b.04

a.05

b.05

a.06

b.06

a.07

b.07

a.08

b.08

Ildefons Cerdà i Sunyer, Catalan Spanish urban planner, proposed the Eixample (extension) to Barcelona, 1. Cerdà proposed a square block measuring 113.3m (372ft) with chamfered corners at 45 degrees. Two block layout each accompanied by eight variations were propagated across a superimposed grid, a.00, b.00, 2. Cerdà proposed a block with 50% plot ratio constructed on the perimeter of the square and the remaining area to be used for open public space, natural ventilation and natural lighting (page 40-41, 1-8). The passive systems were to be accomplished by limiting the height of the block to 20m (65ft) approximately four stories. The following pages will decode the plan and explain the typologies.

Ildefons Cerdà Original Map

Preliminary Design

75

2.1.2 | Barcelona L’ Eixample Cerda’s Proposal

1

2

1. Museu d’Historia de la Ciutat, Barcelona.

Block Analysis

Preliminary Design

76

2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

1. Block Type A

Bounded Public Space:

Open Public Space:

Private Space:

2. Block Type B

Created by the configuration of the structures on the block.

The streets created by the space in between the blocks.

The actual habitable space created by the buildings on the block

2

3

4

These are the base types which create all of the variety on the previous page.

1

Block Analysis

Preliminary Design

77

2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

Natural Ventilation:

Natural Lighting:

Viewing:

Safe Access:

The geometric configuration of the block structures is a function of natural ventilation.

The height of the structure and the rotation of the grid 45 degrees allows for sufficient sunlight.

The bounded public space provide opportunities for neighborly viewing.

The bounded public space provide safe pedestrian access between structures and blocks.

5

6

7

8

Mapping

Preliminary Design

78

2.1.2 | Barcelona L’ Eixample Cerda’s Proposal

The Ildefons Cerdà i Sunyer plan was designed on an octagonal geometric pattern which was transformed by a set of operations (page 44-45, 1-6). To begin the iterative process of transformations, the plan was mapped through four filters: grid, block, boulevard, and district. Then the mapped plan was decomposed into it constituent parts (the four filters), 2, 3, 4, 5. The four filters were then applied in order to each transformation until reaching an approximation of the original plan.

1

1. Museu d’Historia de la Ciutat, Barcelona.

Decomposition

Preliminary Design

79

2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

2. Decomposition: Grid

3. Decomposition: Block

4. Decomposition: Boulevard

5. Decomposition: District

Operational Transformations

Preliminary Design

80

2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

1. Shifting

2. Superposition

3. Interference: Street

Operational Transformations

Preliminary Design

81

2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

4. Distortion

5. Transformation: Grouping

6. Final

Route Identification

Preliminary Design 2.1.2 | Barcelona Lâ&#x20AC;&#x2122; Eixample Cerdaâ&#x20AC;&#x2122;s Proposal

Further decodification of Cerdaâ&#x20AC;&#x2122;s proposal reveals that the geometric operations which transformed the pure superimposed grid were primarily disciplined by exiting trade routes and other paths of access, going back to the medieval city or the roman city. The diagram on the left identifies the edges and routes that gave Cerda the rules to transform the superimposed grid.

83

L’Eixample

Preliminary Design

84

2.1.3 | L’Eixample District Present Decoding

Section 2.1.3 — L’Eixample District Present Decoding

The original proposal of Ildefons Cerdà i Sunyer was implemented over time; however, time and changing ordinances have caused a drastic change to the block structure and respectively the entire master plan. 1, 2. The originally proposed courtyard for open public space was built up for the increase of floor area causing high congesting in the cores. Further, ordinances allowed for the blocks to build higher and deeper resulting in denser plans and varied building typologies. The following pages will explore these analyze these changes and explain how much the blocks deviated from the original proposal.

1

1. Google Maps; Barcelona, Spain

Lâ&#x20AC;&#x2122;Eixample

Preliminary Design

85

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

2

2

2. Google Maps; Barcelona, Spain

Cerdà Block

Preliminary Design

86

2.1.3 | L’Eixample District Present Decoding

The figures on the left shows the block evolution since the original proposal in 1859 up to the last major ordinance change in 1976. Each figure displays gross floor area (GFA) and floor area ratio (FAR).

1859 Cerdà Block GFA 21,654 sqm 233,082 sqft FAR: 1.8

Block Development

Preliminary Design

87

2.1.3 | L’Eixample District Present Decoding

1860

1891

1924

1976

Ordenanca de parcel-la

Ordenances d’illa de cases

Ordenances congestives

Ordenanca d’el PGM

GFA

GFA

GFA

GFA

34,885 sqm

69,835 sqm

98,343 sqm

57, 436 sqm

375,500 sqft

751,698 sqft

1,058,545 sqft

618,236 sqft

FAR: 2.8

FAR: 5.7

FAR: 8.0

FAR: 4.7

Mapping

Preliminary Design

88

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

The present Lâ&#x20AC;&#x2122;Eixample district was constructed on an octagonal geometric pattern which was transformed by a set of operations (page 52-53, 1-6). To begin the iterative process of transformations, the plan was mapped through four filters: grid, block, boulevard, and district. Then the mapped plan was decomposed into it constituent parts (the four filters), 2, 3, 4, 5. The four filters were then applied in order to each transformation until reaching an approximation of the original plan.

1

Decomposition

Preliminary Design

89

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

2. Decomposition: Grid

3. Decomposition: Block

4. Decomposition: Boulevard

5. Decomposition: District

Operational Transformation

Preliminary Design

90

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

1. Shifting

2. Superposition

3. Interference: Street

Operational Transformation

Preliminary Design

91

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

4. Distortion

5. Transformation: Grouping

6. Final

Site and Context Model

Preliminary Design

92

2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

The site and context model material include acrylic and MDF. The acrylic was fabricated using the laser cutter. There are four layers that represent the evolution of the city from the original Roman wall and town square to the Medieval wall to the Idefons Cerda expansion and lastly to what was actually built. The model is also a representation of the geometric operations that were performed to construct the octagonal grid city.

