think, script, build Architectural engineering through parametric modelling of intelligent systems in architecture.
Rasmus Holst s061860 M.Sc. Architectural Engineering
ResumĂŠ  This is a project about programming of intelligent systems by implementing information of physical behaviour, materialproperties and connection design regarding advanced architectural projects. Focus is on utilisation of the geometric behaviour of elastic elements in connection with deformation for optimization, fabrication and inspiration. The purpose is to create a process from ideas on sketch to a realizable project through scripting, parametric design and algorithmic modelling. The motivation and inspiration for this project are the smooth shapes that come from bending and deforming simple elements. Furthermore, the fact that these shapes come from the very nature of minimizing internal potential energy, makes this approach very interesting in terms of both architecture and engineering. Scripting and parametric modelling allow for generation of complex geometry. In combination with engineering knowledge of geometry, material behaviour, constraints, external influences etc., parametric design is a great source of opportunities to fulfil creative ideas. Almost everything can be generated digitally and digital fabrication allows for production in most cases. However, sometimes these procedures become costly, material intensive and therefor often not sustainable. When an advanced shape needs to be cut out of sheets or blocks, there will be material waste, some of which can be recycled, however, this also uses energy. By thinking and scripting, complex and optimised projects can be built simply and sustainably. This thesis aims to demonstrate how.
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Title sheet DTU  MSc. Architectural Engineering Master thesis  Autumn 2012 (E12)  30 ECTS Points Subjects:
Architectural Engineering
Parametric design Optimization
Supervisor:
Henrik Almegaard  DTU  hal@byg.dtu.dk
Business partner:
Henning Larsen Architects
Jakob StrĂ¸mann Andersen  JSTR@henninglarsen.com
Project period:
The project work is carried out during the period from 03.09.12  25.02.13. This period includes 3 weeks of vacation.
Handin: 25.02.13 All rights Rasmus Holst, Autumn 2012. Thesis done by:
Date:
Signature:
Rasmus Holst, s061860 page 2
Preface This thesis is the conclusion of my Master program in Architectural Engineering at the Technical University of Denmark (DTU) At the beginning of this thesis the direction was set out by the belief that a combination of curiosity, programming skill and craftsmen experience held the potential of interesting and optimised design. The amazing collective around the parametric forum of Grasshopper3d has been an amazing resource. Especially the likes of Daniel Piker (Kangaroo), the team behind Karamba3d and many others have been very helpful and inspirational. The collaboration with Henning Larsen Architects has proven very interesting and useful. Especially thanks to Jakob StrĂ¸mannAndersen who has been great at putting my solutions to the test in ongoing projects. Also thanks to some really nice colleagues and to the entire firm for setting me up with computers, software, modelling room, laser cutter, printers etc. Last but certainly not least, a big thanks to Henrik Almegaard for great guidance, advice and inspiration throughout this project and my entire study time at DTU. It has been very interesting doing this thesis and I hope that you will enjoy reading it as much as I have enjoyed working with it.
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Report layout This report is divided into 8 chapters. Firstly an introduction to the thesis problem statement, motivation and method are given. To guide the reader through the process of this project, the report is set up in more chronological order than the actual iterative process. This way chapters 12 introduces the thesis and the principles of parametric design. Case[0] in chapter 2 exemplifies parametric design in action. Chapter 3 is an initial research into structural principles from simple elements. This sets up an outline for overview and inspiration. One principle is chosen for further investigation in the following chapters. (think) Chapter 46 sets up the theory of physical modelling and investigates scripting results. The method build in chapter 4 and 5, is tested on Case[1] in chapter 6. (script) Chapter 7 goes through the final case work, using the theory, methods and scripts build through the project in case[2]. The case is a pavilion design and focus is on fabrication and build ability. (build) Finally chapter 8 reflects upon obtained results and perspectives. When ever this logo is shown close to an illustration, a corresponding animation can be found on www.vimeo.com/rasmusholst/albums. Click on the album â€œThink, Script, Buildâ€?.
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think script build
1
Introduction
2
Case [0]
3
Initial research
4
Modelling
5
Investigation
6
Case [1]
7
Case [2]
8
Reflection
page 5
table of contents 1 Introduction
page 16
2 Case [0]: EPO
page 26
3 Initial Research
page 46
4 Modelling
page 68
1.1 Architectural engineering 16 1.2 Historical perspective 16 1.3 Parametric design 18
2.1 European Patent Office  Introduction 26 2.2 Parametric facade design. 28 2.3 Optimization 30 2.4 Scripting 32 2.5 Parametric studies 34 2.6 Discussion 38 2.7 Perspective 40 2.8 Part conclusion 42
3.1 Simple advanced structures 46 3.2 Gridshells 64
4.1 Paper play 68 4.2 Theory 70 4.3 Physical modelling 72 4.4 Definition breakdown [Spline] 76 4.5 Plate modelling. 78 4.6 Connection modelling 87 4.7 Definition breakdown [Mesh] 90 4.8 Part conclusion 92
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5 Investigation
page 96
5.1 Jukbuin pavilion 96 5.2 Jukbuin pavilion  Script 100 5.3 Jukbuin pavilion  Build 130
6 Case [1]: Nordea Bank
page 134
7 Case [2]: Pavillion
page 166
8 Reflection
page 180
9 Bibliography
page 186
6.1 Nordea bank ørestad  Introduction 134 6.2 Method 136 6.3 Form studio 138 6.4 Example 140
7.1 Introduction 166 7.2 Concept 168 7.3 Form studio 170 7.4 Structure 172 7.5 Fabrication 174 7.6 Part conclusion 176
8.1 Discussion 180 8.2 Perspective 182 8.3 Conclusion 184
9.1 Resources 187
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Background Background The digital age has given access to tools for architects and engineers that allows for creation of complex geometry and advanced systems. At the same time there is a great demand for affordable solutions, sustainability and optimization. In the process of optimization one looks for the best solution of weighted parameters within a given space of solutions. When it comes to efficiency of buildings, there is a great geometrical challenge for the architect and the engineer in choosing the right solutions from the beginning. Parametric design In a design process, the solution space within the concept is often endlessly large. This gives thousands of possible combinations of angles, lengths, heights etc. The normal procedure is to boil this solution space down to a few proposals, chosen on the basis of intuition, aesthetics, analysis and/or experience. Scripting allows for designing parametrically, which enables the setting up of intelligent systems. These systems become intelligent by adding information to geometry, often points, nodes and lines. Through iteration processes, in which equilibrium of the stored information in the system is searched for, the system becomes selfemergent*. In some literature this is compared to ant hills, mould fungi, bird flocks and schools of fish. Here each individual has a simple local knowledge about its needs and tasks. This is what makes the global system work and achieve its goal. This theory of selfemergence is a matter of big discussion, research and investigation. The basic insight in this phenomena, â€œwisdom of the crowdsâ€?, is in short that useful informations can be obtained via many shots in the dark. Scripting allows the designer to make the computer go through many solutions and output wanted results and consequences. *
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Emergent  Developing. Under development.
Mimicking nature As engineers we are used to analyse proposals for structures of different kinds. That being energy efficiency of facades, structural behaviour of building elements etc. We have to predict the output of our project before it is build. For centuries theory has been built on this subject, and there is a lot of methods and software that help us predict consequences of more and more advanced systems. When it comes to structural performance we usually take our starting point in the undeformed state of elements. One exception is the prestressed concrete beams and slabs. The prestressing has the advantage of creating opposite stresses to those coming from imposed loads. Therefor it increases the element performance, while it decreases the resource usage. Maybe the reason why engineers and designers do not utilize this behaviour more, is the complexity that lies behind the bending and twisting of elements. Mostly we strive towards linearity, planarity and thereby try to avoid bending, all the while the design of architects tend to get more organic. At the same time it is known that simple nonrigid elements can gain stiffness from deformation into double curvature. By setting up an intelligent system, using scripting, it is possible to mimic the behaviour of these elements and to setup tools for formfinding, simulation and analysis.
Form finding
Fig. 1 Behaviour in nature
Form finding is the abstract modelling of material organisations as the actively negotiate internal and external influences. These can be laws of physics as well as architectural affect, spatial requirements and performance demands. The traditions has a long history that stems not only from the physics, but
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Fig. 2 Gaudi chain model
also from the desires of designers and engineers to invent and innovate within the space of possibilities. Amongst others in this history is the invention of a description of the catenary curve and the reciprocal funicular curve by Antonio Gaudi, used for designing compression only structures. Frei Otto’s study and abstraction of tendencies and behaviours of minimal surface organisation found in soap film, led the way to novel knowledge of membrane structures. The expansive learnings of Frei Otto’s distillation and abstraction of materiality has also enhanced the knowledge and ability to design large span shell, lamella and lattice structures, even though they are constructed from seemingly contradictory materials.
Algorithm Knowledge of a given material’s properties and/or molecular anatomy needs to be established as a computational description of the selforganisation. In Frei Otto’s soap film the setup can be understood as the process of one molecule acting upon another while negotiating influences.
Fig. 3 Frei Otto  Optimized path experiments
A contemporary method is the use of particlespring systems that digitally simulates soap film structures. In terms of architectural geometry, such as surfaces, volumes etc., it is represented as a network of particles hitched to one another by springs. Equilibrium is found via numerical iterations. Two common strategies are dynamic relaxation or the force density method. Both uses the mathematics of Hooke’s laws of elasticity. The algorithm allows for implementing architectural schema also. Such as to define for example the front door. This becomes another layer of influences or motivations that is to be negotiated within the system. Digital form finding is not to replicate the work that has preceded it, but rather to seek out new territories.
Wood These methods can be used to describe many different materials and behaviours. The focus of wood in this thesis is purely a case of interest and admiration of the sustainable, flexible and aesthetic nature of the material. Fig. 4 Wood materiality
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Problem statement How can parametric design be implemented in an architectural design process and become added value in terms of modelling speed, time, analysis, optimization and development. From this perspective, the intention of this project is to discover if and how we can use simple, regular building elements in the construction of the increasing numbers of complex shaped architecture projects. The solution to this problem is interesting to all parties in a building project; designers, engineers, investors, entrepreneurs and so on, in terms of economy, optimization, sustainability and buildability. The title of this report  think, script, build  is the short description of the approach that will be taken in discovering the above problem: How can scripting be used to simulate the deformation of simple elements and networks and thereby become a tool for formfinding, analysis and building descriptions of complex geometries. The interdisciplinary method of architectural engineering will be used in combination with parametric modelling of digital systems, that become intelligent by scripted information, for architectural projects.
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Method process  disposition At the beginning of the research, this project looks at interesting structures and principles of more or less simple and regular elements. An outline of this initial research is set up for inspiration and overview. One of these principles are chosen for more thorough explorations within the think, script, build approach. To begin with, an introduction to parametric design and scripting is given by brief explanation and exemplification through case studies together with Henning Larsen Architects. By doing so, common ground for further collaborative investigations are established. Then the script for parametric design of the chosen structure principle is build in an iterative process consisting of theory, exploration/experimentation and casework. The idea behind think, script, build is to set up an effective and interactive workflow between the three parts. This means working in and out of the computer, sketching and creating ideas, scripting and building digital parametric models. The building of the parametric model strives towards real time user control, simulation and analysis. To be able to analyse the model real time and find responses to any changes made, the model is linked with internal and external calculation engines. The chosen main software is Rhinoceros3D, from Robert McNeel & Associates, a 3d modelling software with many plugin and extension possibilities. Grasshopper3d is a visual programming language plugin for Rhinoceros3D, developed by David Rutten. Grasshopper3d also implements more common scripting languages, such as Python, VB, C# etc, as well as many application plugins for different use.
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Preliminary time schedule  proces
paper case01 theory
test/explore
case02 theory
case03 test/explore
writing
conclusion
model
1:1 part model
study
september
october
november
Fig. 5 Preliminary timeline
report
december
january
february
2013
AAG12 Conference and Solar decathlon:
Summation of method: 25.09.12  08.10.12
 Introduction to parametric design.  Gathering of construction principles 
Wood structures.

Aggregate/module structures
Newer installation structures.

 Building of script(s).

Theory: Materials, properties, physical model.

Experimentation and comparison

Connections

Intelligent system.
Linkup with other analysis software

 Implementation. 
In collaboration with Henning Larsen Architects method and principles are tested on case studies of competetion projects.
 Model/pavilion

Building of 1:1 scale model.
The described method is not chronological, but should be seen as an iterative process. This process is illustrated as the time line on Fig. 5 setup at beginning of this thesis.
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[INTRODUCTION]
page 14
[Architectural engineering] Rhino
+ [Parametric design]
Grasshopper
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1  Introduction 1.1 Architectural engineering Architectural engineering is intended to bridge the gap between architects and engineers. Meaning that the architectural engineer is involved in the project from the early design stages, working closely with the architect to evaluate and contribute in choosing the best solution of a given design. “The role of architectural engineers can overlap with that of the architect and other engineers. Like architects, architectural engineers seek to achieve optimal designs within the overall constraints, but mainly use engineering tools to attain their goal.” (DTU MSc. Architectural Engineering 2012) Using the technical knowledge in a creative way is meant to lift the architectural visions in a way that optimise and implements the ideas of the architect. In doing this, one strives towards creating a fusion between art and science. Early considerations of structural behaviour, energy consumption, material etc. helps in optimising the design. Small changes to the geometry or materiality, can have a big impact on the structure, amount of daylight, heat transfer, buildability etc. There are many parameters to adjust and therefor an almost endless amount of solutions. Parametric design allows changing parameters of the project and quickly review the consequences without having to redo everything. This becomes the ultimate tool for architectural engineering, combining aesthetics, science, math and analysis in the same model. Every change made to the design has an impact on the performance and can easily be reviewed realtime.
1.2 Historical perspective A significant inspiration for this mentioning of architectural engineering are the likes of important people, amongst others Antonio Gaudi, Mies van der Rohe, Buckminster Fuller, Pier Luigi Nervi and Frei Otto. Each of them combining arts and sience using different approaches.
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1.2.1 Mies van der Rohe and Bauhaus. Ludwig Mies van der Rohe was one of the three architect directors at Bauhaus. The German school combined crafts and the fine arts and became one of the most influential currents in modernist architecture. One of the main objectives of this school was to unify art, craft and technology.
1.2.2 Buckminster Fuller and the Dymaxion World. Richard Buckminster Fuller spent his life working across multiple fields, such as architecture, design, geometry, engineering, science and education. He described himself as “a comprehensive anticipatory design scientist”*. The concept of Dymaxion  DYnamic, MAXimum, tensION is the idea of the most efficient overall performance per unit of input.
Fig. 1 MIes van der Rohe
1.2.3 Pier Luigi Nervi Italian architect and engineer who did great innovative research in applications of reinforced concrete structures, especially working with thin shell structures. He stressed that intuition should be used as much as mathematics in design. Like Heinz Isler, Felix Candela and Eduardo Torroja, Nervi looked towards Gaudi’s funicular models.
Fig. 2 Buckminster Fuller
1.2.4 Frei Otto Frei Otto is the leading authority on lightweight tensile and membrane structures and has pioneered the advances in structural mathematics and civil engineering. His work on optimization of structures and formfinding by looking at nature, is still a great inspirational source in the parametric environment. ** Fig. 3 Pier Luigi Nervi
Common to them all is that they are working with the beauty within knowledge of science. This enables them to work and design across multiple fields and thereby create interesting and optimised structures. However, they did not have the same access to digital tools and emerging technologies as we have today, therefor it is interesting to look at future developments of their ideas and knowledge. * http://bfi.org/aboutbucky ** Finding From  Frei Otto (1996)
Fig. 4 Frei Otto
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introduction 1.3 Parametric design 1.3.1 What is parametric design? When googling the word “parametric design”, it is one of the few searches where the first hit is not a wikipedia explanation. This is probably because their is no precise definition, rather there is a lot of related terms; generative design, algorithmic design, nodebased design, logical modelling, programmatic modelling etc. But these notions are not very good and a bit latent  basically it is a far more sophisticated way of modelling digitallly. Many times when modelling a concept, often certain operations are monotone and repetitive, operations that can be considered as algorithms. A good example of such is the “array” tool in most CAD software*  a way to repeatedly move and copy elements. The physical aid of “array” would be the Linex line spacer (Fig. 5). Instead of doing it manually and use time on drawing, erasing and redoing, we can use the abilities of computers to work with algorithms.
Fig. 5 space ruler.png
Imagine that you are drawing the facade of a skyscraper. Each floor has 8 windows, divided evenly along the length of the facade. It is simple, but after drawing it in the morning, you find out that you need bigger windows, then you find out that you need more windows, then the facade size is decreased, then it is rotated, then skewed  with maybe a thousand windows, you are going to use a lot of time. Algorithms will make the computer do your calculations and draw your geometry. Making the computers do your calculations and run algorithms has been around for a long time. Describing rules for geometry started with Euclid approximately 300bc. Far back the first calculator, the abacus, is believed to have been used. The age of computation started when William Shockley invents the transistor in 1947. This leads the way to circuit boards, electronic calculators and computers.
*
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CAD  Computer aided design.
2008
1982
77
78
19
19
196
3 300 BC: Euclid  Elements.
1947
500 BC: Ancient greek Abacus. 1947: William Shockley invents the Transistor. 1963:
Ivan Sutherland writes SketchPad. Graphics.
1977:
Dassault wirtes Catia. 3D drawing
1978:
Hewlett Packard  First PCâ€™s.
1982:
Autocad first release.
2008: David Rutten invents Grasshopper3d.
C B 0
30
0B
C
50
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introduction The concepts of the abacus and the knowledge of Euclid was combined by Ivan Sutherland in 1963, when he wrote the first CADapplication SketchPad, where the designer interacted with the computer graphically by using a light pen to draw on the monitor. SketchPad even included a solver that allowed assigning constraints, such as lines being individually perpendicular**. Ivan Sutherland described himself as being a visual thinker, therefore he had his interest in computer graphics, saying: â€œ...if I can picture possible solutions, I have a much better change of finding the right one.â€?***
Fig. 6 SketchPad Demo (1963)
In the late 1970s simpler operating systems and the release of desktop computers encouraged engineers to experiment with programming and became the start of workstation computing. The development in CAD grew fast during the 1980s, during which the software Pro/Engineer (1987) by PTC and more famously  AutoCAD (1982) by Autodesk was released. In 1992 the McNeel group integrated their NURBs geometry library in AutoCAD and in 1994 McNeel released the first beta version of Rhinoceros3D (Rhino). Pro/Engineer (1987) was the first software to fully implement the concepts of SketchPad (1963), where constraints and solvers created the basis for parametric design. FInally in 2008 David Rutten invents Grasshopper for Rhino Programming and finite element analysis has been used for a while by engineers, but the visual programming interface of the plugin Grasshopper for Rhino has taken programming to a familiar place and this has created a common playground for architects and engineers. In the following, the use of Grasshopper will be explained.
** SketchPad on youtube: http://www.youtube.com/watch?v=mOZqRJzE8xg *** Ivan Sutherland. http://historycomputer.com/ModernComputer/Software/Sketchpad.html
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“Hello John. We are going to show you a man actually talking to his computer.”  Prof. Steven Coons, MIT (TV Show 1963). Fig. 7 Youtube  Skethcpad demo
“..if I can picture possible solutions, I have a much better change of finding the right one.”  Ivan SUtherland
Fig. 8 Ivan Sutherland
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DATA
ACTION
Fig. 10 Step 1. Pts + line
Fig. 9 Points + line comp.
1.3.2 Rhino + Grasshopper Grasshopper was written by David Rutten in 2008 for McNeel. David studied architecture at TUDelft, where he got tired of the lack of scientific approach in design. All designs where based on emotional or philosophical considerations. He wanted to see numbers and proof that some solutions where better than others. Similarly to the likes of Frei Otto and Nervi, he implements technical knowledge as well as emotions, philosophy in the evaluation of design using Grasshopper. There are many different descriptions of Grasshopper; a visual algorithmic interface, visual programming, Visual Basics without the Basics and many more. Briefly how it works : There are two basic elements  data and actions. Every step of connecting components is like lines in a source code, except there is no code. All boxes can be considered to represent small pieces of code. As with coding, all commands take inputs and creates output through the requested action. These Grasshopper scripts will be referred to as definitions throughout this report. Fig. 11 Grasshopper Infacade. Rhino Viewport
This method is very handy in speeding up the drawing process and having ultimate precision, but it is the access to all data, that makes it really interesting. Especially for engineers, treating the data mathematically has a great potential. Similarly to the visual approach of Ivan Sutherlands (cf. p. 20), Grasshopper enables realtime visualization of solutions and consequences and it thereby becomes a great media within a competition team.
