Parametric Modelling for Point Support Optimisation of Plate and Shell Structures

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M.Sc. Thesis Master of Science in Engineering

Parametric Modelling for Point Support Optimisation of Plate and Shell Structures

Daniel Kolling Andersen (s080068)

Kongens Lyngby Juli 2015

Abstract This thesis deals with the development of a tool intended for automatic optimisation of point support distributions for plate and shell structures. In an architectural context point supported plate or shell constellations are typically represented by doubly-spanning, in situ concrete systems carried by columns. Correspondingly, the tool is developed with in situ concrete systems in mind. However, the tool appliance is not restricted to concrete as material choice, although it is designed for a doubly-spanned load carrying structure consisting of an isotropic material. The tool is targeted appliance in an early, architectural design setting, and therefore generation of multiple candidate solutions are in focus in order to enhance design freedom. The script will handle any given surface geometry, which may be designed by the architect, as representative for the plate/shell structure itself. Regarding the support conditions, the tool operates with free and fixed supports as well as support restricted areas. Free supports will undergo optimisation in their distributions whereas fixed supports will always be kept in their given positions. Restricted areas can be defined on the surface domain where supports are not wanted for practical or architectural reasons. The kinematic boundary conditions may be set freely by the user, who can choose among six degrees of freedom (DOFs) per support; three translations and three rotations. The proposed tool is based on parametric modelling, the finite element method (FEM) with appliance of linear elastic theory and optimisation through genetic algorithms, such algorithms use stochastic search methods and are not gradient based. All parts are merged together in one parametric modelling environment to ensure a smooth flow between geometry generation, structural analysis and optimisation. However, the smooth flow is challenged along the way, why necessary compromises are developed. The optimisation routine optimises the point support distribution on basis of a fitness function, which is defined by the user, this literally means that anything, which may be represented by a number, can be made target of optimisation. Among more obvious choices for fitness functions one could choose the deflection or internal energy of the structure. Furthermore the routine does not only work for single objective optimisation, but also for multiple objective optimisation meaning that more fitness functions may be dealt with simultaneously in the optimisation routine, for instance, one could run an optimisation having both the deflection and internal energy as fitness functions at once. This multiple objective functionality in particular is reckoned a potential catalyst for discovery of various, relevant design alternatives. The tool performance is thoroughly tested before finally applied to a relevant, architectural case for demonstration of its practical potential.

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Preface Project type:

Master’s Thesis

Field of study:

Architectural Engineering

University:

Technical University of Denmark Department of Civil Engineering Brovej, Building 118 2800 Kgs. Lyngby, Denmark

Comments:

This report is a part of the requirements to achieve the MSc degree in Architectural Engineering at Technical University of Denmark. The report represents 30 ECTS points.

Project period:

02.02.2015-02.07.2015

Author:

Daniel Kolling Andersen, s080068

Academic supervisor:

Jan Karlshøj

Industry cooperation:

Søren Jensen Rådgivende Ingeniørfirma A/S

Industry supervisors:

Andreas Bak Andreas Castberg

Class:

1 (Public)

Edition:

2nd Edition

Rights:

© Daniel Kolling Andersen, 2015

Acknowledgements I would like to thank my academic supervisor Jan Karlshøj for his competent guidance, and for allowing me to follow my interests with this project. Further gratitude goes out to my two external supervisors Andreas Castberg and Andreas Bak from Søren Jensen for taking me under their wings during my thesis work, and for welcoming me into their office with the rest of the friendly and helpful employees at Søren Jensen.

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Contents Abstract ...................................................................................................................................................... 1 Preface ........................................................................................................................................................ 2 Acknowledgement ...................................................................................................................................... 2 1. Introduction ............................................................................................................................................ 5 Motivation ............................................................................................................................................... 5 Problem Statement .................................................................................................................................. 6 2. Methodology........................................................................................................................................... 8 3. Primer ..................................................................................................................................................... 9 Computational and Parametric Modelling in the AEC industry.................................................................. 9 Concept and Application of Parametric Modelling...................................................................................14 Genetic Algorithms .................................................................................................................................16 4. Inspirational, Existing Studies.................................................................................................................20 Point Supports for Telescope Mirrors ......................................................................................................21 The Groningen Twister ............................................................................................................................23 A Gradient Based Method of General Application for Beams and Plates..................................................26 5. Survey of Horizontally Spanning Concrete Structures ............................................................................29 Prefabricated Systems ............................................................................................................................30 Hollow Core Slab (HCS) .......................................................................................................................31 TT-Slab (TTS) .......................................................................................................................................34 SL-Deck (SLD) ......................................................................................................................................35 In Situ Systems ........................................................................................................................................39 Flat Slab ..............................................................................................................................................40 Waffle Slab..........................................................................................................................................45 BubbleDeck (BD) .................................................................................................................................47 Free Form Moulding Systems ..................................................................................................................48 6. Form/System/Configuration-Dependency .............................................................................................49 Form .......................................................................................................................................................50 System ....................................................................................................................................................51 Configuration ..........................................................................................................................................52 Form/System/Configuration-Diagram .....................................................................................................53 7. Script Scope and Platform ......................................................................................................................54 Scope ......................................................................................................................................................54 3

Platform..................................................................................................................................................56 8. The Script ...............................................................................................................................................57 Design Domain Input...............................................................................................................................58 Grid Generation ......................................................................................................................................60 Generation of Free Point Support Positions.............................................................................................61 Gathering Support Points ........................................................................................................................63 FEA Computation ....................................................................................................................................64 FEA Input ............................................................................................................................................65 FEA Routine ........................................................................................................................................66 FEA Results .........................................................................................................................................67 Optimisation Routines ............................................................................................................................69 IFC Model Export ....................................................................................................................................71 9. Results ....................................................................................................................................................72 Test A – Comparison to Gradient Based Optimisations ............................................................................72 Support number, n=4 ..........................................................................................................................73 Support number, n=8 ..........................................................................................................................76 Test B – Study of FEM Computed Field Quantities ...................................................................................78 Test C – Karamba to Robot ......................................................................................................................80 Test D – Study of Correlation between Internal Energy and Peak Deflection ...........................................82 Case – The Flying Carpet .........................................................................................................................84 Left, Column Supported Surface Part...................................................................................................86 10. Discussion...........................................................................................................................................102 11. Conclusion ..........................................................................................................................................103 Bibliography .............................................................................................................................................104 List of Figures ...........................................................................................................................................107 List of Videos ............................................................................................................................................111

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1. Introduction Motivation During my enrollment at DTU, as a student of architectural engineering, the concept of ‘parametric modelling’ has regularly cropped up, with the frequency of its appearance increasing in most recent years. Although I never quite grasped the implication of this design paradigm, it somehow still managed to imprint itself on my psyche and focus my interests. An acute fascination with the interactive ability, of the parametric model, to undergo geometric change, coupled with an innate satisfaction in programming were convincing drivers of interest. The additional belief that parametric modelling tools could present a new environment for combining architectural and engineering disciplines only strengthened my interest as an architectural engineer with an inherent call to this multidisciplinary design field. Writing my master’s thesis would finally provide me with the opportunity to unfold the parametric design approach, but with no previously gained experience within the area, it was necessary for me to approach a qualified counterpart having the required skill set to get me going and keep me on track. Consequently, it was decided that I get in touch with Søren Jensen consulting engineers1, for whom implementation of computational design methods is an important strategy. Søren Jensen sees the main potential of parametric modelling lying within the early design phase of building projects. This phase is the natural habitat of the architectural engineer, for it is here where the architectural design can be influenced the most. Søren Jensen acknowledges the potential benefits of employing parametric design tools in the early stages. Most particularly, they enable the analysis of a large variation of different design possibilities and, with this happening at the beginning of the process, the design is of such calibre that it can be carried on through the later project stages in an unproblematic manner. At my first meeting with Søren Jensen one of their recent projects was introduced. It was a competition project for a new urban space encompassing 2000 new bike stands for the University of Copenhagen. Søren Jensen collaborated with Polyform architects, and the design fell upon an undulating concrete shell structure, which would shelter the bike stands underneath its surface and at the same time provide a recreational area on top 2. With the geometry of the shell fixed, the architects expressed to Søren Jensen how they wanted the columns carrying the shell to depict an irregular pattern in order to enforce a more dynamic feel. This architectural request, along with my curiosity for the parametric modelling field, became the starting ground for my thesis. The aim of the thesis was to develop a parametric design script in collaboration with Søren Jensen. With the intention that this script could be applied within the company, in order to support future architectural requests of same character.

1 2

http://en.sj.dk/ ; https://vimeo.com/user39987959 http://www.universitetstorvet.dk/mapper/Polyform.pdf

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Problem Statement Optimization by means of appropriate placement of the supports of a structure is the least studied of all the different trends in the optimal design of structures. The reasons why it has not been extensively studied may be explained by the fact that in a real design process, it is possible to control the placement of supports only on comparatively rare occasions. Meanwhile, the effectiveness of such an approach exceeds only slightly that of other approaches. Problems involving point supports occupy a special place among these types of problems. (Dekhtyar’ 1997) The above reflection figures as the opening lines of A. S. Dekhtyar’s article ‘Optimal Point Support of Shells and Plates’. I find that this quotation is particularly apt with regards to my thesis; it deals exactly with the task of finding optimised point support distributions for shell and plate structures. For any kind of load-carrying system the internal force distribution evolving through the structural members is fostered by the equilibrium interaction between applied loads and support reactions, consequently choosing support conditions has a significant say on overall structural performance. However, usually there is no straightforward mathematical relation between positions of supports and structural response for 2D and 3D structures, i.e. plates and shells, why finding analytical solutions as a general approach to these kinds of problems is deemed out of the question. This fact may pose another explanation as to why this specific topic has not received more scientific attention; the mathematical difficulty behind setting up deterministic approaches might prove too big an obstacle (Wang 2004). For the same reason interest seems to be growing around the use of non-gradient based problem solving techniques instead. As opposed to deterministic approaches, which rely on mathematical gradients for setting up sensitivities between objective functions and input design variables, a non-gradient based method does not have this dependency. This means that it is applicable for analysis on any type of objective function no matter how discontinuous the relation between variables and objective function may be. The drawback of using nongradient based methods is that they do not guarantee finding optimal solutions in addition to the fact that they are also time-consuming, yet, it seems they may still help find good solutions. Currently, the most popular non-gradient based approach seems to be the use of genetic algorithms, the application of these algorithms have already given qualified results for optimisation of support conditions in some specific and relatively simple cases found in literature (Cheng 2013; Marcelin 2012; Marcelin 2001; Wang & Chen 1996), and also in complicated projects, however, not dealing with support optimisation (Turrin et al. 2011; Hofmann et al. 2008). In continuation hereof Cheng summarises that the applicability of the genetic algorithm to a complicated problem of same character still needs investigation. Part of this investigation is dealt with in this thesis, which ultimately intends to design a script capable of performing point support optimising routines for arbitrary numbers of supports on arbitrarily shaped plate and shell concrete structures. This script will revolve around the application of computational design by linking geometry, finite element analysis (FEA) and genetic algorithms together in a single platform constituted by a parametric design modelling software. The script is targeted for application in early design stages, with generation of multiple design alternatives as key.

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The specific intension of the thesis is to clarify the following question: •

Can parametric modelling in combination with optimisation routines based on genetic algorithms provide a competent tool for generation of point support configurations with both architectural and structural design interests in focus?

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2. Methodology The question defined in the Problem Statement is sought answered through building up knowledge based on engagement in a number of diverse activities. Through the entire project phase, I have been in daily contact with the employees of Søren Jensen, as I was offered a work station in the company. This has provided a unique opportunity of being able to trail the working methods of the company and their practical implementation of parametric modelling. Being allowed this insight helped me passively increase general awareness of the subject, and my own, active dialogue with Søren Jensen provided valuable guidance on relevant literature, modelling tools and specific application cases for the script. However, most importantly they helped me with defining the conceptual outline of the script as well as with the hands on script coding, which followed afterwards. The coding part itself has also been a defining aspect for the thesis, as the particular coding language applied for script was foreign to me at the outset. Therefore, the communication with Søren Jensen has also helped me progress faster in learning this language than I could have, had I been on my own. The initial studies undertaken in order to define the direction and application of the script, are presented in Chapters 3-6. These chapters summarise studies of literature and evaluation of concrete construction systems. Concrete systems have been given focus simply because, in the AEC industry, continuous slab and shell structures are mainly built from this material. However, this focus actually has not limited or defined the general application of the script in how it was developed or came out in the end.

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3. Primer Computational and Parametric Modelling in the AEC industry Before being accepted into the AEC industry, parametric modelling was first rooted in the product-design and aerospace industries. In 1994, the Boeing 777 aircraft became the first of its kind to be completely designed in a digital model using parametric modelling software, thereby the Boeing 777 was also the first aircraft that’s design was solely reliant on computer technology (Shepherd 2011). The AEC industry is notoriously known as conservative, and has a long history of using the paper drawing as preferred design communicator. The industry has, however, transitioned to the use of Computer Aided Design (CAD) during the last some 40 years. In spite of this, and although implied in the statement “computer aided”, the potential of computational design power seems not fully unleashed in the traditional notion of the CAD term. The original intention of introducing CAD was to create drawings in digital files instead of on paper for the sake of drawing efficiency. Nevertheless, the core drawing mechanism stayed very much the same in the sense that in a CAD drawing, geometry is drawn in absolute coordinates similarly to what is happening when drawing on a piece of paper, the drawn geometry is fixed in itself and also to a specified position. So when changes are to be made in a digital CAD drawing, it is also necessary to delete and redraw, which is still a time consuming discipline although quicker than erasing and redrawing on paper. With this philosophy in mind one might argue that traditional CAD serves as nothing more than a substitute for the pen to paper drawing. This argumentation further implies that CAD is merely used for communicating design as a tool responding to biased user intentions rather than being an interactive medium contributing new possibility for aiding design decision making. This is why parametric modelling branches out from traditional CAD to take the computational design evolution of the AEC industry further. When the parametric design approach was first applied in the AEC industry, it was because of the fact that traditional CAD systems were not suited for handling certain geometrically complex building projects, i.e. the completion of Gaudi’s Sagrada Familia and some of Frank Gehry’s creations (Shepherd 2011). Since then parametric modelling tools and the increasing amount of complex building projects (Mitchell 2005) have helped pave the way for each other in an inseparable partnership. For this reason it is also common to associate parametric modelling with design projects of high complexity.