Site and Context Model

Preliminary Design 2.1.3 | Lâ&#x20AC;&#x2122;Eixample District Present Decoding

93

Blocks

Preliminary Design

Rule Categorization

Section 2.1.4 | (Block) Geometric Value System

RULE.00

RULE.01

94

Section 2.1.4 â&#x20AC;&#x201D; (Block) Geometric Value System

RULE.02

RULE.03

RULE.04 RULE.05

RULE.00 - Base RULE.01 - Doubling (Two Grouped) RULE.02 - Scaling (Four Grouped) RULE.03 - Displacement (Increased Chamfer)

RULE.06

RULE.04 - Superposition (Concave) RULE.05 - Projection (Straight Cut)

RULE.07

RULE.06 - Transformation (Directed Dissection) RULE.07 - Superposition (Convex)

RULE.08

RULE.08 - Intersection (Angular Street Cut) RULE.09 - Interference (Percentage Subtraction) RULE.10 - Distortion (Single Stretch)

RULE.09

RULE.11 - Doubling and Distortion (Double Stretch) RULE.12 - Distortion and Intersection (Cut and Stretch)

RULE.10

RULE.11 RULE.12 RULE.13

RULE.13 - Interweaving (Random Dissection)

Value Scale

Preliminary Design Section 2.1.4 | (Block) Geometric Value System

The rules are further categorized into three sections: 1. Positive (+) section groups RULE.07-RULE.13, this section identifies the blocks that were transformed through random usage by the inhabitants.

2. Negative (-) section groups RULE.01-RULE.06, this section identifies the blocks that were transformed strategically by a master plan.

3. Origin (0) section identifies the base block which is RULE.00.

95

Parametric Definition

Preliminary Design

98

Section 2.1.4 | (Block) Geometric Value System

The parametric model on the right requires the input to be the rule based geometry of the blocks. Then the model weaves the data through a user based pattern defined by the controller. The final output is the number of blocks color coded in context of the whole city.

Parametric Definition

Preliminary Design Section 2.1.4 | (Block) Geometric Value System

99

Rule Based Axons

Preliminary Design

100

Section 2.1.4 | (Block) Geometric Value System

The following diagrams were derived from the parametric model shown on the previous page. With one controller, the parametric model switches between the different rules and outputs the number of blocks in context of the whole city.

RULE.00 - Base 202 BLOCKS

RULE 01: Scaling (Four Grouped) 3 BLOCKS

Rule Based Axons

Preliminary Design

101

Section 2.1.4 | (Block) Geometric Value System

RULE 02: Doubling (Two Grouped)

RULE 04: Superposition (Concave)

3 BLOCKS

13 BLOCKS

RULE 03: Displacement (Increased Chamfer)

RULE 05: Projection (Straight Cut)

10 BLOCKS

17 BLOCKS

Rule Based Axons

Preliminary Design

102

Section 2.1.4 | (Block) Geometric Value System

RULE 06: Transformation (Directed Dissection)

RULE 08: Intersection (Angular Street Cut)

1 BLOCKS

39 BLOCKS

RULE 07: Superposition (Convex)

RULE 09: Interference (Percentage Subtraction)

4 BLOCKS

44 BLOCKS

Rule Based Axons

Preliminary Design

103

Section 2.1.4 | (Block) Geometric Value System

RULE 10: Distortion (Single Stretch)

RULE.12 - Distortion and Intersection (Cut and Stretch)

24 BLOCKS

23 BLOCKS

RULE 11: Doubling and Distortion (Double Stretch)

RULE.13 - Interweaving (Random Dissection)

2 BLOCKS

13 BLOCKS

Block Placement

Preliminary Design

104

Section 2.1.5 | Sagrada Familia Block Placement

Section 2.1.5 â&#x20AC;&#x201D; Sagrada Familia Block Placement

The floor plan of the church is the traditional Latin cross.

Gaudi, however, through several additions transformed the Latin cross to the geometry of city block there by internalizing the courtyard typology, 1. The Latin cross floor plan which is the internalized courtyard is enclosed by the chapels and the cloister, 2, 3, 4. The two elevated entrances of Passion and Nativity facades extend to the block geometric edge to complete the comprehensive floor plan, 5.

1

Decomposition Maps

Preliminary Design

105

Section 2.1.5 | Sagrada Familia Block Placement

2

3

4

5

Preliminary Design

106

Section 2.2 | Space Frame Analysis

Section 2.2

Table of Contents

Section 2.2.1 | Space Frame Types Section 2.2.2 | Space Frame Joint Systems Section 2.2.3 | Space Frame Geometric Decoding

Description

Preliminary Design

107

Section 2.2 | Space Frame Analysis

Section 2.2 â&#x20AC;&#x201D;

Section 2.2 will survey space frame systems. The first section will exhibit space frame type classifications and connection operations, the second section will classify the different types of jointing systems, and lastly the third section will decode the space frame geometry to explain how can geometric operations construct the space frame grids.

Space Frame Analysis

Definition

Preliminary Design

108

Section 2.2.1 | Space Frame Types

Section 2.2.1 —

space frame

single layer

double layer one way two way three way four way

direct offset differential lattice 1

Space Frame Types

“A space frame is a structure system assembled of linear elements so arranged that forces are transferred in a threedimensional manner. In some cases, the constituent element may be two dimensional. Macroscopically a space frame often takes the form of a flat or curved surface.”

Space frames are generally single or double layered grids that possess one, two, three, or four way symmetry. The double layered grids are spaced apart and interconnected by members in one of four ways: direct, offset, differential, or lattice. And single layered grids connect only in lattice forms which construct lattice shells, 1.

2,3,4 Way Symmetry

Preliminary Design

109

Section 2.2.1 | Space Frame Types

The space frame used for the geometric analysis are of only two and three way symmetry of regular and semi-regular polygonal grids.

The first five grids possess two-way symmetry, 2. And the second set of five grids possess three-way symmetry, 3.

2

3

Direct and Offset

Preliminary Design

110

Section 2.2.1 | Space Frame Types

1. Direct Two grids similar in geometric configuration are spaced apart; therefore, directionally they are the same and connected in a oscillating pattern for structural rigidity.

2. Offset

1

2

Two grids similar in geometric configuration are space apart and offset each other so that the vertices of the grid below are centered within the geometric pattern of the grid above. Then they are braced (connected) by the separating members.

Differentia and Lattice

Preliminary Design

111

Section 2.2.1 | Space Frame Types

3. Differential Two grids different in geometric configuration are spaced apart; therefore, one is the differential of the other. Then they are interconnected to form the triangulation required for stability.

4. Lattice

3

4

Two grids similar in geometric configuration are space apart and interconnected such they form girders. The girders are pre-fabricated and assembled on site. Lattice differs from direct in that if the grids are placed closely to each other it could create a stiffened single space frame.