1.3.3 Parametric design in action. So apart from drawing spheres on a line, how is parametric design an added value to an architectural company like Henning Larsen Architects? The thesis statement is that parametric design will speed up the modelling processes. It might take longer to set up a good script, than doing the first drawing. From
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ACTION
DATA
ACTION
DATA Fig. 12 Step 2. Divide line
Fig. 13 Step 3. Draw spheres
here on however, the script will create changes within seconds. On top of this, the open platform, that Rhino3d and Grasshopper is build upon, enables linking the model with analysis tools for optimization of e.g energy use and structural efficiency.
1.3.4 Software confusion Because this thesis is based on a modelling software, Rhino, the Grasshopper plugin and plugins and extensions for Grasshopper, it is very easy to be confused as a reader. Therefor explanation icons will be presented at the introduction of each chapter. The plugins for Grasshopper are really important as they add different extended abilities of the scripting interface. All extensions will be explained when used, but here in bulletform: Rhinoceros:
3D modelling software. Basis for modelling. The builder.
Grasshopper3D (GH):
Visual scripting interface plugin. Tells the Rhino what to do.
Python:
Scripting language plugin for GH. Intelligent helper for the Grasshopper. (Mentioned Python scripts are coded by the author)
Kangaroo:
Physics engine plugin for GH. Simulates physical behaviour for the Grasshopper.
Galapagos:
Evolutionary solver plugin for GH. Goes through solutions for the Grasshopper.
[DIVA]
DIVA:
Daylighting and energy modeling plugin. The Grasshopper’s energy advisor
[Karamba3d]
Karamba3D:
Finite Element program fully embedded in GH. The Grasshopper’s structural advisor.
In the following chapter the principles of parametric design will be tried out and explained by implementing these methods on an, at the time being, ongoing competition project together with Henning Larsen Architects. The case is meant to shed light on the possibilities of interactive analysis and optimization of a facade, using parametric design.
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CASE [0]
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[European Patent Office  New main office] [Hague]
Rhino
Grasshopper
Python
Galapagos
DIVA
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2  Case [0]: EPO 2.1 European Patent Office  Introduction 2.1.1 About the case The European patent office in Rijswijk  the Hague, is building a new main office. The competition team is Henning Larsen Architects and Arup. At the time of the work carried on this case, the project is in its 2. phase and a new and more innovative approach is asked for. The European patent office stands for innovation and new inventions as well as looking backwards in history. Therefore they want this to be noticeable in their new main office. The people working at the European patent office are working very individually and needs to be very focused on their assignments, as well as doing very thorough research. Therefore a proposal of individual cell offices is chosen. These offices are where the employees can focus in quiet. Then they can meet up with their colleagues in open interactive common spaces. As these cells take up the majority of the collected office area and of the facade, these are of main interest in the energy optimization. Some of the important keynotes for the project are:
 Maximum individual user comfort.
High individual control of comfort.
High degree of concentrated work with great views.
 Optimal functionality.
Flexibility in the use of spaces.
Quiet workspace vs. interactive areas.
Strong central interactive connection areas between departments.
 Optimal design development and implementation.
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Future proof design solutions.
Fig. 1 New main office  Birdview. Henning Larsen Architects ©
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Fig. 3 Rhino Viewport
2.2 Parametric facade design. 2.2.1 Math surface An idea of individual working cells within a bigger organism was the driving force for the proposals. The important aspects of the facade is to be energy efficient in terms of solar shading, natural daylight, glare, air quality, heating x(u,v) = u and cooling for the individual cell workspaces.
y(u,v) = v To introduce the method of parametric design, the following example uses z(u,v) = u3 3*u*v2 an algebraic surface as an attractor for the solar shading. In this case the idea was that the variation of the surface could enable selfshading and an interesting expression. Here the algebraic surface called a “monkeysaddlesurface” is chosen. This has the algebraic equation:
z=x(x2 3y2 ) To create shading, relating to the saddle surface, a pointgrid is set up on the surface for each cell office. The x and yvalues of each point in the planar grid is extracted. From these the corresponding zvalue is calculated. In this case the operation is done in the xzplane, meaning it is the yvalue that is calculated. The rectangular edge of each cell on the planar surface is then extruded towards the algebraic surface.
2.2.2 Variations
Fig. 4 Sketches
Parameters in this script are variables multiplied with the x and zvalues, the interval of the representation of the surface and a damping coefficient. The sliders in the script represents these variations and every change can be seen in the Rhino viewport  see Fig. 3. For each solution an evaluation can be carried out. To make an example, here the material usage is extracted. In an Excel spreadsheet, a specific solution to an equation can be found by use of the goal seek utility. A similar tool is the extension Galapagos Evolutionary Solver for Grasshopper.
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Fig. 6 Render
The script is set up to input the variables of the surface as input to the solver . These inputs are called genomes. Then a â€œgood enoughâ€? value or fitnessvalue is defined in order for the solver to be able to find a solution. The solver basically choses a combination of genomes within a certain range and determines if this combination is good or bad. There is a lot of theory behind the evolutionary problem solving and a lot more to explain about Galapagos, but it will not be explained in detail here. For more info see: http://www.grasshopper3d.com/group/galapagos The important thing to know, is that Galapagos enables automation of the evaluation and optimization process. In this example the surface variables are the genomes, and the goal is to fulfil the fitness function best possibly, within a chosen interval of the variables. Next step is to define a good fitness value defining the actual goal. In this case the goal is the minimum amount of material and the minimum of solar gain. A numerical value of minimum solar gain is simplified to being the minimum area of solar radiation for optimization. Using this method on the design chosen by the architects, is shown in the following.
Fig. 5 Grasshopper definition.
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Case [0]: EPO 2.3 Optimization 2.3.1 What is optimization? Optimization is the selection of the best solution. Selection with regard to some criteria from some set of available alternatives. In the case of parametric design it is all about setting up the script so that it matches the idea and allows for the necessary variations. Especially for geometry, this means setting up a logic for interaction and interconnection of points, lines and surfaces. The chosen design for the shading is a irregular cassette (Fig. 8). So here the exercise is to look at best variations of the corners in terms of maximum shading and material consumption. It is clear that the best solution for minimum material usage is the smallest cassette possible. At the same time one of the most efficient solutions to minimum solar gain is obtained by a maximum amount of shading or cassette size. The graph below shows the obvious  as the size of the casette increases, material usage increase and solar gain decrease.
Fig. 7 Materials / solargain relationship
However there are solutions with different amounts materials, that gives a similar result in solar gain. So a defined fitness value is needed to use the Galapagos solver and this is possibly the most important part of using an evolutionary solver.
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Fig. 8 Cassette principle
2.3.2â€‚â€‚Fitness functions Darwinâ€™s Theory of Evolution states survival of the fittest. But it can be very difficult to say what it means to be fit. In evolutionary computation, however, fitness is a much easier concept. It can be what ever it is wanted to be, because we are trying to solve a specific problem. Therefor we know what it means to be fit. The designer needs to figure out which parameters and goals are the most important ones. Lets try and set up the fitness function in this case. The material consumption is named A and the solar radiation B. The distance from the outer corners to the corresponding inner points is set A= matCons. to an interval of {1 > 1000}mm. The interval starts at 1 for computational reasons. Firstly the best and A= worst possible fitness values for A and B is matCons. B= radArea set up. This is calculated by using the script and gives: B= matCons. radArea A= Best fitness: {A=4, B=86}
Best fitness: {A=4, B=86} B= radArea {A=4202, B=4667} Worst fitness: In table form:
{A=4202, B=4667} Worst fitness: A= matCons. fitness: {A=4, B=86} A {min=4; Best max=4202; range=4198; target=4}
A {min=4; max=4202; range=4198; target=4} B= radArea {A=4202, B=4667} Worst fitness: B {min=86; max=4667; range=4581; target=86} max=4667; range=4581; target=86} BA{min=86; fitness: {A=4, {min=4; Best max=4202; range=4198; f=A B B=86}target=4}
f=A B parameters {A=4202, B=4667} Worst fitness: range=4581; target=86 B {min=86; The fitness function: formax=4667; minimizing both is: } / 4198)  ((B86) / 4581) f=((A4) / 4198) B ((B86) / 4581) f=((A4) A {min=4; max=4202; range=4198; target=4} f=A
The above function justmax=4667; states that we want/ 4581) to minimize} A and B. Now it range=4581; target=86 B {min=86; / 4198)  ((B86) f=((A4) needs to be normalized, meaning only having values from 0 to 1, and to beweighted considering the range.*f=A ThenB this gives:
f=((A4) / 4198)  ((B86) / 4581) The value of this fitness function is for an extended version of the cassette shown here (see Fig. 10 on page 33), by adding an extra variable point in the centre. * http://www.grasshopper3d.com/forum/topics/galapagosmultiplefitness
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Case [0]: EPO 2.4 Scripting 2.4.1 Grasshopper definition The definition is build up in two parts. The calculation of the sun position (cyan) and the generation of geometry (magenta). The sun calculation part, calculates the sun position at a specific time at a specific location and is based on a slightly refined version of a definition by Ted Ngai Jan and his visual basic script (VB). This script is build upon the solar position algorithm by NOAA*. The second part is where the shading geometry is created and the radiated area is calculated. The engine of this part is the Python script** which turns the facade in to an intelligent system. In this system each cell knows about its facade mesh, normal vector, inner and outer vertices, geometry of the fins and it runs the radiation calculation internally. The output of this script is the fin geometry and the radiated area as line geometry, for faster computation speed. This can then be visualized as solid geometry and the area data is used for optimization using the fitness function mentioned in 2.3.2. The variable distances (grey) for the outer vertices become the genomes for the Galapagos Solver and the fitness function is used for the to look for the best combination. The definition can of course be used without Galapagos. Instead of letting the solver go through all combinations, one can set up different extremes and in between values. These can be evaluated in terms of aesthetics and performance to give an impression of the direction to take. This more “manual” evaluation has the advantage of control and speed, but one might very easily miss the most optimal combinations. *
National Oceanic and Atmospheric Administration  U.S. Government
** See appendix A
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Fig. 9 Grasshopper definition.
12:00
north
sunrise
sunset
Fig. 10 Rhino viewport.
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Case [0]: EPO 2.5 Parametric studies 2.5.1 Script calculation  example This way of systematic modelling with an intelligent system behind, allows the designer to evaluate aesthetics and performance at the same time and get an idea of in which direction to move forwards. In this case the performance output is material consumption, radiated area and the relation between the two.
s u n 5 ,6 3 m 2 m a t e ri a l 3 141 m 2 r e l. 0 ,9 8 Fig. 11 Evaluation step  example.
Studies like these are a good way of gaining common ground for collaboration between architects, engineers and other consultants. One the right hand side is an example of an evaluation schedule for discussion. Having numbers on the performance, while studying design solutions, enables designers and engineers to decide whether something is better than something else  both in terms of aesthetics and performance. From here they can move on together. A similar manual method is common practice at Henning Larsen Architects. In such a practice, a model is normally build in a modelling program to visualize and it is then transferred or rebuild in an analysis program. The difference is the speed and precision that comes with the parametric modelling. The following will show how linking software enhances this procedure.
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EUROPEAN PATENT OFFICE Parametric optimization of facade structure
Summer solstice, 21st June
Equinox, 31st March
10 AM
10 AM
s u n 5 ,2 9 m 2
sun 10 m2
m a te r i a l 2 8 0 0 m2
material 1,2 m2
re l. 1 ,0 4 1
rel. 0, 06
12 AM
12 AM
s u n 5 ,5 7 m 2
sun 0.66 m2
m a te r i a l 3 0 70 m2
material 16, 6 m2
re l. 1 ,0 2 5
rel. 13,1
14 AM
Fig. 12 Example of schedule
14 AM
sun 5 m2
s u n 5 ,6 3 m 2 m a t e r i a l 3 1 41 m2
material 10 m2
r e l. 0 ,9 8
rel. 1
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Case [0]: EPO 2.5.2â€‚â€‚DIVA Analysis As mentioned in part 1.3.3, the open platform of Rhino3d and Grasshopper, as well as the increasing interest in the software, seems to make it an interesting place for software developers. There is many different kinds of plugins that will extend the abilities of Grasshopper. Some plugins work with an internal calculation engine, meaning that everything is done within Grasshopper and Rhino. Other plugins acts as a translator between external software. These plugins work like adding specialist tools to the toolbox. DIVAforRhino is a daylightning and energy modelling plugin, initially for Rhino. DIVAforGrasshopper plugin extends these tools to Grasshopper. Instead of having the script doing simple geometric calculations of the radiation area, this software extension allows for much more advanced analysis. DIVA links validated simulation engines like Radiance, Daysim and Energy+.* In this case it was important to find a shading configuration that allowed enough natural daylight into the room, while minimizing the solar radiation on the glass, in order to minimize overheating. That means that an algorithm that optimise the geometry, in relation to both the daylight factor and the solar radiation, is needed. To do this a single cell office is modelled in Rhino and the parametric shading is created in Grasshopper through the same custom Python script as before. This time the calculation code is taken out of the script, so that it only creates the geometry.
Fig. 13 DIVA data output.
The dynamic shading geometry is then analysed in e.g. Radiance, through DIVA. DIVA uses the analysis engine of, but does not open, these programs and gives fast feedback. Depending on the numerical goal, the fitness function is defined and using the Galapagos solver, the best solutions can be found.
* See www.diva4rhino.com for documentation
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Fig. 14 GH Def. DIVA + Geometry script
Fig. 15 VIsual representation of daylight factor and radiation on window. See animation on www.vimeo.com/rasmusholst
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Case [0]: EPO 2.6â€‚â€‚Discussion As a part of the research area of this thesis, a discussion of how far to take the written code is interesting. Here the Python script for the geometry was setup in collaboration with the competition architects. This script divides the facade into the individual cell offices and creates a parametric geometry of the solar shading as explained previously. Then two different approaches was used: (1) Calculation of radiated area and material usage inside the script (2) Link to external analysis software through DIVA The first approach (1) has the advantages of simplicity and interaction speed. This means that each iteration is executed fast and therefor can be executed in higher numbers  e.g. on the whole facade at once. This can be an advantage as the aesthetics of the facade can be evaluated while going through the optimization. But at the other hand this simplicity means that only the direct sun on the facade is taken into account. The link with external software (Radicance, Energy+ etc.) makes it possible to see all the effects of each iteration step  this being solar irradiation, illuminance, daylight factor, thermal performance etc., but these more detailed processes takes longer to simulate. Here approximate 10 seconds pr. change It is clear that in order to be able have full control in approach (2), the designer needs to know about the software that DIVA links to. It is necessary to know the setup of the models in order to understand the output that comes back into Grasshopper.
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EPO

SCRIPTING
Fig. 16 Scripting only. Rhino + Grasshopper screenshot
EPO

SCRIPTING + DIVA
Fig. 17 Script linked with DIVA engine. Rhino + Grasshopper screenshot
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Case [0]: EPO 2.7 Perspective The geometry here was relatively simple and the focus was on energy concerns. It is not hard to imagine doing the modelling and the analysis manually. In cases of more complex geometry, the strength of intelligent systems and parametric design would come to show. The idea of individual cells within a working organism was the driving force for these other proposals:
2.6.1 Origami cell Origami is folding of a single element into smaller individual compartments. This principle has the properties of using planar and relatively simple elements as well as having an interesting spatial expression. The idea is to somehow figure out and list the building DNA and map the needs of each part by the folding origami structure. Meaning that needs for solitary confinement, shading, views, acoustics and so on are accounted for by the size, direction, angles etc. in each origami compartment. (Fig. 18)
2.6.2 Voronoi cell When thinking about cells from a mathematical point of view, the Voronoi diagram is the first thing that comes to mind. The Voronoi principle can be used in two or three dimensions, and can be applied in one or more layers. As for the origami principle, the distribution of Voronoi cells accommodate for the needs of each individual office. Some cells might spread across offices because of similar needs or external impacts. Especially when adding a third dimension and thereby different angles out of the plane of the facades, this principle becomes really interesting in terms of structural and shading optimization. (Fig. 19)
2.6.3 Karamba3d  Finite Element Analysis. As the geometry becomes more complex, alternative analysis tools are necessary. DIVA proved handy for energy analysis. When it comes to structural analysis, plugin Karamba3d, provides the same kind of flow as DIVA. Here tested on a parametric freeform truss. (Fig. 20)
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CELL 
ORIGAMI FOLDING
Fig. 18 Initial sketches and references.
CELL 
VORONOI_math
Fig. 19 Initial sketches and references.
FREEFORM 
TRUSS_analysis
Fig. 20 GH definition + Karamba. Freefom truss example (RH).
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Case [0]: EPO 2.8â€‚â€‚Part conclusion Using the parametric model and analysis tools, enables teams to optimise the shading design dynamically. Together with the Galapagos solver the process can be automated. This means that the designer or engineer can work on other things, while the computer solves the algorithms set up. So on top of saving a lot of time on drawing, redrawing, evaluating and starting over, it can be done automatically and perfectly precise. This immense increase in speed allows going through large number of iterations  which is the strength of the computer  and see some results that might else have been missed. Case [0] serves as an introduction and an example of parametric design in architecture projects.
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Fig. 21 New main office  Collaborative zone. Henning Larsen Architects ÂŠ
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INITIAL RESEARCH
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[Construction principles] + [Overview] + [Inspiration]
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3  Initial Research 3.1 Simple advanced structures 3.1.1 Introduction The strength of parametric modelling and scripting is the complete control over all aspects in the modelling. All inputs, outputs and constraints are accessible. This control can be used to create complex systems, where constraints might be size of elements, connections, material strengths and so on. These are very rational things. Things that are meant to bridge the gap between drawing and building. In the book “Translations from Drawing to Building”*  Robin Evans talks about great inventions happening in this very gap. When a simple logic is combined with other simple logics, is twisted slightly or something similar, unexpected consequences might arise. This makes it possible to narrow and explore the cap, at the same time. Instead of defining a method to execute a project, the method ends up defining the project. This is a way of manipulating the tectonic method to define the design. Set up correctly, construction information can be generated directly from the design information. Rather than having to figure out how to manufacture some complex shapes, it is already held within the design. Seemingly chaos can be nothing more than simple logics put into system. It is structures like these that will be explored in this chapter. It has been tried to define categories and to explain these in short, even though some principles will consist of overlapping tectonics, such that they might be put into more than one category. At the end of this chapter, one of these principles will be chosen for further investigation and exploration.
*
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Translations from Drawing to Building and Other Essays  Robin Evans. (1997)
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system
Sectioning
logic
Sectioning
Any 3D geometry can be constructed from planar elements by slicing through the geometry in two or more directions.
connection examples
model
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Normally the geometry is sliced in two orthogonal directions and a grillage structure is created. This means that the elements edges will have to be cut along the curvature of the geometry. A series of profiles are intersected and connected. This principle is known from air plane building and shipbuilding. The profiles acts as the structural ribs and can be clad after assembly.
references 2
1
3
1 2 3 4
Fig. 1 Indigo Deli. Sameep Padora & associates. Fig. 2 Serpentine Pavilion 2005. Siza, Moura & Balmond. Fig. 3 Metropol Parasol, Sevilla. J端rgen MayerHermann. Fig. 4 Olympic stadion Beijing. Herzog & de Meuron.