Figure 1: Guggenheim Museum (left) & La Sagrada Familia (right). Photos: Guggenheim Bilbao Museoa & Temple expiatori de la Sagrada Familia

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Frank Gehry’s office set out into the parametric design era embracing the very same software, which was applied in the design of the Boeing 777, the specific software used was CATIA by Dassault Systèmes. Gehry Technologies have since customised the original CATIA platform for the AEC industry resulting in the offspring software Digital Project (DP) in 2004. Other high-profile vendors of CAD software soon followed adding parametric capabilities to their own product repertoires. In 2007, Bentley Systems released the software GenerativeComponents (GC) as a competitive counterpart to CATIA/Digital Project, a beta version of GC had been around since 2003 when it was namely used by London based architect firms (Davis 2013a). Grasshopper (GH) is another tool in league with GenerativeComponents and CATIA/Digital Project when it comes to more advanced parametric application. Grasshopper is developed by David Rutten for Robert McNeel & Associates and functions as a plug-in engine for their 3D CAD software Rhinoceros (Rhino). David Rutten was assigned to develop Grasshopper by McNeel as a direct result of McNeel and Bentley’s failed negotiations on making GC available for Rhino (Tedeschi 2010), the first version of GH was released in 2007. Graphisoft has added simple parametric relations in ArchiCAD, and the same has been done for Autodesk products AutoCAD and Revit (Holzer et al. 2009). However, Autodesk is currently focused on developing a plug-in for Revit named Dynamo, which is supposed to undertake a similar relation to Revit as GH provides for Rhino. The here mentioned AEC software products can be categorised based on their underlying Programming Paradigm which relates to how they are operated. There are two main types of programming paradigms; imperative and declarative. Digital Project, Rhino, ArchiCAD, AutoCAD and Revit belong to the imperative paradigm. The imperative software tools are operated by the user executing imperative actions, i.e. geometry is processed directly through calling manual commands. Digital Project stands out from the other mentioned imperative CAD tools by being a so called history-based modeller, while the others classify as direct modellers. In a history-based modeller the graphical user interface (GUI) holds a data tree saving every step of the modelling process in a chronological order by listing all executed commands and intertwined model relations. Therefore this kind of modeller will accommodate for designing parametrically, as parameters can quickly be altered from accessing the modelling subroutines presented in the structured data tree. Direct modellers do not keep track of command history, but focuses merely on spontaneous and untangled creation of explicit geometry offering larger freedom in some areas while at the same time limiting other applications, like the fore mentioned (Sweeney 2009; Evans 2013). In programming paradigm terminology the imperative CAD modellers can be classified between the two imperative subcategories named object oriented and procedural. The direct CAD modellers, Rhino, ArchiCAD, AutoCAD and Revit, would fall under the object oriented subcategory and the history-based modeller, Digital Project, would find itself somewhere in between the object oriented and the procedural mechanisms (Davis 2013b). GenerativeComponents and Grasshopper are representatives of the declarative programming paradigm. These tools are operated by processing data as opposed to pure geometry, and they are both based on Visual Programming Language (VPL). Visual programming is based on manipulating graphical elements instead of textually writing code like in a Textual Programming Language (TPL). Compared to textual programming VPLs have the advantage of being highly intuitive, and therefore award a steep learning curve (Leitão et al. 2012). This naturally appeals to newcomers to programming and explains their popularity. In addition GC and GH support real time modification display, this means that making script modifications will

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immediately update the displayed output geometry. This is possible due to their data-flows being organised via the Directed Acyclic Graph (DAG) principle.

Figure 2: Directed Acyclic Graph (DAG). Illustration: HAMRTech.com

The general DAG programming scheme is depicted in Figure 2. The system consists of nodes (circles) and directed edges (arrows) passing data from one node to another. The system does not support cyclic behaviour between nodes (hence acyclic) so performing iteration loops, for instance, seem out of scope for the DAG, which is of course a limitation compared to a TPL. The DAG is absolved from the usual “EditCompile-Run” rigidity, which presents the three step action required for generating output from TPL scripts. When the functionality of a node is changed in a DAG, only the affected nodes need updating, and it is therefore not necessary to compile and rerun the entire script. Thus minimising computational effort and making it possible to perform the “Edit” and “Run” routines simultaneously. This is why the response from editing is instantaneously seen as an update in the displayed output geometry, which is the key feature behind triggering the interactive and intuitive design environment.

Figure 3: Google search interest since 2004 for the four parametric modelling tools; GH, DP, GC and Dynamo. By the looks of this chart GH seems trendsetting and both the interest for DP and GC has slowly decreased since the advent of GH. Interest for Dynamo is steadily rising. Graph: Google.com/trends/GH+DP+GC

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The very data-flow of the VPLs already represent a degree of design documentation, because as with the history based modellers, the entire modelling process can be decoded through analysing the associated data-flow. However, the discipline of decoding may not be as intuitive, since the DAG does not back any overall coding structure, one can easily end up with a so called spaghetti code, which is the common word used for a chaotic code structure tangled up by intersecting wires resembling a bowl of spaghetti, see Figure 4. Therefore, whether or not to make a structured data-flow is a question that can only be addressed by the programming designers.

Figure 4: Severe case of spaghetti code in GH. Illustration: Daniel Davis

Up to this point, focus has been kept on the beneficial aspects of parametric design. It may appear, consequently, that ArchiCAD and Revit have been displayed in a harsh light. This was not the intention, and so to justify these products the principles of Building Information Modelling (BIM) will be utilised to broaden the discussion and provide a necessary layer of nuance. BIM represents another innovation spectrum of the CAD field and the BIM practice is currently more prevalent within the AEC industry than parametric modelling, it could be suggested though that parametric modelling features as a part of BIM. In theory, this could very well be justified, but nevertheless the tendency arising in praxis elucidates a different story. Some of the most notable initiatives concerning BIM are standardisation of building processes and working methods, interoperability and exchange of information, embedded specification data within model objects/components and direct extraction of construction documentation. Generally speaking these initiatives are not supported in parametric design software and, importantly, on the contrary, what Revit and ArchiCAD lack in compatibility with parametric application and doubly-curved freeform geometry, they make up for with their devotion to BIM. Consequently, BIM and parametric modelling seem to have facilitated each their own, separate community of advocates (Boeykens 2012). Bridging this gap could very well be the next big move within CAD development and why practitioners are also excited about the development of Dynamo for Revit.

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Figure 5: Google search interest since 2004 for GH and the two BIM tools Revit and ArchiCad. Graph: Google.com/trends/GH+Revit+ArchiCAD

A comparison between Figure 3 and Figure 5 helps indicate the current power balance between building information modelling and parametric design modelling. The comparison demonstrates that the interest for BIM tools is significantly higher. It is also interesting to see how Revit has gained an edge over ArchiCAD. When further comparing Figure 5 to Figure 6 a strong hint is given that traditional CAD remains the dominating design approach.

Figure 6: Google search interest since 2004 for GH, Revit, ArchiCad, Rhino and AutoCAD. Graph: Google.com/trends/GH+Revit+ArchiCAD+Rhino+AutoCAD

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Concept and Application of Parametric Modelling The generic form of a mathematical function f(x) and its mechanism is commonly known. The free input parameter x defines outputs according with the typology of the function f(x), be it a straight line f(x)=x, a parabola f(x)=x2, a sine curve f(x)=sin(x); the amount of function typologies is infinite. Likewise, the number of permissible input parameters is endless, and therefore one function may depend on several parameters, f(x1, x2, ‌ , xn). In mathematics differential calculus offers a tool for solving optimization problems. The derivative function f’(x) can be applied for the sake of determining values of x yielding optimum function solutions, e.g. function minima or maxima solving for f’(x)=0. Figuratively speaking, an analogy comparing the mathematical function and the concept of parametric modelling exists to further shine a light on the workings of the latter. Like a mathematical function, parametric modelling is all about typology and input parameters. The design typology is defined based on the relevant design case, be it an aircraft, a multi storey car park, a doubly-curved freeform structure etc. So the design typology corresponds to what the designer would actually model geometrically, and when building up a parametric design model the modelling work consists of programing rather than traditionally executing manual commands in an imperative Computer Aided Design (CAD) software. The model script is built around input parameters, and, in this sense, the design becomes a function of its input parameters. Thus, a parametric model can undergo quick design alterations, as long as the alterations are kept within the boundaries of the programmed parameters. The parameters may, for instance, govern base dimensions, room heights, roof inclination, number of storeys, glass areas etc. The possibility of making quick design changes is one advantage of the parametric model in itself, but where its parametric characteristic becomes yet more interesting is when the model is coupled with performance measuring applications like structural FEM computations, or simulations on building physics. Such a model link will allow the designer to receive design feedback based on adjusting parameters. This feedback may be given instantly, making it possible to play around with parameters and exploring related performance effects in real time. A more intuitively driven design process is thus accommodated for, since the engineering performance state is constantly documented. Having the connection between geometric presentation and quantifiable performance also introduces a new opportunity for collaboration between architects and engineers, as their professional domains can then be integrated under one setting. Comparable to the role of differential calculus in optimising mathematical functions, optimisation tools are also applicable for parametric design where its potential is hereby further increased. These optimisation routines interact with parametric models, where model parameters will be optimised based on targeting a desired design performance. These optimisation methods are commonly based on stochastic methods, which can be generally used for treating any optimisation problem, as opposed to analytical methods which are always problem specific and of course cumbersome, if not impossible, to express depending on the complexity of the specific optimisation case. A direct analogy is therefore not present between the derivative function and the optimisation tools available for parametric design, besides using stochastic methods for optimisation is not bulletproof as the optimum solution found will always come from a limited amount of investigated solutions set up from initial boundary conditions. As a consequence it may be problematic to gain proper control of the stochastic approach and its solution frame why caution must always be applied for verification of computed results.

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Developing a digital model while actively using performance based criteria in the design process is called informed modelling. This approach is also referred to as performance based design, and some other widespread keywords linked to the topic of parametric design are; genetic design, generic/generative design, algorithmic design and morphogenesis, see Figure 7. A parametric model will, depending on its complexity, contain numerous different design parameters, some of which will normally counteract each other, and therefore it will most likely not be possible to optimize one design for all parameters and performance aspects at once, trying to do so will instead lead to impractical results. Furthermore, it should also be kept in mind that not all performance qualities can be quantified, think for instance of aesthetics. As a result of the aforementioned trade-off problematic, the responsibility to compromise between different parameters and their representing performance disciplines falls to the designer. As any design will undergo trade-offs between different design qualities along the way, this is not an unusual issue, nonetheless one might still argue that the exact design trade-offs become more evident in a parametric design scenario, and consequently require bigger focus. In addition the successful parametric designer must possess a strong perception of design constraints and requirements in order to outline the right set of variable design parameters for the useful model to revolve around. This demand is considered a significant pitfall related to parametric designing because it is necessary for the parametric model designer to foresee the potential conceptual directions in which a project should go before even starting to define geometry, in order to avoid remodelling later on and the extra effort involved in this endeavour.

Figure 7: David Rutten’s subjective understanding of terms commonly related to the notion of parametric design. Illustration: David Rutten

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Genetic Algorithms A genetic algorithm (GA) is a stochastic search method inspired by the same rules governing Darwinian genetic evolution of species, cf. Figure 8.

Figure 8: Charles Darwin. Illustration: simplecapacity.com

For a genetic algorithm to run it needs two inputs; an objective function and a genome set. The objective function, also commonly known as the fitness, represents the value targeted for optimisation, and the genome is the gene holder affecting the fitness. As the fitness is a direct function of the genome, one can once again refer to the principle of the mathematical function for describing their relation, in which the fitness represents f(x) and the genome represents the variable x. The genetic algorithm will hence seek to minimise, maximise, or aim for a specified value of the fitness by altering the genome through sequential generations of genome reproduction. Based upon the performance of genes in each generation, the gene pool is constantly filtered applying the mechanisms of natural selection, mating and mutation of genes. A textbook example of genetic distribution through the course of a GA run is depicted in Figure 9, where genome values will eventually gather around the global optimum. The two variables x1 and x2 represent the two variable genomes of the GA and each unique value of x1 and x2 represent a certain gene in their corresponding genome, the fitness function is drawn in the area spanned out between the two genome axes, and this area is commonly called the search space or the fitness landscape. This landscape can only be illustrated on paper for one- and two-dimensional fitness functions. When more than two genomes, n>2, are influencing the fitness function the fitness landscape becomes n-dimensional, and thereby a more abstract measure, but still the same principles of local and global optima will apply throughout the ndimensional landscape, and the disturbing bumps along the way will only increase with the number of n.

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Figure 9: Seamless search progress illustrated for a two dimensional fitness landscape. Illustration: (Sarkisian et al. 2009)

A figurative example showing the principles of mating/crossover and mutation strategies of a GA is given in Figure 10, where genes are representing different truss layouts.

Figure 10: Principles of crossover and mutation of GAs. Illustration: (Sarkisian et al. 2009)

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Finally, a full schematic flow chart of a traditional GA is presented in Figure 11. The survivor operation has not been explained yet, but as alluded to in the name this operation simply lets the gene, typically a topperforming one, of a previous generation slip directly through the generation gap and into the next generation. The constant k represents the number of generations produced, and K is then a threshold value for the maximum allowed number of generations. The constant h is defined to stop the algorithm when stagnating behaviour has been going on for too long, i.e. h defines a maximum allowed number of consecutively generated generations showing no improvement in the fitness function. Stagnation is a sign that the algorithm has converged around a single optimum in the fitness landscape, when this happens, it is unlikely for the algorithm to produce better results when letting it continue to run.

Figure 11: GA procedural flow chart. Illustration: (Sarkisian et al. 2009)

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Figure 12: GA procedural flow chart with equivalent gene dynamics depicted. Illustration: (Scheurer & Stehling 2011)

The flow chart of Figure 12 is less detailed than the one in Figure 11, but it provides an explanatory illustration of the actual gene mechanisms associated to each step of the flow chart. Two parent genes are found based on the two highest fitness scores of previous generation, they are then allowed to recombine their genes fostering the next generation, which will then undergo a degree of mutation before the fitness is again evaluated. The process will then continue in this cyclical manner until termination.

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4. Inspirational, Existing Studies This chapter intends to present and elaborate on a number of studies that are particularly pertinent in relation to the outline of this thesis.

Telescope mirror

Figure 13: Meridian cross-sectional drawing of the 200-inch Hale Telescope and dome. Drawing: Russell W. Porter

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Point Supports for Telescope Mirrors The search for literature on point support optimisation led down an unforeseen path with the discovery that most research within this area seems to be focussed on the design of telescope mirrors, cf. Figure 13. Telescope mirrors have strict deflection demands as mirror deflections have the potential to lead to optical image distortion. The articles in focus, (Nelson 1982; Craig & Boulet 1999), analyse the circular mirror with constant thickness using the theory of thin plates, the mirror is subjected to a uniformly distributed transversal load caused by gravity of the mirror. Minimising the transversal deflection is the sole target of optimisation, and the prevailing support distribution strategy is to place a number of equally spaced point supports along rings offset with certain radii from the plate centre. The articles in focus thoroughly present interesting observations and results for a generous number of different support configurations. Naturally, their findings are strictly related to the circular plate geometry, and therefore are not applicable in a broader context of point support distribution. Furthermore both articles set up symmetry constraints on the support distribution to narrow down the mathematical search space for optimisation. However, in (Nelson 1982) an introductory survey discusses point support distribution in a general perspective by considering infinite plates. As such plates are infinite in extent and no boundary edge effects will have impact on the deflection. The three archetypes of regular point distribution are compared; these are the triangular point grid (Figure 14 left), the square point grid (Figure 14 middle) and the hexagonal point grid (Figure 14 right).

Figure 14: Thee three only point symmetric grid structures; here shown with equal distances between neighbouring grid points.

As can be seen from Figure 14 the three topologies prompt different tessellation shapes. In the infinite triangular grid system each separate point has six neighbouring points yielding a hexagonal tessellation, the infinite square grid system provides four neighbouring points yielding a square tessellation and lastly the hexagonal grid structure exhibits a triangular tessellation having three neighbouring points per point. A relevant factor in relation to these tessellations is the tessellation area per point. Let the square grid have a reference area of 1, then the tessellation area per point corresponds to 0,87 for the triangular grid and 1,31 for the hexagonal grid.