Defintion

Preliminary Design

112

Section 2.2.2 | Space Frame Joints Systems

Section 2.2.2 â&#x20AC;&#x201D;

Space Frame Joint Systems

The jointing system is an extremely important part of a space frame design. An effective solution of this problem may be said to be fundamental to successful design and construction. The type of jointing depends primarily on the connecting technique, whether it is bolting, welding, or applying special mechanical connectors. It is also affected by the shape of the members. This usually involves a different connecting technique depending on whether the members are circular or square hollow sections or rolled steel sections.

The effort expended on research and development of jointing systems has been enormous, and many different types of connectors have been proposed in the past decades.

The joints for the space frame are more important than the ordinary framing systems because more members are connected to a single joint. Furthermore, the members are located in a three-dimensional space, and hence the force transfer mechanism is more complex. The role of the joints in a space frame is so significant that most of the successful commercial space frame systems utilize proprietary jointing systems. Thus, the joints in a space frame are usually more sophisticated than the joints in planar structures, where simple gusset plates will suffice,1.

1. Structural Engineering Handbook, Space Frame Structures

Classification

Preliminary Design

113

Section 2.2.2 | Space Frame Joints Systems

Space frame jointing systems are divide in two types; the preparatory and the purpose made system.

The preparatory system is a standardized system that is mass produces.

While, the purpose made system is customized for specific structural loading and geometric criteria.

The diagram on the left illustrates the proprietary system classifications which yields all of the different parameters ultimately effecting the design of the space frame.

with node

without node

sphere

cricular

cylindar

rectangluar

disc

square

prism

rolled steel

form of member addition of member bolting geometric solids

prefabricated

system

2d components

welding

3d components

mechanical connectors

sub-system

connection

section

Typical Joints

Preliminary Design

114

Section 2.2.2 | Space Frame Joints Systems

1. Mero System: TP: Double SYS: Proprietary Joint SSYS: With Node - Sphere CN: Bolting MS: Cicular Hollow MM: 18 MM: 10

2. Disc Node: TP: Single SYS: Proprietary Joint SSYS: With Node - Disc CN: Bolting MS: Square/Rect. Hollow MM: 10 MM: 4

1-4. Structural Engineering Handbook, Space Frame Structures

3. Bowl Node: TP: Double SYS: Proprietary Joint SSYS: With Node - Sphere CN: Bolting MS: Square/Rect. Hollow MM: ?

MM: 1

5. Block Node: TP: Double SYS: Proprietary Joint SSYS: With Node - Disc CN: Bolting MS: Square/Rect. Hollow MM: 4

Typical Joints

Preliminary Design

115

Section 2.2.2 | Space Frame Joints Systems

5. Block Node: TP: Double SYS: Proprietary Joint SSYS: With Node - Disc CN: Bolting MS: Square/Rect. Hollow MM: 4

6. Space Deck: TP: Double SYS: Proprietary Joint SSYS: PreFab/Solids CN: Bolting MS: Square/Rect. Hollow MM: 1

5-8. Structural Engineering Handbook, Space Frame Structures

7. Triodetic System: TP: Single & Double SYS: Proprietary Joint SSYS: With Node - Cylinder CN: Bolting MS: Square/Rect. Hollow MM: 8

8. Unistrut System: TP: Double SYS: Proprietary Joint SSYS: With Node - Prism CN: Bolting MS: Rolled Steel MM: 8

Typical Joints

Preliminary Design

116

Section 2.2.2 | Space Frame Joints Systems

9. Oktaplatte System: TP: Single SYS: Proprietary Joint SSYS: With Node - Sphere CN: Welding MS: Circular Hollow MM: ?

10. Unibat System: TP: Double SYS: Proprietary Joint SSYS: PreFab 2D-3D CN: Bolting MS: Square/Circular Hollow MM: 1

9-12. Structural Engineering Handbook, Space Frame Structures

11. Nodus System: TP: Double SYS: Proprietary Joint SSYS: W/Out Node Add. CN: Welding/Mech.Connect MS: Circular Hollow MM: 8

12. Nippon Steel (NS): TP: Double SYS: Proprietary Joint SSYS: With Node - Sphere CN: Bolting MS: Circular Hollow MM: ?

Operational Transformation

Preliminary Design

117

Section 2.2.3 | Space Frame Geometric Decoding

Section 2.2.3 â&#x20AC;&#x201D;

Space Frame Geometric Decoding

The space frame grid geometry takes 5 geometric operations to created the corresponding grid to ultimately created a double layer space frame. The vertices on the grid (GI) are connected (VC) to identify the 3 dimensional joint (3DN) and by connecting the 3DN the semi-regular (SR) is constructed.

Lastly, overlay all of the systems, the structural members of the space frame (SFSM) are identified.

GI

3DN

Legend: GI - Grid R - Regular SR - Semi-Regular VC - Vertex Connection NID - Node Identification 3DN - 3 Dimensional Node

VC

SR

RD - Dual Regular SRD - Dual Semi-Regular SFSM - Space Frame Structural Members

NID

SFSM

GI.R.001

GI.R.002

GI.SR.001

GI.SR.002

GI.SR.003

GI.SR.004

GI.SR.005

GI.SR.006

GI.SR.007

GI.SR.008

Grid.Regular.001 (GI.R.001)

Preliminary Design

120

Section 2.2.3 | Space Frame Geometric Decoding

GI.R.001

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planer diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.R.001)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.RD.001

121

SFSM

Grid.Regular.002 (GI.R.002)

Preliminary Design

122

Section 2.2.3 | Space Frame Geometric Decoding

GI.R.002

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.R.002)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.RD.002

123

SFSM

Grid.Semi-Regular.001 (GI.SR.001)

GI.SR.001

Preliminary Design

124

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.001)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.001

125

SFSM

Grid.Semi-Regular.002 (GI.SR.002)

GI.SR.002

Preliminary Design

126

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.002)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.002

127

SFSM

Grid.Semi-Regular.003 (GI.SR.003)

GI.SR.003

Preliminary Design

128

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.003)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.003

129

SFSM

Grid.Semi-Regular.004 (GI.SR.004)

GI.SR.004

Preliminary Design

130

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.004)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.004

131

SFSM

Grid.Semi-Regular.005 (GI.SR.005)

GI.SR.005

Preliminary Design

132

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.005)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.005

133

SFSM

Grid.Semi-Regular.006 (GI.SR.006)

GI.SR.006

Preliminary Design

134

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.006)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.006

135

SFSM

Grid.Semi-Regular.007 (GI.SR.007)

GI.SR.007

Preliminary Design

136

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.007)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.007

137

SFSM

Grid.Semi-Regular.008 (GI.SR.008)

GI.SR.008

Preliminary Design

138

Section 2.2.3 | Space Frame Geometric Decoding

VC

NID

Beginning with the top grid and connecting the vertices yields the structural members connecting the top and bottom grids and the point could of the 3-dimensional node, VC, NID, 3DN. The bottom grid is derived by connecting the 3-dimensional nodes bilaterally and through the center point of the members above, GI.RD. Overlaying all of the diagrams results in the planar diagram of the space frame, SFSM.