4
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system
Folding
Folding
connection examples
logic Folding turns a flat surface into a 3dimensional structure. When folds are introduced into otherwise planar elements, those elements gain stiffness and rigidity. They increase their span distance and can often be selfsupporting. The logic is that by introducing curves or lines on one or more planar sheets of material and rotating around these curves, a new 3dimensional structure and thereby new spaces are generated. Folding is the further consequence of bending. In materials that allow plastic behaviour, the folding can plastic deformation of the material. Folding concepts can be used together with those of tessellation. Meaning that the folds are rather cuts and connections, in materials too brittle, for complete folding. Gregory Epps and ROBOFOLD uses Grasshopper + plugins to generate and simulate curved folding behaviour. While noncurved folding is much simpler to simulate.
model
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references 1
2
3
1 2 3 4 5
4
Fig. 5 Richard Sweeney  Paper sculpture Fig. 6 Andrea Russo  Origami tessellation Fig. 7 Juergen Weiss  Barcelona Block Fig. 8 rvtr  Resonant chamber. Fig. 9 Ryuichi Ashizawa Architects  Folded Plate Hut
5
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system
Shells
Shells
connection examples
model
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logic A shell structure is a spatial structure that is economical in terms of materials if designed correctly. The curved shell surface geometry can be made out of one or more curved elements, in concrete for example, like the structures of Heinz Izler and Felix Candela. However the production of such elements are expensive and time consuming. The shell can also be divided into facets of planar elements. These facets can be modified to create certain properties in terms of light, structure, aesthetics etc. This division of the shell into planar elements without cap or overlap is also called tesselation. This tesselation can be divided into layers. Meaning that each division can be futher divisioned and modified in or out the original division plane.
references 1
2
1 2 3 4
Fig. 10 Roskilde dome  Tejlgaard + Almegaard Fig. 11 Roskilde dome  Tejlgaard + Almegaard Fig. 12 BOWOSS Bionic Pavilion  Saarland University Fig. 13 ICD/ITKE Research Pavilion 2011
3
4
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system
Bending
Bending Twisting
connection examples
logic Bending or flexure is movement out of one or more planes of a planar element. Bending occurs when applying external loads perpendicular to an element or in the longitudinal direction of the element. It also occurs from temperature and humidity differences. In engineering bending is referred to in 3 axis  bending of rods is 1axis, for beams it is 2axis and plates and shells it is 3axis. If in beams we have â€œbendingâ€? around the longitudinal axis, we refer to this as torsion or twisting. To utilize bending of slender elements, an external load is applied and the elements are held in the deformed state by fixation. Elements can either be fixed to a predefined form work or in systems where internal relations create the bending. Otherwise nonrigid elements gain stiffness and structural stability by this deformation through bending  similarly to the description from folding. This can occur in the element locally and therefor it become even more significant in networks. Gridshells are probably the most known structure where bending is utilized for an optimised structure.
model
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references
1
1 2 3 4 5 6
Fig. 14 Hermes Boutique  RDAI Fig. 15 Digital Weave  IwamotoScoot Fig. 19 Timber Fabric  IBOIS Fig. 16 Stripmodel test Fig. 17 Eclaireur Paris  Arne Quinze Fig. 18 ICD/ITKE research pavilion 2010
2
3 4
5
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system
Network
Network
connection examples
logic All structures work in some kind of network. Most of them are in practical systemized order. This order is of course to ease the drawing and building process. Such systems or networks are often orthogonal linear systems, as grids for example. The simplicity of these linear systems can be very attractive aesthetically. Nonetheless, chaotic and seemingly random structures has an intriguing appearance. Examples of such structures in nature are nests and beaver dams. The structural stability is created by interlocking of the elements. In architecture the interlocking is not restricted to the elements themselves, as the systems can be modelled with mechanical connections where ever. There are two main approaches: 1  Elements, connections, logics etc. define the shape more or less randomly. 2  Attraction of an element network towards a predefined surface defines the shape.
model
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Logic and algorithm needs to ensure structural stability. Rules might be: Each element needs to connect to at least two other elements, but not more than four. At least two of each connecting elements need to connect in a way that ensures triangles. All connecting elements need to be touching in the same plane, but not intersecting. Many more and other rules might be necessary. Here an iterative process using Galapagos might be useful.
references
1
“There Is No Chaos Only Structure”  Arne Quinze (2011).
2
3
4
1 2 3 4
Fig. 20 Uchronia  Arne Quinze Fig. 21 Roof installation  Arne Quinze Fig. 22 CityScape  Arne Quinze Fig. 23 Aggreation Anenom  Dave Vu and David Pigram
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system
Modules
Aggregates Modules
logic An aggregate is a collection of items that are gathered to form a total quantity. Most structures are made of a number of elements. In most planar steel structures, beam and column elements are used to create a skeleton. In concrete structures, often precast modules in practical size are used. Somewhat related to the network system, it is important that the internal relations between adjacent modules. These need to have a common side, edge or some something else to connect them. It is this relation, the shape of each aggregate and potentially the transformation of these, that creates the overall shape and properties of the structure. This is what we now from many structures in nature  the backbone for instance. Each aggregate can be a module with individual properties, other than the structural. This might be light and thermal properties. Similarly to a window in a facade, each aggregate can be more less dynamic and respond to surrounding impacts. Fractals are a similar phenomenon, known from both nature and maths. Like aggregate structures, fractals show a repeating, selfsimilar structure.
model
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references
1
2
3
1 2 3 4 5 6
4
5
6
6
Fig. 24 InOut Curtain  IvamotoScott Fig. 25 Paper Art  Richard Sweeney Fig. 26 Voussoir Cloud  IvamotoScott Fig. 27 Harpa Concert Hall  Henning Larsen Architects Fig. 28 Bent Wood Exoskeletons  .Joel Letkemann Fig. 29 Differentiated Wood Lattice Shell Huang + Park
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system
Gridshell
Gridshell
connection examples
logic Gridshells are the most known construction where the bending nature of wooden elements are utilized for an optimised structure. The gridshell structure derives its strength from its double curvature, in the same way that a fabric structure derives strength from double curvature, but is constructed of a grid or a lattice. Large span timber gridshells are commonly constructed by laying the laths on top of a sizeable temporary scaffolding in the wanted shape, which were removed in phases to let the laths settle into the desired curvature. A recent project example is the Savill Garden gridshell Another approach is initially laying out the main lath members flat in a regular square or rectangular lattice, and subsequently deforming this into the desired doubly curved form. This can be achieved by imagining to push the members up from the ground. Similar approach is used in the Mannheim Multihalle. Here Frei Otto used hanging models. Two different approaches are: 1. From wanted shape to grid and connections. 2. From planar grid to shape by moving of constraints.
model
Gridshells might be made out of any elastic material, but is mostly made out of wood. The material properties, joints, grid and constraints are defining factors for the shape. Members need to be slender enough to bend, so in some cases, the structure has to be constructed in multiple layers.
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references
1
1 2 3 4 5
Fig. 30 Centre Pompidou, Metz  Shigeru Ban + Arup Fig. 31 The Savill Gardens  Glen Howells Architects Fig. 32 Centre Pompidou, Metz  Shigeru Ban + Arup Fig. 33 Gridshell Digital Tectonics  Smart Geometry â€˜12 Fig. 34 Centre Pompidou, Metz  Shigeru Ban + Arup
2
3
4
5
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Research 3.1.2â€‚â€‚Part conclusion Many of the structural principles overlap in the catergorization. Many of these are closely related and combinations of them hold very interesting potentials. For instance combining bending + modules + network. Some of the referenced elements in the module part can be bend plate elements and worked in a more or less complicated network. Gridshells are mostly considered to construct a single three dimensional surface and sometimes offsets of this surface. A combination with principles from the more multidimensional network part, could prove to be an interesting kind of weave structure. Here elements can move in and out of the surface and start and end wherever, on or off the surface. The common thing for the presented structures are that they are different. Different from what is normally built and advanced in the looks of it, but pretty simple in the logic. Most of them are impossible or at least very time consuming to create without computation and scripting. Most examples are built as studycases, casual pavilions, installations or furniture. The transformation from these principles to building elements, wall, floors, roofs etc. are very challenging, but also very exciting. It is tempting to put the conventional thought to the back for a while and think of buildings as multidimensional structures  not necessarily divided into floors, walls and roofs, but as a coherent environment.
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Fig. 35 Tripudio Bestia  Matthias Pliessnig
Fig. 39 Lamella flock  CITA
Fig. 36 Allotropic System  Nicholas Bruscia
Fig. 40 Plasti(K) Pavilion  Marc Fornes THEVERYMANY
Fig. 37 Centre Pompidou, Metz  Shigeru Ban + Arup
Fig. 41 Dorian Pattern Facade  Khiem Nguyen
Fig. 38 Resoloom  Peter Vikar
Fig. 42 Migration  Matthias Pliessnig
3
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3.2â€‚â€‚Gridshells From the initial research, the gridshell structure is chosen for further investigation. The reason for choosing this principles is the direct utilization of bending in networks of slim regular elements. The thought is that when knowing how to construct more regular lattice gridshells, it is possible to combine this with some of the more random looking structures from the network part. This combination might end up somewhat like the pictures on the right hand side. The further investigation approach will look into parametric tools for form finding and the simuation of element behaviour. The first step is to setup an intelligent system that resembles the properties and geometry of real elements. Afterwards the elements will be put into networks, such as the lattice in the gridshell. Constraints in relation to connections and supports will be investigated. A really important part of setting up such a system, is the ability to evaluate in terms of curvature, stresses, moment distribution etc. and in the end be able to translate the model to building elements. The prospects are that such a working method will enable the construction and analysis of complex geometry surfaces for whole building envelopes, roof, facades interiors etc. At the same time the thesis is that by setting up this scripting system, one will be able to create optimised structures, using similar approach as mentioned in chapter "2.3 Optimization" on page 30.
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Fig. 43 Hello wood
page 65 Fig. 44 Eclaireur Paris  Arne Quinze
MODELLING
page 66
[Interactive physics]
Rhino
Grasshopper
Python
Kangaroo
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4  Modelling 4.1 Paper play 4.1.1 Physical investigation. Physical models are used for investigation in order to setup the needed script. By simple physical studies of shapes created by deformation, an idea of the behaviour is obtained. In the case of gridshells, it is interesting to look at what effect the initial grid has on the behaviour of the deformation and stability. Investigations include the shapes that is created from changing fixing points of the planar grid, the significance of connections and the deformation of individual elements. Another approach of the investigations is to lay a grid of strips on a already defined/built surface, fixing these at intersections on the surface and thereby finding the necessary grid configuration. The grid created by this method, is not necessarily planar. It is a way of constructing a predefined shape. This is particularly interesting for architectural projects in search of a certain shape.. When laying and interlocking strips on top of a surface, the relationship between element connections, material flexibility etc., creates 3dimensional structure. This interesting effect that will be investigated later on.
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Fig. 1 paperPLay  first physical test with regular paper. From grid to form.
Fig. 2 paperPLay  First test: From form to grid.
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Modelling 4.2 Theory 4.2.1 Compression members To create the deformations of the paper model, the members are subjected to external forces. The basic theory of this behaviour is that of compression members. A compression member  or a column  can be defined as a beam subjected to a compression force acting in the beam axis (local xdirection). Basically the theory can be divided into short columns and long columns. Short columns has the property that a column of the same cross section has the same load carrying capacity. If subjected to a bending moment, by e.g. eccentric loading, the stresses are found by Navier’s equation:
Normal stresses : σc =
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N A
M Moment stresses : σm =±N Normal stresses : σc = W A N M N M N Combined stresses : σt =  : σc = M σc = + Normal stresses A W: σ =±A A W Moment stresses m N W Normal stresses : σc = M A Long columns or slender columns areNcharacterized Moment stresses M: σm =± W byN being M forced to deCombinedtostresses : σt = large = + If a simple linear flect when subjected a sufficiently normalσforce. M(x)=N*u(x) A W: σ =± M c A W Moment stresses m M N, that N Mthe deflection elastic member is subjected to normalNforce Wσ gives Combined stresses : σt = 2c= + u(x). Moment equilibrium at any given point x is: Ad u(x) W A W M(x)=EI* N 2M N M Combined stresses : σt = dxσc = + M(x)=N*u(x) A W A W d2 u(x) 2 For a member of linear elasticM(x)=N*u(x) material =κ d with u(x)small deflections: dx2 M(x)=EI* dx2 2 M(x)=N*u(x) d M(x)u(x) M(x)=EI* 2 ⇒κ=d u(x) EI dx2 =κ d2 u(x) 2 dx M(x)=EI* Curvature κ can be defined as: d2 u(x)1 dx2 =κ κ= 2 M(x) dx 2 R ⇒κ=d u(x)EI =κ M(x)=κ*EI dx2 M(x) ⇒κ=1 EI κ= 1 M(x) M(x)= ⇒κ=R1 *EI κ=R EI M(x)=κ*EI R 1 σ = E*ε = yEκ κ= 1 M(x)=κ*EI M(x)= MR*EI σm =±R1 M(x)=κ*EI W*EI σ =M(x)= E*ε =RyEκ
(1)
(2)
(3)
(4)
N A N Normal stresses : σc = M N Moment stresses : σm =±A Normalstresses stresses: W:σσ=c =N Normal c N M AA Normal stresses : M:σσc =m =±N Moment stresses N N M Normal stresses Combined stresses : σN =  : σcA=NAW σc =MM+ Normal stresses : σMoment c =N t Astresses W: σ = : :σσ=± m =±A W Normal c N Normal stresses : Moment σcstresses = A Nstresses M m WNW M M A Normalstresses stresses c = M σc = + Combined Moment stresses :A σtM = : :σ: σmσ=± AW =± Moment stresses A W A W NN MM Moment stresses : stresses σm =±M : σ =NNm MM MW Combined σ= + t W Combined stresses : σ = =± Moment stresses : σ M N σMc =c AA+ WW Moment stresses : σM(x)=N*u(x) m =± N Mt AmA WW W Moment :stresses Combined stresses σ =W N :Mσm =± σc = +N M N M :tσtA= W Combined stresses  N M W σcA= W + Combined stresses : σt =N M σ2cu(x) =MW +M NA NA MW N d A WM(x)=N*u(x) Combined : σt = Nσ =MA+ W σc = N + M Combined stresses : σt =stresses M(x)=EI* A cdxWA2 W A W : σt = M(x)=N*u(x) Combined stresses σc = A + W M(x)=N*u(x) Ad2 u(x) W A W 2 M(x)=EI* M(x)=N*u(x) u(x) dx2 2 2 dM(x)=N*u(x) u(x) =κ u(x) Which then gives the relation between momentd dand curvature: M(x)=N*u(x) 2M(x)=EI* dx M(x)=EI* 2 dx 22 d u(x) 2 M(x)=N*u(x) M(x)=N*u(x) 2 dx d u(x) d u(x) M(x)=EI* M(x)=N*u(x) 2 =κ d2 u(x) M(x)=EI* M(x) dx 22 2 dx2 d2ddu(x) (5) u(x) dx M(x)=EI*d2 u(x) ⇒κ=u(x) 2 =κ EI2 u(x) M(x)=EI* =κ 2 2 d M(x)=EI* dx 2 2 2d u(x) dx dxdx M(x) dxM(x)=EI* d2 u(x) =κ 2 be found geometrically by: ⇒κ=dx 1 Local curvature of ad2spline geometry can also 2 u(x) =κ dx 2 2 EI M(x) κ= =κ d dx u(x)R⇒κ=M(x) d2 u(x) dx2 =κ d2 u(x)⇒κ==κ EI dx2 M(x) 1=κ dx2 M(x) EI ⇒κ=(6) 2 κ= dx REI M(x) M(x)=κ*EI ⇒κ=1 EI ⇒κ=M(x) 1 M(x)κ= EI ⇒κ= M(x) κ= ⇒κ= radius. 1 1 where R is the curvature EI RR M(x)=κ*EI EI M(x)= ⇒κ=κ= 1*EI 1 EI κ= RR M(x)=κ*EI Equations (5) and (6) means κ= 1 that bending 1R moment can be found by: M(x)=κ*EI 1 κ= *EI κ= R σM(x)=κ*EI =M(x)= E*ε =RR1yEκ R κ= 1 M(x)=κ*EI RM(x)=1 *EI M(x)=κ*EI or M(x)= 1= MyEκRR*EI (7) M(x)=κ*EI σM(x)= =M(x)=κ*EI E*ε 1 =± σ *EI m M(x)=κ*EI 1 *EI M(x)= RW Stresses can then beM(x)= found *EI by either: =E*ε E*ε= =yEκ yEκ σσ=1RM 1R *EI M(x)= 1 M(x)= *EIσ = E*ε σm =± = yEκ or by Navier (1): R σM(x)= = E*ε R=W*EI yEκ M R σm =±M σ = E*ε = yEκ σ m =± W =M yEκ (8) W σ = E*ε = yEκσ =σE*ε M =± m = yEκ M σ = E*ε σm =± W m =±M Euler At the same time the σcritical loadMWcan be found by putting (2) and (3) d2 u(x) =± σ m M σm =± W EI* +N*u(x)=0 W (9) dxσ2 m =± W d2 u(x) W EI* +N*u(x)=0 N2 together: dx2k2d=2du(x) u(x) EI* EI 2 +N*u(x)=0 +N*u(x)=0 2 EI* 2 d u(x) 2 Ndx dx 2 d u(x) EI* u(x) +N*u(x)=0 2 k = +N*u(x)=0 d2 u(x) EI*d2dx (10) 2 2 dx2 +kEI N ⇒ *u(x)=0 EI*d2 u(x) +N*u(x)=0 2 N d u(x) 2 2 2 k2k= = EI EI* d2dx +N*u(x)=0 u(x) EI* dx2 +N*u(x)=0 2 N EI equation: dx 2 d u(x) dx N and rewriting this as a second EI* order 2differential +N*u(x)=0 2 ⇒ dx2k2k=+k =2π2*u(x)=0 EI *EI 2 N dx d=2du(x) u(x) EI k =N ⇒N 2 el 2 N 2 +k *u(x)=0 2 EI 2 ⇒⇒ = k ls2 +k2 *u(x)=0 (11) N k = d u(x) 2 dx 2 EI 2 dx EI⇒ d u(x) 2π *EI = k +k EI *u(x)=0 2 2 el = +k ⇒N d2 u(x) 2 ⇒ dx *u(x)=0 2 2 dx2 2 2 *EI Solving this gives: ⇒d2 u(x) +k *u(x)=0 d u(x) 2 ls ππ *EI 2 ⇒N = dx 2⇒ 2 +k ⇒N = 2l2 2 *u(x)=0 d dx u(x) ⇒ +k *u(x)=0 elel 2 π *EI 2 2 2 dx ⇒ ⇒N 2 = +kπ*u(x)=0 *EI ls s (12) el 2 π2 *EI dx ⇒N = l el 2 s 2 ⇒Nel = π2 *EI π l*EI s 2 ⇒Nel = l2s ⇒Nel = π2l*EI 2 s ⇒Ndetermine ls to el = The critical load might be used what force is needed for any l2s deformation. This theory will be used later to find stresses, forces and moments in the bend members. Normal stresses : σc =
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Modelling 4.3 Physical modelling 4.3.1 Particle system The digital modelling will be carried out using addon Kangaroo. It allows the user to interact in real time with the model as the simulation is running. The Kangaroo engine is based on relaxation of a particle system. Particles are points or objects in a system that can have mass, position and velocity. They respond to forces, but have no spatial extent. Despite their simplicity, these particle systems can be set up to exhibit a wide range of physical behaviours. A system built by connecting particles with simple damped springs can resemble a wide variety of nonrigid structures. “Macroscopic properties of materials such as the behaviour in bending, shear and torsion can actually be seen as emergent* on a molecular level from simple interaction between pairs of particles” (Kangaroo Manual) This means that it is the interaction and behaviour of the millions of particles/molecules that gives the overall bending behaviour that we know from a plate for example. It is clear that the amount of particles in these computational simulations are vastly smaller than in the real world. With the right knowledge, real world behaviour can be translated into the distribution of points and springs, as well the internal relations between particleparticle and particlesprings. The results can be used for comparison between the two. This approach can give a good approximation of the real world physical behaviour. Even though this particle system has its limitations, it has the advantage of being fairly easy to understand and control. The theory behind the system just described, will be explained in the following stepbystep.
*
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Developing / under development.
4.3.2â€‚â€‚Dynamic relaxation The aim of dynamic relaxation is to find a geometry where all forces in the system is in equilibrium. As mentioned in the introduction chapter, the likes of Gaudi and Frei Otto did the dynamic relaxation using hanging chains and weights, soap films etc.
p0
p1 s0
p2 s1
p3
p4
s2
s3
p4
So by discretizing the continuum, meaning dividing an element into smaller pieces, the particle system is setup. In the example on Fig. 3 a steel rod is simulated.
s4
p5
rest length
p3
When a force is introduced to the system, an iterative process follows by simulating a pseudodynamic process in time, with each iteration based on an update of the geometry. The basis is to trace the motion of each node of a structure, stepbystep for small time increments, âˆ†t. Due to artificial damping, the structure can come to rest in static equilibrium.
p5
p4
v5
p4
v5
v5
p4
v3
v3
v3
The first influence in the system is the desire of each part element  spring [si]  to reach its rest lenght. In most materials and in this example the rest lenght of [si] is the initial length as the material does not want to shrink. When particle [p5] is moved to towards left, the spring stiffness for [s4] introduces a force vector [v5] acting on particle [p4]. At the same time the stiffness of the spring [s3] between [p4] and [p3] introduces an opposite reaction force vector [v3] to [p4]. This then happens throughout the particle system  [p4] pushes on [p3], [p3] > [p2], [p2] > [p1] etc.