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Figure 15: The hexagonal bee honeycomb allocates the best ratio between storage space and cell wall material. Photo: homefurn.com/blog/decorate-hexagons/

The hexagonal tessellation therefore represents the optimal strategy for area enclosure in terms of unit cells per area. One of nature’s very own examples of applying this strategy is the honeycomb structure presented in Figure 15. Accordingly, using the triangular grid system for point support distribution is also the best solution for minimising deflections of infinite plates. A formula for comparing the deflection efficiency between different support configurations neglecting shear effects is given as:

where

đ?‘ž

đ??´ 2

đ?‘¤ = đ?›žđ?‘ ďż˝ ďż˝ đ??ˇ đ?‘

đ?‘¤ = đ?‘Ąâ„Žđ?‘’ đ?‘™đ?‘Žđ?‘Ąđ?‘’đ?‘&#x;đ?‘Žđ?‘™ đ?‘‘đ?‘’đ?‘“đ?‘™đ?‘’đ?‘?đ?‘Ąđ?‘–đ?‘œđ?‘›. đ?‘ž = đ?‘Ąâ„Žđ?‘’ đ?‘Žđ?‘?đ?‘?đ?‘™đ?‘–đ?‘’đ?‘‘ đ?‘“đ?‘œđ?‘&#x;đ?‘?đ?‘’ đ?‘?đ?‘’đ?‘&#x; đ?‘˘đ?‘›đ?‘–đ?‘Ą đ?‘Žđ?‘&#x;đ?‘’đ?‘Ž. đ??¸â„Ž3 đ??ˇ= , đ?‘Ąâ„Žđ?‘’ đ?‘?đ?‘™đ?‘Žđ?‘Ąđ?‘’ đ?‘“đ?‘™đ?‘’đ?‘Ľđ?‘˘đ?‘&#x;đ?‘Žđ?‘™ đ?‘&#x;đ?‘–đ?‘”đ?‘–đ?‘‘đ?‘–đ?‘Ąđ?‘Ś. 12(1 − đ?œˆ 2 ) đ??´ = đ?‘Ąâ„Žđ?‘’ đ?‘?đ?‘™đ?‘Žđ?‘Ąđ?‘’ đ?‘Žđ?‘&#x;đ?‘’đ?‘Ž. đ?‘ = đ?‘›đ?‘˘đ?‘šđ?‘?đ?‘’đ?‘&#x; đ?‘œđ?‘“ đ?‘ đ?‘˘đ?‘?đ?‘?đ?‘œđ?‘&#x;đ?‘Ą đ?‘?đ?‘œđ?‘–đ?‘›đ?‘Ąđ?‘ .

Using this formula đ?›žđ?‘ represents a fixed constant. For a small number of supports edge effects will be significant making đ?›žđ?‘ a function of the support number, đ?‘ . For an increasing number of supports đ?›žđ?‘ will decrease towards an asymptotic value đ?›žâˆž . Poorly supported plates will have larger values of đ?›ž, why đ?›ž can hence be denoted as a measure of the support point efficiency. The following point support efficiencies were found for infinite plates using the grid systems presented in Figure 14: đ?›žđ?‘Ąđ?‘&#x;đ?‘–đ?‘Žđ?‘›đ?‘”đ?‘˘đ?‘™đ?‘Žđ?‘&#x; đ?‘”đ?‘&#x;đ?‘–đ?‘‘ = 1,19 ∗ 10−3 đ?›žđ?‘ đ?‘žđ?‘˘đ?‘Žđ?‘&#x;đ?‘’ đ?‘”đ?‘&#x;đ?‘–đ?‘‘ = 1,33 ∗ 10−3 đ?›žâ„Žđ?‘’đ?‘Ľđ?‘Žđ?‘”đ?‘œđ?‘›đ?‘Žđ?‘™ đ?‘”đ?‘&#x;đ?‘–đ?‘‘ = 2,36 ∗ 10−3

The support efficiency of the triangular grid is then reckoned as the ultimate measure for achievable deflection efficiency, and the support configurations analysed later in the article would then be assessed by comparison to this target value.

22

The Groningen Twister Progressing from the talk on tessellation strategies, leads to the next study, which deals with the column support design approach for a real architectural project initiated in Holland, 2003. The project was carried out in collaboration between Kees Christiaanse Architects & Planners (KCAP), ARUP and the chair of Computer Aided Architectural Design (CAAD) at ETH Zurich. This project was the design for the Groningen Stadsbalkon in the city of Groningen. The project embodies a public square with bicycle parking underneath, see Figure 16. The articles in focus are (Scheurer 2003; Scheurer 2005; Scheurer 2007). Fabian Scheurer was approached by the project architects enquiring after a ‘forest of columns’ to carry the concrete slab of the pedestrian area above the bicycle parking. The architects used this metaphor to portray their wish for a dynamic, non-repetitive column layout as a feature of the architectural idea, see Figure 21 and Figure 22. What the architects were exactly asking for was to have varying column diameters, varying column spacing and varying column tilt angles.

Figure 16: The Groningen Stadsbalkon. Photo: tunnelvariant.nu/stationsgebied.html

This request presented an intangible amount of degrees of freedom if explicit design decisions were to be made. So for the solution of this task it was then chosen to develop a CAD-tool for facilitating the design. The specific tool was named the Groningen Twister. This software was programmed in Java, and is actually still available from the website of ETH 3, but unfortunately the Java version of the program is outdated, yet screenshots of its GUI can still be seen, and some viewport examples are also shown in Figure 20. To put it briefly, what the software did was to perform a self-organising circle packing with interdependent dynamics of interaction. The circles represented columns and were sized according to the load carrying capacity of the represented column.

3

http://wiki.arch.ethz.ch/twiki/bin/view/Extern/GroningenTwister.html

23

This is why a link can be drawn between the tessellation talks of Figure 14 and this project, because circle packing constitutes another tessellation principle closely related to the hexagonal tessellation. The circle itself is more compact than the hexagon, but when used for tessellation will not cover up areas quite as well because of the leftover areas in between circles. Figure 17 illustrates how tight circle packing leads to the triangular grid pattern, and hence the hexagonal tessellation principle. There were two kinds of input given for the Twister, these were of functional and structural character.

Figure 17: Tight circle packing.

The functional requirements concerned constraints on column locations and these constraints can be seen from Figure 18; where the coloured areas are column restricted zones. Listed from top to bottom these zones are; circular holes in slab, bike stands and slab areas resting on ground. The structural requirements provided by ARUP defined the column specifications, these were based upon rules of thumb. The design allowed for three different column types with diameters of 150mm, 250mm and 300mm. Their corresponding load carrying capacities were then represented by circles of respective radii; 2m, 3m and 4m. A column tilt angle of 10째 could be ignored in the structural calculations and was hence set as the maximum allowed inclination. The circle agents were programmed to dynamically fill out the slab domain by growing, splitting, shrinking and dying, these mechanisms and their triggering local effects are illustrated in Figure 19. Step no 1 depicts a column growing between the three prescribed circle sizes when having enough space. Step no 2 depicts how a column is splitting when it cannot grow any further and there is still surplus of space. Step no 3 depicts a column shrinking when exposed to local pressure from neighbours. Step no 4 depicts how a column dies when it cannot shrink any further.

Figure 18: Column habitat functional constraints. Illustration: (Scheurer 2003)

Figure 19: Agent dynamics. Illustration: (Scheurer 2003)

The software enabled the architects to produce a selection of possible solutions, these solutions then had to be handed over to the engineers, who would import the models to their FEA software, so the models could be undergo structural analysis, and the best performing model could be picked. This step also caused some minor changes to the models. In the discussion section of (Scheurer 2003), Scheurer remarks that the Groningen Twister software lacks a quality assuring ability for determining whether generated output configurations actually comply with structural demands. He, therefore, suggests adding a fitness measure for quantifying structural performance internally within the software as the obvious next step of development. With the inclusion of such a fitness quantity he then goes on further suggesting the implementation of a genetic algorithm for fitness optimisation.

24

Figure 20: Viewport examples of the Groningen Twister software. Illustration: (Scheurer 2003)

Figure 21: The Stadsbalkon taking form. Photo: designtoproduction.com

Figure 22: The bicycle parking area. Photo: designtoproduction.com

25

A Gradient Based Method of General Application for Beams and Plates The last article to bring up is (Jang et al. 2009). This research deals with a gradient based optimisation method of finding optimal point support locations of beams or plates under self-weight. Once again, the motivation for this study does not originate from application in the AEC industry, but this time from the production of LCD panels, see Figure 23. In similarity with the telescope mirrors, LCD panels have strict deflection demands as well. This is due to an imprinting process using nanoscale imprinted lines, for this process an uneven deflection across the panel will deteriorate the quality of those imprints.

Figure 23: LCD panel. Photo: news.softpedia.com/news/Korea-Leads-the-Way-in-LCDPanel-Construction

The employed approach is numerical and builds upon FEM computation meaning that the beam or plate to be analysed is discretised into a number of finite elements. The concept is then to start with an initial support configuration which prescribes elastic springs at all nodes in the FE mesh. Running this initial configuration through an iterative simulation routine should then yield a final configuration where only distinct, rigid supports are present. The outcome number of rigid supports wanted is given as input by the user. The final, distinct support configuration is achieved, because the continuous design variables (here the intermediate spring stiffness values) are forced towards either extinction, or infinite stiffness by the use of a penalisation strategy. This algorithmic strategy is indeed recognisable from the same kinds of mechanisms applied in the more established field of topology optimisation, which deals with optimising 2D domains for plane stress situations 4. The original intention for the optimisation routine was to minimise the overall deflection field using the L2 norm of the deflection as objective function. However, using the L2 norm as objective function did not turn out a successful approach, as it yielded poor convergence of the springs to distinct supported/unsupported states. This is shown in Figure 24, where it can be seen, how only intermediate spring stiffness values are present (the weaker the colour, the weaker the support stiffness).

Figure 24: Optimised support layout for beam structure using the L2 norm and two distinct support points as target. Illustration: (Jang et al. 2009)

Hence, the objective function was taken up for revision, and eventually the mean compliance (internal energy) of the system was chosen instead. Tests were carried out for comparison of support solutions derived from the L2 norm and the mean compliance. Results for the L2 norm were found using extensive search amongst all possible solutions. Such a search strategy is commonly referred to as brute force.

4

http://www.topopt.dtu.dk/?q=node/863

26

Figure 25: Beam solutions found for the two objective functions; L2 norm vs mean compliance. Table: (Jang et al. 2009)

2

Figure 26: Plate solutions found for the two objective functions; L norm vs mean compliance. Table: (Jang et al. 2009)

27

The results of the comparison tests performed are presented in Figure 25 and Figure 26. For the beam solution identical support configurations are found for all cases except n=8, but here it turned out that the best solution found for compliance figures as the second best solution found for the L2 norm minimisation problem. Next a plate analysis was performed on a mesh of 24x16 elements for n=4 and n=8. The foursupport case showed identical results, and for the eight-support case the best solution for compliance figures as the third best solution amongst the solutions found for the similar L2 norm minimisation problem. Based on the obtained results it is assumed that the compliance will constitute a good substitute for the L2 norm in terms of deflection minimisation also in general cases with arbitrary plate shapes and for arbitrary numbers of supports. The article finally presented one solution showing intermediate iteration steps for a plate structure of less simple shape, see Figure 27.

Figure 27: Support convergence history of five-support locations for the G-shaped plate problem after (a) 35 iterations, (b) 70 iterations, (c) 100 iterations, (d) 130 iterations and (e) 150 iterations.

The article never mentions anything about time consumption for running these optimisations, but based on what is known about similar optimisation procedures for 2D domains in plane stress situations, it must be assumed that computation times are similar meaning that the time aspect will never be an issue for practical application (Sigmund 2011).

28

5. Survey of Horizontally Spanning Concrete Structures The following chapter covers the most relevant present-day construction methods for building horizontally spanning concrete structures. The focus is set on concrete structures as these pose the most relevant plate and shell structures within architecture. The chapter introduces common prefabricated and in situ systems for building planar slabs, as well as special solutions for building free form concrete structures.

29

Prefabricated Systems Prefabricated systems are composed of factory produced elements. The elements covered in this subsection are the hollow core slab, the TT-slab and the SL-Deck.

30

Hollow Core Slab (HCS) The hollow core slab is a factory produced uniaxialspanning slab element. It is mounted in single spans between load-carrying walls or beam-column systems by the use of cranes. Due to its inability to take up negative moments, the HCS accommodates only for single spans, so using significant cantilevering or continuous spans is not an option. The HCS has voids running along its length in order to save material and weight. The void diameter is typically within 66% - 75% of the slab height and the slab bottom can be either softly reinforced or prestressed with steel for taking up tension forces. The slab comes in standard widths of 1,2 and 2,4m Figure 28: Hollow core slabs lined up after production. Photo: Concrete Technology L.L.C and typical heights in Danish context are 180, 220, 270 320 and 400mm. An idea of possible span lengths is given in Figure 29 with the vertical axis representing the value of the dimensioning load, qd [kN/m2], the horizontal axis representing span length, L [m], and the drawn curves representing HCSs of different heights, h [mm]. The slabs are dimensioned using ordinary beam theory.

Figure 29: Guiding load carrying capacity of HCS. Illustration: (Betonelement-Foreningen 1995)

The use of the HCS is mainly popular in Northern and Eastern European countries where low seismic activity grants the right conditions for using prefab concrete element systems, which due to many construction joints and element connections cannot obtain the same robustness and stability as in situ systems.

31

The HCS has a strong tradition in Denmark, where it was first introduced during the 1950s with the postwar building boom. By the 1960s it really manifested itself through the governing modular building fashion (Dansk Byggeskik n.d.), when construction plans were popularly based on transverse, load carrying walls and slab elements spanning parallel with the longitudinal faรงades, see Figure 30.

Figure 30: The Jespersen System. Illustration: Modul og Montagebyggeri, Nissen 1973

32

The uniaxial-span direction combined with the fact that the HCS comes in prefabricated elements and that it cannot cope with negative moments and hence continuous spans, limit the design freedom connected with this slab system. Slab recesses and grid irregularity can, however, be accommodated for, but only to a certain degree. Recesses are limited in size due to transport and mounting of the element, which requires a slab strip of 650mm to be uninterrupted through the element length. Caution must also be applied regarding how many reinforcement strings one can actually cut for the load-carrying in a permanent situation. Irregularity from a rectangular grid layout can be achieved by cutting edges off at an angle and openings can be realised by putting in a slab hanger transferring forces to neighbouring elements. But if the opening runs through too many elements this type of beam support will not be sufficient and an additional wall or beam-column installation must be added underneath the irregular elements.