Space Frame

Preliminary Design

(GI.SR.008)

Section 2.2.3 | Space Frame Geometric Decoding

3DN

GI.SRD.008

139

SFSM

Part Th Schem Design Sectio

hree matic n on 01 Part Three

Schematic Design

In this part you will find the fabrication proposal of the 12 point star structural column and the design of the space frame structure to support the column.

Table of Contents:

Section 3.0 | Structural Column Section 3.1 | Space Frame

Schematic Design

142

Section 3.0 | Structural Column

Section 3.0

Table of Contents

Section 3.0.1 | Construction Axonometric Section 3.0.2 | Internal Geometries Section 3.0.3 | Parametric Mapping Section 3.0.4 | Geometric Possibilities Section 3.0.5 | Geometric Selection

Description

Schematic Design

143

Section 3.0 | Structural Column

Section 3.0 â&#x20AC;&#x201D; Structural Column

Section 3.0, Structural Column, will display the actual construction of the 12 point star column, the internal geometries of the column and geometric possibilities explored. And lastly this section will show the geometric selection for the internal geometries and all of the contours to be fabricated at full scale.

Polar Coordinates and Axonometric

Schematic Design

144

Section 3.0.1 | Construction Axonometric

Section 3.0.1 â&#x20AC;&#x201D;

The 12 point star column is constructed out of tambours which are 1 meter in height and 12 sections around the diameter. Each tambour in plan holds polar coordinates which identifies where it is situated along the perimeter, 1.

The tambours are built up like drums and the void is filled with reinforced concrete, 2, 3.

90 120

60

150

30

180

0

210

330

240 1

300 270

Construction Axonometric

2

3

(1.05,30,21.00) CS.04 (1.05,30,20.00) (1.05,30,19.00) (1.05,30,18.00) CS.03 (1.05,30,17.00) (1.05,30,16.00) (1.05,30,15.00) (1.05,30,14.00) (1.05,30,13.00) (1.05,30,12.00) CS.02 (1.05,30,11.00) (1.05,30,10.00) (1.05,30,09.00) (1.05,30,08.00) (1.05,30,07.00) (1.05,30,06.00) (1.05,30,05.00) (1.05,30,04.00) (1.05,30,03.00) (1.05,30,02.00) (1.05,30,01.00) (1.05,30,00.00) CS.01

1

2

An axonometric displays the tambours that build up the entirety of the 21 meter high column, 1.

Spherical coordinated are used to identify the tambours around the columns perimeter and along the columnâ&#x20AC;&#x2122;s height, 2.

The individual tambours ready for CNC fabrication, right page.

Tambours

Schematic Design

147

Section 3.0.1 | Construction Axonometric

(1.05,30,00.00) CS.01

(1.05,30,00.01)

(1.05,30,00.02)

(1.05,30,00.03)

(1.05,30,00.04)

(1.05,30,00.05)

(1.05,30,00.06)

(1.05,30,00.07)

(1.05,30,00.08)

(1.05,30,00.09)

(1.05,30,00.010)

(1.05,30,00.11)

(1.05,30,00.12) CS.02

(1.05,30,00.13)

(1.05,30,00.14)

(1.05,30,00.15)

(1.05,30,00.16)

(1.05,30,00.18) CS.03

(1.05,30,00.19)

(1.05,30,00.20)

(1.05,30,00.21) CS.04

Parametric Definition

Schematic Design

148

Section 3.0.1 | Construction Axonometric

The parametric definition on the right, asks for an two inputs: first, the cross sectional profile curves (CS) at each stretch (a total of four) and second, number of contours along the height. The parametric model then lofts the curves, contours the height, and divides the diameter into 12 equal arc lengths, resulting in the individual tambours.

Parametric Definition

Schematic Design Section 3.0.1 | Construction Axonometric

149

12.PS.CS

Schematic Design

150

Section 3.0.2 | Internal Geometries

Section 3.0.2 â&#x20AC;&#x201D; Internal Geometries

The geometries the original geometry used to construct the compound curves which result in the construction of the tambours. From these base geometry the internal geometries were extracted, 1, 2, 3, 4.

12.PS.CS Geometry

1

Schematic Design

151

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS Base Geometric Discrption

Schematic Design

152

Section 3.0.2 | Internal Geometries

From the base geometries on the previous page, the star system geometries were extracted. For each cross section (CS) the internal 12,24,48,96-gon regular convex (RC) is represented and the external 12,24,48,96-gon regular complex (RX) is represented. Between the internal and external shape-gon the star system is constructed and it is represented in 12,24,48,96-points, 1, 2, 3, 4.

The next two pages display the 6 different geometric possibilities for the interiority of the column.

12.PS.CS Base Geometry

1

Schematic Design

153

Section 3.0.2 | Internal Geometries

2

3

4

TYPE.00

TYPE.01

TYPE.02

12.PS.CS.03

12.PS.CS.03

12.PS.CS.03

TYPE.00

TYPE.01

TYPE.02

12.PS.CS.02

12.PS.CS.02

12.PS.CS.02

TYPE.00

TYPE.01

TYPE.02

12.PS.CS.01

12.PS.CS.01

12.PS.CS.01

TYPE.00

TYPE.01

TYPE.02

12.PS.CS.00

12.PS.CS.00

12.PS.CS.00

TYPE.03

TYPE.04

TYPE.05

12.PS.CS.03

12.PS.CS.03

12.PS.CS.03

TYPE.03

TYPE.04

TYPE.05

12.PS.CS.02

12.PS.CS.02

12.PS.CS.02

TYPE.03

TYPE.04

TYPE.05

12.PS.CS.01

12.PS.CS.01

12.PS.CS.01

TYPE.03

TYPE.04

TYPE.05

12.PS.CS.00

12.PS.CS.00

12.PS.CS.00

12.PS.CS TYPE.00

Schematic Design

156

Section 3.0.2 | Internal Geometries

EG: 96 POINT CURVE

Type.00 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

IG: 12-GON RX

Legend:

OFFSET: 21 METERS

EG: External Geometry

12.PS.CS.03

IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star 12.PS.CS.02 EG: 48 POINT CURVE IG: 12-GON RX OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 12-GON RX OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RX OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.00