Fig. 3 1D dynamic relaxation
To reach equilibrium each particle node moves, so that all vector forces acting on it is equal. If [p0] is fixed particles equally divide. Fig. 4 Spring component
page 73
Modelling p0
p1 s0
p2 s1
p3
p4
s2
p5
s3
s4
p5
As soon as an outofplane force is introduced to the system, the particles move in different directions corresponding to the direction of the force vectors introduced. In this example; when moving [p5], the stiffness of [s4] introduces a force on [p4]. As [p4] moves in direction of vector [v5], it moves closer to [p3] and therefor the spring force of [s3] acts on [p4] as [v3].
p4 th
g
t es
len
r
p3
p4 p4
To find equilibrium in [p4], it has to move in the direction of the resultant of [v3] and [v5].
v5 v3
p3 p4
Previously the system was only introduced to forces acting in line, lets call it planar force, this means that there are no outofplane forces in the system. This needs to be introduced in order to cause bending of the rod. This is one of the big differences from the real world to the idealized system.
r35 v5
Newtonâ€™s third law tells us that every action has an equal reaction:
Faction = Freaction
v3
r35
Fig. 5 2D dynamic relaxation
p4
This mean that in the same way that forces acted on each other throughout the system in the previous example, so does it in this example. Lets say [p5] is fixed (cannot move) in the new position, then [v3] acts on [p3] as well as on [p4]. So once again equilibrium is found through the iterative process based on the update of the geometry. When particles move out of line, in one or two dimensions, the bending resistance is what tries to keep the adjacent particles in line. Up till now, this did not exist in the system, but will be added now.
page 74
bending of a curve principle
algorithm:  draw lines between points  setup up the three lists in correct order  set list of bending resistance result = intellingent system knowing the internal bending relation
p0
The logic of the bending resistance of the particles is that three adjacent points wants to be in a straight line. This means that a force vector is added to the particles. The direction of this force vector is towards the straight line betweenlist0 the two neighbouring list1 list2 particles. See Fig. 6.
p1
p2
p3
p4
algorithm:
bending of a curve list0 principle
 draw lines between p  setup up the three l  set list of bending r
list1 list2
The algorithm for the script is as follows:
 Draw the lines between points [pi] > [pi+1].
 Setup the three lists in the correct order.
 Set the list of bending resistances.
bending of a curve principle
The result is a system knowing the internal bending relations. As well as for the rest length, one also defines the rest angle of the springs. This angle define the desired angle between two adjacent springs  in the case of a straight element, this angle is of course 0.
p5
result = intellingent sy
algorithm:  draw lines between points  setup up the three lists in correct order  set list of bending resistance result = intellingent system knowing the internal bending relation
p0
list0 list1 list2
list0 list1 list2
p1
p2
p
list0
list0 list1 list2
It will be shown later, how other rest angles can be useful. The BendStrength input (Fig. 6) is a numerical value that defines the weight of the desire to resist bending, in comparison to other forces acting on a particle.
p3
Fig. 6 Bending resistance principle.
Further proof for this numerical method can be found in: Tensigrity spline beam and grid shell structures [1].
Fig. 7 Particle movement.
page 75
Modelling
1 2 3 4 5 crv
4.4â€‚â€‚Definition breakdown [Spline] Reference curve is divided into 20 parts and 21 points.
Three particle lists are set up: List0 = [p0] > [pn2],
List1 = [p1] > [pn1],
Connected as inputs to set the bending strength.
Lines are drawn between particles: Line from [pi] to [pi+1]. Converted into springs with their initial length as spring rest length. Referenced control points. Controlled through Rhino interface.
Kangaroo settings. Global settings for the simulation.
3
1
page 76
4 6
2 Fig. 8 GH definition
List2 = [p2] > [pn].
5
pts
Kangaroo main engine. Inputs: Force objects  Springs, constraints (e.g. bending resistance), attraction, re pulsion etc. Anchor points  For fixed points and thereby for controlling objects as they can be moved in Rhino for interaction. Settings 
Input from settings component.
Geometry 
Geometry to transform: Lines, points, meshes etc.
6
Simulation reset  Boolean toogle: False = run, True = reset. A timer component is attached to recompute the solution at a certain time interval. Here a solution is found every 1 millisecond and the geometry is updated. The output is either all the particles or the transformed geometry. This can then be used for further work.
pt
crv
7
pt
7 z y x
Fig. 9 Rhino viewport illustration
page 77
z y x
Modelling 4.5 Plate modelling. 4.5.1 Oneaxial bending
Fig. 10 Plate axis
In the previous chapter, behaviour of a rather linear element, such as a rod was modelled. In this case it is investigated how to best model a plate. The rod resembles isotropic material properties, meaning e.g. the same bending strength in all directions. The plate can have anisotropic properties, meaning different properties for each direction. Normally we define bending as rotation around the yaxis or zaxis and torsion as rotation around the xaxis. So it is required to find a configuration of springs and rotation constraints that resemble the properties of a plate.
Fig. 11 Bending resistance principle
First of all, bending resistance is set for rotation around the x and yaxis. A comparison of the particle system and real world behaviour was carried out  see Fig. 12 on opposite site. A model of a planar element 30 mm x 200 mm is modelled using the particle system. Comparison is made with a same size, 0,5 mm plastic sheet. Testing is carried out by moving two endpoints in a straight line from right to left. Firstly the points are moved 40 mm. The particle model shows a deflection of 5,29 cm  the real world model had a deflection measured to roughly 5,4 cm. Next the endpoints are moved 60 mm more. The particle model shows a deflection of 7,45 cm and the real world model is measured to 7,6 cm. These pretty simple comparisons shows good results. The difference is considered to be due to inaccuracies in the real world model, such as the pinned ends. The elastic bending behaviour does not depend on the numerical values compared to the actual strength of the material. The strength will determine the reactions or forces required for the bending to happen. The values describes the weighting or the size of the vector forces from bending or stiffness  in other words, whether the element will change length or bend first.
page 78
page 79 Fig. 12 Validation  first tests.
Modelling 4.5.2â€‚â€‚Multiaxial bending Fig. 13 Stiffness principle
Fig. 14 Twisting in square grid.
So as long as the imposed forces act only in either the x or yaxis the model is realistic. But the square grid does not resist shear forces very well. If a force is introduced in the ydirection the square will deform, as nothing keeps the corners perpendicular. This deformation also happens when rotating the endpoints and thereby introducing torsion to the plate. The simplest solution for the deformation is to add diagonals and create rigid triangles. Different ways of resisting this torsion are tested. The diagonals changes the deformation of the plate dramatically (Fig. 15). The impact of the direction in which the diagonals are placed is tested and compared in Fig. 17 and Fig. 18. For plates with opposite diagonals there is a difference in the deformation, but the overall shape is similar. However there are no particlespring resistance for bending around the diagonal axis. The diagonals gives an inplane stiffness, but no resistance for outofplane movement of particles.
Fig. 15 Rectangular grid with diagonals.
A way of resisting this movement is to try and keep the particles in plane. By adding a hingeresistance between adjacent triangles, vector forces will try an keep the four points in the same plane. The Kangaroo hingecomponent takes the four points, calculates the deviation from planarity and adds a force vector. Point 1 and 2 are the start and endpoints of the common edge, point 3 and 4 are the tip points of each triangle. It is clear that this approach takes more scripting logic, in order to create these 4 point lists in the correct order. This will be explained later.
Fig. 16 Hinge component
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The most important thing, is how we translate material properties to the particle system.
plate algorithm:  Same algorithm as for lines, but now in x and ydirections.  For twisting resistance diagonals are added.  Setup â€œhingeâ€? resistance, between adjacent triangels. Fig. 17 Diganoal opposite direction.
Fig. 18 Dual diagonal grid.
Fig. 19 Hinge principle
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Modelling 4.5.3â€‚â€‚What is torsion? The bending resistance of 1axial bending is considered simple and correct. Now it is necessary to be able to resist multiaxial bending or torsion. By going through physical paper tests, a concept was build, that the torsion can be described as bending around an axis in diagonal direction of the considered sheet. Fig. 23 + 24 shows the tests carried out: Looking at a sheet bend into a circle and applying opposite loads P at the quarter points normal to the plane of the circle, torsion is applied. The main bending axis is sketched out on the deformed paper sheets. Folding around these alternating diagonal axis shows that it is possible to construct both the bending My and the torsion Txy.
Fig. 20 Bending axis
When transferring this to a particle system, a better approximation can be achieved by adding straight cross segments between the alternating diagonals (Fig. 22). It is clear that if the diagonals are not alternate, the circular and torsional deformation of the bend ring would not be possible. For even better approximation a higher dimension of segmentation could be used, meaning to divide the sheet into even more triangles. Similarly to the division in the previous chapter (4.5.2). The consequences will be higher degrees of calculations and thereby longer computation times.
Fig. 21 Circle torsion
Fig. 22 Particle system
page 82
The strips in a gridshell are relatively thin and therefor not in high risk of folding  this means bending moment around the xaxis Mx is neglected and it is considered sufficiently accurate to have one division along the width of the sheets. Also it has to be decided whether the resistance of bending around the zaxis Mz, that is neutralized by the diagonal squares and an the spring stiffness, is a wanted property.
Fig. 23 Circular bending and torsion
Fig. 24 Teisting appoximation.
page 83
Modelling 4.5.4â€‚â€‚Diagrid mesh The particle system described previously in (4.5.3) is named diagrid mesh for convenience. In order to prepare this for dynamic relaxation with Kangaroo, a Python script with the following algorithm is set up:  0: Select a grid of curves.  1: Turn each curve into an intelligent member. (A class member*)  2: Member Intelligence: ID, curve, divisions, offset points, centre points, geometry/lines, mesh vertices, hinges, intersection curves, connections.  3: Offset all curves a given distance on two sides in a perpendicular direction.  4: Divide the offset curves by a given number. List0 and List1  5: Go through all offset points in the two lists and draw the corresponding diagonals. For n=0,2,4.... draw lower left to top right. For n=1,3,5.... draw lower right to top left. (Even  odd) Fig. 25 Rhino viewport. Diagrid mesh.
 6: Scripting function with similar approach to find mesh vertices and hinge force data.
* Fig. 26 Connection.
page 84
Python reference. http://docs.Python.org/2/tutorial/classes.html
To begin with a connection between intersecting strips is created by sharing four particles (Fig. 26). Such a connection will counteract rotation and is considered to be fixed. The line geometry is turned into a pretty simple particle system. As mentioned previously the bending My is resisted by the threeparticlesinline system (chapter 4.5.1) along the offset lines. Bending Mz is resisted by the spring stiffness and diagonals. Direct bending Mx is neglected and only components Mxy is accounted for as torsion by the hinge force resistance.
Fig. 27 Test example grid. Fixed support.
The diagrid mesh is tested and compared to a cardboard model. The test example is an extended 5x5 square grid (see Fig. 27 + Fig. 28). In the example on (Fig. 25), two anchor points at the end represent a simple support. In the cardboard model test, the ends of the strips are fixed with tape, creating a fixed support. To represent this, four adjacent particles are used as anchor points. The cardboard model is created by bolted connections which does not counteract rotation. This is the biggest difference, but the fixing of the ends makes it possible to compare the physical and the digital models. Results are very similar.
Fig. 28 Card modelling. Validation.
page 85
Modelling 4.5.5â€‚â€‚Diagrid02 mesh Another version of a simple grid with diagonals is the diagrid02 mesh. By adding another diagonal to each segment, the stiffness of the square is increased. An important property is that each segment now has the ability of having up to four resistance hinges instead of only one. This means that each segment can allow and resist torsion around two axis instead of only one. Another property is the creation of a centre point. This centre point allows for a more realistic material behaviour around a bolted connection. An explanation of investigated connection types follows in the next chapter. In the original diagrid mesh, it would be possible to use the same centre points as the midpoint on the diagonal. However in the real world pulling in a centre point of plate section, would result in small deflections around both the xand yaxis. In other words the centre point moves a little out of the plane. The algorithm here is pretty similar to the previous:  0: The intelligence of the curve determines what mesh to draw, based on the inputs: Diagrid or Diagrid02.  1: If Diagrid02, then draw a line from pt01i to centre point (cpt). Then from pt02i to cpt, from pt02i+1 to cpt etc.  2: Determine the number of hinges and use same order as in step [1].
Fig. 29 DIagrid2 mesh. Fixed support.
page 86
Modelling 4.6 Connection modelling The aim of this thesis is to look at utilizing the nature and behaviour of these elements subjected to deformation. In a gridshell structure, as in many other structures the connection has a big influence on the overall behaviour and shape. The connections presented here applies to both spline and mesh systems. By reducing the constraints to the minimum that will still create the desired shape, the minimizing of internal energy will have the best conditions. This follows equilibrium of the work equation:
W = Wint  Wext = 0 According to the principal of least action nature will always find the most efficient course from one point to another. This states that particles will follow paths of least action, meaning paths in which the total energy needed to get from A to B is minimised.* The structure itself will seek to minimise the internal work and therefor also minimising the external work.
4.5.6 Bolted connections The bolted connection will allow elements in a system to rotate around their local zaxis (tangent plane). This means that the constrained element can rotate to a certain degree, in order to get as close to the path of least action as possible. I many cases this will minimize the torsion and moment in the elements around the connection. To script this connection logic into the particle system, intersecting curves are offset in elevation. Between intersection points on each curve a spring is attached. In the real world, the elements cannot bend freely around their y and xaxis close to a connection because of collision with connected element.
Fig. 30 Bolted connection sketch
Fig. 31 Bolted connection. Model + detail
* http://www.principlesofnature.net/principle_of_least_action.htm
page 87
Fig. 32 Bolter connection.
In the particle system, this rotation resistance is generated by a three points logic as mentioned earlier (Fig. 6 on page 75). In this case, however, the rest angle between these three points is set to Pi/2  90 degrees. This means that the spring member is kept perpendicular to the element. Fig. 32 illustrates how rotation is resisted in the pink and blue planes. This constrains 5 out of 6 degrees of freedom. Translational movements (Tx, Ty, Tz) is resisted by the spring and rotation around y and xaxis (Rx, Ry) by the 3 point resistance. As in the real world, this bolted connection this allows principally only rotation around the zaxis (Rz). This connection logic is coded into the Python script component explained on the previous page. However, this connection calls for a another mesh than the diagrid mesh to be able to allow for proper behaviour of the connection. In order to create the points needed for resistance in the pink and blue plane, two diagonals pr. segment is more appropriate, i.e, diagrid02 mesh,
4.5.7â€‚â€‚Flexible bolted connections The idea with this type of connection is to allow the connections to be flexible. Meaning that they can move, within in a given interval, along intersecting elements.
Fig. 33 Flexible connection sketch
The flexible connection allows for the overall structure to find a shape with least internal work and thereby an optimised structure for the given conditions. These connections are loose and will allow for movement in the erection process and can then be fastened when the structure has found its best shape. In terms of scripting this connection requires a logic that allows for the intersecting curves to know which they are to be connected to, but not at a specific point/particle. This means that a spring logic would not be applicable.
Fig. 34 Flexible connection. Model + detail.
page 88
On the other hand the elements has to be kept in the correct order, so that two curves in the same direction does not intersect during relaxation. This can be accounted for by adding a spring between initially parallel curves. At the time of making this thesis attraction between two curves as well as collision between line segments is not yet implemented in Kangaroo. After talks with Daniel Piker, the inventor of Kangaroo, this will be implemented in a future release. In the mean time, it will be accounted for in another way. This will be explained in the next chapter.
4.5.8 “Free” connection In a way similar to the flexible bolt connection, this connection is created by the collision between elements in grid. This means that connections are based on a logic of how the curves/elements are arranged rather than actual connection points at intersections. The logic is somewhat similar to weaving, meaning that connections are dependent on the arrangement of the curves. The order of how curves are internally placed on top or below each other creates the interlocking connections. This kind of connection has another logic and requires the same kind of attraction and collision forces mentioned in previous part. For each intersection of curves the script needs to be able to decide whether the element should be above of below the other intersecting element.
Fig. 35 Spring between neighbour curves.
The advantage of this connection type is that the elements can move even more freely to obtain minimum internal work and that in principle no mechanical connections are needed.
Fig. 36 Free connection. Model + idea
page 89
Modelling 4.7â€‚â€‚Definition breakdown [Mesh]
1 2
Incoming geometry. Curves in groups corresponding to their direction. Python script. (appendix B  code)
The inputs are width of strip, the minimum edge of segments (determines number of divisions), and the drop down menu for Diagrid or Diagrid02. Outputs all information for further use.
3 4 5 6 7 8 page 90
Connection and element springs Bending resistance of My. Connection bending resistance. (4.5.6 on page 87) Torsion hinges (Fig. 19 on page 81) Support conditions. Bound01.1 + Bound02.1 = Simple support. Bound01.1 + Bound01.2 + Bound02.1 + Bound02.2 = Fixed. Kangaroo relaxation engine.
1
4 5 6 2
3
8
7 Fig. 37 Mesh GH definition
page 91
Modelling 4.8â€‚â€‚Part conclusion Increasing the number of force objects  springs, hinges etc.  will also increase the number of calculations with an exponential power of up to 4. This is a big increase in a system like this. The Diagrid mesh has 20x50x1 = 1000 springs and the Diagrid02 has 20*50*4 = 4000 springs. Its is the same for all other force objects. So it has to be decided what is needed for the individual case and compare it to the computation time. Because Kangaroo for the time being is a 3 DOF* system and not a 6 DOF system, it has to be chosen what properties are needed and wanted in the specific case. In this modelling chapter two substantially different element particle systems has been set up: Spline: Treating elements as space curves or 3Dsplines will resemble isotropic material behaviour, meaning same bending properties in y and zaxis. Torsion, as well as torsionresistance, is neglected in simulation. This mean that the spline geometry will be the centre line of the element and cross section rotation will then be decided from the approximate surface that the elements in the gridshell forms. (Explained in the next chapter) Fig. 38 Spline particle system. [10 springs]
Mesh: Treating elements as meshes allows for anisotropic material behaviour. This means that different material properties in different directions can be resembled. As mentioned in this chapter, different ways of setting up the mesh and the relationship between particles and springs gives different material behaviour.
Fig. 39 Mesh particle system. [52 springs]
Higher degrees of detailing of elements means more computations, more calculations and a more complex system for errorfinding. In the next chapther, the spline system will be investigated, looking at a build pavilion case.
*
page 92
DOF = Degrees Of Freedom
Fig. 40 Freeform physical model with flexible connections.
Fig. 41 Lasercut plastic elements.
Fig. 42 Freeform physical model with free connections.
page 93
INVESTIGATION
page 94
[Jukbuin pavilion]
Rhino
Grasshopper
Python
Kangaroo
Karamba3D
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5  Investigation 5.1 Jukbuin pavilion 5.1.1 Introduction CODA* is a design consulting agency. Together with students from The Polytechnic University of Catalonia, Barcelona (UPC), CODA designed this experimental pavilion. The aim was to create an innovative construction solution, wich efficiency in cost and energy. Main characteristics are weaving connections and bending of thin elements. The 90 m2 pavilion is constructed with only 15 birch boards, sliced into 5 cm planks and it has a budget of 1500₠. Through the images and illustrations presented by CODA and UPC on this page, a similar pavilion is modelled using principles from the previous chapter. The research here is to investigate the behaviour of a system with different constraints, properties, edge conditions, supports etc. The aim is to set up a general method and approach for other gridshell projects.
Fig. 1 Illstration  CODA + UPC
*
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CODA: Computational Design Affairs  www.codaoffice.com
Fig. 2 Jukbuin Pavilion  CODA + UPC
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5.1.2â€‚â€‚Main approach When working with gridshells, a common approach is to lay curves on top of a predefined surface. The creation of these curves are based on the theory of geodesics from differential geometry. This thesis works with a main approach strategy and a closely related secondary approach. The main approach  1. From grid to shape  differs from the common approach, by thar the shape is found by adjusting planar grids and controls. The second approach is more similar to the common procedure this is named 2. From shape to grid. The difference in the two approaches are clear, but both have properties that can be advantageous in different situations. This will be explained and exemplified later on. The illustration on the right hand side shows the overall idea of the main approach [1]. This is divided roughly into seven steps:
 Grid, Form finding, Analysis, Elements, Structure, Energy, Fabricate.
These steps goes from idea to building, analysing behaviour along the way. The form finding is done by dynamic relaxation using Kangaroo. The analysis part calculates the degree of curvature compared to the maximum bending strength of each element. Here a coloured plot is shown real time while running the form finding procedure. Crosssections are applied to the curve geometry and the structure of elements are setup. Then the structure is analysed and optimised by changing parameters in previous steps. Simultaneously the structure is analysed in terms of energy concerns. At the end a fabrication sheet for production and assembly is setup. This will be explained in terms of scripting in the following.
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GRID
3
FORM
2
Form finding through dynamic relaxation of the system with Kangaroo.
ANALYSE
1
Setup grid and particle system for relaxation.
Real time analysis of curvature, moment and stresses of elements in the found form.
6 7
ELEMENTS
5
FABRICATE ENERRGY STRUCTURE
4
Construct the elements within the structure from curve geometry.
FEM  Finite Element analysis of the structural behaviour combined with calculations of moment from bending. Optimization.