Figure 31: Irregular opening in HCS element using slab hanger. Illustration: (Betonelement-Foreningen 1995)

In Denmark the HCS is renowned as a cheap, fast and easy solution. Accordingly it is also very often the preferred solution and in particular by contractors. Stating that the HCS can provide a fast and easy building process, can hardly be argued, one only has to order the elements from the manufacturer in time for them to be produced and delivered for the scheduled start of mounting on site. The mounting can be facilitated by a crane operator and three workmen, one at each end of the element ready for guiding it into position along with another workman standing by on ground responsible for rigging the next element, i.e. preparing it for lifting by connecting the crane hooks to the element lifting devices. Some in situ concrete work is, however, still necessary for casting construction and element joints. The economic aspect is however relying on local market and in the case of a HCS vs. an in situ slab solution the question is a trade-off between added price for the HCS element procurement and the price for labour force on site. Compared to mounting of HCSs casting an in situ slab is more labour and time consuming work, it requires instalment of formwork construction, laying out reinforcement nets and pouring concrete, which will need time for hardening. In Denmark the price of labour force is relatively high, why contractors see an interest in having as few man hours as possible, which then balances out the extra price charged for production of HCS elements because of the speed in which they can be mounted.

33

TT-Slab (TTS) The TT-slab is a prefabricated element similar to the HCS in many ways. Like the HCS the TTS has modular widths of either 1,2m or 2,4m with different heights and also it is only suited for single spans. The crosssection geometry looks like a double T-profile, which also explains the name of the element. The TTS is designed for large spans and often finds its use in industrial buildings and sports centres, because of its size safety regulations dictate a stricter mounting procedure than for the HCS. The two web beams contains prestressed steel wires in their bottom zones and the compression zone in the Figure 32: Piled TT-slabs. Photo: Moldtech S.L. relatively thin top plate can be extended by adding an extra 60mm layer of concrete topping on site. An idea of span lengths is given in Figure 33, the legend numbers for the drawn curves indicate (height/width + concrete topping) with height and width given in cm and concrete topping in mm.

Figure 33: Guiding load carrying capacity of TTS. Illustration: (Betonelement-Foreningen 1995)

The TTS has very limited freedom in design, it has the same element related issues as the HCS but on top of that its structural function governed by the cross-section geometry of the slab puts up some additional limitations for this type of element. In the top plate recesses are not allowed to intersect with the web beam transition zone and in the web beams themselves minor openings for installation ducts can be placed in their top zones just below the top plate.

34

SL-Deck (SLD) Although building demands and regulations have changed and increased since the standardisation of the HCS in the 1960s, this kind of prefabricated element type has not undergone significant innovation, only the production process has been subjected to inventions. Just recently an alternative to the HCS, called the SL-Deck, saw the light of day at DTU. The SL-Deck is invented by Kristian Hertz, professor at DTU, in line with his new concrete technology named Super-Light Structures and patented in 2009. In 2010 the spin-out company Abeo was founded with the mission of commercialising the SL-Deck and putting it into industrialised mass production, which was achieved in 2013 in collaboration with Danish concrete element manufacturer Perstrup Beton Industri. The SL-Deck comes in three heights; 220mm, 270mm and 320mm, the standard width is 2,4m, but smaller widths can also be produced.

Figure 34: SL-Deck anatomy. Illustration: Abeo

The SL-Deck combines blocks of light-weight concrete with a stronger, prestressed concrete in a structurally beneficial way. The light-weight aggregate blocks serve as a moulding base for the concrete layer going on top and also leave grooves for both longitudinal and cross reinforcement. The blocks have no primary loadcarrying function but they do offer stability against buckling of the load-carrying concrete layer. Additionally, in the support zones blocks must be omitted and substituted with the higher strength concrete in order to transfer shear forces, which might also require shear reinforcement. As seen in Figure 35 the structural composition is somewhat similar to the principle of the TTS with a through-going top plate and “web beams� with prestressed reinforcement in the bottom.

Figure 35: SL-Deck cross-section. Illustration: (Hertz et al. 2014) Photo: Abeo

35

The presence of the light-weight blocks provides the deck with superior quality in fire resistance and sound damping compared to the HCS. Furthermore ,the SL-Deck also sets the bar higher in terms of flexibility regarding both structural solutions and production wise. The extra structural possibilities are rooted in the fact that the SL-Deck can accommodate for cantilevered, free ends and continuous spans. When going for the cantilevering option one would go for a custom fabricated deck with prestressed wires in the top zone as well for taking up the negative bending moments. A principle solution may look as shown in Figure 36.

Figure 36: Longitudinal section in continuous, cantilevered SL-Deck.

Note also how it will be possible to create integrated “concrete beams� lying internally within the deck by leaving out straight rows of aggregate blocks. The other opportunity provided for creating continuous spans is illustrated in Figure 37, where it is seen how grooves can be left open in the supporting zones for manually adding reinforcement bars on site, which will then allow moments to be transferred between elements. However, the degree of fixation will not be as strong as the solution provided in Figure 36, but the connected decks will still benefit from this solution in terms of achieving higher eigenfrequencies and longer spans.

Figure 37: Moment transferring connection between adjoining SL-Decks. Illustration & Photo: Abeo

Openings and recesses can also be arranged under consideration of load carrying capacity and placement of the prestressed wires. Where the SL-Deck once again distinguishes from the HLS is with the ability of having curved edges.

Figure 38: Openings and recess in SL-Deck. Illustration: Abeo

36

Figure 39: SL-Deck with curved ends. Illustrations & Photo: Abeo

To sum up the performance of the SL-Deck Abeo presents the table shown in Figure 40.

Figure 40: SL-Deck vs. Hollow Core Slab. Table: Abeo

Three different spans are shown for the SL-Deck because of the fixation possibility already discussed and depicted in Figure 37. So for a design load of 4kN/m2, when having simple supports at both ends the deck will span 8m, when having one fixed and one simple support the deck will span 11m and when having two fixed supports the element will span 14m. As regards the spans of the HCS, they seem remarkably underestimated when compared to the spans indicated by the HCS load-carrying graphs in Figure 29.

37

Along with the Super-Light Structures technology followed yet another concrete technology from Kristian Hertz named Pearl-Chain Reinforcement. With Pearl-Chain Reinforcement the idea is to create discretised curvature by fixing precast concrete components together along tensioning cables, similar to how a pearl necklace finds its form through the pearls lining up against each other directed by the curve of the penetrating string. Thus far the focus of this technology has been kept on developing Pearl-Chain Bridges, a concept consisting of single arches shaped from segments of SL-Decks pulled together along posttensioning cables. More freeform and undulating structures have not been experimented with as of yet, but in theory the technology could hold a potential for creating developable surfaces and as can be seen from Figure 42, the technology even aspires to create doubly-curved shell-like structures as well.

Figure 41: First realised application of the Pearl-Chain Bridge system [Skjern, Denmark – March 2015]. Photo: Innovationsfonden

Figure 42: Conceptual principles of the Pearl-Chain technology. Illustrations: Abeo

38

In Situ Systems In situ concrete systems are cast on site in formwork moulds. The systems covered in this subsection deals with flat slab solutions including waffle and bubble slabs.

39

Flat Slab The flat slab is an in situ cast concrete slab resting on columns. As opposed to prefabricated slab elements basically working as wide beams, the flat slab benefits from plate behaviour allowed to carry bending in multiple directions. The flat slab uses orthogonal reinforcement to facilitate the biaxial bending and in addition the demand for shear reinforcement is heavy at column supports. In principle there is no limit to the region size which can be covered by a flat slab solution. During the pouring process one would not cast zones exceeding 30x30m at a time due to cracking, but after initial concrete shrinkage, separate zones can be joined by casting out their bordering joints and combined zones can then be structurally calculated as one continuous region, dependent on the joint type. Cracks may still occur, but they will be below the restricted crack width. The possibility of utilising plate behaviour and having continuous spans lowers the required slab height. Moreover, the embodied in situ feature provides flexibility in plan layout since irregular column positioning is enabled together with the fact that the flat slab can be adapted into any planar shape by the casting in moulds. Additionally the flat slab is very generous to slab openings and recesses because of the robustness sustained by its comprehensive reinforcement net. The flat slab solution comes in many constellations and to introduce a selection, the American organisation Concrete Reinforcing Steel Institute (CRSI) gives their say on the following examples, which they present in text and figures as follows 5. The pure flat slab constellations: This is the basic flat slab layout, which also has the highest degree of freedom regarding service distribution because it is not necessary to redirect installation ducts around downstand beams or element joists. Importantly, partition walls can be placed anywhere. The columns can use drop panels or column heads to increase shear and negative moment capacity and to reduce the effective slab span.

Two-way Flat Slab There are no beams between the columns. Instead, the floor is heavily reinforced in both directions. In addition there is reinforcing steel in the floor at the columns to transfer the loads.

Two-way Flat Slab with Drop Panels This system is similar to the two-way flat plate system except there is a drop panel to provide extra thickness around the columns. This strengthens the column-floor connection. The rest of the floor can be slightly thinner with less dead weight.

5

http://www.crsi.org/index.cfm/building/floor

40

The edge beam constellations: These examples integrate beams as supporting lines between columns, which is structurally beneficial to the slab, but unfavourable concerning service distribution and partition walls.

Flat Slab with Beams The joists act like small beams. This floor system is very economical because the formwork is readily available, and less reinforcing is needed. Because there's only a small span between each joist, the slab can be thinner. One-way Flat Slab and Beam The floor loads are transferred to the beams, which are then transferred to the columns thus making it ideal for heavy load areas. It is a common system for parking structures, elevator and stair areas. Flat Slab with Banded Beams This system is not only economical but can use flying form systems. It also utilizes smaller columns than the traditional flat plate system. Provides uniform clear space below slab as well as providing flexible layout of columns/partitions.

41

The added effect constellations: These solutions integrate structural modification initiatives throughout the actual slab geometry, introducing among others the waffle slab and the bubble deck.

Joist/Wide Module One-way wide module joist slab in the lower middle is a variation on the one-way joist slab. Provides the depth required for stiffness and readily accommodates HVAC and floor penetrations.

Two-way Waffle Slab Two-way joist slab, also called a waffle slab. Because there are joists in both directions, this floor system is the strongest and will have the least deflection. It's typically used when stiffness is important or if there are abnormally heavy loads.

Voided Slab The concept of the voided slab is simple in that concrete mass is removed from the areas in the slab where it is not needed to resist load and as a result of the reduced dead load, voided slabs can span further. Because of this efficiency, this system can withstand an increased load capacity, create larger spans without beams and larger open floor areas, is inherently earthquake resistant and is resource efficient.

42

As just shown, the flat slab solution comes in many constellations, which are always custom-made solutions developed independently in every project. Accordingly it is hard to track down general load-carrying tables like for the prefabricated elements, which are standard products. The Danish handbook for structural engineering titled Teknisk StĂĽbi advises following rules of thumb for plates (Jensen 2015). Thickness Strength requirement, ULS â„Ž = ďż˝800 đ?‘šđ?‘’đ?‘‘ + 15 â„Ž is the slab height in đ?‘šđ?‘š đ?‘šđ?‘’đ?‘‘ is the design moment pr. unit length in đ?‘˜đ?‘ Deflection requirement, SLS Singly-spanning plate: 1

30

ℎ>�1

30

đ?‘˜

3

đ?‘ž

for q < 5đ?‘˜đ?‘ /đ?‘š 2

đ?‘˜ ďż˝ 5 for q > 5đ?‘˜đ?‘ /đ?‘š2

Doubly-spanning plate: 1

40

ℎ>�1

40

Span

đ?‘˜

3

đ?‘ž

for q < 5đ?‘˜đ?‘ /đ?‘š 2

đ?‘˜ ďż˝ 5 for q > 5đ?‘˜đ?‘ /đ?‘š2

đ?‘˜ is the shortest span đ?‘ž is the characteristic load in đ?‘˜đ?‘ /đ?‘š2 â„Žđ?‘šđ?‘–đ?‘› = 60đ?‘šđ?‘š for roof plates â„Žđ?‘šđ?‘–đ?‘› = 80đ?‘šđ?‘š for other plates Singly-spanning plate: đ?‘™ ≤ 5đ?‘š Doubly-spanning plate: đ?‘™ ≤ 8đ?‘š

Main reinforcement

Reinforcement diameter: đ?‘™ ≤ 5đ?‘š Reinforcement distance: 100đ?‘šđ?‘š đ?‘Ąđ?‘œ 2 â„Ž

Distribution reinforcement

Reinforcement area > 1/5 main reinforcement Reinforcement distance 400đ?‘šđ?‘š or 10â„Ž The formulas directed by Teknisk StĂĽbi are warrant on a very general basis. Another table (Wight & MacGregor 2012) developed by the American Concrete Institute (ACI) and incorporated in the American concrete norm, ACI Code 318-08 Section 9.5, advises following minimum slab thicknesses for some of the more specific constellations presented above.

43

Figure 43: Minimum slab heights for ιfm ≤ 0,2. (Wight & MacGregor 2012)

Figure 43 deals with slabs having a ratio β≤2 of longer to shorter span and with beams between all column supports. fy is the reinforcement yield strength and the listed strengths can be converted to MPa as follows; 40000psi ≈ 275MPa, 60000psi ≈ 415MPa, 75000psi ≈ 520MPa. The value of Îąf defines the beam-to-slab stiffness ratio, đ??¸đ?‘? đ??źđ?‘? /đ??¸đ?‘ đ??źđ?‘ , the exact calculation method can be examined in (Wight & MacGregor 2012), but is out of the scope in this context, Îąfm defines the average values of Îąf for all four panel sides. The tabulated height relations are valid for values of Îąfm ≤ 0,2, i.e. constellations, where slabs are without edge beams or with edge beams only adding relatively little stiffness to the system. Additionally the minimum slab thicknesses given for Îąfm ≤ 0,2 are hmin=5in. (≈130mm) for slabs without drop panels and hmin=4in. (≈100mm) for slabs with drop panels. For constellations with edge beams yielding values in the interval 0,2 < Îąfm < 2,0, the following function is advised. â„Ž=

đ?‘™đ?‘› (0,8 + đ?‘“đ?‘Ś /200000) 36 + 5đ?›˝(đ?›źđ?‘“đ?‘š − 0,2)

but not less than 5đ?‘–đ?‘›.

â„Ž=

�� (0,8 + �� /200000) 36 + 9�

but not less than 3,5đ?‘–đ?‘›.

For constellations with edge beams yielding values Îąfm > 2,0, the following function is advised.

The above given rules of thumb do not handle the effects of the waffle or bubble deck. Subsequently, these two solutions will be handled separately and explored here.

44

Waffle Slab For the waffle slab CRSI gives a table of equivalence between dimensions of standard flat and waffle slabs.

Figure 44: Standard Dome Dimensions and Other Data. Table: (Concrete Reinforcing Steel Institute 2008)

45

Figure 45: Estimate of waffle slab load carrying capacity. Illustration: (Concrete Reinforcing Steel Institute 2008)

46

170

230

p

t h

g

n

t r e

s

t

n

e

m

o

M

S

BubbleDeck (BD) For the bubble slab following graph (BubbleDeck Group n.d.) gives an overview of achievable spans and coherent slab heights.

280

340

390

450

Deck height

Figure 46: Moment strength and span of BubbleDeck. Illustrations: BubbleDeck

The execution sequence for the BubbleDeck may be facilitated in two ways, by a complete in situ solution or by using filigree elements as depicted below.

Figure 47: Execution sequences for BubbleDeck. Illustrations: BubbleDeck

47

Free Form Moulding Systems Moulding systems used for designing free form concrete structures are customised and handcrafted by skilled workers with special training in this art. Using free form moulding is therefore labour intensive, and another concern is the reinforcement fitting. An example of a free form, doubly-curved concrete structure under construction is seen in Figure 48.