1

Schematic Design

157

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS TYPE.01

Schematic Design

158

Section 3.0.2 | Internal Geometries

12.PS.CS.03 EG: 96 POINT CURVE

Type.01 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

IG: 12-GON RC

Legend:

OFFSET: 21 METERS

EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star

12.PS.CS.02 EG: 48 POINT CURVE IG: 12-GON RC OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 12-GON RC OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RC OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.01

1

Schematic Design

159

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS TYPE.02

Schematic Design

160

Section 3.0.2 | Internal Geometries

12.PS.CS.03 EG: 96 POINT CURVE IG: 12-GON RC OFFSET: 21 METERS

Type.02 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star

12.PS.CS.02 EG: 48 POINT CURVE IG: 12-GON RC OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 12-GON RC OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RC OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.02

1

Schematic Design

161

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS TYPE.03

Schematic Design

162

Section 3.0.2 | Internal Geometries

12.PS.CS.03 EG: 96 POINT CURVE IG: 96-GON RX OFFSET: 21 METERS

Type.03 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star

12.PS.CS.02 EG: 48 POINT CURVE IG: 48-GON RX OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 24-GON RX OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RX OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.03

1

Schematic Design

163

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS TYPE.04

Schematic Design

164

Section 3.0.2 | Internal Geometries

12.PS.CS.03 EG: 96 POINT CURVE IG: 96-GON RC OFFSET: 21 METERS

Type.04 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star

12.PS.CS.02 EG: 48 POINT CURVE IG: 48-GON RC OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 24-GON RC OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RC OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.04

1

Schematic Design

165

Section 3.0.2 | Internal Geometries

2

3

4

12.PS.CS TYPE.05

Schematic Design

166

Section 3.0.2 | Internal Geometries

12.PS.CS.03 EG: 96 POINT CURVE

Type.05 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4.

IG: 96-GON RC

Legend:

OFFSET: 21 METERS

EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star

12.PS.CS.02 EG: 48 POINT CURVE IG: 48-GON RC OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 24-GON RC OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RC OFFSET: 00 METERS

CS: Cross Section

12.PS.CS TYPE.05

1

Schematic Design

167

Section 3.0.2 | Internal Geometries

2

3

4

Description

Schematic Design

168

Section 3.0.3 | Parametric Mapping

Section 3.0.3 — Parametric Mapping

A parametric definition was created to generate all of the column geometries in the previous section. The definition’s basic input is the ‘type’ and after a type is received the definition creates all of the required calculations for the geometric procedures.

Grasshopper Components

Schematic Design

169

Section 3.0.3 | Parametric Mapping

WEAVE: Takes the input from the Slider and “weaves” all of the CS profiles that pertain to that type.

ROT3D: The input form “eval” is received at the rotation component and the CS is rotated accordingly.

LOFT: All of the CS profiles are collected and lofted to create form of the column.

CONTOUR: The form of the column is contoured to produce the 21 CS profiles.

12.PS.CS TYPE.01,03,04

Schematic Design

172

Section 3.0.4 | Geometric Possibilities

Section 3.0.4 â&#x20AC;&#x201D; Geometric Possibilities

Other configurations of the column were discovered through the parametric model. The lines of extrusion of both the external and internal geometries can twist about the center axis of the column. The twist transformations is a derivative of the rotation which occurs at each CS (cross section). The follow pages will display these configurations on Type.01, Type.03, and Type.04

12.PS.CS TYPE.01,03,04

Type.01

Schematic Design

173

Section 3.0.4 | Geometric Possibilities

Type.03

Type.04

12.PS.CS TYPE.01.P

Schematic Design

174

Section 3.0.4 | Geometric Possibilities

12.PS.CS.03 EG: 96 POINT CURVE

EGR: -107.496

IG: 12-GON RC

IGR: -107.496

OFFSET: 21 METERS

12.PS.CS.02 EG: 48 POINT CURVE

EGR: -19.057

IG: 12-GON RC

IGR: -19.057

OFFSET: 18 METERS

Type.01.P receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4. The primary difference between Type.01 and Type.01.P is that Type.01.P applies a rotation at the four primary CS (cross sections) resulting with twist.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star CS: Cross Section P: Possiblity

12.PS.CS.01 EG: 24 POINT CURVE

EGR: 31.303

IG: 12-GON RC

IGR: 31.303

OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE

EGR: 31.304

IG: 12-GON RC

IGR: 31.304

OFFSET: 00 METERS

EGR: External Geometry Rotation IGR: Internal Geometry Rotation

12.PS.CS TYPE.01.P

1

Schematic Design

175

Section 3.0.4 | Geometric Possibilities

2

3

4

12.PS.CS TYPE.03.P

Schematic Design

176

Section 3.0.4 | Geometric Possibilities

12.PS.CS.03 EG: 96 POINT CURVE

EGR: -166.875

IG: 96-GON RX

IGR: -166.875

OFFSET: 21 METERS

12.PS.CS.02 EG: 48 POINT CURVE

EGR: 64.511

IG: 48-GON RX

IGR: 64.511

OFFSET: 18 METERS

Type.03.P receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4. The primary difference between Type.03 and Type.03.P is that Type.03.P applies a rotation at the four primary CS (cross sections) resulting with twist.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star CS: Cross Section P: Possiblity

12.PS.CS.01 EG: 24 POINT CURVE

EGR: 43.212

IG: 12-GON RX

IGR: 43.212

OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE

EGR: 0.000

IG: 12-GON RX

IGR: 0.000

OFFSET: 00 METERS

EGR: External Geometry Rotation IGR: Internal Geometry Rotation

12.PS.CS TYPE.03.P

1

Schematic Design

177

Section 3.0.4 | Geometric Possibilities

2

3

4

12.PS.CS TYPE.04.P

Schematic Design

178

Section 3.0.4 | Geometric Possibilities

12.PS.CS.03 EG: 96 POINT CURVE

EGR: -166.875

IG: 96-GON RC

IGR: -166.875

OFFSET: 21 METERS

12.PS.CS.02 EG: 48 POINT CURVE

EGR: -91.744

IG: 48-GON RC

IGR: -91.744

OFFSET: 18 METERS

Type.04.P receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively, 1, 2, 3, 4. The primary difference between Type.04 and Type.04.P is that Type.04.P applies a rotation at the four primary CS (cross sections) resulting with twist.