FEM  Finite element analysis of the environmental impacts, such as solar radiation, dayligth, thermal conditions etc. Optimization.
Fabrication description of all elements and connections. Ready for printing, production and assembly.
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Investigation 5.2 Jukbuin pavilion  Script 5.2.1 Script setup The principles of particle systems has been explained in chapter 4. Therefor an understanding of these are assumed in the following, to be able to give a quicker basic understanding of the investigations. The Jukbuin pavillion seems to be a gridshell created by moving edge points of a very geometrical planer grid towards a common centre. Also it is presumed that it is possible to generate this behaviour with a spline geometry. As the system is rather big, in terms of number of grid elements, it will decrease the number of calculations needed with a spline model as simple as possible. To show investigations and discuss the behaviour of this pavilion, the final definition will briefly be broken down into steps in chronological order. Throughout the creation of the script, each step is setup in a way that allows for a general parametric model. Meaning that the script can be used for many other variations than this pavilion. The figure shows that the procedure roughly consists of 7 parts. The approach briefly: Create a planar grid  set up particle system for dynamic relaxation  evaluate the shape, curvature etc.  set up input for FEM analysis  analyse the structure.
2 1
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3
SIMU GRID
3
EVAL
1 2
A geometrical regular GRID can be created of triangles, squares and hexagons. Controls: type, side length, rotation, boundary.
EVALUTION of the curvature of the curves. Section and material properties determine the allowed curvature and evaluates the actual max. curvature in each element. Controls: Cross section, material strength.
7
PREP STRU
6
PREPARATION of geometry for FEM analysis by dividing splines into straight elements and finding normals of each part element. These normals, are also used for creating solid geometry in Rhino. Controls: Geodesic curves or Kangaroo output. STRUCTURE. Sweep of crosssections along the element curves with the correct rotation, corresponding to the normals at the given points. Here also the different sets of elements are offset to the correct layer. BENDING stresses from the initial bending M0 is calculated in the centre of each part element. This can then be superpositioned with moment from external loads.
FEM.A
5
κmax = (fy / (z∙E))
κ = 1/(r)
BEND
4
SIMULATION of member behaviour, through dynamic relaxation with Kangaroo. Controls: Anchor points, controlpoints, properties, internal and external forces.
My
My
My
M = EI/r
Mz
My
FINITE ELEMENT ANALYSIS are carried out using plugin analysis software Karamba3d. Section and element properties follows settings earlier in the script. Support, materials, loads and combinations are setup and analysis are carried out.
5 4
6
Fig. 3 GHDefinition
7
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Investigation 5.2.2â€‚â€‚Grid User input
This script lets the user chose the setup of the planar grid with variations of polygon type. From the selected type, side length and number of arrays, a Python script* calculates the radius of the polygon, needed as input to draw the polygon. The script also defines the offset array distance and the range in which to offset all curves. After setting up the array of curves, the boundary, or an offset of this, is used to trim the curves. This boundary can be whatever curve or spline geometry that suits the given project. After trimming, the curves are sorted corresponding to their direction by comparing the tangent vectors of each curve. This order is important in order to offset the elements in correct layers after relaxation. Elements in the same offset layer must not intersect. The last step is to find all intersection points of the curves and sort them along the curve. This is to clean out small line segments, see last step on Fig. 5, and to prepare line segments for the particle model. To find the points along the lines and sort them appropriately, two Python scripts are set up**. If the side length of the chosen polygon is too big, it can be necessary to divide each line segment further for precession. Fig. 6 illustrates the grid variation parameters and Fig. 7 shows examples of different settings.
*
Python script appendix C1
** Python scripts. findCrvPts app: C2 and sortAlongCrv: app. C3
page 102 Fig. 4 GH def. Part 1. (see Fig. 3 on page 101)
Type
Extend
Fig. 6 Grid parameters: Type, sidelength, rotation, boundary.
Array
Boundary
Trim
Finish Fig. 5 Grid method
Fig. 7 Grid variations.
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Investigation 5.2.3 Simulation To simulate the strength of this grid, the particle model is set up according to the approach mentioned earlier. The structure gains its strength from the bending resistance of each element. x y
Using only 3 degrees of freedom as Kangaroo does currently, axial stresses and bending without torsion can be calculated, accounting for Young’s modulus and sectional area. This follows the paper Tensigrity spline beam and grid shell structures  S.M.L Adriaenssens, M.R. Barnes [1].
z
x y z
Fig. 8 Particle translation
The 3 translational degrees of freedom describes that the particle can move in x,y and z axis. When thinking of the local element coordinate system, translation in xaxis is resisted by the spring. Movement in y and zaxis is resisted by the bending stiffness which seeks to keep three adjacent particles in a straight line.
x y
Rx z
x y
To be able to describe and resist torsion of the particle system, the 3 rotational degrees of freedoms needs to be available. The torsion can then be described as rotation around the xaxis.
Rx z
Fig. 9 Particle rotation
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As it is now the Kangaroo focuses on geometrical optimization. The numeric values does not resemble real world units. However what is important here is the form finding. As long as the numeric value of bending stiffness is the same along the whole element, the geometric behaviour will be correct. The paper mentioned [1] describes calculating bending around axis as EIi. Having only one option for the value of bending stiffness for a springelement, the same stiffness will be given for bending around the yand zaxis. The spline model is too simple to account for bending around the xaxis, buckling, and is neglected. In assymetric crosssections the moment of inertia will be different for each axis, and thereby the bending stiffness will not be the same in all directions.
Fig. 10 GH def. Part 2. (see Fig. 3 on page 101)
Fig. 11 Shell behaviour
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Investigation If elements with a big difference in stiffness is sought, which e.g will mean no, or very little bending around the zaxis, the result using this spline, 3dof approach will be less correct. In the case of the Jukbuin pavilion, with a hex/triangle grid, stiffness is secured more geometrically. Illustrations on Fig. 11 on page 105, show good results using this approach. By moving the 12 supports towards the centre, the gridshell is formed.
Fig. 12 Triangles in grid
The Jukbuin pavilion is constructed using element cross sections with dimensions of 5 x 1,5 cm. This means that resistance Mz is higher than My. At the same time, on the pictures it is noticeable that the longitudinal connections of elements allow for some rotation around the zaxis. In the simulation, the hexagon grid contains changing triangles along all elements. The adjacent triangles along and between elements will ensure some geometrically resistance of movement in the local element ydirection, meaning resistance of Mz. In this case, results are good because of the geometrical constraints of the grid, but in another situation this might not be the case. One can imagine the square grid presented on Fig. 7 on page 103. Here the geometry of the grid would not resist the deformation. If the elements intended for construction could however resist inplane rotation (Mz), the real world behaviour would be different than the simulated spring model. Therefor in some cases another way of insuring resistance of inplane rotation will be needed. This will be discussed at the end of this investigation.
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Other than different behaviour, the problems arise in connection with the fabrication. As the idea is to be able to create elements out of simple straight parts, it is necessary that the curve geometry can indeed be obtained by straight elements. In common gridshell approaches this is insured by using geodesic curves along the surface.
Fig. 13 Square grid configuration
Fig. 14 Deformation in square grid
Fig. 15 Behaviour spline vs. element
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Investigation Fig. 16 Curvature of a curve
5.2.4â€‚â€‚Geodesic problematic The theory of curvature of curves on a surface from Springer: Elementary Differential Geometry [11], says that a curve, Ď’, must have zero geodesic curvature everywhere to be a geodesic: (1)
đ?‘˜đ?‘˜đ?‘”đ?‘” = đ?›žđ?›žĚˆ . (đ?‘ đ?‘ Ă— đ?›žđ?›žĚ‡ )
Where đ?›žđ?›žĚ‡ is the tangent of curve đ?›žđ?›ž and đ?‘ đ?‘ is the unit normal
This is the requirement of proposition 8.2 [11]. (đ?‘ đ?‘
)
đ?‘˜đ?‘˜đ?‘›đ?‘› = đ?›žđ?›žĚˆ . đ?‘ đ?‘ , đ?‘˜đ?‘˜đ?‘”đ?‘” = đ?›žđ?›žĚˆ . đ?›žđ?›žĚ‡ For a surface the curvature is normallyĂ— described by principle curvatures, kCurvature ofisathen given point given by: on the surface. The product of the principle cur1 and k2 đ?œ…đ?œ… vatures is the Gaussian curvature K. Looking at Fig. 16, Ď“ is a unit vector đ?‘˜đ?‘˜ 2 = đ?‘˜đ?‘˜đ?‘›đ?‘›2 + đ?‘˜đ?‘˜đ?‘”đ?‘”2 and is by defintion tangent vector the surface Ďƒ. The tangent vector is perpendicular to the normal N of Ďƒ, therefor the binormal vector is the cross product of N x Ď“. đ?‘˜đ?‘˜đ?‘”đ?‘” = đ?›žđ?›žĚˆ . (đ?‘ đ?‘ Ă— đ?›žđ?›žĚ‡ )
Normal curvature of the curve is described in the direction of the normal as Where đ?›žđ?›žĚ‡ is the tangent of curve đ?›žđ?›ž and đ?‘ đ?‘ is the unit normal the scalar kn, meaning that it is the magnitude of the normal curvature. Similarly the geodesic curvature kg is the scalar in the binormal direction N x Ď“. đ?‘˜đ?‘˜đ?‘›đ?‘› = đ?›žđ?›žĚˆ . đ?‘ đ?‘ , đ?‘˜đ?‘˜đ?‘”đ?‘” = đ?›žđ?›žĚˆ . (đ?‘ đ?‘ Ă— đ?›žđ?›žĚ‡ )
Curvature đ?œ…đ?œ… is then given by:
đ?‘˜đ?‘˜ 2 = đ?‘˜đ?‘˜đ?‘›đ?‘›2 + đ?‘˜đ?‘˜đ?‘”đ?‘”2
Fig. 17 shows the approximate surface of the curve network geometry. The approximate surface is a NURBS surface generated by a patch function in Rhino3d, based on all intersection points of the network. This is done by firstly finding the best fit plane, then the surface deforms to match the points. In the bottom of the figure the strip is magenta and the corresponding geodesic on the surface is cyan. This strip is drawn as the shortest path along the approximate surface between endpoints of the stripcurve. The geodesic will unroll to a straight line in plane, and the strip from the
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Fig. 18 Calculation of geodesic curvature following (1).
Kangaroo output, will be slightly nonlinear. The measure of nonlinearity can be seen as a measure of the bending around the zaxis that the element is subjected to. This should be taken into account and stresses from Mz can be calculated from the geodesic curvature. Fig. 18 shows how geodesic curvature is found for a space curve using formula (1).
Fig. 17 Approximate surface and geodesic comparison.
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Investigation 5.2.5â€‚â€‚Evaluation Real time analysis of the curvature is a visual measure of how much each element bends. This enables the designer to evaluate, in the forming process, whether the intended element can indeed bend the amount necessary.
Fig. 19 Curvature evaluation.
Fig. 20 Reduced cross section
Fig. 21 Reduced cross section. GH def.
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As mentioned previously the script lets the user chose between the Kangaroo output curves or the geodesic curves. By evaluating the curvature at 510 points along the curve, the corresponding curvature circles are found. By measuring the length of the circle, the radius can be found by: đ??śđ??ś đ?‘…đ?‘… = 2đ?‘ƒđ?‘ƒđ?‘ƒđ?‘ƒ đ??śđ??ś 1 where C is the circumference đ?‘…đ?‘… of =the 2đ?‘ƒđ?‘ƒđ?‘ƒđ?‘ƒcircle. The curvature and the bending đ?œ…đ?œ… = đ??śđ??ś stresses can be found by: đ?‘…đ?‘… = đ?‘…đ?‘… 1 2đ?‘ƒđ?‘ƒđ?‘ƒđ?‘ƒ đ?œ…đ?œ… = 1 đ?‘…đ?‘… đ?œŽđ?œŽđ?‘?đ?‘? = âˆ’đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§ =1âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ ďż˝ ďż˝ đ?‘…đ?‘… đ?œ…đ?œ… = 1 đ?‘…đ?‘… đ?œŽđ?œŽđ?‘?đ?‘? = âˆ’đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§ = âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ ďż˝ ďż˝ đ?‘“đ?‘“đ?‘Śđ?‘Śđ?‘Śđ?‘Ś đ?‘…đ?‘… 1 đ?œ…đ?œ…đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š = đ?œŽđ?œŽđ?‘?đ?‘? =the âˆ’đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§ =âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ ďż˝ ďż˝ of the material the maximum By substituting the Ďƒb with bending strength đ?‘“đ?‘“đ?‘Śđ?‘Śđ?‘Śđ?‘Ś đ?‘…đ?‘… allowed curvature becomes: đ?œ…đ?œ…đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š 1 = âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ 3 đ??śđ??ś đ??źđ??ź = âˆ— đ?‘?đ?‘?đ?‘“đ?‘“âˆ—đ?‘Śđ?‘Śđ?‘Śđ?‘Śâ„Ž đ?‘…đ?‘… = 12 = 2đ?‘ƒđ?‘ƒđ?‘ƒđ?‘ƒ đ?œ…đ?œ…đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š 1 âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ 3 đ??źđ??ź = âˆ— đ?‘?đ?‘? âˆ— â„Ž 1 12 đ??¸đ??¸đ??¸đ??¸ đ?‘€đ?‘€ = âˆ’ đ?œ…đ?œ… = 1 The maximum curvature of each element is previewed 3 đ?‘…đ?‘…âˆ— â„Ž đ?‘…đ?‘… with a coloured shade. đ??źđ??ź = âˆ— đ?‘?đ?‘?đ??¸đ??¸đ??¸đ??¸ 12 The colour scale is defined inđ?‘€đ?‘€the interval from [0Îşmax]. Green being zero =âˆ’ 1 đ?‘…đ?‘… 3 utilization1 percentage of f and red being Îşmax. Crosssection properties and đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; = đ??źđ??ź âˆ’ ( đ?œŽđ?œŽđ??¸đ??¸đ??¸đ??¸ âˆ—đ?‘?đ?‘? đ?‘‘đ?‘‘=â„Ž âˆ’đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§đ?‘§ âˆ—â„Ž ) = âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ ďż˝ ďż˝ yd đ?‘…đ?‘… used through all đ?‘€đ?‘€ sliders. = 12 âˆ’ These settings are can easily be changed using the 1 đ?‘…đ?‘… đ??źđ??ź âˆ’ (đ?‘€đ?‘€sense âˆ— đ?‘‘đ?‘‘ âˆ— â„Ž3 ) furter analysis. Here itđ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; also=makes stresses in the reduced 12 â„Žto look atđ?‘“đ?‘“đ?‘Śđ?‘Śđ?‘Śđ?‘Ś đ?œŽđ?œŽ = 1 âˆ— đ?‘§đ?‘§ đ?œ…đ?œ…đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š = 3 cross section around connections. hole for the bolt: đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; = đ??źđ??ź âˆ’đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; (Inđ?‘€đ?‘€ this âˆ— đ?‘‘đ?‘‘ case âˆ— â„Ž the ) âˆ’đ?‘§đ?‘§đ?‘§đ?‘§ 12 â„Ž đ?œŽđ?œŽ = âˆ— đ?‘§đ?‘§ 1 đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; đ?‘€đ?‘€ đ??źđ??ź = âˆ— đ?‘?đ?‘? âˆ— â„Ž3 đ?œŽđ?œŽ = âˆ— đ?‘§đ?‘§ 12 đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; đ??¸đ??¸đ??¸đ??¸ đ?‘€đ?‘€ = âˆ’ đ?‘…đ?‘… 1 đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x; = đ??źđ??ź âˆ’ ( âˆ— đ?‘‘đ?‘‘â„Ž âˆ— â„Ž3 ) 12 đ?œŽđ?œŽ =
đ?‘€đ?‘€
đ??źđ??źđ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;đ?‘&#x;
âˆ— đ?‘§đ?‘§
Fig. 22 GH def. Part 3. (see Fig. 3 on page 101)
Fig. 23 Curve analysis
Fig. 24 Realtime curvature coloring
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Investigation 5.2.6â€‚â€‚Preparation As mentioned in previous chapter the script allows the user to easily change between geodesic curves (cyan) and the Kangaroo output curves (magenta). All curves are divided into part segments of straight lines. This is in order to translate the model to finite element analysis. In this case, curves are divided at all intersection points. More divisions can be chosen if it is decided that distances between intersections are too great. It is simply a matter of detailing of the FEM model.
Fig. 25 Drop down menu
For the geodesic curves, the approximate surface is generated and surface normals are found at each intersection point and set up in a data structure corresponding to the points along the surface. The Python scripts, coded for the grid setup (5.2.2 on page 102) are used again to find and sort intersection points along the curves. For the Kangaroo output curves the triangular mesh between the curves is generated. In a similarly way, used with the geodesics, the mesh normals are set up in a data tree. These normals are needed for offsetting elements in the correct structure order. The normals at each part element is also needed to describe the orientation of the each part element in the finite element analysis.
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Fig. 26 GH def. Part 4. (see Fig. 3 on page 101)
Fig. 27 Kangaroo output curves and corresponding mesh
Fig. 28 Geodesic curves and appromixate surface
Fig. 29 Surface normals at intersection points
Fig. 30 Comparison. Difference between curve types.
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Investigation 5.2.7â€‚â€‚Structure The logic of offsets for each element is setup by defining an algorithm that sets the correct value. In chapter 5.2.2 it was explained that curves are sorted by tangent direction. This was done so that a series of numbers can be set up to define a different numerical offset for each curve group.
Fig. 31 Section of elements
All through the script it is important to know the data structures of Grasshopper3d. In short, there are lists and there are trees. A list is a simple list of data and a tree is a list containing lists. Each list being a branch on the data tree. As with trees, all branches can have more branches growing on them.* In this case, each of the three curve groups consists of 20 curves. So a data tree with 3 branches, consisting of 20 numbers are created. Branch[0] = [0,0,0,....0], branch[1] = [1,1,1,...1], branch[2] = [2,2,2,...2]. The following function is set: x*(height/1000)*amount. Where x is the data branches and amount is an adjustable factor. Height is divided by 1000 to convert from mm to m. This function creates the new data tree with branch[0] = [0,0,0,...0], branch[1] = [h,h,h,....h], branch[2] = [2h, 2h, 2h,....2h], with the amount value of 1. These numbers defines the length of lines created in the direction of each normal vector (See Fig. 33). So that the offset curves can be drawn from the endpoints of these lines. The reason for offsetting the elements this way, is to avoid clashes between intersecting elements, such that they can be connected with a bolt or a similar connection type. The element cross sections are swept along the curves and the zaxis of these are rotated to align the normals. The script allows for multiple offsets of each curve group, meaning that more layers can be added to the structure. This can be necessary to gain strength while still being able to bend the elements. *
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see more on: http://digitaltoolbox.info/grasshopperbasic/trees/
Fig. 32 GH def. Part 5. (see Fig. 3 on page 101)
Fig. 33 Offset vectors
Fig. 34 Offset elements
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Investigation 5.2.8â€‚â€‚Bending The bending moment that comes from bending the initially straight elements, is calculated for the offset curves. This is done by evaluating curvature at the parameter along the curve closest to the middle of each straight part element. This prestressing from the bending moment is added to the moment from self weight and imposed loads after finite element analysis by superposition. The calculations are the same as for the part 3  Evaluation, but in order to superposition the moments it has to be done at all parameters on the curve corresponding to all part elements from finite element model. Also most curves are offset at these points, and therefor the curvature radius is different from the original curve evaluated in part 3. As mentioned in chapter 5.2.4, the geodesic curvature is evaluated alongside the principal curvature, so decide if bending around z, Mz, is to be taken into account. Here the geodesic curvature is calculated to be very close to zero. Only by using very high precision, the geodesic curvature becomes approx. 1,114e16 and is therefor not taken into account. What might be more interesting could be to find the torsion Ď„* of the curve elements:
đ?œ?đ?œ? =
(đ?›žđ?›žĚ‡ Ă— đ?›žđ?›žĚˆ ). đ?›žđ?›žâƒ› â€–đ?›žđ?›žĚ‡ Ă— đ?›žđ?›žĚˆ â€–2
Where is torsion of curve Theđ?œ?đ?œ? shear stresses fromđ?›žđ?›ž.torsion can then be calculated and added to the shear found by finite element analysis. However this is not done here.
* From Elementary Differential Geometry [11] page 38.
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(1)
Fig. 35 GH def. Part 6. (see Fig. 3 on page 101)
My
My
My Fig. 36 Bending momoent calculated from curvature radius in middle of each part element.