Figure 48: Rolex Learning Centre. Photo: archicentral.com

48

6. Form/System/Configuration-Dependency This chapter serves to define the scope of overall design solutions, which can be realised by combining different geometrical forms, concrete construction systems and column distribution strategies. The outcome of the chapter will be a diagram depicting possible combinations between the forms, systems and support distribution strategies described.

49

Form Three geometrical surface classes are considered; the planar, the singly-curved and the doubly-curved. The symbols illustrating these three geometric types are depicted chronologically below.

Figure 49: Planar, singly-curved and doubly-curved form symbols.

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System The construction systems considered are a selection of the systems already discussed in Chapter 5. The following systems will be given further attention and consideration. Prefabricated systems:

HCS

TTS

The hollow core slab

The TT-slab 6

SLD

The SL-Deck

In situ systems:

Flat Slab

The pure flat slab

Waffle Slab

The two-way waffle slab

BD

Free Form

The BubbleDeck

The free form moulding systems 7

6 7

http://prefcat.com/htmleng/soluciones/pi.htm http://arcaro.org/ffshell/index.htm

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Configuration Three main support configuration cases for allowable support distribution are considered. Each main case contains four different sub-configuration classes based on hypothetical design scenarios. The surface geometry is the design domain, for which a support distribution is sought. This domain is represented by a bounding circle. The green colour represents points where supports are be placed. The blue colour represents prescribed support positions, could be existing columns, or merely columns placed based on design ideas. Red colour represents restricted zones where supports are not allowed to be placed, these voids could e.g. represent undisturbed flows or open spaces of any character.

Colour legends: Free support Fixed support Restricted zone

The locked grid layout: Supports are mandatory at all intersection points between orthogonal grid lines. The grid lines represent global modular lines for construction plans.

Figure 50: Locked grid layout.

The free grid layout: Columns may be placed at intersection points between orthogonal grid lines. The grid lines could represent local construction patterns based on the specific concrete system, the grid could e.g. be given by the joists of the waffle slab or the reinforcement net of the flat slab.

Figure 51: Free grid layout.

The free layout: Columns are absolved from any grid layout and may be placed anywhere within the concrete surface domain.

Figure 52: Free layout.

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Form/System/Configuration-Diagram The Form/System/ConfigurationDiagram (FSC-Diagram) seeks to illuminate realisable combinations between form, system and configuration types through line connections. Keep in mind however that a diagram of this type is not uniquely quantifiable, but based on my subjective rationale. It has been attempted to assess all combinations from a general point of view, which is a delicate discipline. This has also resulted in the use of weaker line weights for combinations, which are not completely applicable in all thinkable cases, but still might be for some specific instances. The outcome of this assessment is seen in Figure 53.

Figure 53: The FSC-Diagram

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7. Script Scope and Platform Scope With the background obtained from the initial exercises performed in Chapters 4, 5 and 6 the scope of the script is ready to be outlined. Form GH is able to represent any kind of surface geometry, therefore, the script sets no restrictions on the surface form to be analysed. System It is evident that the prefabricated concrete slab systems are strictly relying on column grid regularity. This feature simply comes as part of the element DNA, for this reason no prospect of optimising column positions for these systems is seen. Also, singly-spanning slab systems will not be treated any further, the structural analysis of a continuously singly-spanning slab field is equivalent to performing beam analysis, a task which can be solved with an analytical approach by hand. This purge leaves only the doubly-spanning in situ systems for further consideration. Configuration It has already been decided to abandon the prefabricated concrete systems, this simultaneously eliminates the idea behind the locked grid layout. Based on Scheurer’s suggestions, for further development of the Groningen Twister presented in Chapter 4, in combination with the possibilities provided by the use of GH, it is decided to base the script upon FEM computation. This integration will provide the script with both the quantification of structural performance and a fitness measure for the further potential of optimisation; two aspects which Scheurer was missing in the Groningen Twister. Now as a FEM approach is chosen, discretisation of the surface into a FE-mesh is implicit. The support conditions of a FE-mesh are enforced at node points and so to enforce any support conditions, a list of candidate support points are required for referencing the support conditions on to the FE-mesh, these candidate support points are extracted from the surface geometry. Note that the free layout contains an infinite number of candidate support points since all points on the surface are in fact candidates, retrieving a list of all the distinct candidate support points on the surface geometry is therefore impossible, why this configuration is abandoned as well. Thereby, the choice of support alignment to be applied in the script falls upon the free grid layout. This layout can be scaled how the user wants, so both coarse and fine meshes can be applied. It can also be noted that letting the grid fineness go towards infinity will make this configuration layout converge towards the free layout.

Figure 54: Free grid configuration layout.

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FSC-Diagram To summarise the previous decisions of the present chapter, the concrete solutions which can be handled by the script are presented in the following, reduced FSC-Diagram.

Figure 55: Part of FSC-diagram (see Figure 53) which can be covered by script.

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Script Application Range The FEM calculation performed by the script is based on the theory of linear elasticity, which implies linear relations between applied load and deformation and between stress and strain given by Hooke’s law: σ=E·ε. The linear elastic analysis is a valid strategy when only small structural deformations arise. Large deformations will disrupt the linear relation between applied load and resulting deformation, as large deformations may lead to stability problems where structural equilibrium may be lost at intermediate stages. An example of this is buckling, and handling such cases requires geometrically non-linear analysis where load is applied in iterative steps. However, large deflections of slab solutions in construction should be avoided in any case, why the linear analysis is not considered a limitation to the script, but rather a prerequisite. Material non-linearity such as plastic behaviour of yielding steel is not valid for the linear elastic theory either, as such behaviour does not comply with Hooke’s law. The surface geometry will be analysed as a shell structure made of homogeneous, isotropic material, e.g. steel, aluminium or concrete. This means that any isotropic material can actually be chosen for the analysis, the script is not restricted to concrete structures. What this also means, is that composite cross-sections, such as reinforced concrete, which is of course the material structure of main interest to this thesis, cannot be calculated as standard. However, custom material specifications can be set up meaning that one can edit and input material stiffness manually. But to use the FEA routine directly for reinforcement design will not be an option, yet the section forces found from the FE computation will still be valid for the reinforcement dimensioning, it just cannot be done internally within the script. See also the section FEA Computation of Chapter 8. Documentation Extent The tool is meant for the early design phase providing well informed guide lines. Any kind of final structural documentation should not be based upon the tool.

Platform As parametric modelling environment the constellation of Rhino and Grasshopper is chosen for the script development. By reputation GH offers a steep learning curve, which was an essential element in the early consideration of software choice. The constellation of GH+Rhino is also the in-house parametric modelling environment used at Søren Jensen. This means that technical support would be within reach, and presents a simply practical aspect as regards future application of the tool within Søren Jensen. In addition GH has a well-established and open online community 8 with both users and the program developers themselves standing actively by with assistance.

8

http://www.grasshopper3d.com/

56

8. The Script This chapter will explain the basics behind the algorithm of the programmed script. It is desired to structure the script by organising it into different modules along the way, hence the explanation of the script will be given chronologically step by step. The script modules are presented schematically in Figure 56 and its detailed “spaghetti� equivalence as unfolded in GH is shown in Figure 57.

Figure 56: Schematic flow chart of developed script.

Figure 57: Script in Grasshopper.

In the following sub-chapters the modules of Figure 57 will be presented one by one. It is intended not to go into too much detail during this walk-through, but rather to give an idea of the overall flow and interdependencies. If reading a digital version of this thesis and interested in getting a better chance of reading the specific component names one might benefit from zooming in on the screen shots, as these are taken in high resolution.

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Design Domain Input

Figure 58: Design Domain Input.

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The Design Domain Input is an input module which therefore relies on user interaction. This module contains four sub segments; {Design Domain} This part holds the direct reference to the surface geometry for which a support distribution is sought. {Free Supports, Grid and Number} This part contains two number slider components. The slider values define the density of the free grid layout as well as the number of point supports available. The value of the grid density represents the distance between neighbouring grid points in meters. {Prescribed Supports} This part defines prescribed support points which are all referenced under the first point |Pt| component. {Restricted Areas} Restricted areas are set up by projecting closed, boundary curves onto the surface geometry. The part of the surface geometry lying within the projection of the closed boundary curve will be defined as a restricted area.

59

Grid Generation

Figure 59: Grid Generation.

This module generates the free grid layout and collects a set of activated grid points using components of PanelingTools, which is a GH add-on free for download. This module uses the inputs given for surface geometry, grid density and restricted areas. First a set of continuous grid points is computed on the surface geometry and later in the data flow the sub sets of these points lying within restricted areas are dispatched so only candidate support points are passed on to the support point generation module. The module also contains a set of components used for visual tagging of the activated grid points in the Rhino viewport, these components are shown in the bottom right corner of Figure 59. The only user interference possible in this module regards the slider value for the Mode-input of the |ptTrim| component. This input defines the grid trimming strategy around the surface geometry edges, three strategies are possible and represented by the three numbers 0, 1 and 2. The trimming strategy represented by the number 1 should not be used in this context, why only strategy 0 and 2 are valid. It is possible to orientate the grid direction, however not directly through the script. An instruction for controlling grid orientation is given in the GH forum 9. 9

http://www.grasshopper3d.com/group/panelingtools/forum/topics/controlling-grid-direction-usingsurface-distance-component 60

Generation of Free Point Support Positions

Figure 60: Generation of Free Point Support Positions.

This script step computes the locations of the free point supports. The two components holding a set of numbers and appearing with a pink frame are called gene pools, these components constitute several number sliders compiled under one component. The setup involving these particular two gene pools does not come as a standard solution. The standard gene pools of GH are strictly output components meaning that their output can only be generated by setting their slider values manually. The gene pools here, however, use customised VB coding so that they can actually be controlled by input parameters. This allows for setting up parametric relations to the number of supports and the grid size, which is a key feature to the applicability of this script. The customised gene pool was found in the GH forum 10. Two gene pools are necessary because grid points are referred to using coordinate indexing, i.e. two numbers per point. A coordinate indexing system rather than a sequential indexing system will be used, as this will be

10

http://www.grasshopper3d.com/forum/topics/request-parametric-genome-list

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beneficial to the later optimisation routine, see my discussion with David Rutten for elaboration 11. As the support coordinates are generated from the gene pools, their number of sliders should correspond to the number of supports, therefore the slider controlling the number of supports is given as input for count. It is seen that six supports are given in Figure 58 which corresponds to the amount of sliders in the gene pools of Figure 60, where sliders 0 to 5 are present. The definition of the gene pool slider ranges comes next. The i-index will be referred to as rows and the j-index as columns. Both the row and column coordinate should start from 0, this is simply because all computed indexing in GH starts from the base of 0. The relation between rows and columns will be that the amount of rows in the grid will be fixed whereas the amount of column coordinates might vary from row to row based on the shape of the surface geometry. Therefore it is obvious that all row sliders should have the same range starting from zero and going to the highest row index, the value of the highest row index is also parametrically computed based on the grid size in the top |Cluster| component of Figure 59 (a |Cluster| component clusters a collection of other components into one, therefore the logics behind a |Cluster| component are not directly visible). Since all sliders in a gene pool must have same range, the highest column index must be chosen for the maximum value of the range for the column index as well. This explains why the sub group for the Support j-index has some more strings attached than the Support i-index sub group, because the j-index slider values need remapping to assure sense of output. The i- and j-indexes are combined into coordinate sets using the |ptItem| component and the indexes are matched slider wise, meaning that support no 0 will receive its coordinates by combining the values from slider no 0 of the two gene pools, etc. A situation may occur where duplicate support points are present, this is because identical coordinate sets can be outputted from the gene pools. This cannot be avoided, but duplicate points can instead be removed and this is done using the |CullPt| component. In such cases a lower number of supports may actually be present than the target number originally set back in Figure 58. User interference in this module is allowed for dragging the slider values of the gene pools, which ultimately control the support positions. Hence, supports can be moved around manually whenever a simulation routine is not running.

11

http://www.grasshopper3d.com/forum/topics/does-galapagos-prefer-some-genome-data-structures-over-others

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Gathering Support Points

Figure 61: Gathering Support Points.

No real action is taken here, only that all support points are collected in one component for convenience. This means that input is taken from the Support Position Generation module, see Figure 60, and the {Prescribed Supports} sub segment of the Design Domain Input module, see Figure 58.

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FEA Computation

The FEA computation module of the script uses a commercial plug-in for GH named Karamba12. This tool is developed by Clemens Preisinger in collaboration with Bollinger + Grohmann Engineers. Karamba is custom-made for implementation in GH, it therefore offers a fluent connection between geometry, FEA and structural optimisation by use of GAs already developed for GH. The fundamental attribute behind a FE core, which can computationally keep up with the interactive standards set by GH and provide efficiency when linked to a GA, is speed. For this reason the FEM computation engine of Karamba runs significantly faster than standard off-the-shelf FE programs (Preisinger 2013). The range of structural calculation varieties offered by Karamba is, however, not (yet) as comprehensive as other off-the-shelf FE programs, which are typically used directly for structural documentation purposes. Subsequently, Karamba is primarily intended as a design tool relevant for the early design stages, structural optimisation and application amongst architects where the interactive aspect allows for a more intuitive approach to structural design. Karamba already has several benchmark examples provided with the download, but a verification of Karamba for the specific problem types to be analysed in the context of this thesis will be conducted in Chapter 8 by comparison of results obtained with other FE software. Karamba uses shell elements developed on Kirchhoff plate theory meaning that shear deflections are neglected in the analysis, this should however be a fair assumption for the problem types faced in this thesis. See my GH forum discussion for further detail on FE formulation of the shell element 13. The FEA computation module of the script is divided into three segments; an input module, an analysis module and a result module.

12

http://www.karamba3d.com/ ; http://www.karamba3d.com/wp-content/uploads/gh/Install/Karamba_1_1_0_Manual.pdf 13 http://www.grasshopper3d.com/group/karamba/forum/topics/fem-definition-for-the-shell-element-of-karamba

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FEA Input FEA Input contains all inputs relevant to the FEA computation and is divided into five sub categories. {FE-Mesh Fineness} The mesh resolution sets the target element size in m. The fineness or resolution of the FE-mesh determines both the accuracy and the computation speed of the analysis. The finer the mesh becomes, the closer the FE-discretisation approximates its physical continuum structure, and therefore the accuracy of the results obtained will increase for finer resolutions. But a finer resolution also requires more computation effort, as more nodes and elements will have to be calculated in the analysis. The definition of mesh resolution should therefore keep the right balance of getting acceptable accuracy in results in acceptable time. The computation time aspect is especially relevant when the optimisation routine is considered, as these routines rely on performing many analyses. As Karamba uses Kirchhoff theory, it should never be necessary to use elements smaller than the shell height (Blaauwendraad 2010). {Support BCs} The support boundary conditions (BCs) hold the degrees of freedom (DOFs) of the supported nodes. The locations of the supported nodes are passed on as input from Figure 61. Each supported node has six DOFs; three translations (Tx, Ty, Tz) and three rotations (Rx, Ry, Rz). The translation and rotation indices refer to the axes of the global coordinate system. The definition of support BCs is a problem specific task and a delicate discipline, as false settings of support conditions might have severe, structural consequences. Therefore this discipline requires both focus and flair and should be undertaken by a qualified structural engineer. {Load} This is where the load specifications are defined. Uniformly distributed loads and point loads can be defined and their magnitudes are set on the number sliders. {Surface Cross-Section Height} This slider defines the height of the shell cross-section. {Material Selection} This component holds a library of standard materials to choose amongst for the structural material. The predefined materials of Karamba are steel, concrete, wood and aluminium. As previously mentioned custom materials can alternatively be computed manually.