Legend: EG: External Geometry IG: Internal Geometry RX: Regular Complex RC: Regular Convex PS: Point Star CS: Cross Section P: Possibility

12.PS.CS.01 EG: 24 POINT CURVE

EGR: -33.3002

IG: 24-GON RC

IGR: -33.302

OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE

EGR: 24.310

IG: 12-GON RC

IGR: 24.310

OFFSET: 00 METERS

EGR: External Geometry Rotation IGR: Internal Geometry Rotation

12.PS.CS TYPE.04.P

1

Schematic Design

179

Section 3.0.4 | Geometric Possibilities

2

3

4

Selection TYPE.03

Schematic Design

180

Section 3.0.5 | Geometric Selection

Section 3.0.5 â&#x20AC;&#x201D; Geometric Selection

Type.03 was selected as the final geometric application for the interior of the column. The selection was based on three factors: First, the cross sectional area for each contour is the lowest given the complexity of the geometry; second, the internal geometry is a direct representation of the star point system used to create each compound curve of the external geometry; and third, there are more direct points of contact to attach the space frame structure to.

70 Contours TYPE.03

Schematic Design Section 3.0.5 | Geometric Selection

181

12.PS.CS TYPE.03 CS Data

Schematic Design

182

Section 3.0.5 | Geometric Selection

12.PS.CS.03 EG: 96 POINT CURVE IG: 96-GON RX OFFSET: 21 METERS

Type.03 receives the cross sectional (CS) geometries on the left and lofts them to create the 21 contours, the solid exterior and interior form, the exterior form, and the interior form respectively.

Type.03 was contoured to produce the 70 CS profiles which will be used for the space frame design analysis and fabrication.

Legend: EG: External Geometry IG: Internal Geometry

12.PS.CS.02 EG: 48 POINT CURVE IG: 48-GON RX OFFSET: 18 METERS

12.PS.CS.01 EG: 24 POINT CURVE IG: 24-GON RX OFFSET: 12 METERS

12.PS.CS.00 EG: 12 POINT CURVE IG: 12-GON RX OFFSET: 00 METERS

RX: Regular Complex RC: Regular Convex PS: Point Star CS: Cross Section

Schematic Design

184

Section 3.1 | Space Frame

Section 3.1

Table of Contents

Section 3.1.1 | Space Frame Iteration 1 Section 3.1.2 | Space Frame Iteration 1 Joint Analysis Section 3.1.3 | Space Frame Iteration 2 Section 3.1.4 | Space Frame Iteration 2 Joint Analysis Section 3.1.5 | Space Frame Iteration 3 Section 3.1.6 | Space Frame Final Selection Section 3.1.7 | Column Stressed Skin Space Frame

Description

Schematic Design

185

Section 3.1 | Space Frame

Section 3.1 â&#x20AC;&#x201D;

Section 3.1 will examine the geometric construction of the space frame and the joint. Through three different iterations the design is geometrical distilled and synthesized into a final composition.

Space Frame

Description

Schematic Design

186

Section 3.1.1 | Space Frame Iteration 1

Section 3.1.1 â&#x20AC;&#x201D;

Space Frame Iteration 1

The first iteration of the space frame design derives its geometry from the 12 point star system of Guadiâ&#x20AC;&#x2122;s Column at the Sagrada Familia. By using the integral of the external curvature the internal point star system is derived. The internal geometric shapes are then taken through a series of operations which derive the space frame. The series of operations are listed and explained in Section 2.2.3.

The idea for the space frame is to act as a core substitution for existing column, 1. The concrete core is extracted and substituted with the space frame, leaving the tambours to act as skin only, 2, 3. The tambours will be fabricated based on 70 contours representing the geometric deformations, 4.

1

2

3

4

Existing and Altered Geometry

Schematic Design

188

Section 3.1.1 | Space Frame Iteration 1

The following series of axonometrics illustrate the transformation of the internal geometry and the substitution of the core material.

1. Existing Column 2. Internal Geometry 3. Space Frame Substitute 4. Substitution Axonometric 5. Space Frame 6. 70 CS Contours

1

2

3

Core Substitution and Space Frame

4

Schematic Design

189

Section 3.1.1 | Space Frame Iteration 1

5

6

Base Geometry

Schematic Design

190

Section 3.1.1 | Space Frame Iteration 1

1, The base geometry of the four cross section (CS) displayed from bottom to top (base to capital). 2, A regular convex 12-gon is created by connecting the points of the internal geometry. 3, The vertices are connected resulting in 12 internal triangles. 4, An octagon is positioned such that the length of each side is either 12 or divisible by 12. 5, The geometry is trimmed to give the final geometric resultant. 6, The joints are identified as joints with 3 extensions.

1

2

Geometric Connections

Schematic Design

191

Section 3.1.1 | Space Frame Iteration 1

3

4

5

6

Members (Struts)

Schematic Design

192

Section 3.1.1 | Space Frame Iteration 1

The geometric analysis on the previous two pages derived the space frame component shown on the right, 2. Plan, 3. Right Elevation, 4. Front Elevation, 5. Perspective. The space frame component consist of 36 members of 3 varied lengths carrying two contours, 1. If the component propagate along the length of the column it begins to form and the number of members will increase by 24 every foot.

1

Space Frame Component

Schematic Design

193

Section 3.1.1 | Space Frame Iteration 1

2

3

4

5

Space Frame Joint

Schematic Design

194

Section 3.1.2 | Space Frame Iteration 1 Joint Anaylsis

Section 3.1.2 â&#x20AC;&#x201D; Space Frame Iteration 1 Joint Anaylsis

For the first iteration of the space frame the joint was analyzed as sets of systems that are transformed to create the configuration required for the space frame. The diagram on the right show how each joint behaves with its connected members and how all the joints and members triangulate together to create the configuration.

Joint Structural Placement

Schematic Design Section 3.1.2 | Space Frame Iteration 1 Joint Anaylsis

195

Joint Codex

Schematic Design

196

Section 3.1.2 | Space Frame Iteration 1 Joint Anaylsis

The table of joints above illustrates how only 3 different types of joints can be transformed via Operation Transformation (O.T.) such as rotation and reflection and result in 36 joint configurations. The 36 joints are configured on the right to form the space frame component, 1, 2, 3, 4.

Joint Distribution

Schematic Design

197

Section 3.1.2 | Space Frame Iteration 1 Joint Anaylsis

1. First pass (Joint AA) contains 4 joints each connecting 3 members. All members are perpendicular to each other and each joint successively rotates 90 degrees counter-clock wise.