My M = EI/r
Mz
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Investigation 5.2.9 Finite element analysis Karamba3d* is a Finite Element software like many others. However it has some significant features that makes it really interesting in regards to parametric design. Karamba3d is fully embedded in the parametric environment of Grasshopper. This makes it easy to combine parameterized geometric models, finite element calculations and optimization algorithms like Galapagos. The first step is to take the centre line geometry and turn it into beam elements. To be able to identify beam elements, an ID can be added to elements as they are defined. In this case it was chosen to create a Python script that defines an ID data tree with same data structure as the line geometry data tree. This data tree then contains a string for each part element. Fig. 37 Karamba3d Beam componont
IDTree = [branch[0] = [element0, element1..., elementn], branch[1] =.[element0, element1..., elementn], branch[n]=...] This gives each beam element the ID structure example: “beam(33): E(10) ” It is common in finite element software to define layers in a construction as offset or eccentricities of the cross section centroid from the centre line geometry. This means that the original onelayer curve geometry is used as beam element reference, and then offsets from these are defined later on.
Fig. 38 Eccentricity definition. (Karamba3d manual fig. 25)
As for most structural analysis programs, cross sections, materials, support conditions and loads are setup. In this case all elements are wood, but if there are different materials in a model, element ID becomes important. The material selected at this stage in the overall script, is also the properties that are fed back to the initial curvature, moment and stress calculations in part 3. At the same time the settings of cross section dimensions are being fed from part 3 to here to setup the Karamba3d cross section. This creates a coherent parametric model. *
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See Karamba3d manual [3] for documentation
Fig. 39 GH def. Part 7. (see Fig. 3 on page 101)
Fig. 40 FEM model  Loads and supports. Perspective
Fig. 41 FEM model  Element cross sections. Elevation.
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Fig. 45
Investigation z
x
y
Fig. 42 Section axis
As default setting, the cross section along each part element is that the local xaxis follows the centre line and the zaxis follows the global zaxis. This mean that the finite element model needs the rotation of the elements as input. This rotation is normally referred to as the beta angle and defines the orientation of the elements. Either this angle of rotation or the local zaxis, can be fed into the orientate beam component. The local zaxis of the element must correspond to the normal of the surface.
Fig. 43 Karamba3d Orientation Component
Because each bend element is divided into smaller straight part elements, only one beta angle rotation is defined per part element. It is chosen to define this rotation or corresponding local zaxis at the beginning of each part element. The normal of the surface at the intersections, corresponding to the beginning and endpoints of these part elements, will then equal the local zaxis. These normals are found in part 4 of the script. Here the model is simplified so that each part element is rotated and therefor no stresses from initial torsion is accounted for. This can be done by evalu1 đ?‘¤đ?‘¤ 2 ating the Gaussian curvature of the strips generated đ?œ€đ?œ€ = ďż˝ ďż˝ in part 5.2.7. From this 1 đ?‘¤đ?‘¤ 2 2 đ?œŒđ?œŒ ďż˝strip ďż˝ edges can be found by: the stresses from elongation ofđ?œ€đ?œ€ =the 2 đ?œŒđ?œŒ
1 đ?‘¤đ?‘¤ 2 đ?œŽđ?œŽ2đ?‘Ąđ?‘Ą = đ??¸đ??¸ ďż˝ ďż˝ 1 đ?‘¤đ?‘¤ 2 đ?œŒđ?œŒ đ?œŽđ?œŽđ?‘Ąđ?‘Ą = đ??¸đ??¸ ďż˝ ďż˝ 2 đ?œŒđ?œŒ 2 đ?œ?đ?œ? = of â„Žđ??şđ??şđ??şđ??şGaussian curvature. Shear where w is width and Ď 1 =đ?‘¤đ?‘¤1/âˆšK is radius đ?œŽđ?œŽđ?‘Ąđ?‘Ą = đ??¸đ??¸ ďż˝ ďż˝ 1 đ?‘¤đ?‘¤ 2 đ?œ?đ?œ? = â„Žđ??şđ??şđ??şđ??ş 2 đ?œŒđ?œŒ stresses are found by: đ?œŽđ?œŽđ?œ€đ?œ€đ?‘Ľđ?‘Ľ = = 2đ?œŽđ?œŽđ?‘?đ?‘?ďż˝ đ?œŒđ?œŒ+ďż˝đ?œŽđ?œŽđ?‘Ąđ?‘Ą đ?œ?đ?œ? =1â„Žđ??şđ??şđ??şđ??ş đ?‘¤đ?‘¤ 2 đ?œŽđ?œŽđ?‘Ľđ?‘Ľ = đ?œŽđ?œŽđ?‘?đ?‘? + đ?œŽđ?œŽđ?‘Ąđ?‘Ą đ?œ€đ?œ€ = ďż˝ ďż˝ 1 đ?‘¤đ?‘¤ 2 đ?œŽđ?œŽđ?‘‰đ?‘‰đ?‘‰đ?‘‰ 2 đ?œŒđ?œŒ đ??¸đ??¸ đ?‘Ľđ?‘Ľ2ďż˝+ďż˝3đ?œ?đ?œ? 2 đ?œŽđ?œŽđ?‘Ąđ?‘Ą = ďż˝đ?œŽđ?œŽ đ?œŽđ?œŽđ?‘Ľđ?‘Ľ = đ?œŽđ?œŽđ?‘?đ?‘? + đ?œŽđ?œŽđ?‘Ąđ?‘Ą 2 đ?œŒđ?œŒ 2 + 3đ?œ?đ?œ? 2 đ?œŽđ?œŽmodulus where h is height, G is shear and Î¸ is the angle of rotation pr. length. đ?‘‰đ?‘‰đ?‘‰đ?‘‰ = ďż˝đ?œŽđ?œŽđ?‘Ľđ?‘Ľ 1 đ?‘¤đ?‘¤ 2 ďż˝ đ?œŽđ?œŽđ?‘Ąđ?‘Ą = đ??¸đ??¸and2ďż˝ torsion đ?œ?đ?œ? = â„Žđ??şđ??şđ??şđ??ş Stresses from bending 2đ?‘Ľđ?‘Ľ + đ?œŒđ?œŒ 3đ?œ?đ?œ? 2 is combined and von mises yield criterium đ?œŽđ?œŽđ?‘‰đ?‘‰đ?‘‰đ?‘‰ = ďż˝đ?œŽđ?œŽ 1 đ?‘¤đ?‘¤ 2 đ?œ€đ?œ€ = ďż˝ ďż˝ 2 đ?œŒđ?œŒ
Fig. 44 Analysis Gaussian Curvature
can be used for combining stresses and shear: đ?œŽđ?œŽđ?‘Ľđ?‘Ľ = đ?œŽđ?œŽđ?‘?đ?‘? + đ?œŽđ?œŽđ?‘Ąđ?‘Ą đ?œ?đ?œ? = â„Žđ??şđ??şđ??şđ??ş From [13].
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đ?œŽđ?œŽđ?‘Ľđ?‘Ľ = đ?œŽđ?œŽđ?‘?đ?‘? + đ?œŽđ?œŽđ?‘Ąđ?‘Ą
đ?œŽđ?œŽđ?‘‰đ?‘‰đ?‘‰đ?‘‰ = ďż˝đ?œŽđ?œŽđ?‘Ľđ?‘Ľ2 + 3đ?œ?đ?œ? 2
đ?œŽđ?œŽđ?‘‰đ?‘‰đ?‘‰đ?‘‰ = ďż˝đ?œŽđ?œŽđ?‘Ľđ?‘Ľ2 + 3đ?œ?đ?œ? 2
5 section of part 7.
Using the Karamba3d software, it is convenient to use the normal vectors as the zvectors. If the model is to be transferred to other external analysis software as e.g Robot Structural Analysis, it is necessary to use the calculated beta angle and then transfer this with the model. Calculation is easily done by having the data structure of both the normals and the zaxis.
Z Z
β β
Fig. 46 Default rotation of part element. Following global z.
β β N N
Fig. 43 Rotated part element. Local zaxis follows normal N of the surface.
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Investigation As mentioned in the beginning of this chapter, the offset of elements in their corresponding layers, is defined as eccentricities of the cross section centroid from the centre geometry.
Fig. 47 Karamba3d Eccentricity component
These eccentricities are defined by the size and direction of a vector corresponding to local directions of the element. The direction and size of these local eccentricity vectors are defined from the calculations done in part 5. Here the length and direction of the vector equals that of the line drawn in the normal direction of the surface. This finite element analysis part of the script allows the user to chose between one or two layers. This is easily changed by use of the drop down menu.
Fig. 50 Eccentricity definition. (Karamba3d manual fig. 25)
Fig. 48 Custom dropdown menu
Letâ€™s make a simple analysis of the Jukbuin pavilion as it is build in Barcelona. We will assume that the cross sections of the plywood strips are 15 x 50 mm. As there is no real function of the pavilion other than being a proofofconcept installation, there exists no enclosure. This means that no or very little wind and snowload will affect the structure. Firstly looking at this onelayer structure with only self weight (loadcase 0) acting on it. Then deformations will be as shown on Fig. 54, with largest nodal translation = 0,026 m = 26 mm. If imagining a more specific function of this pavilion, some kind of covering will be needed. By creating a mesh from the nodes in the structure, snow and wind area loads are transformed into equivalent nodal loads.*
h = 15mm W = 50mm
Fig. 49 Assumed cross section.
In the case of a snowload of 1 kN/m2 and selfweigth (LC1) acting on the structure, max. nodal displacement is 1,25 m. (Fig. 55). By adding an extra layer the max. nodal disp is 0,14m = 140mm. Weight of onelayer structure = 3,66 kN = 366 kg, two layer structure = 732 kg.
*
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This is done by calculating the resultant load on each meshface and distributing it among the mesh vertices. Then vertex loads are distributed to structure nodes, by considering distance between mesh vertex and structural node.
Fig. 51 GH Def section of part 7.
FEM 
Fig. 52 One layer structure. (OLS)
Elements + Loads
Fig. 53 Two layer structure (TLS)
FEM 
Deformation Lc0
FEM 
Deformation Lc1
Fig. 54 Deformation  Loadcase0. One layer structure.(OLS)
Fig. 55 Loads (orange) + deformation (green)  LC1. (OLS). Red: Undeformed
Fig. 56 Loads (orange) + deformation (green)  LC1. (TLS). Red: Undeformed
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Investigation A deformation of 14 cm will cause problems with covering elements etc, this means that the element cross sections will have to be increased. As well as looking at the overall structure deformation, moments and stresses has to be evaluated as well. The stresses are output in kN/cm2 = 10 N/mm2 (MPa) and it is possible to adjust the colour scale. This needs to have a small range of green around 0 MPa, otherwise the visual output does not give a realistic output for evaluation. The range is defined for all output results, so if there is a high moment or stress distribution in a very small part of the structure, this will define the upper colour range. This can lead to high amount of moments or stresses being plotted as a shade green, even though the values are too high.
Fig. 57 Resultant section forces. Data output.
As the elements are small finite elements, it can be difficult to evaluate the structure visually with this output. Also it does not contain the stresses of the initial bending. To simplify the visual output, the maximum resultant of each part element is extracted from the FEmodel. The moments at each part element from initial bending, calculated in part 6, is named M0 and is here added to the moments calculated by Karamab3d M1. The element meshes are also extracted from Karamba3d and a new colouring scale is applied. Here it is chosen to define the colour range within [Mmax > +Mmax]. Mmax is blue, Mmax is red and zero is green. Now it is easier to see in which elements, moment distribution is too high. In this case bending strength Mmax = 0,0433 kNm and the maximum resultant moment = 0.189 kNm. This means that in many of the blue shaded elements, moments are far too high.
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FEM 
Axial stresses (Karamba3d)
FEM 
Resultant moments
Fig. 58 Axial stresses + legend (Karamba3d Output)
Fig. 59 GH, def.. Section of part 7.
Fig. 60 Resultant moment. Custom script output.
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Investigation The last part of this script finds those elements in which the moments are too high and shows only these. At the same time this part sorts the resultant moments and shows the maximum values of the resultant moments. A simple calculator of the maximum bending strength is set up to be able to give an idea of what kind of material strength and cross section is needed to be able to resist the maximum moments in the visualized elements. The figures on the right hand side shows how the number of elements with moment distribution higher than the allowed, changes with the changing cross section dimensions. In these examples, all elements are changing cross sections, but it is also be a possibility to just add strength, add material or use a different cross section for the highlighted elements. Here the data structure and element IDâ€™s come into play, as this makes it possible to identify which elements needs to be changed and makes it possible to change these elements in the input. These finite element calculations with Karamba3d is not meant as structural engineering documentation, rather it is meant to give an idea of possibilities and constraints at early stages in the project. Also it works as a great platform for further documentation and collaboration between architects and engineers. Data, geometry, forces etc. can be exported to other finite element programs depending on the engineers preferences. These initial analysis does not take into account factors like connection shear forces, reduced load carrying capacity in joints, realistic supports conditions etc.
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Fig. 61 GH def. Section of part 7.
FEM 
20x50mm
FEM 
25x50mm
FEM 
30x60mm
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5.2.10 Optimization Optimization may be interpreted in many ways and can be carried out in regards to many different parameter. The shape of the shell can be adjusted to find a configuration that will fulfil project parameters best possibly. By adding a 2 or 3dimensional space for movement of the control points, the relaxed elements can be evaluated in an iterative process, until the most optimised configuration is achieved.
Karamba3D
A simple example is set up. The movement vectors of the control points are defined as genomes for Galapagos, Kangaroo iteration setting is increased* and the fitness value is set to be the minimum of internal energy calculated by Karamba3d. This allows for optimization of the shape configuration within the given parameters, in regards to the internal energy.
Here the three groups of control points are able to move in an interval of [0  1,5m] in direction towards the centre. The result was that the internal energy was at a minimum (1,63 kNm) close to all values being 1. In a more detailed setup, fitness value would probably be a combination of moment distribution and deformations. *
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To have a realtime visual output, Kangaroo is normally set to show 10 iteration pr. given interval. By removing the interval timer and changing iterations pr. run to 1000, only “steady” states will be output from Kangaroo
Karamba3d also contains predefined optimization algorithms. One of these is the Optimize Cross Section algorithm. By defining a list of cross sections for the algorithm to chose from, it defines the optimal cross section for each part element.*
*
Karamba3d Manual  section 6.5.7, page 58.
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5.3â€‚â€‚Jukbuin pavilion  Build The last part of the investigation is to build a scaled model. In terms of fabrication, the symmetry of the grid makes it fairly simple. By using the script (5.2.2), element numbers and lengths are extracted and used for production of wood strips. This is a 1:10 model made out of masonite strips 1.5mm thick and 8mm wide. Behaviour is as expected and corresponds to the particle model. The flat grid can be adjusted along flexible supports. Lessons learned from making this model, is that reinforcement of support elements are necessary and that the structure becomes stabile even though it consists of very flexible members.
The masonite strips where flexible in bending around both the yand the zaxis, therefor the Kangaroo spline elements proved to be a good approximation. Tests where also carried out with mesh elements, but computation time was much higher and results not as good as with the spline geometry. In the following chapter, the approach build here, will be tested and developed, in a design case with Henning Larsen Architects.
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CASE [1]: NORDEA BANK
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Nordea Bank  Ă˜restad
Rhino
Grasshopper
Python
Kangaroo
Galapagos
DIVA
Karamba3D
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6  Case [1]: Nordea Bank 6.1 Nordea bank ørestad  Introduction Nordea’s new office building in Ørestad, Copenhagen will house approx. 2000 employees and be ready for opening in 2016. Stage one covering a total of 40.000 m2 consists of two light, sculptural buildings placed on a base. To reflect the functionality of the buildings in the architectural concept, office floors are placed on the top and the shared, outward and more public functions are placed in the base. In terms of sustainability the new Nordea office should achieve the highest score in the international certification system in green building, LEED. This certification evaluates the building in terms of water treatment, energy, materials and indoor climate. Therefor the energy consumption has been continuously simulated through studies of volumes, materials, room heights, light, shadow etc. Each building volume consists of a big covered atrium and in this chapter the thesis will look at implementing the principles of the parametric gridshell. Parameters for optimization in this case include daylight, solar cells, struture, lifecycleanalysis and build ability. This project case contains obvious demands in terms of waterproofing, that the Jukbuin pavilion (Chapter 5) did not have. It is clear that being part of a building envelope, there are different demands compared with to demands of pavilion. The approach defined in part 5, will be used to generate the gridshell as the cover for these atrium spaces. The idea is that the gridshell structure will be using less material, than a planar beam structure. As mentioned the daylight factor inside, and especially in the bottom of, the atrium is a very important parameter. This is of course a matter of both the atrium roof, but also the facade design of the building volumes. The parametric model to be set up here will only focuse on the influence of the atrium roof.
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Fig. 1 Nordea bank  Birdview. Henning Larsen Architects ÂŠ
Fig. 2 Sketch  Nordea bank. Henning Larsen Architects ÂŠ
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Case [1]: Nordea Bank 6.2â€‚â€‚Method As mentioned in the introduction, the approach from chapter 5 is used as the basis of this case work. Additional parameters are implemented in the parametric model. This is done by further development of the Grasshopper definition. Extensions include panelling, solar cell optimization, daylight analysis and fabrication. The many different ways of panelling structures is worth its own thesis. In this case a simple triangular panelling creates the basis of operations as the main objective is to set up a parametric model that contains all parameters.
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Fig. 3 Principle
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Case [1]: Nordea Bank 6.3â€‚â€‚Form studio One of the main concerns in the form finding is to achieve closure of the covering roof. This means a tight connection between building volume and roof structure. On the right hand side a small form studio is carried out. In the Jukbuin pavilion, the grid was very symmetric and a few control points where moved to create the structure. In this case the atrium outlines are all asymmetric. The gridshell structure can be created without regards to the atrium outline, by setting up a regular boundary for the grid. This can be a square, triangle, circular or other boundary. Another way is to create a more irregular grid boundary, that follows the outline of the atrium . By moving control points or single supports towards a centre, it is clear that the edges of the structure will elevate. This leaves an opening and will concentrate forces on the single support elements (Fig. 6). Fig. 7 shows restraining of all edge points in the grid. To obtain more enclosure, all edge points are restrained to only move in the plane of the top of the building volume. This means that the system only moves the grid edges in this plane during relaxation. Acceleration is done by moving the two control points in each corner. In the test in Fig. 8, springs are added from all edge points to their initial position in the planar grid. The spring stiffness can then be adjusted, this is shown on Fig. 9. Fig. 10 and forward shows different configurations of curve attraction. There is a huge variety of ways to control the shape relaxation. Forces, attractions, repulsions, boundaries, properties etc. all have different effects on the structure. The main idea in this case is to try out the concept throughout the whole process. So a simple shape is chosen for further work.
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Fig. 4 Planar grid
Fig. 7 restrain edge
Fig. 10 Curve attraction. Twisted rectangle
Fig. 13 Curve attraction. Two circles
Fig. 5 1 support movement
Fig. 6 2 support movement
Fig. 8 edge springs
Fig. 9 edge springe  less resistance.
Fig. 11 Curve attraction. Circle
Fig. 12 Curve attraction. Deformed circle
Fig. 14 Curve attraction. Two Squares
Fig. 15 Render of Fig. 14.
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Case [1]: Nordea Bank 6.4 Example 6.4.1 Shape The shape chosen for process example is generated by laying out a planar triangular grid. The boundary of the grid is an offset of the atrium outline. All particles along the edge of the grid are attracted to the boundary curve (magenta). This boundary curve is then offset inwards and when the outer particles tries to get nearer to this, the structure is forced to move upwards. Fig. 17. This creates a smooth surface with edges attached to the building volume. As the boundary curve is offset towards the centre, the corresponding maximum curvature in the elements are visualized (Fig. 18).
6.4.2 Construction As mentioned earlier the shape is constructed from slender wood laths in the three directions. Each direction is defined as a group of elements. Here it is chosen to have each group in two layers, creating a total of six layers (Fig. 16).
Fig. 16 Connection.
Because of the simplicity of the triangular mesh, the panelling of the shape can be directly translated. This means that each element can be connected to the wooden structure by the corner points. As each connection follows the surface normal at the intersection, it is imagined that the bolt going through all layers, will also connect to the triangular panels. The structural behaviour will be analysed in the same way explained in the previous chapter, by using Karamba3d and prestressing moments from the initial bending.
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Fig. 17 Generation principle.