Figure 62: FEA Input.

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FEA Routine

Figure 63: FEA Routine.

This segment contains the necessary components for Karamba to process its FEA computation. Most importantly, it generates the FE mesh on the surface geometry other than that it sets up the structural selfweight and collects the data given in Figure 62 for assembly. No user interference is required for this module, except when wanting to scale the self-weight for specific load combinations. This can be easily done by plugging in the value of the relevant partial coefficient as input for the gravity load component. Two results are outputted at this stage; these are the values for the maximal structural displacement and the internal energy computed within the structure. The maximal displacement is given as a vector length, i.e. using the L2 norm. But here it is referring to the peak value of all the node displacements, and not the entire deflection field as was the case for the gradient based method back in Chapter 3. The internal energy is commonly referred to as the compliance in FEM literature, here it is also most often the target of topology optimisation. Unlike the maximal deflection the internal energy is a measure describing the entire structure, why this value can be used for judging the structural efficiency from a global point of view.

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FEA Results

Figure 64: FEA Results.

The result module should not be considered as a fixed module, it is really open to user configuration. All results have been computed at this stage, so no changes to this module can possibly ruin the core algorithmic flow, and results can be set up as needed for the specific design purpose. The two components |ModelView| and |ShellView| are however always handy for visually displaying fundamental results and contour plots in the Rhino viewport. Other Karamba result components used in this setup are |React|, |NodeDisp| and |S-Forces|. |React| returns support reaction forces and moments, |NodeDisp| returns the node displacements and |S-Forces| returns the section forces as the principal forces and principal moments. Two parametric data flows have been manually set up for this module, one concerning the maximum deflection and one concerning the maximum moment.

67

{Max Deflection Routine} The upper component row of this sub routine checks the structural deflection criteria, a deflection demand is therefore needed as input for the |Deflection Demand| panel, which for the current setup is set to 500 yielding the deflection criteria wmax<L/500, where the span, L, is parametrically computed as the shortest distance found between two support points. The bottom component row identifies the node undergoing the largest deflection and tags it visually in the Rhino viewport. {Max Moment Routine} This subroutines identifies the largest, absolute moment value and tags the element for which this value is computed in the Rhino viewport.

[A video14 has been put online showing the core functionality of the script modules elaborated so far.]

14

https://youtu.be/AF-565YbvyE

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Optimisation Routines

Figure 65: Optimisation Routines.

Like the FEA Results module, the optimisation routines are also completely open for configuration and the range of fitness measures to optimise for are boundless, still a number of evident routines dealing with maximal deflection, internal energy, maximal moment and reaction force distribution have been set up in the current situation. The optimisation routines can be performed using either Galapagos15 or Octopus 16, both engines run on GAs. Galapagos is the standard GA optimiser of GH, and like GH Galapagos is also developed by David Rutten. Octopus is a free, add-on component for GH developed by Robert Vierlinger, who is also working for Bollinger + Grohmann and on board the Karamba development team as well. Galapagos runs on a single fitness function whereas Octopus handles up to five at a time. As any function can be described for the fitness function, in principle Galapagos can handle multi objective optimisation17, but this requires normalisation of the combined element values, which further requires pre-existing knowledge of the range of values for the individual, computed elements. Therefore, Octopus is the convenient choice for multi objective optimisation as it lets multiple fitness values be presented in a threedimensional coordinate system where separate fitness values can be presented along the three axes and by range in size and colour. Octopus can also perform single objective optimisation, but here Galapagos will be preferred for its simplicity compared to Octopus.

15

http://www.grasshopper3d.com/group/galapagos ; http://www.grasshopper3d.com/profiles/blogs/evolutionary-principles 16 http://www.grasshopper3d.com/group/octopus 17 http://www.grasshopper3d.com/forum/topics/galapagos-multiple-fitnesses

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Along with the fitness function/functions, the optimisation routine also needs an input of the variables allowed to change for optimising the fitness score. This set of variables is called the genome. The genome input can be number sliders and gene pools. As an optimised support distribution is sought in all optimisation cases, the two gene pools in Figure 60 determining the support locations should be chosen as genome at all times.

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IFC Model Export

Figure 66: IFC Model Export.

It is possible to export the model resulting from the GH algorithm as an IFC file. This makes it possible to import the computed model into other software such as e.g. Revit or Robot for onward processing. The IFC export module uses another commercial GH add-on called GeometryGym. Karamba does have an export component allowing for export to either Robot or RSTAB. However, the Robot import is not compatible with shell structures generated by Karamba, why this component cannot be used for the specific script. The RSTAB export has not been checked because at Søren Jensen, Robot, and not RSTAB, is the FEA software used.

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9. Results Test A – Comparison to Gradient Based Optimisations This test of the script will compare computed results to the optimised support configurations found for the gradient based method presented in Figure 26: Plate solutions found for the two objective functions; L2 norm vs mean compliance. Table: (Jang et al. 2009)Figure 26 of Chapter 3. The plate was modelled as a 1,2m × 0.8m LCD Panel with a thickness of 0,01m and the following material properties; E=730MPa, ν=0.23 and ρ=2370kg/m3. For the FE discretisation 24 × 16 square elements were used giving 25 × 17 node points. Now as Karamba uses triangular finite elements the mesh is represented with two triangular elements per square element. 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figure 67: Equivalent Karamba Grid for LCD Panel.

In (Jang et al. 2009) only support configurations located symmetrically, with respect to x and y axes were investigated to discover the results presented in their shown table. This is why brute force search could be applied for calculating the values of the L2 norm, which they failed to build their optimisation routine for. With a look at the combinatorics behind the solution space it is seen that without symmetry constraints the number of different support combinations are (17·25)4 ≈ 3,3·1010 for four supports, and (17·25)8 ≈ 1,1·1021 for eight supports. Both these numbers are astronomical, and for setting them into perspective, it can be mentioned that with my PC, running on an intel i5 core, a brute force search for a similar plate problem, with only three supports and a space of 363 ≈ 47000 solutions, took three entire days, see GH forum for the problem details18. So doing some simple math comparing to the four-support problem, it is seen that (17·25)4/363 ≈700000, meaning one could expect a pc runtime of 3·700000 days or 5750 years. Now if applying symmetry by dividing the rectangular plate into four equal segments with symmetry lines through row 8 and column 12 of Figure 67, suddenly the solution spaces become 9·13=117 for the four supportproblem, and 92·132=13689 for the eight-support problem. 18

http://www.grasshopper3d.com/forum/topics/does-galapagos-prefer-some-genome-data-structures-over-others

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Support number, n=4 Best, symmetrical configuration for mean compliance/internal energy (IE) by brute force search:

Figure 68: Best support configuration generated for mean compliance, IE=2,00路10-4kNm.

The best solution obtained in terms of total mean compliance is identical to the one found in Figure 26. Best, symmetrical configuration for L2 norm of deflection by brute force search:

Figure 69: Best support configuration generated for L2 norm, |L2|=0,94m.

For the L2 norm, results are not the same. The green support configuration represents the one computed by my script, and the red one represents the best configuration of Figure 26. However, the red configuration came in as second best of all for the script computation with|L2|=1,02m. An explanation to the deviation might be the use of different elements, triangles vs squares. 73

Best found, free configuration for mean compliance/internal energy by Galapagos optimisation:

Figure 70: Best support configuration generated for mean compliance when not constrained by symmetry, -4 IE=1,64路10 kNm.

Comparing Figure 70 to Figure 68 shows that Galapagos manages to find a better solution than the best, symmetric solution for internal energy. Best found, free configuration for L2 norm of deflection by Galapagos optimisation:

2

2

Figure 71: Best support configuration generated for L norm when not constrained by symmetry, |L |=0,642m.

The configuration of Figure 71 beats the L2 norm of Figure 69. The symmetry of this configuration goes against human nature by breaking any kind of internal symmetry amongst the supports. But then again for these types of complex problems one can only rely on intuition, and it is therefore difficult to judge from a factual point of view. Because of mistrust the configuration of Figure 71 has been mirrored in the following figure. 74

Mirrored version of best found, free configuration for L2 norm of deflection by Galapagos optimisation:

2

Figure 72: Mirrored version of best support configuration generated for L norm when not constrained by symmetry, 2 |L |=0,644m.

There is a slight deviation in the L2 norms of Figure 71 and Figure 72. Ideally this should not be the case, but it must be due to using triangular elements, as their diagonal directions will inevitably introduce dissimilarity upon the mesh, see Figure 67. [Videos showing the principles behind the brute force search 19 and the Galapagos optimisation20 have been added to my youtube channel 21.]

19

https://youtu.be/ZysIgKFwxEQ https://youtu.be/r7ExqHHK6hc 21 https://www.youtube.com/watch?v=AF-565YbvyE&list=PLdkQOU6SqqjnccGaBKTVveEWj8J6R_gFF 20

75

Support number, n=8 Best, symmetrical configuration for mean compliance/internal energy (IE) by brute force search:

Figure 73: Best support configuration generated for mean compliance, IE=3,90路10-5kNm.

The obtained best solution in terms of total mean compliance is not entirely identical to the one found in Figure 26, however the fitness score for that solution came in second best with a value of IE=4,10路10-5kNm. Best, symmetrical configuration for L2 norm of deflection by brute force search:

Figure 74: Best support configuration generated for L2 norm, |L2|=0,150m.

It turns out that the best configuration for the L2 norm was also the best one for mean compliance, see Figure 73. There is a slight deviation from the solution found in Figure 26, this solution came in at a 9th place with a value of |L2|=0,18m.

76

Best found, free configuration for mean compliance/internal energy by Galapagos optimisation:

Figure 75: Best support configuration generated for mean compliance when not constrained by symmetry, IE=3,50路10-5kNm.

The solution found beats the best, symmetrical solution, see Figure 73. Best found, free configuration for L2 norm of deflection by Galapagos optimisation:

Figure 76: Best support configuration generated for L2 norm when not constrained by symmetry, |L2|=0,136m.

The solution found beats the best, symmetrical solution, see Figure 74. An interesting thought concerning both Figure 75 and Figure 76 could be to consider, if the support configurations have in fact morphed towards triangular grid distributions only distorted by the plate edge effects, see Figure 14 and Figure 15. 77

Test B – Study of FEM Computed Field Quantities

Test A deals with the L2 norm of entire structural deflection fields as well as structural mean compliance, these are fitness measures used for global assessment of structures. However, as a structural engineer one’s main concerns are most often structural peak values, as these values are typically governing global dimensions of a given structure. For example, and seen strictly from a structural point of view, a singlyspanning slab does not need its full height over the entire length, but only at mid span. Nevertheless, the required height at mid span sets a global, uniform height of the slab for practical convenience. When designing load carrying slabs, the structural peak values of main interest concern moment and deflection. Depending on the slab span length the dimensioning factor will be either the maximum moment or the maximum deflection, cf. plate formulas from Teknisk StĂĽbi given in Chapter 5 for flat slabs. So, as the peak moment and deflection will normally be the values of main interest to the structural designer, these values are also obvious targets of the optimisation routine. Therefore, the circumstances of computing these two particular values using Karamba should be investigated. For this investigation, the results for a circular plate of radius 2m with a fixed point supports in its centre and uniformly distributed load is chosen. The material is concrete having E=34000MPa, the height is h=0,05m and the load is p=3kN/m2. The analytical solution is found in (Johansen 1949): 1 1 đ?‘€ = đ?‘?đ?‘&#x; 2 = ∗ 3đ?‘˜đ?‘ /đ?‘š 2 ∗ (2đ?‘š )2 = 4đ?‘˜đ?‘ đ?‘š 3 3 đ?‘¤~

1 đ?‘€đ?‘&#x; 2 1 = ∗ 4 đ??¸đ??ź 4

4đ?‘˜đ?‘ đ?‘š ∗ (2đ?‘š )2 = 0,013đ?‘š (0,05đ?‘š)3 3,4 ∗ 107 đ?‘˜đ?‘ /đ?‘š 2 ∗ 12

Following numerical values were obtained, see Figure 77 and Figure 78 for an idea of the mesh resolutions. Mesh Resolution 0,14 0,13 0,10 0,09 0,08 0,06

NElements 1238 1432 2500 3025 3919 6990

IE [kNm] 0,1534 0,1534 0,1534 0,1534 0,1533 0,1534

wmax [m] 0,0130 0,0129 0,0129 0,0129 0,0129 0,0129

Mmax [kNm/m] 8,58 8,82 9,84 10,19 10,62 13,45

Table 1: FEM field quantities for circular plate with fixed point support at centre.

As seen in Table 1 the deflection values correspond to the analytical result. Both the value of the slab internal energy and maximum displacement show very quick convergence. This means that relatively coarse FE meshes can be used when computing these values, and therefore they will be efficient targets for the stochastic optimisation routines because of savings in computation time. The moment values, however, are nowhere near the analytical result, and only show a disturbing behaviour by increasing as a function of used elements in the FE mesh. This unfortunate tendency of increasing peak moments at point supports is a classic pitfall already known for FEM computation, and in (Blaauwendraad 2010) Chapter 14 is solely devoted to dealing with this issue. So, there are ways of handling this error practically, which will be demonstrated later in the report. As assessing the numerically computed Mmax is useless, so will the assessment of principal stresses be, as these stresses are calculated directly from the principal moments.

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Figure 77: FE mesh for a resolution of 0,14 yielding 1238 elements.

Figure 78: FE mesh for a resolution of 0,06 yielding 6990 elements.

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Test C – Karamba to Robot As Karamba cannot handle detailed, structural analysis at the same level of heavier stand-alone FE packages, having the possibility of importing the Karamba model to one of these packages could be a fruitful facility. As earlier mentioned, Søren Jensen uses Robot, why a link between Karamba and Robot is in focus. A planar surface with the shape of an amoeba and an area of 1940m2 was drawn in Rhino and structurally analysed using the script with Karamba. The slab is supported at seven points, one support is placed in the centre of gravity for ensuring horizontal stability with 0 horizontal translations (Tx & Ty) and in-plane rotation (Rz), the six other supports only take up vertical reactions (Rz). The surface is modelled as a concrete slab of h=0,5m, E=34000MPa, ν=0.2 only exposed to self-weight, p=25kN/3. Reactions and peak deflection in Figure 79 was found using Karamba with a mesh resolution of 2m yielding 855 elements.

Figure 79: The amoeba mesh, support reactions and peak deflection found in GH environment.

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The model was exported to an IFC file compatible with Robot by using ggGHKaramba from GeometryGym 22. In the current version the concrete self-weight gets badly scaled why this value must be altered manually within Robot. Furthermore support BCs might also change during the import. But, when overcoming these minor imperfections similar results as in Figure 79 are produced in Robot.

Figure 80: The amoeba mesh, support reactions and peak deflection found in Robot.

Comparing Figure 79 and Figure 80 it is seen that reaction forces are found within acceptable deviation ranges. The peak deflection location seems coinciding to the naked eye, and their magnitudes are also okay, 71mm vs 67mm.