2. Second pass (Joint BB) contains 8 joints each connecting 3 members. All members are either 135, 105, or 135 degrees to each other and each joint successively rotates 90 degrees and reflects about the member axis.

3. Third pass (Joint BB) contains 12 joints and each connecting 3 members. All members are 75 degrees to each other and each joint successively rotates 30 degrees counter-clock wise.

4. A total of 24 joints connecting 36 members to form the space frame component carrying 2 contours (24 tambours)

Description

Schematic Design

198

Section 3.1.3 | Space Frame Iteration 2

Section 3.1.3 â&#x20AC;&#x201D;

Space Frame Iteration 2

The second iteration of the space frame design derives its geometry from the octagonal grid of Barcelona. By using the 8 sided figure (octagon), lines are traced to meet the points of the internal geometries. The figure is then taken through a series of operations which derive the space frame. The series of operations are listed and explained in Section 2.2.3.

The idea for the space frame is to act as a core substitution for existing column, 1. The concrete core is extracted and substituted with the space frame, leaving the tambours to act as skin only, 2, 3. The tambours will be fabricated based on 70 contours representing the geometric deformations, 4.

1

Core Replacement

Schematic Design Section 3.1.3 | Space Frame Iteration 2

2

3

4

Existing and Altered Geometry

Schematic Design

200

Section 3.1.3 | Space Frame Iteration 2

The following series of axonametrics illustrate the transformation of the internal geometry and the substitution of the core material.

1. Existing Column 2. Internal Geometry 3. Space Frame Substitute 4. Substitution Axonometric 5. Space Frame 6. 70 CS Contours

1

2

3

Core Substitution and Space Frame

4

Schematic Design

201

Section 3.1.3 | Space Frame Iteration 2

5

6

Base Geometry

Schematic Design

202

Section 3.1.3 | Space Frame Iteration 2

1, The base geometry of the four cross section (CS) displayed from bottom to top (base to capital). 2, A regular convex 12-gon is created by connecting the points of the internal geometry. 3, The vertices are connected resulting in 12 internal triangles. 4, An octagon is positioned such that the length of each side is either 12 or divisible by 12. 5, The geometry is trimmed to give the final geometric resultant. 6, The joints are identified as joints with 3 extensions.

1

2

Geometric Connections

Schematic Design

203

Section 3.1.3 | Space Frame Iteration 2

3

4

5

6

Members (Struts)

Schematic Design

204

Section 3.1.3 | Space Frame Iteration 2

The geometric analysis on the previous two pages derived the space frame component shown on the right, 2. Plan, 3. Right Elevation, 4. Front Elevation, 5. Perspective. The space frame component consist of 36 members of 3 varied lengths carrying two contours, 1. If the component propagate along the length of the column it begins to form and the number of members will increase by 24 every foot.

1

Space Frame Component

Schematic Design

205

Section 3.1.3 | Space Frame Iteration 2

2

3

4

5

Space Frame Joint

Schematic Design

206

Section 3.1.4 | Space Frame Iteration 2 Joint Anaylsis

Section 3.1.4 â&#x20AC;&#x201D; Space Frame Iteration 2 Joint Anaylsis

For the second iteration of the space frame the joint was analyzed as sets of systems that are transformed to create the configuration required for the space frame. The diagram on the right show how each joint behaves with its connected members and how all the joints and members triangulate together to create the configuration.

Joint Structural Placement

Schematic Design Section 3.1.4 | Space Frame Iteration 2 Joint Anaylsis

207

Joint Codex

Schematic Design

208

Section 3.1.4 | Space Frame Iteration 2 Joint Anaylsis

The table of joints above illustrates how only 4 different types of joints can be transformed via Operation Transformations (O.T.) such as rotation and reflection and result in 13 joint configurations. The 13 joints are configured on the right to form the space frame component, 1, 2, 3, 4.

Joint Distribution

Schematic Design

209

Section 3.1.4 | Space Frame Iteration 2 Joint Anaylsis

1. First pass (Joint AA) contains 1 joint each connecting 8 members. All members are respectively perpendicular to each other.

2. Second pass (Joint BB) contains 4 joints each connecting three members. All members are either 133, 137, or 90 degrees to each other and each joint successively rotates 90 degrees.

3. Third pass (Joint CC and DD) contains 8 joints, 4 of each and each connecting two members. Each type of joint rotates 90 degrees counter-clock wise.

4. A total of 13 joints connecting 20 members to form the space frame component carrying 2 contours (24 tambours)

Description

Schematic Design

210

Section 3.1.5 | Space Frame Iteration 3

Section 3.1.5 â&#x20AC;&#x201D;

Space Frame Iteration 3

Iteration 3 of the space frame is designed around the construction of a stressed skin versus a standardized strut and joint fabrication. The codification of the entire system is thought of as one planar shape material which could be connected at various orientations to produce an assembly of space. On the right is a lexicon of 6 different types of geometries with their respective spatial assembly given in perspective, plan, and elevation.

Base Geometry

Schematic Design

212

Section 3.1.5 | Space Frame Iteration 3

The stressed skin is designed from a triangular geometric shaped. The triangular shape is taken through a set of operations to produce the final profile of the stressed skin.

The triangle is constructed and by connecting its vertices, the axes of symmetry are identified. And from with the tesselation that the axes of symmetry produce a spline is constructed using the tesselated triangleâ&#x20AC;&#x2122;s vertices as control points, 1, 2, 3. The spline is then offsets to yield varying material optimizations.

The 6 different iterations of the stressed skin vary in the geometric profile and the placement of the connection joint, 4-9.

1

2

3

Stressed Skin

Schematic Design

TYPE.00-05

Section 3.1.5 | Space Frame Iteration 3

4

5

6

213

7

8

9

TYPE.00

Schematic Design

Base Geometry

Section 3.1.5 | Space Frame Iteration 3

214

TYPE.00 is constructed by having the connection joint on the vertices and the spline is created using two axes of symmetry, 1, 2.

On the right are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a fractal growth, 3, 4.

1

2

Plan, Elevation, Isometric

Schematic Design

TYPE.00.A/B

Section 3.1.5 | Space Frame Iteration 3

3

215

4

TYPE.01

Schematic Design

Base Geometry

Section 3.1.5 | Space Frame Iteration 3

216

TYPE.01 is constructed by having the connection joints on the center of each side of the triangle. And the spline is created using three axes of symmetry, 1, 2.

On the right are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a fractal growth, 3, 4.