Fig. 18 Curvature analysis
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Case [1]: Nordea Bank Dominant snowload:
6.4.3â€‚â€‚Structural analysis Karamba3D
The analysis structure is modelled as described in 5.2.9. To begin with elements with a cross sectionâˆ— 1,0 of 70x70mm All âˆ—endpoints of the đ??şđ??şđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ â„Žđ?‘Ąđ?‘Ą + đ?‘†đ?‘†đ?‘›đ?‘›đ?‘›đ?‘›đ?‘›đ?‘› âˆ—are 1,5 analysed. + đ?‘Šđ?‘Šđ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘– âˆ— 1,5 0,3 wooden laths are considered as fixed supports. Load combinations:
Only very simple load combinations are looked at in this and snow and wind loads are set to 0.8 kN/m2. This is to focus on method and not as much the more detailed documenting analysis. The shape of a structure makes the detailed calculation of especially wind loads complicated, and are therefor neglected here. The following simple load cases are looked at in this example. Dominant snowload: LC1: Only selfweigth LC2: Dominant snow load Dominant windload:
đ??şđ??şđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ â„Žđ?‘Ąđ?‘Ą âˆ— 1,0 + đ?‘†đ?‘†đ?‘›đ?‘›đ?‘›đ?‘›đ?‘›đ?‘› âˆ— 1,5 + đ?‘Šđ?‘Šđ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘– âˆ— 1,5 âˆ— 0,3
LC3: Dominant wind load
đ??şđ??şđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ â„Žđ?‘Ąđ?‘Ą âˆ— 1,0 + đ?‘†đ?‘†đ?‘›đ?‘›đ?‘›đ?‘›đ?‘›đ?‘› âˆ— 1,5 âˆ— 0,3 + đ?‘Šđ?‘Šđ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘– âˆ— 1,5
Selfweight G is calculated by Karamba3d from material and section selection. Snow and wind loads are added as the mesh loads, as mentioned in 5.2.9. This is done by calculating the resultant load on each mesh face and distributing it among the mesh vertices. Then vertex loads are distributed to the structure nodes by considering distance between mesh vertex and structural node. Karamba3d has three different ways of orientating the given load vectors option (c) is chosen, as this prowind loads option (a) is chosen as the force here will act perpendicular to the faces. As it will be shown later, đ??şđ??şđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ â„Žđ?‘Ąđ?‘Ą 1,0each + đ?‘†đ?‘†đ?‘›đ?‘›đ?‘›đ?‘›đ?‘›đ?‘› âˆ— 1,5 enables âˆ— 0,3 + đ?‘Šđ?‘Š âˆ— 1,5 distribuknowing the normal vectorsâˆ— of element, more đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘– detailed tion of loads. This is in connection with wind force coefficients and shape coefficients for snow load.
Dominant Fig. 19 Karamba orientation of loads on mesh. windload: (Fig. 19). For the snow load the orientation (a) Local to mesh (windload) jects the global plane load to the mesh. For (b) Global direction (additional weigth) (c) Projected load (snowload)
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Fig. 20 Structural model
Fig. 21 Utilization
Fig. 22 Displacement
Fig. 23 Resultant moment
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Case [1]: Nordea Bank 6.4.4â€‚â€‚Structural optimization As mentioned throughout this report, optimization depends on the parameters to optimise for. In this case structural optimization determines the shape of the gridshell and the forces acting in this shell determines the size of the elements. Firstly the shape is changed to obtain the smallest maximum span of the elements. This means moving the edge close to the atrium boundary. Compared with the first shape on Fig. 23 on the previous page, the number of elements subjected to moment forces greater than their strength (red) is smaller. Fig. 24 shows that most elements in the structure are able to carry the load needed with the original cross section of 70x70mm. The second part of the optimization is then to select those elements that are subjected to a moment higher than their strength. The forces in these elements are not the same, but by identifying the greatest one, the needed section area can be determined. In this case, adding 30 mm to the outer elements, makes all elements able to withstand the external forces. Other ways of optimization has already been mentioned in the previous chapter 5. In this case it is likely that the approximate expression of the shape is determined by, or in collaboration with, the architect. The principles of generating this shape can then be set up in a script that changes:  Grid: type and size  Boundary: outline, controls This is to evaluate and find the shape that generates the least amount of internal energy in the structure and the smallest maximum moment of all elements.
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70mm Fig. 24 Optimised structure. Resultant moment distribution
70mm
70mm
120mm
30mm Fig. 25 Elements with critical moment distribution
70mm
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Case [1]: Nordea Bank 6.4.5â€‚â€‚Energy analysis DIVA
Sustainability is a really important part of this case project. Therefor when generating complex geometry, it is necessary to be able to evaluate the indoor climate. As it was shown in chapter 2  Case[0], the DIVA plugin for Grasshopper can be used to evaluate the effects of the generated geometry. In a case like this, the daylight factor in the atrium and the rooms at the bottom are important. The model shown only consists of very simple building volumes, but more detailed geometry is easily added to the simulations set up in this script. The shape, type and density of the grid, as well as the size of the elements has a great influence on the analysis of daylight and energy. Therefor the parametric model now contains both the energy and the structural aspects. This is the strength of the collected parametric model. Each change in geometry can easily be analysed from different points of view and decisions can be made on the basis of these. As mentioned in the introduction to this case, the utilization of solarcells is one of the main parameters. The daylight and solar radiation that enters the atrium is highly influenced by placing solar cells on the shell. The first part of the definition on Fig. 27 calculates and analyses the effect of the solar cells. The second part analyses the collected energy effect. Solar cell optimization Energy analysis
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Fig. 26 Daylight analysis
Fig. 27 GH definition  Solarcell + Energy analysis
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Solar cell optimization
6.4.6â€‚â€‚Solar cell optimization The idea of the work with this case is to show how it is possible to combine many aspects of a building project into one parametric model. This enables a precise and coherent workflow. When working with many different models, chances of differences are high. This often leads to incongruence and mistakes. In order to obtain the highest score in the LEED certification, a certain percentage of the energy use, must be covered by sustainable energy. After approximating energy use of the building volume, the amount of energy that is to be gained by solar cells are approximated. Firstly all the panelling on top of the wood lath structure is setup. For each of the panel faces the normal is found (blue arrow on Fig. 30). Then the radiation on each panel is defined by calculating the tilt and azimuth angle. The tilt angle is how vertical or horizontal the panel is and the azimuth is the orientation in regards to south (yellow arrow on Fig. 30) The radiation on each panel is then found by setting up a Python script* that finds the value, based on tilt and azimuth on the background of the table shown on Fig. 30**. The panels are sorted by efficiency and these are used as input in the next script*. By setting the amount of energy gain wanted and the efficiency of the cells, the needed panels are selected. Selection is made in order of most efficient panels first.
Fig. 28 Slider settings  GH def.
*
panelEff (Python) and numOfPanels (Python) see appendix D.
** Data from www.solenergi.dk
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Fig. 29 Sketches  Considerations
Fig. 30 Optimization principle Gridshell.
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Fig. 31 Optimization start
Fig. 32 Optimization00
Fig. 33 Optimization01
Fig. 34 Optimization02
37393 kWh Fig. 35 Optimization02  Reduction random seed 1
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Fig. 36 Optimization02  Reduction random seed 2
Case [1]: Nordea Bank Fig. 37 Parameters
The illustrations on the left hand side shows screen captures. The slider values are changed in the Grasshopper script and the response can be evaluated immediately. Evaluation is based on energy gain, daylight, solar radiation and aesthetics. The main settings for the solar cell optimization is the 6 sliders shown on Fig. 37. VecX and VecY describes the south vector, thereby the orientation of the building. The slider Energygain (kWh) is to set the amount of energy gain wanted and the slider Cell efficiency (%) is to set the efficiency of the solar cells. Fig. 32  Fig. 34 shows how the amount of selected panels increases as the setting of the needed energy gain is increased. As the amount of cells increase, the covered area of the grid increases. Both in terms of aesthetics and daylight distribution, a large closed surface is not desirable. Therefor a random reduction is introduced. This picks holes in the surface and lets light in. The second last slider, Reduce by (num), is the number of cells to remove. Removed cells are replaced by glass panels. This lets light in and the daylight can be evaluated in real time. Seed slider is a setting that controls the randomness of the reduction. If one pattern is not desirable, another one is generated by changing the seed value. It is clear that by setting the Energygain value to 50.000 kWh and reducing the number of chosen panels, the final energy gain is going to be less than 50.000. In the example  Optimization02  when reducing by 137 panels, the final energy gain becomes 37393 kWh pr. year.
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Case [1]: Nordea Bank 6.4.7â€‚â€‚LCA  Life cycle analysis The LEED certification has certain demands to life cycle analysis (LCA) that has to do with material usage, production, disposal, emission, durability, lifetime, etc. As all informations about the geometry are quickly extracted from the model, it can be exported to spreadsheets, schedules, LCA software etc. This gives an extra parameter to evaluate and design for. Sustainability, lifecycle and economy is greatly influenced by the shape, structure, connection design etc. and it is therefor of great importance to know the influence of these parameters. If a spreadsheet is set up with fixed values for cost of materials, connection types, labour pr. unit etc. fast approximate values for economy and time can be viewed. This data might come from V&S Prisdata* or other national indexes. 5D BIM management software like VICO office links 3D models from e.g. Revit or ArchiCad and also imports other file types available from Rhino3D. This means that in terms of planning and evaluating, the workflow might look like this: Sketch Karamba3d
DIVA Rhino3D + Grasshopper
BE10
VICO office Revit
Robot SA
IESve
* http://www.byggecentrum.dk/produktadgang/vsprisdatastartside/?L=0
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WOOD STRUCTURE Length
7014,55 m
Volume
34,7 m3
Density
600 kg/m3
Total weight
20820 kg
Lifetime
... years
Cost
... Dkr
GLASS STRUCTURE Area
1117,5 m
Total weight
...kg
Lifetime
... years
Cost
... Dkr
SOLAR CELLS Area Energy gain Lifetime Cost Payback time
270 m2 29349 kWh/y ... years ... Dkr ... years
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6.4.8â€‚â€‚Fabrication Throughout this project the principle of think, script, build has been kept in view. The main issue is to maintain the data structure, so that information about the final elements can be used. To generate a fabrication sheet, the elements has to be unrolled and ordered in plane. After this an ID is attached to the unrolled geometry. Next the intersections from the 3D geometry needs to be interpreted to the unrolled elements and for each element a short description of which other elements to connect at this location. In terms of scripting: The GH multiple intersection component outputs 2x2 lists that describes each intersection, by listing the first intersecting curve in list[1], the second curve of the intersection in list[2], the corresponding parameter at this first curve in list[3] and the parameter for the second curve in list[4]. This information is translated into a datatree, where each branch holds information about the given intersection; elementID of the actual element, elementID of the other element and parameter of the intersection along the actual element. See appendix 5 for details and Python scripts. That means that a complete fabrication sheet is generated for each solution. The idea is that the detailed drawing makes it easy to produce, assemble and construct the gridshell. In this case the shape comes from deforming a planar grid. Depending on the size of the gridshell and the elements, the forces needed to deform the grid, might be hard to generate. If not possible by hand, some kind of manual or hydraulic guide rail system can be used. If it is too complicated to setup the required system to generate the deformation of the planar grid, joining and weaving can be done over a scaffolding system. A similar procedure as used in the construction of The Savill Gardens Gridshell* *
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The Savill Gardens Gridshell  Glen Howells Archiects. Wood for good [2]  Chapter 2
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Case [1]: Nordea Bank 6.4.9â€‚â€‚Second approach Earlier it was mentioned in chapter 5.1.2, that this thesis focus on the method of approach 1. From grid to shape. The second approach  2. Shape to grid will be explored shortly here. It is very likely that the architect has a clear idea of the shape that is wanted. So another way of utilizing the bending of simple elements, is to lay curves on top of a given surface. These curves can be geodesic curves, which have the property of unrolling to a straight line. This means, as mentioned previously (chapter 5.2.4), that the elements will not be subjected to local multiaxial bending in erection of the structure. Curves can also be principal curvature lines, projected curves or curves pulled towards the surface, but the torsion of the curve has to be evaluated as well as the curvature. These curves can be drawn in regular or random patterns. After laying the grid based on the surface, the procedure is the same as for approach 1. Similarly to this approach, optimization can be carried out by changing the shape, the distribution of the curves etc. The fitness in this case can be structural stability, daylight distribution, build ability and so on.
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5
6
7
ANALYSE
3 4
ELEMENTS
CURVES
2
FABRICATE ENERRGY STRUCTURE
SHAPE
1
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Here is a short example of the workflow for the second approach: A freeform surface is created by use of 4 parametric edge curves.
By dividing opposite edges into a number of points and shuffling the order in these point lists, seemingly random geodesic curves are drawn on the surface. Here the Galapagos solver looks for a configuration with the largest distances between intersections. This is for fabrication purposes. The curves are turned in to structural members and analysed by using Karamba3d.
Stresses in the elements are analysed real time, when changing the surface or configuration.
Here the Galapagos solver is set up with the fitness value as the smallest max. resultant stresses in the elements. This is done by looking at different configurations of the geodesics.
Here the shape is changed and the structural response is visualized real time.
Fig. 38 Second approach. Freefrom surface elements.
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Fig. 39 Render of final shape from example.
A third possible approach is the combination of the two different approaches. In the below example, random geodesics are drawn on top of the approximate surface created by the deformed square grid.
Fig. 40 Approach 2.1. Combination of approach 1 and 2.
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Fig. 41 Interior render  original Henning Larsen Architects ©. Roof edited by Rasmus Holst
6.4.10 Part conclusion This case study has explored how many different parameters can be combined in one model. This concept can be used for optimization in regards to different aspects. When adding a solver as Galapagos, many iterations can be done automatically and best solutions can be pointed out.
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The result of the chosen example is a different, curved and lightweight cover for the atrium. It is made out of a very sustainable structure, in terms of materials, production and economy. With a construction weight of approx. 21000 kg* and elements that are very simple to produce, it is a worthy competitor to a more regular steel beam solution**. *
Analysis carried according simplified load cases and not considering detailed connections etc.
** 1/30 of span (30 m) â‰ˆ 1m CS height. One beam approx. 9400 kg. with 2m between a total of 141.300 kg. HE..B Profile  Teknisk stĂĽbi.
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Case [1]: Nordea Bank
Fig. 42 Birdview render  3D model volume Henning Larsen Architects ÂŠ. Roof and render illustration by Rasmus Holst
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CASE [2]: PAVILLION
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Pavillion  Proof of concept
Rhino
Grasshopper
Python
Kangaroo
Galapagos
DIVA
Karamba3D
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7  Case [2]: Pavillion 7.1 Introduction 7.1.1 Proof of concept The aim of this third and last case is to test and prove the concept worked out through the thesis. In this case focus is on build, which means that a detailed fabrication preparation is to be generated. There are two main interests of investigation:
1. Fabrication. The efficiency of fabrication without special production tools by detailed and systemized fabrication sheets.
2. Construction. Elements behaviour in a 1:1 model with real materials.
The idea is to design and construct a small pavilion, with a complex geometry created by straight wooden plywood boards. Often the concept of gridshells is considered as roofs and domes. Shell structures however, does not necessarily have to be considered like this. They do have a the advantage of being very material economic, if constructed correctly, in terms of avoiding bending moment. From this perspective it is clearly interesting to look at shells as roofs and horizontal covering. Here the shell will be considered in creating vertical enclosure. This is done by looking at how the concept can be used for the construction of curved facades and/or internal walls, etc. The procedure will be to follow a combination of approach 1 and 2. A planar grid is generated and deformed using particle attraction to control curves. After deforming the grid into a desired shape, geodesic curves are layed on top of the approximate particlespring surface and elements will be a mixture of geodesics and particlespring elements.
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Fig. 1 Pavilion variation render.
Fig. 2 Outline of stabile and stiff shel configurations  Henrik Almegaard. [12]
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7.2â€‚â€‚Concept Structure and threshold is generated by strands. This creates a diffuse division between rooms or between inside and outside. Structural stability comes from a network of flexible elements. The idea is to create a dynamic double curved shape that creates a transparent enclosure. The differentiation between wall, floor and roof is sought to be eliminated by deforming the smooth surface into multiply functions. The transition between e.g floor and wall is washed out by the curvature of the surface. By deforming a planar rectangular grid, a basic structural stability is ensured. The initially straight elements in the grid are bend round to create the rounded enclosure. Fig. 3 Planar grid
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Longitudinal elements are considered stiff, in the sense that they resemble thin wooden laths with a certain resistance of bending. The shorter cross elements are considered as flexible springs. These have much less spring stiffness and bending resistance. They are to be thought of as soft elements that allows the longitudinal elements to behave more freely, but ensures coherence in the network.
Fig. 4 Animate networks (UTS)  Chris Bamborough and Rasmus Holst
Fig. 5 Sketches
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Fig. 7 Relaxed grid
Fig. 8 Curvature
Fig. 9 Curve surface
7.3â€‚â€‚Form studio Parametric shape and stability investigations are carried out. Three control curves are chosen as attraction curves. The attraction is created by a force between particles along the top element and a top curve. The same goes for particles in the middle element and bottom element.
Fig. 6 Grid. Control curves
This attraction allows for control of the deformation of the grid. Each of the three curves consists of a number of control points. By moving the control points, the grid is wrapped around itself to create an enclosure that allows for entrance. All shape configurations are evaluated firstly in terms of bending curvature. This means only shapes with valid curvature in the elements are considered. Four main variations are considered: var[0]  the first variation creates an inviting vertical enclosure and a space along the long edge coming out of the volume. Only very little horizontal cover is created. The two short edges stand alone and curvature of the surface is only ensured by the relationship between element length and connections. Support is possible along the bottom only and the two top corners are not supported. var[1]  In the second variation, the shape wraps around and the top element connects to itself. Here three corners are supported. var[2]  The third variation has the same top wrapping and connection. But at the edge of the long side, the top is flipped down and creates something of a seating furniture. At the same time all four corner points are supported. var[3]  In the fourth variation the far short edge is flipped over to create opening and seating. In this variation three corners are supported.
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Fig. 10 Short grid elements replaced by geodesic curves (magenta)
PAV 
PAV 
Fig. 11 Random interlocking geodesics (cyan) are added
PAV 
var[0]
PAV 
var[2]
Fig. 12 Wood elements
var[1]
var[3]
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7.4 Structure As mentioned, the aim of this case if to look at buildablity. Therefor no detailed load cases are considered. First of all the structure has to be able to carry its self weigth, when constructed in one layer. If the function of the pavilion changes after construction, another layer can be added.
7.4.1 Loads For a thin construction like this, vertical loads, like snow load, on parts of the structure might be critical. Here the vertical loads are defined by a mesh. The mesh faces that take up loads are the ones that are mostly horizontal. To define these, the normal of each face is evaluated. The algorithm firstly removes the faces with normal vectors below horizontal from the load list. Afterwards the angle Φ (Fig. 13) is calculated. A conditional script determines which mesh faces that shall receive loads. The Eurocode* for snow load on structures state that the shape coefficient of an angle of pitch on a roof above 60° equals zero. See Fig. 14  as the angle of the normal changes, the number of considered meshes changes.
7.4.2 Asymmetric load When analysing structures like this during external vertical loading, cases of asymmetric loading has to be considered. This means that only certain areas of the structure are loaded. Here the algorithm allows users to set points in the Rhino viewport, then mesh faces in an adjustable distance are selected for loading (Fig. 15).
7.4.3 Random geodesic The distribution of the random curves locks the structure. The interlocking comes from intersection of the rectangular grid. The distribution is carried out by using a algorithm for removing a random pattern of a datalist. By using the Galapagos solver, the random distribution with least close connections is found. This is done out of construction concerns. For a given load case a similar approach can be used to obtain best structural result. These curves are put on two separate offset layers to avoid element clashes. *
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Eurocode 1 [4]
ÎŚ
Fig. 13 Mesh normal evaluation
Fig. 14 Load area,
Fig. 15 Custom load areas.
Fig. 16 Geodesic distribution optimisation
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7.5 Fabrication 7.5.1 Fabrication sheet All elements in the structure is unrolled and put into system. As described in chapter 6.4.8, all elements contain an element ID, all connections are placed at the correct parameter of the element and a description at each connection of which other element(s) it connects to. Now this is taken a bit further, as the elements are to be made out of strips cut from plywood boards of 1,20m x 2,40m, with a thickness of 9mm. 2 sheets are generated for printing. Sheet 1 is an overview of all elements. Sheet 2 is a description of each element and all part elements. Part elements have a maximum length of 2,40m and is connected to create the full element. Each part of an element is connected by an overlap of 15 cm with two bolts.
7.5.2 Sticker sheet For each connection a small illustration is printed out on a sticker sheet. The script places the groups of text describing each connection in an array that fits with a A4 sticker sheet (Fig. 20). Each sticker contains information of which element it belongs to, which part element, the connection ID and the main info is which other elements that are connected to this (Fig. 21).
7.5.3 Procedure Sufficient number of strips in lengths of 2,40m are cut out of the plywood boards. The craftsmen uses the element sheets (sheet 2) to determine lengths and cut out part elements, drill holes, attach stickers at each hole and finally connect part elements into whole elements. Long elements may have to be assembled continuously in step with connecting and erecting the structure.