22

https://www.youtube.com/watch?v=j6KLIqK0pDI

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Test D – Study of Correlation between Internal Energy and Peak Deflection As previously mentioned in Test B, peak values were of greater interest to the structural engineer than the global, structural measures. With the moment peak value out of the picture due to the findings of Test B, it would be interesting to see, if there is a correlation between the peak deflection value and the internal energy of the slab in the same way that there appeared to be one between the internal energy and the L2 norm of the deflection, as discussed in Test A. With basis, once again, in the amoeba slab a couple of results have been produced with Karamba, and they seem dignified for drawing a conclusion upon the subject. The values computed for the investigation are tabulated in Table 2, and the corresponding support distributions are depicted in Figure 81, Figure 82 and Figure 83. Subject Specimen 1 Specimen 2 Specimen 3

IE [kNm] 327 272 415

wmax [m] 0,070 0,087 0,069

Table 2: Study of correlation between IE and wmax.

Based on the computed results in this test, it should be fair to state that there is no direct correlation between obtaining minimum peak deflections and internal work.

Figure 81: Specimen 1 – found using Galapagos optimisation with IE as the fitness measure. Same configuration as in Figure 79

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Figure 82: Specimen 2 – found using Galapagos optimisation with wmax as fitness measure.

Figure 83: Specimen 3 – found using Galapagos optimisation with wmax as fitness measure.

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Case – The Flying Carpet This case will deal with the actual triggering project of this thesis. The project baseline has been shortly described in the Motivation, but to elaborate, the Flying Carpet was thought as an undulating concrete shell structure with singly-curved rises at its ends for bike parking, and a flat middle piece in ground level, see Figure 84 and Figure 85 for an idea of the shape.

Figure 84: Rendering of the Flying Carpet. Illustration: Polyform Architects

Figure 85: Architect’s model of the Flying Carpet in Rhino.

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Based on the surface geometry, Søren Jensen proposed an initial column layout based on rules of thumb. A 6x6m orthogonal column grid was directed in the principal curvature direction of the two shell peaks, extra attention was given around the two surface voids. A slab thickness of 250mm was estimated, and the in situ columns were suggested as 300x300mm. See Figure 86 for details.

Figure 86: Early structural principles developed by Søren Jensen. Drawing: Duncan Horswill

The planar layout is given a makeover following the colour symbolism introduced in Chapter 6. This layout is presented in Figure 87.

Figure 87: The Flying Carpet plan with configuration colour legends.

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Left, Column Supported Surface Part For the left surface part a configuration of 35 columns was suggested. A column restricted area is established in the narrowing end space along the diagonal boundary edge. This boundary edge is in fact a retaining wall planted in soil. It will be considered load carrying as well as stabilising in both x and y directions. Normally, a wall should only be considered stabilising along its longitudinal direction, but due to the direct soil contact of the retaining wall, it is assumed that additional structural arrangements can be facilitated underground incorporating wall and soil to ensure out of plane bending strength for the wall as well. Speaking of horizontal stability the friction between the middle surface part and the ground should not be considered a stabilising element for the left surface part, as these two surface parts are separated by a movement joint. This ultimately means that the columns need to be part of the structurally stabilising system, so frame action between slab and column can be activated. Therefore, the point supports will act as fixed supports restrained against rotation and, naturally, lateral movement, the fixed DOFs of the column point supports are therefore Tz, Rx and Ry. For the retaining wall all DOFs are fixed.

Figure 88: Left surface part configuration using 35 columns.

Before an irregular column pattern can be found through optimisation, the initial suggestion is first analysed. Figure 89 is converted from paper to Rhino, and referenced into the GH script, see Figure 89 and Figure 90. The structure was to be analysed for a live load of:

The self-weight of the concrete is:

đ?‘ž = 5đ?‘˜đ?‘ /đ?‘š 2

đ?‘” = đ?›ž ¡ â„Ž = 25đ?‘˜đ?‘ /đ?‘š3 ¡ 0,250đ?‘š = 6,25đ?‘˜đ?‘ /đ?‘š 2

Only the results of the characteristic loads are investigated at first, so no partial coefficients are applied in the load combination.

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Figure 89: Søren Jensen’s initial column layout using 35 columns imposed in GH script.

Figure 90: FEM deflection results computed for initial column layout, wmax=0.005m.

Next, for handling the moment insecurities and determining the actual section forces of the shell, the model is exported to Robot.

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Figure 91: Vertical reactions in Robot (FZ)

First a quick check of the reaction forces is performed. As the column grid is 6x6m one should expect reaction forces for the internal columns around: 𝑅𝑧 = 6𝑚 · 6𝑚 · (𝑔 + 𝑞) = 6𝑚 · 6𝑚 · (6,25 + 5)𝑘𝑁/𝑚2 = 405𝑘𝑁

Most of the internal columns seem to have reactions a little larger, but they are still within range, so the model is acceptable. One should also to keep in mind that the structure is not a planar slab, but a curved shell, this must have some impact on the computed results.

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Now concerning moments, the immediate distribution given by Robot is as shown in Figure 92.

Figure 92: Initial peak and field distribution of 1st principal moments.

Some quite serious peaks are spotted all around. (Blaauwendraad 2010) advocates treating this problem, by not looking at the very peak moment values, but at an average moment value in the surroundings of the peak. The specific surrounding column area to be taken into account is defined in Figure 93.

Figure 93: Smearing out of moment peak. Illustration: (Blaauwendraad 2010)

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The method presented in Figure 93 is motivated by the fact that even though the extreme peak moments are highly dependent on the number of finite elements, the average area of the moment diagram actually is not. So, the next step will be to apply this knowledge in Robot. Robot has a build in function in the result module called ‘Reduction of forces above columns and walls’ 23 and furthermore one can choose to perform a global smoothing of values24. For the ‘reduction of forces’ function, one needs to alter the support conditions and define them as columns and walls with geometric dimensions. The point supports were assigned as 300x300mm rectangular columns and the retaining wall point supports were defined as a wall of 300mm thickness. What the reduction of forces command actually does, mathematically speaking, is unknown to the user, however, the results obtained using this function in combination with global smoothing seem to be in correlation with another rule of thumb provided in (Blaauwendraad 2010). This rule of thumb states that the moment on top of a column in an orthogonal grid, as in the current case, can be expressed relatively from the column reaction Rz within the range of M=Rz/6 - Rz/4. A practical rule of thumb is M=Rz/5, and a safe rule is M=Rz/4. If applying the safe rule and comparing results of Figure 91 and Figure 94 the relation seems to hold, as globally representative one can use Rz≈430 (see Figure 91) yielding M≈430/4≈110, which corresponds well with the peak moment values found in Figure 94. Hence, the FE model is accepted and therefore suitable for further analysis.

st

Figure 94: Reduced and smoothened 1 principal moment values.

23 24

http://help.autodesk.com/view/RSAPRO/2016/ENU/?guid=GUID-E4E8BE33-DCD2-4343-824C-447A1AB5ECA0 http://help.autodesk.com/view/RSAPRO/2016/ENU/?guid=GUID-E73EA3A2-AEC8-40EA-8AC3-E7EDCA7644C2

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When reviewing the indicative formula for flat slabs of Teknisk StĂĽbi with the max moment found in Figure 94, one gets â„Ž = ďż˝800 đ?‘šđ?‘’đ?‘‘ + 15 = ďż˝800 ¡ 125,82 + 15 ≈ 335đ?‘šđ?‘š, hence the height of 250mm seems rather low set, and even more so because the moment used is in fact only the characteristic moment, đ?‘šđ?‘˜ , and not the real design moment, đ?‘šđ?‘’đ?‘‘ . On the safe side, the design moment can be set as:

đ?‘šđ?‘’đ?‘‘ = 1,5 ¡ đ?‘šđ?‘˜ ≈ 190, this would then give a plate thickness of â„Ž = √800 ¡ 190 + 15 ≈ 405đ?‘šđ?‘š. The actual point of interest behind all of this investigation is to determine whether or not the concrete has cracked, and consequently the tension stresses are in focus in this final Robot analysis step.

Figure 95: 1st principal stresses calculated at top surface layer.

As seen from Figure 95, all tension stresses (positive) at column supports are well above concrete tension strength, fctm=1,6 – 4,1MPa (for concrete strengths C12-C50), and accordingly deflections and strengths should be calculated for the cracked concrete cross-section, which only allows for taking the reinforcement and concrete compression zone into account. However, before moving on a quick check of the computed stresses is carried out.

Figure 96: Three representative stress check points have been chosen.

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The normal forces found for the shell are presented in Figure 97.

Figure 97: 1st principal forces.

With basis in Figure 94, Figure 95, Figure 96 and Figure 97 a check of the selected peak stresses is carried out by use of Navier’s formula: đ?‘€ đ?‘ đ?œŽ= + đ?‘Š đ??´ Check 1: 56,09 ¡ 103 đ?‘ 18,09đ?‘ /đ?‘šđ?‘š đ?œŽ = 6¡ + = 5,38đ?‘€đ?‘ƒđ?‘Ž + 0,07đ?‘€đ?‘ƒđ?‘Ž = 5,46đ?‘€đ?‘ƒđ?‘Ž ≅ 5,46đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! 250đ?‘šđ?‘š (250đ?‘šđ?‘š)2

Check 2: 112,22 ¡ 103 đ?‘ 42,84đ?‘ /đ?‘šđ?‘š đ?œŽ = 6¡ + = 10,77đ?‘€đ?‘ƒđ?‘Ž + 0,30đ?‘€đ?‘ƒđ?‘Ž = 10,94đ?‘€đ?‘ƒđ?‘Ž ≅ 10,85đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! 250đ?‘šđ?‘š (250đ?‘šđ?‘š)2 Check 3: 85,30 ¡ 103 đ?‘ −111,13đ?‘ /đ?‘šđ?‘š đ?œŽ = 6¡ + = 8,19đ?‘€đ?‘ƒđ?‘Ž − 0,44đ?‘€đ?‘ƒđ?‘Ž = 7,74đ?‘€đ?‘ƒđ?‘Ž ≅ 7,74đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! 250đ?‘šđ?‘š (250đ?‘šđ?‘š)2

As seen from the above checks, the stresses seem correct, why the assumption that concrete cracking will occur, must be legitimate. Another conclusion to draw when comparing the stress components resulting from bending (M/W) and normal force (N/A), is that bending is still the much dominant structural behaviour. Even though the structure has curvature, there is not a lot of arch effect going on.

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This study illuminates yet another obstacle emerging between the developed GH script and the structural nature of reinforced concrete. Because, as the GH script offers no capability of checking the actual section forces, neither does it have a chance of analysing whether the concrete has cracked, which ultimately decides if the full cross-section can be used for the deflection calculation, or only the cracked cross-section. There will be substantial differences in calculated deflections between using an uncracked and a cracked cross-section, why this is a significant problem. However, the problem only relates to the magnitude of the calculated deflection, and not its location in the structure, why the peak deflection can still be applied as fitness value for the support distribution using linear analysis. For this reason the basic functionality of the support positioning optimisation routine is not affected. An useful follow up would be to check, if the 405mm slab, hinted by the formula of Teknisk Ståbi, could prevent the slab from cracking, however, this potential can be immediately turned down by a quick hand calculation. Reusing the max moment value 𝑚𝑒𝑑 = 190 of the 225mm slab for the 405mm slab must now be considered on the unsafe side due to increased concrete self-weight, but even so, cracking of the concrete will still occur: 𝜎𝑀 = 6 ·

190 · 103 𝑁 = 6,95𝑀𝑃𝑎 > 𝑓𝑐𝑡𝑚 (405𝑚𝑚)2

Consequently, the structure must be considered cracked, and the specifications of the cracked crosssection needs to be established. For doing so a new module, presented in Figure 98, has been added to the script.

Figure 98: Calculation of transformed moment of inertia for cracked cross-section.

This module assumes the top and bottom reinforcement layer to be the same, and furthermore symmetrically placed around the geometric centre line of the cross section. The module takes input on reinforcement diameter, spacing and cover. Also the modulus of elasticity is needed for both the reinforcement and concrete. The cross-section height is referenced directly from the FEA Input module. Based on these inputs the module runs parametrically, and with it introduced, the deflection of the 250mm slab presented initially in Figure 90 can be correctly scaled.

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Figure 99: FEM peak deflection computed for initial column layout with cracked cross-section, wmax=0.014m.

As can be seen when comparing Figure 90 and Figure 99 the maximum deflection has been scaled by almost a factor three due to the cracked cross-section for a preliminary reinforcement choice. Following the advice of Teknisk St책bi, the slab thickness is changed to 400mm giving a peak deflection as illustrated in Figure 100.

Figure 100: FEM peak deflection computed for initial column layout with cracked cross-section and h=400mm, wmax=0,006m.

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Following these findings, the time for optimisation has now come. The optimisation goal is set as follows: Minimise column number, slab deflection and peak moment using multiple fitness optimisation. One might be surprised to see the moment figure as fitness value, when taking into account the previous error discussion, but I have computed the moment by rule of thumb with inspiration in Blaauwendraad’s findings of M=Rz/6 - Rz/4, and being conservative, it was chosen to only divide the maximum, vertical reaction by two, hence why the max moment is computed as Rz,max/2. [The videos linked25 in the footer introduces Octopus along with the search strategy applied for finding candidate support configurations.] A candidate design is found using Octopus, minor local alterations have been made manually afterwards. The candidate support distribution is presented in Figure 101 and Figure 102.

Figure 101: Candidate solution using 32 columns, wmax=0.009m.

Figure 102: Candidate solution using 32 columns with free grid point layout of 2,5m density. 25

https://youtu.be/euOkFmzUPDY https://youtu.be/WniZpvbMRkY

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As the moment calculation has been set up using a rule of thumb, it is still necessary to verify the model using Robot. Following moments have been calculated.

Figure 103: Reduced and smoothened 1st principal moment values.

nd

Figure 104: Reduced and smoothened 2 principal moment values.

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As can be seen from Figure 103 and Figure 104, the 1st principal moments are dominating, why these will be dimensioning for the reinforcement. The largest moment found is 395,15kN for the outermost, lone column placed at the bottom right corner, the reinforcement will, however, not be dimensioned for this value, instead it would be suggested to increase this column size further, to lower the moment. Looking away from the just mentioned moment, the 2nd largest moment found is 208,93kN (at the top left boarder). With that in mind, the guess of M=Rz/2 was conservative as intended, but, I believe, still not way off, because the largest column reaction force in GH was found as Rz,max=582kN (in Robot it was found as 588kN), yielding Rz/2=291kN.

Figure 105: Cracked cross-section of concrete slab.