1

2

Plan, Elevation, Isometric

Schematic Design

TYPE.01.A/B

Section 3.1.5 | Space Frame Iteration 3

3

217

4

TYPE.02

Schematic Design

Base Geometry

Section 3.1.5 | Space Frame Iteration 3

218

TYPE.02 is constructed by having the connection joints on the vertices and on the center of each side of the triangle.

And the spline is created using three axes of symmetry, 1, 2.

On the right are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a fractal growth, 3, 4.

1

2

Plan, Elevation, Isometric

Schematic Design

TYPE.02.A

Section 3.1.5 | Space Frame Iteration 3

3

219

TYPE.03

Schematic Design

Base Geometry

Section 3.1.5 | Space Frame Iteration 3

220

TYPE.03 is constructed by having 3 connection joints be around the vertices. One joint directly on the vertices and two joints one on each side of the vertices. And the spline is created using two axes of symmetry, 1, 2.

On the right are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a fractal growth, 3, 4.

1

2

Plan, Elevation, Isometric

Schematic Design

TYPE.03.A/B

Section 3.1.5 | Space Frame Iteration 3

3

221

4

Plan, Elevation, Isometric

Schematic Design

TYPE.03.C/D

Section 3.1.5 | Space Frame Iteration 3

1

222

2

Plan View

Schematic Design

TYPE.03.D

Section 3.1.5 | Space Frame Iteration 3

On the left are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a fractal growth, 1, 2.

TYPE.00.D is further growing to created a spatial matrix. On the right is a plan view of the spatial matrix, 3.

3

223

TYPE.04.05

Schematic Design

Base Geometry

Section 3.1.5 | Space Frame Iteration 3

224

TYPE.04 and TYPE.05 are mirror images of each other.

Each one is constructed by having 2 connection joints be around the vertices. One joint directly on the vertices and one joint on one side of the vertices. And the spline is created using two axes of symmetry, 1, 2.

On the right are two systems resulting from connecting the shape to its corresponding joint. This system can grow infinitely yielding a spiraling fractal growth, 3, 4.

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Plan, Elevation, Isometric

Schematic Design

TYPE.04.05

Section 3.1.5 | Space Frame Iteration 3

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Description

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Section 3.1.6 | Space Frame Final Selection

Section 3.1.6 â&#x20AC;&#x201D;

Space Frame Final Selection

The final selection of the space frame was driven by a wide range of variables. Constraints in budget and methods of fabrication eliminated the first two iteration. The first two iterations required more usage of standard power tools to construct the joints which adds more time for fabrication and allows more opportunities for error in precision. The third iteration considered all of these factors and the resultant was a planar sheet of material cut through the laser cutter or CNC mill with the joints included. The final outcome will only have to rotated about the required axes to configure into its proposed form. The geometric figure on the right exhibits the variable placement of the third iteration by rotating it to complete the circle.

Base Geometry

Schematic Design

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Section 3.1.6 | Space Frame Final Selection

1, The base geometry of the four cross section (CS) displayed from bottom to top (base to capital). 2, A regular convex 12-gon is created by connecting the points of the internal geometry and the vertices are connected resulting in 6 triangles and 6 squares. 3, connecting the intersections yields a hexagon in the middle. 4, the geometry is trimmed to give the final geometric resultant. 5, The joints are identified as joints with 3 extensions. 6, The joints are replaced with the stressed skin.

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Geometric Connections

Schematic Design

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Section 3.1.6 | Space Frame Final Selection

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Joint Codex

Schematic Design

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Section 3.1.6 | Space Frame Final Selection

The table of joints above illustrates how only 2 different types of joints can be transformed via Operational Transformations (O.T.) such as reflection and result in 18 joint configurations.

The 18 joints are configured on the right to form the space frame component, 1, 2, 3, 4.

Joint Distribution

Schematic Design

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Section 3.1.6 | Space Frame Final Selection

1. The geometric shapes composing the space frame which include 6 triangle, 6 squares and 1 hexagon.

2. First pass (Joint AA) contains 6 joints each connecting 4 members. All members are either 120, 60, or 90 degrees to each other and each joint successively reflects.

3. Second pass (Joint BB) contains 12 joints, each and each connecting three members and each joint successively reflects.

4. A total of 18 joints connecting 30 members to form the space frame component carrying 2 contours (24 tambours)

Joint Symmetry Models

Schematic Design

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Section 3.1.6 | Space Frame Final Selection

The basic joint symmetry model was the analysis used to create the stressed skin planar shape. The joint for space frame iteration 3 required 6 axes of symmetry, 3. That means a spherical joint will have to be tesselated 6 times across and once through its cross section to yield the proper number of flat surfaces to weld, bolt, or mechanical connect struts, 1. To replicated the same 3 dimensional complexity with a planar stressed skin shape requires the same number of axes of symmetry. The stressed skin shape has 3 axes of symmetry, but since it is also planar symmetrical that yields another 3 axes upon reflection, 2, 4

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Joint Symmetry Analysis

Schematic Design Section 3.1.6 | Space Frame Final Selection

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Description

Schematic Design

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Section 3.1.7 | Column Stressed Skin Space Frame

Section 3.1.7 â&#x20AC;&#x201D; Column Stressed Skin Space Frame

The second iteration of the space frame design derives its geometry from the octagonal grid of Barcelona. By using the 8 sided figure (octagon), lines are traced to meet the points of the internal geometries. The figure is then taken through a series of operations which derive the space frame. The series of operations are listed and explained in Section 2.2.3.

The idea for the space frame is to act as a core substitution for existing column, 1. The concrete core is extracted and substituted with the space frame, leaving the tambours to act as skin only, 2, 3. The tambours will be fabricated based on 70 contours representing the geometric deformations, 4.

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Core Replacement

Schematic Design Section 3.1.3 | Space Frame Iteration 2

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Existing and Altered Geometry

Schematic Design

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Section 3.1.7 | Column Stressed Skin Space Frame

The following series of axonametrics illustrate the transfomation of the internal geometry and the substition of the core material.

1. Existing Column 2. Internal Geometry 3. Space Frame Subtitue 4. Subtitution Axonometric 5. Space Frame 6. 70 CS Contours

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Core Substition and Space Frame

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Schematic Design

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Section 3.1.7 | Column Stressed Skin Space Frame

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Members (Struts)

Schematic Design

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Section 3.1.7 | Column Stressed Skin Space Frame

The space frame column integration shown on the previous two pages derived the space frame component shown on the right, 2. Plan, 3. Right Elevation, 4. Front Elevation, 5.

Perspective. The space frame component consist of 72 planar sections.

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Space Frame Component

Schematic Design

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Section 3.1.7 | Column Stressed Skin Space Frame

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