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Fig. 17 Fabrication elements
Fig. 18 Sheet 1  All elements overview
Fig. 19 Sheet 2  Element description pr. element.
Fig. 21 Sticker closeup
page 175 Fig. 20 Sticker sheet. Connection description
7.6 Part conclusion 7.6.1 Final design Variation 3 was chosen as the final design. This solution has an interesting formal language and the shape challenges the use of simple elements. The four corners of the original grid is supported and that means a higher degree of stability of the shape than the other proposals. The final design fulfils the demands set up to begin with. It ensures a transparent enclosure. At the same time the geometry of the pavilion acts as both walls, to some extent as roofing and as a seating furniture at the low end. Structural stability is obtained for self weight by the one layer construction. The structure of the pavilion in one layer is meant as a proofofconcept and is thereby intended for indoor use. The corresponding two layer structure has been evaluated an here structural stability is obtained for two simple load cases of dominant snow load and dominant wind load. Thereby a second layer can be added to the structure and it can be evaluated more thoroughly if the function of the pavilion changes. The pavilion is sponsored by the sustainable building festival Det grønne pakhus, Bornholm, and will be a part of the exhibition in the summer of 2013. First tests of the physical model showed promising results using 0,5mm plastic strips in a 1:10 model (Fig. 22). The curved geometry her comes from the internal relations between the elements. Here no edges are fixed and curvature is obtained by the changing length between connections.
Fig. 22 Physical model test var[00]
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The production of the elements and the building of the pavilion is to be carried out after delivery of this report. It is going to be an experimental prototype in 1:1 to see if the any unforeseen consequences is to be taken into account in future work.
Fig. 23 Render final design
Fig. 24 Connection detaildesign 1:5
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REFLECTION
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Perspective Conclusion
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8  Reflection 8.1 Discussion 8.1.1 Gridshell structures Many building structures tend to be heavy and inefficient, but sometimes lightweight structures are sought. In these situations, the geometry is crucial. For gridshells, other shellstructures and domes the idea is to find a geometry that achieves lightweight members with no or only small amounts of bending stresses. The principle of gridshells utilizes simple elements and can achieve geometries for optimised structures. The variations of the geometry are many and can be adjusted to fit many different purposes. To obtain bending free members, only certain geometries apply, but the principle still has advantages in solutions where bending moment will occur in the elements. The advantages lies in the use of simple elements and the prestressing of these from the initial bending, if constructed in the correct way. The method used in this thesis allows for an expansion and investigation of the theory of shell structures. By taking the starting point in research results like Skalkonstruktioner* [12], the theory can be used for a wide range of architectural investigations. That leads to the idea that gridshell structures, does not only have to be thought of as roofs, but could also be interesting to look at as facades, floors, walls, whole building envelopes and/or a mixture of more functions. This way organic shapes can be created out of simple regular elements. The use of geodesic curves, can set limits to a desired structural pattern. By being able to evaluate and restrict both normal and geodesic curvature of elements, the limits of the patterning are pushed. This means that we can allow some torsion, because we are able to evaluate consequences. Fig. 1 Phoenix international Media center.jpg *
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Eng: Shell structures  Henrik Alemgaard Phd.
8.1.2 Architectural engineering. One of the advantages of architectural engineering is the ability to bridge the cap between architects and engineers. It is the way of speaking both languages that allows for understanding, collaboration and therefor better projects. In form finding for example; for the architect form follows function, for the engineer form follows force. In reality form follows function follows force. As mentioned in the background chapter (page 8), nature’s structures are very optimised both in terms of structure and function. Therefor it is a great source of inspiration. Looking very simply at a tree for example, it spreads out its leafs by growing branches to obtain the function of photosynthesis. To accommodate for the growing of branches structurally, it grows networks of roots that resist wind forces.
Fig. 2 Mesonic fabrics  Biothing
8.1.3 Parametric design These optimised structures of nature are constructed through evolution and emergence. They react to the surrounding influences over time and adapt. In the construction industry, we do not have time to let our projects grow and slowly adapt. Fig. 3 Sadic (T)ropisms  Juan Francisco
However by simulating the wanted behaviour and the influences of a given project, it is possible to find an adapted and optimised solution. This simulation can be done through scripting. Many amazing projects, that have arisen from the digital age, by talented architects have a hard time being turned into buildings and/or building components. The people behind are often very good at reaching an appropriate level of abstraction into the architecture, but the projects often lack the buildability aspects. Therefor these project often ends up as 3D print models and art installations.
Fig. 4 Encoded behaviour  Digital crafting
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8.2â€‚â€‚Perspective These architectural thinkers are very much needed to setup these abstractions and to create ideas and vision. If the knowledge of the engineers can be taken to the same level of abstraction and combined with the ideas of the architects, very beautiful, interesting, optimised and buildable projects can arise. Innovative engineers such as Rohe, Fuller, Nervi, Otto etc. are good examples of this at their time. In recent days the likes of Jan Knippers (KnippersHelbig) and Cecil Balmond (Arup) combines their knowledge of materials, physic behaviour, math and geometry to create amazing projects together with architects. By adding more technical aspects in the parametric model, these digital projects can evolve further and adapt to more influences. This will make such projects very interesting in terms of programme, aesthetics, buildability and optimization. To be able to create this synthesis of collaboration, there needs to exist a motivation, inspiration and understanding of abstractions from the engineer. If this exists, scripting and parametric design is a great tool for developing exciting projects. The scene of scripting and the whole area around parametric design is in a rapid development. Rhinoceros 5, that contains the Rhino.Python scripting language, used in this thesis, was only finally released january 2013. There is a constant development of software and plugins. Much of the development can lead to cross platform collaboration. Within the scripting community there is a constant energy and an open source idea. This allows for everyone with an interest to think, share, discuss and develop.
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The plugins for Grasshopper used in this project are still very much work in progress and the people being are very helpful and open to ideas on how to develop their software. There has been a lot of contact with Daniel Piker of Kangaroo and the team behind Karamba3d on many occasions. Discussion between users and developers will lead to future versions of Kangaroo that will contain output of reactions, implementation of real world material units, 6DOF systems, curvecurve attraction, lineline collision etc. Some of these ideas might have been on the drawing board for a while, others have arisen through discussion throughout this thesis. Also a closer collaboration between Kangaroo and Karamba3D has been discussed. The future of parametric design is extremely interesting and will keep developing, as users and developers join in developing useful tools. Already Grasshopper holds tools of Geometry Gym that allows for interaction with BIM modelling software such as Revit and other external analysis software. This opens up to a very interesting workflow between collaborative partners. This thesis has focused on gridshells using spline geometries. However the setup of the mesh particlesystem for plates in chapter 4, leads to the idea of using bend planar elements for interesting curved modules. The development is not only happening on the software scene, also the production industry is changing and in many ways to accommodate and utilize the new digital tools. New phenomenons like augmented reality*, and interactive hardware tools such as Arduino** and software Firefly** will change the way we communicate and present our projects. Firefly allows for real time adjustments of the parametric model by use of hardware remote controls or through applications for Iphone, Android etc. It also works the other way, so that the model can controls physical models. *
Fig. 5 Augmented reality  Iphone
Fig. 6 Hardware slider for GH
Augmented = Increased reality.
** www.arduino.cc and www.fireflyexperiments.com
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8.3â€‚â€‚Conclusion These new tools and methods are great future possibilities of collaboration within the building industry. It takes a slightly different mindset to work with parametric modelling compared to more conventional methods, but also extents the possibilities. By collecting several parameters into one model, optimization can be carried out in relation with structure, energy, architecture, build ability etc. Parametric design does not mean that everything has to be curved and complex geometry. The use of scripting are also very useful for investigation and optimization of more regular geometries as shown on case[0].
Fig. 7 Zollverein  SANAA.
The problem statement was to look at possibilities of simulating bending of elements to be able to construct advanced geometry. In the process from sketch to built project, the behaviour of elements was simulated using dynamic relaxation by using scripting in Grasshopper and Kangaroo. This was the basis for the project to set up a method for using simple elements for advanced structures. Combining simulation of behaviour, differential geometry and engineering theory in scripting enabled the creation of the two approaches  1. From grid to shape and 2. Shape to grid. These two approaches where test on the specific case[1]  Nordea Bank. It was shown that a lightweight gridshell structure can be generated by using this method and that the parametric modelling method can extent to fulfil many different parameters in architectural projects. Finally the scripting of intelligent systems proved to enable the construction of complex geometrical shapes using simple building elements. To conclude the author believes that this thesis has shown and proven the many possibilities of scripting and parametric design.
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STRUCTURE
BUILD ABILITY
ARCHITECTURE
ENERGY
THINK, SCRIPT, BUILD 2012 page 185
9  References 9.1 Bibliography [1] Tensigrity spline beam and grid shell structures  S.M.L Adriaenssens, M.R. Barnes. Engineering Structures 23. (2001) [2] Wood for good  Olga Popovic Larsen, Daniel SangHoon Lee. 2011. Chapter 2, page 15f. ISBN: 9788778302663. [3] Karamba3d manual for version 1.0.1  Clemens Preisinger (2012)  http://www.karamba3d.com/downloads [4] Eurocode 1  Actions on structures  Part 13: General actions  Snow loads. DS/EN 199113. (2007) [5] Kangaroo Manual (Grasshopper version)  Daniel Piker. (2012)
Google docs. (http://www.grasshopper3d.com/group/kangaroo)
[6] Python for Rhinoceros 5. Rhino Python Primer 1st Edition. Rev. 3. S. Tibbits, A. van der Harten and McNeel. (2011) (http://python.rhino3d.com) [7] Digital fabrications  Lisa Iwamoto. (2009) ISBN13: 9781568987903 [8] Advances in Architectural Geometry 2012  Hesselgren, Sharma, Wallner, Baldassini, Bompas and Raynaud. (Editors). (2012) Springer. ISBN: 9783709112502 [9] Form Geometry Structure  Daniela Bertol. (2011) Bentley.
ISBN: 9781934493113
[10] Teknisk ståbi, 21. Edition (2011)  red. Bjarne Chr. Jensen. Nyt teknisk forlag. ISBN: 9788757127294 [11] Elementary Differential Geometry  Andrew Pressley (2008).
Springer. ISBN: 1852331526
[12] Skalkonstruktioner  Henrik Almegaard (2003).
SBI. ISBN: 8786311656
[13] Advances in Architectural Geometry 2010  Cristiano Ceccato, Lars Hesselgren, Mark Pauly, Helmut Pottmann, Johannes Wallner (Editors). (2010) Springer. ISBN: 9783709103081 page 186
9.2â€‚â€‚Resources Grasshopper3D : www.grasshopper3d.com Kangaroo : http://www.grasshopper3d.com/group/kangaroo Galapagos : http://www.grasshopper3d.com/group/galapagos Python : http://www.grasshopper3d.com/group/rhinopython + http://python.rhino3d.com DIVA4Rhino : http://diva4rhino.com/ Karamba3d : http://www.grasshopper3d.com/group/karamba + www.karamba3d.com
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References 9.3 List of figures 9.3.1 Figures 0 Fig. 1 Structures in nature:
www.mantamola.blogspot.com http://www.flickr.com/photos/80682926@N00/21175586/ http://m.videobash.com/picture/view/id/46741/title/lots_of_fish_in_the_sea http://www.flickr.com/photos/kingdomanimalia http://www.flickr.com/photos/suteki/4736358984 http://blogs.discovermagazine.com/notrocketscience/2012/10/08/brainlesscreature solvesproblemswithmemoriesofslime/ http://www.flickr.com/photos/elycefeliz/6792146961/
Fig. 2 Gaudi chain model Fig. 3 Frei Otto  Optimized path Fig. 4 Wood materiality
http://physicsworld.com/cws/article/news/2010/jul/06/birdsflockwithscaleinvariance http://europeanescapades.blogspot.dk/2007/09/catenarymodel.html experiments http://somemarkets.wordpress.com/tag/precedents/ http://texturelib.com/#!/category/?path=/a/wood/ends
9.3.2 Figures  Chapter 1 Fig. 1 MIes van der Rohe Photo courtesy Chicago Historical Society) Fig. 2 Buckminster Fuller http://www.sugarbarons.com/?attachment_id=4141 Fig. 3 Pier Luigi Nervi http://www.popscreen.com/p/MTQ1MDY3NjQw/PierLuigiNervi%E2%80%93 OlympicSportsPalace1960AquaVelvet Fig. 4 Frei Otto http://www.smdarq.net/casestudymannheimmultihalle/ Fig. 5 space ruler.png www.linux.com Fig. 6 Computation time pics: http://zaccus.com/page/2/ http://www.jvindians.com/history%20of%20abacus.php http://www.thocp.net/companies/att/att_company.htm http://ungoodthink.net/category/computing/ http://free.tcad.cz/cad_historie.html http://www.hp9845.net/9845/history/products/index.html http://buybetterinform.com/category/autocad/ Fig. 7 + 8 SketchPad Demo http://www.youtube.com/watch?v=mOZqRJzE8xg Fig. 8 Ivan Sutherland http://amturing.acm.org/photo/sutherland_3467412.cfm
9.3.3 Figures  Chapter 2 Fig. 1 New main office  Fig. 2 Main concept. Fig. 3 New main office 
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Birdview. Henning Larsen Architects © Henning Larsen Architects © Collaborative zone. Henning Larsen Architects ©
9.3.4 Figures  Chapter 3 Fig. 1 Indigo Deli. Sameep Padora & associates. http://www.designboom.com/architecture/sameeppadoraandassociatesindigo deli/ Fig. 2 Serpentine Pavilion 2005. Siza, Moura & Balmond. http://info.elcorreo.com/blogs/fotosactualidad/2011/04/01/descaradoeintimo/ Fig. 3 Metropol Parasol, Sevilla. Jürgen MayerHermann. http://www.dezeen.com/2011/04/26/metropolparasolbyjmayerh/ Fig. 4 Olympic stadion Beijing. Herzog & de Meuron. http://www.readthesmiths.com/articles/travel/Amazing_Olympic_Architecture_ in_Beijing_China Fig. 5 Richard Sweeney  Paper sculpture http://www.core77.com/blog/materials/paper_sculpture_3680.asp Fig. 6 Andrea Russo  Origami tessellation http://www.flickr.com/photos/9874847@N03/2655752110 Fig. 7 Juergen Weiss  Barcelona Block http://rethinkarchitecture.blogspot.dk/ Fig. 8 rvtr  Resonant chamber. http://rvtr.com/research/catalogue/ Fig. 9 Ryuichi Ashizawa Architects  Folded Plate Hut http://www.designboom.com/architecture/ryuichiashizawaarchitectsbamboo forestandhutswithwater/ Fig. 10 + 11 Roskilde dome  Tejlgaard + Almegaard http://www.architizer.com/en_us/blog/dyn/31752/geodesicdomeroskilde/#. USPL06V1F8E Fig. 11 BOWOSS Bionic Pavilion  Saarland University http://www.aasarchitecture.com/2012/09/bowoossbionicpavilion.html Fig. 12 ICD/ITKE Research Pavilion 2011 http://icd.unistuttgart.de/?p=6553 Fig. 13 Hermes Boutique  RDAI http://www.aboutdesign.fr/rubrique/visitesprivees_r5/hermesrivegauche_ a677/1 Fig. 14 Digital Weave  IwamotoScoot http://mikeeggers.com/digital%20weave.htm Fig. 15 Timber Fabric  IBOIS http://ibois.epfl.ch/page20702en.html Fig. 16 Stripmodel test Master Thesis  Jacob Drachmann Fig. 17 Eclaireur Paris  Arne Quinze http://www.dezeen.com/2009/10/06/leclaireurparisstorebyarnequinze/ Fig. 18 ICD/ITKE research pavilion 2010 http://icd.unistuttgart.de/?p=4458 Fig. 19 Uchronia  Arne Quinze http://www.tumblr.com/tagged/arne%20quinze Fig. 20 Roof installation  Arne Quinze http://artistaday.com/?p=2365 Fig. 21 CityScape  Arne Quinze http://www.designverb.com/2007/09/19/cityscapearnequinze/ Fig. 22 Aggreation Anenom  Dave Vu and David Pigram http://davidvu.com/29295/301314/projects/aggregationanenom Fig. 23 InOut Curtain  IvamotoScott http://www.iaacblogbackup.com/2008term01/course05/?cat=70&paged=5 Fig. 24 Paper Art  Richard Sweeney http://www.richardsweeney.co.uk/paper.html Fig. 25 Voussoir Cloud  IvamotoScott http://www.flickr.com/photos/stephalin/3140286500/ Fig. 26 Harpa Concert Hall  Henning Larsen Architects www.henninglarsen.com Fig. 27 Bent Wood Exoskeletons  .Joel Letkemann http://www.joelletkemann.com/ Fig. 28 Differentiated Wood Lattice Shell Huang + Park http://gridshell.net/ Fig. 29 Centre Pompidou, Metz  Shigeru Ban + Arup http://www.architecture.com/Awards/RIBAAwards/2012/EUAwards/CentrePompi douMetz/CentrePompidouMetz2.aspx Fig. 30 The Savill Gardens  Glen Howells Architects http://inhabitat.com/savillbuildingscurvaceouslocallysourcedwoodroof resemblesagiantfallenleaf/ Fig. 31 Centre Pompidou, Metz  Shigeru Ban + Arup http://www.thedailytelecraft.com/2012/07/thecentrepompidoumetzart around.html Fig. 32 Gridshell Digital Tectonics  Smart Geometry ‘12 http://matsysdesign.com/category/projects/ Fig. 33 Centre Pompidou, Metz  Shigeru Ban + Arup http://whoyouwe.wordpress.com/2012/04/18/whoyouweartistspotlightshigeruban/ Fig. 34 Tripudio Bestia  Matthias Pliessnig http://www.matthiasstudio.com/ Fig. 35 Allotropic System  Nicholas Bruscia http://cita.karch.dk/Menu/Projects/Digital+Formations/Lamella+Flock+(2010) Fig. 36 Centre Pompidou, Metz  Shigeru Ban + Arup http://cva.ap.buffalo.edu/mac/?page_id=25 Fig. 37 Resoloom  Peter Vikar http://theverymany.com/publicart/11stlouis/_mg_4888_ps_tvm_s/ Fig. 38 Lamella flock  CITA http://www.centrepompidoumetz.fr/en/twoambitions Fig. 39 Plasti(K) Pavilion  Marc Fornes THEVERYMANY http://www.flickr.com/photos/32333197@N07/3463770257 Fig. 40 Dorian Pattern Facade  Khiem Nguyen http://www.petervikar.com/cd/resoloom/ Fig. 41 Migration  Matthias Pliessnig http://www.matthiasstudio.com/ Fig. 42 Hello wood http://www.hellowood.com/ Fig. 43 Eclaireur Paris  Arne Quinze http://www.dezeen.com/2009/10/06/leclaireurparisstorebyarnequinze/
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9.3.5 Figures  Chapter 5 Fig. Fig. Fig. Fig.
1 Illustration  CODA + UPC 2 Jukbuin Pavilion  CODA + UPC 3 Eccentricity definition. 4 Curvature of a curve.
http://www.designboom.com/architecture/codabarcelonatechjukbuinpavilion/ http://www.designboom.com/architecture/codabarcelonatechjukbuinpavilion/ (Karamba3d manual fig. 25) Elementary differential geometry [11]
9.3.6 Figures  Chapter 6 Fig. Fig. Fig. Fig. Fig.
1 Nordea bank  2 Sketch  Nordea bank. 3 Karamba loads 4 Interior render  original 5 Birdview render  3D model volume
Birdview. Henning Larsen Architects © Henning Larsen Architects © Karamba3d manual Henning Larsen Architects ©. Roof edited by Rasmus Holst Henning Larsen Architects ©. Roof and render illustration by Rasmus Holst
9.3.7 Figures  Chapter 8 Fig. 1 Phoenix international Media center.jpg http://www.designboom.com/architecture/biadufophoenixinternational mediacenter/ Fig. 2 Mesonic fabrics  Biothing www.biothing.org Fig. 3 Sadic (T)ropisms  Juan Francisco www.juanfrans.com Fig. 4 Encoded behaviour  Digital crafting www.digitalcrafting.dk Fig. 5 Augmented reality  Iphone http://www.guardian.co.uk/technology/2010/mar/21/augmentedreality iphoneadvertising Fig. 6 Hardware slider for GH http://www.grasshopper3d.com/photo/hardwareremotecontrolv2 Fig. 7 Zollverein  SANAA. http://www.dac.dk/da/servicesider/presse/pressearkiv/2010/sanaajapansk stjernearkitekturpaadac/
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Thesis Autumn 2012. M.Sc. Architectural Engineering.
Rasmus Holst in collaboration with
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