Setting the dimensions of Figure 105 as follows will make up an acceptable reinforcement design for the slab. â„Ž = 400đ?‘šđ?‘šđ?‘‘đ?‘œ = 35đ?‘šđ?‘šđ?‘‘ = â„Ž − đ?‘‘đ?‘œ = 315đ?‘šđ?‘š đ??¸đ?‘ 210đ??şđ?‘ƒđ?‘Ž = = 6 (đ?‘†235 đ?‘Žđ?‘›đ?‘‘ đ??ś35) đ?›ź= đ??¸đ?‘? 35đ??şđ?‘ƒđ?‘Ž đ??´đ?‘ đ?‘› = đ??´đ?‘ đ?‘œ = 5450đ?‘šđ?‘š2 (ø25 đ?‘¤đ?‘–đ?‘Ąâ„Ž 90đ?‘šđ?‘š đ?‘ đ?‘?đ?‘Žđ?‘?đ?‘–đ?‘›đ?‘”)đ??´đ?‘? = đ?‘? ¡ đ?‘Ľ

The added GH module shown in Figure 98, sets up the equilibrium equation of the cross-section, and solves it for x, which is the height of the concrete compression zone, which again is the only active concrete part in a cracked cross-section. The equation yields: 1 ¡ đ?‘Ľ + đ?›ź ¡ đ??´đ?‘ đ?‘œ (đ?‘Ľ − đ?‘‘đ?‘œ ) = đ?›ź ¡ đ??´đ?‘ đ?‘› (đ?‘‘ − đ?‘Ľ) ⇒ 2 đ?‘Ľ = 109,06đ?‘šđ?‘š đ??´đ?‘? ¡

With x known the transformed moment of inertia can be calculated as follows: 1

đ??źđ?‘Ą = 3 ¡ đ?‘? ¡ đ?‘Ľ 3 + đ?›ź ¡ đ??´đ?‘ đ?‘œ (đ?‘Ľ − đ?‘‘đ?‘œ )2 = đ?›ź ¡ đ??´đ?‘ đ?‘› (đ?‘‘ − đ?‘Ľ)2 = 2,75 ¡ 109 đ?‘šđ?‘š 4 Now, stresses can be determined: đ?‘€đ?‘’đ?‘‘ = 1,5 ¡ đ?‘€đ?‘˜ = 1,5 ¡ 208,93đ?‘˜đ?‘ = 315đ?‘˜đ?‘ (đ?‘?đ?‘œđ?‘›đ?‘ đ?‘’đ?‘&#x;đ?‘Łđ?‘Žđ?‘Ąđ?‘–đ?‘Łđ?‘’) 97

đ?œŽđ?‘? =

đ?‘€đ?‘’đ?‘‘ 315 ¡ 106 đ?‘ đ?‘šđ?‘š đ?‘“đ?‘?đ?‘˜ 35đ?‘€đ?‘ƒđ?‘Ž ¡đ?‘Ľ = = 12,5đ?‘€đ?‘ƒđ?‘Ž < = = 24,1đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! 9 4 đ??źđ?‘Ą đ?›žđ?‘? 1,45 2,75 ¡ 10 đ?‘šđ?‘š

đ?œŽđ?‘ đ?‘› = đ?›ź ¡

đ?‘“đ?‘Śđ?‘˜ 235đ?‘€đ?‘ƒđ?‘Ž đ?‘€đ?‘’đ?‘‘ ¡ (đ?‘‘ − đ?‘Ľ) = 175,7đ?‘€đ?‘ƒđ?‘Ž < = = 195,8đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! đ??źđ?‘Ą đ?›žđ?‘ 1,2

đ?œŽđ?‘ đ?‘› = −đ?›ź ¡

đ?‘€đ?‘’đ?‘‘ ¡ (đ?‘Ľ − đ?‘‘đ?‘œ ) = −50,8đ?‘€đ?‘ƒđ?‘Ž < đ?‘“đ?‘Śđ?‘‘ = 195,8đ?‘€đ?‘ƒđ?‘Ž đ?‘śđ?‘˛! đ??źđ?‘Ą

The new reinforcement calculated also facilitates a change in the scaled deflection computed in GH, this is decreased as less reinforcement was assumed initially, and the new peak deflection is 5mm as illustrated in Figure 106. It is not likely that such a small deflection could ever cause problems.

Figure 106: Updated candidate solution using 32 columns, wmax=0,005m

As a last note on the reinforcement design, an orthogonal net using the same reinforcement distribution in both directions is assumed, both for the lower and upper layer. In normal situations with plates having two spanning in two constant span directions, x and y, the reinforcement is placed accordingly to these directions. However, in the cases arising here forces will flow in principal directions that will not be coinciding with the reinforcement directions, however, this will never be an issue, as it is in fact more beneficial when the force trajectories do not run in parallel lines with the reinforcement. This can be seen by considering the following force projection, and therefore the reinforcement design will always be of conservative nature, as it is assumed that only one reinforcement direction is activated for a given force trajectory.

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Figure 107: Simple force projection with axes representing orthogonal reinforcement net.

No real consideration has been made for the cross-sections of the columns so far, and the thesis will not initiate further detailed analysis of these elements. However, a quick look to the Danish Annex (DK NA) for EC2 Part 1-1, more particularly Annex 1 in the DK NA, which is entitled ‘Design of certain columns cast in situ’ states that eccentric loading, i.e. the moment, might be simply accounted for by scaling the vertical reaction Rz by a factor of 2 when the column is subjected to actions unilaterally in two directions, by a factor of 1,25 when the column is subjected to actions from continuous beams or slabs and by a factor of 1,5 in all other cases. Now, once again being conservative the maximum vertical reaction found, Rz≈600kN, can be multiplied with 2 yielding a normal force for the column of 1200kN, when dividing this force by the column cross-sectional area 1200000N/(300x300)mm2=13,3MPa, this value is rather low compared to the concrete compression strength, 24,1MPa, why I would be surprised if stability problems were to occur. Metaphorically speaking, the model can now ideally be passed on to a next phase for further processing, this could for instance involve further detailing in Revit see Figure 108 and Figure 109.

Figure 108: Left surface part with dimensions in GH.

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Figure 109: Left surface part imported to Revit.

Before leaving the case study behind, it would be interesting to compare the obtained results to the performance of a support configuration based on circle packing, like used in (Scheurer 2003). Setting up a circle packing routine using GH is fairly simple, one only needs the physics engine plug-in named Kangaroo 26 to get going. A circle packing routine is scripted in Figure 110.

Figure 110: Circle packing by Kangaroo.

26

http://kangaroo3d.com/

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Figure 111: Visual result of circle packing.

The circle centre points shown in Figure 111 can easily be referenced into the script using the prescribed column input module. The resulting FE displacement is shown Figure 112, and it is seen that the circle packing strategy is the better performer, cf. Figure 100 and Figure 106. One can also clearly notice the presence of a triangular grid structure in the point distribution, and this characteristic is apparently hard to compete with in terms of structural performance. A new candidate is ‘reborn’.

Figure 112: Cracked cross-section displacement of circle packing, wmax=0,004m

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10. Discussion A realistic, architectural design case with basis in the project of the Flying Carpet has been studied by putting the developed, parametric design tool to the test. The Flying Carpet is very similar in character to the project of the Groningen Stadsbalkon, cf. Chapter 4. This resemblance leads on to a discussion of the resemblance between the scripted tools used for early design generation in the two cases. The Groningen Twister was the name of the tool used for the early design generation of the Stadsbalkon. It was written by Fabian Scheurer, and based on optimised support distribution by use of circle packing dynamics as a rule of thumb. What Scheurer would have wanted to be implemented within the Groningen Twister, was the possibilities of structural verification and presence of quantified values to be used for further optimisation. I set out to develop a script for structural optimisation, which would be driven by exactly these aspects. Doing so, it was simultaneously expected that an additional structural verification step could be avoided. However, things did not turn out as wished for. The FE package incorporated in the script was not capable of defining correct moment values, why an internal, structural verification was suddenly out of sight, and at the same time a very relevant fitness candidate constituted by the moment itself was abandoned. Similarly to the Groningen Twister the developed script then became reliant on an extra step for final verification. Hence, a rule of thumb also needed to be introduced within the developed script, this rule of thumb concerned the magnitude of the moment value. For the case investigated, the invented rule of thumb performed satisfactory, and eventually the script did find an optimised support configuration using 32 columns, and thereby eliminating three of the 35 columns used in the initially suggested layout, which had used a classical column grid approach based on practical rules of thumb. The circle packing strategy was also tested on the Flying Carpet. This strategy was easily adopted within the script, and the FE computed deflection results showed that this strategy facilitated the structurally best performing structure. There can be no doubt that circle packing constitutes a strong and simple tool for support optimisation. However, one could ask, will it in fact always be so? Would circle packing routines for instance be relevant for the optimisation cases investigated in Test A, where relatively few support points are in use? Or does its strength depend on its uniform pattern with many repetitions? And exactly how much does it offer in terms of design variation? – One could move individual circles to fixed domain positions but the grid pattern would always be of same character by converging towards the perfect triangular grid, cf. Chapter 4, and therefore showing no irregularities. Also, what would happen in cases with shell structures where curvatures start to play a bigger role on the force distribution? All of these are open questions, which could be interesting subjects of further study, and which can be performed internally within Søren Jensen as well because of their local competences within parametric modelling tools. Concerning the main script developed, subjects for further development could be implementation of column analysis as well as addition of more load cases. The use of FEM for structural design and analysis is also a relevant discussion point, as studies show that FEM computed results are highly dependent on the user. In (Blaauwendraad 2010) an interesting study has been facilitated for the reinforcement design of a slab exposed to the kind of flawed peak moments dealt with in this thesis. The study shows just how much computed results and final reinforcement design may deviate amongst practitioners even when given exactly the same structural problem to dimension for. Upon reflection, one might also have explored the use of a gradient based method, but taking on the task of having to learn both Grasshopper as well as developing a gradient based method, in addition to coding it for compatibility with Grasshopper, was considered too big a risk, especially when keeping in mind that the three authors of (Jang et al. 2009) did not fully succeed in doing so.

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11. Conclusion A tool has been developed for generating optimised point supports for plate and shell structures in the early design stage. The tool is based on parametric modelling, FEM and genetic algorithms as it was originally intended. Unfortunately, problems related to the FEM computation were encountered, which would eventually limit the applicability of the script. However, the script is still applicable for solving the given task of point support optimisation and has proven handy for the solving of a realistic design case, but it needs assistance from an external verification module, why the workflow is not as smooth as it was initially hoped for.

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Holzer, D., Hough, R. & Burry, M., 2009. Parametric Design and Structural Optimisation for Early Design Exploration. International Journal of Architectural Computing, 5(04), pp.625–643. Jang, G.-W., Shim, H.S. & Kim, Y.Y., 2009. Optimization of Support Locations of Beam and Plate Structures Under Self-Weight by Using a Sprung Structure Model. Journal of Mechanical Design, 131(2), p.021005. Available at: http://mechanicaldesign.asmedigitalcollection.asme.org/article.aspx?articleid=1472495. Jensen, B.C., 2015. Teknisk Ståbi 23rd ed., Nyt Teknisk Forlag. Johansen, K.W., 1949. PLADEFORMLER, Leitão, A., Santos, L. & Lopes, J., 2012. Programming Languages For Generative Design: A Comparative Study. International Journal of Architectural Computing, 10(1), pp.139–162. Marcelin, J.L., 2001. Genetic Search Applied to Selecting Support Positions in Machining of Mechanical Parts. The International Journal of Advanced Manufacturing Technology, 17(5), pp.344–347. Available at: http://link.springer.com/10.1007/s001700170169. Marcelin, J.L., 2012. Optimization of the boundary conditions by genetic algorithms. International Review of Mechanical Engineering, 6(1), pp.50 – 54. Mitchell, W.J., 2005. Constructing complexity. Computer Aided Architectural Design Futures 2005 Proceedings of the 11th International CAAD Futures Conference, pp.41 – 50. Nelson, J.E., 1982. Telescope mirror supports: plate deflections on point supports. Proceedings of SPIE - the International Society for Optical Engineering, 332, pp.212 – 228. Available at: http://rokoszoptical.yolasite.com/resources/Nelson PointDeflection1982.pdf. Preisinger, C., 2013. Linking Structure and Parametric Geometry. Architectural Design, 83(2), pp.110–113. Available at: http://doi.wiley.com/10.1002/ad.1564. Sarkisian, M. et al., 2009. Optimization Tools for the Design of Structures. In 20th Analysis and Computation Specialty Conference. Reston, VA: American Society of Civil Engineers, pp. 219–230. Available at: http://ascelibrary.org/doi/abs/10.1061/9780784412374.020. Scheurer, F., 2007. Getting complexity organised Using self-organisation in architectural construction. Automation in Construction, 16(1), pp.78–85. Available at: http://www.sciencedirect.com/science/article/pii/S0926580505001500. Scheurer, F., 2003. The Groningen Twister: An experiment in applied generative design. Generative Art 2003, pp.90–99. Scheurer, F., 2005. Turning the design process downside-up - Self-organization in real-world architecture. COMPUTER AIDED ARCHITECTURAL DESIGN FUTURES 2005, PROCEEDINGS, pp.269 – 278. Scheurer, F. & Stehling, H., 2011. Lost in Parameter Space? Architectural Design, 81(4), pp.70–79. Available at: http://doi.wiley.com/10.1002/ad.1271.

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Shepherd, P., 2011. Aviva Stadium - The use of parametric modelling in structural design. Structural Engineer, 89(3), pp.28–34. Sigmund, O., 2011. On the usefulness of non-gradient approaches in topology optimization. Structural and Multidisciplinary Optimization, 43(5), pp.589–596. Available at: http://link.springer.com/10.1007/s00158-011-0638-7. Sweeney, S., 2009. Direct Versus History Based Modeling. Available at: http://info.kubotek3d.com/3DEngineering-Software-Tools-Kubotek-Blog/bid/22074/Direct-Versus-History-Based-Modeling [Accessed March 19, 2015]. Tedeschi, A., 2010. Interview with David Rutten. MixExperience Tools1, pp.28–31. Available at: http://content.yudu.com/Library/A1qies/mixexperiencetoolsnu/resources/index.htm [Accessed March 9, 2015]. Turrin, M., von Buelow, P. & Stouffs, R., 2011. Design explorations of performance driven geometry in architectural design using parametric modeling and genetic algorithms. Advanced Engineering Informatics, 25(4), pp.656–675. Available at: http://linkinghub.elsevier.com/retrieve/pii/S1474034611000577. Wang, B.P. & Chen, J.L., 1996. Application of genetic algorithm for the support location optimization of beams. Computers & Structures, 58(4), pp.797–800. Available at: http://linkinghub.elsevier.com/retrieve/pii/004579499500184I. Wang, D., 2004. Optimization of support positions to minimize the maximal deflection of structures. International Journal of Solids and Structures, 41(26), pp.7445–7458. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0020768304002707. Wight, J.K. & MacGregor, J.G., 2012. Reinforced Concrete: Mechanics & Design 6th ed., Pearson Education.

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List of Figures Figure 1: Guggenheim Museum (left) & La Sagrada Familia (right). Photos: Guggenheim Bilbao Museoa & Temple expiatori de la Sagrada Familia ........................................................................................................ 9 Figure 2: Directed Acyclic Graph (DAG). Illustration: HAMRTech.com ..........................................................11 Figure 3: Google search interest since 2004 for the four parametric modelling tools; GH, DP, GC and Dynamo. By the looks of this chart GH seems trendsetting and both the interest for DP and GC has slowly decreased since the advent of GH. Interest for Dynamo is steadily rising. Graph: Google.com/trends/GH+DP+GC ..................................................................................................................11 Figure 4: Severe case of spaghetti code in GH. Illustration: Daniel Davis......................................................12 Figure 5: Google search interest since 2004 for GH and the two BIM tools Revit and ArchiCad. Graph: Google.com/trends/GH+Revit+ArchiCAD.....................................................................................................13 Figure 6: Google search interest since 2004 for GH, Revit, ArchiCad, Rhino and AutoCAD. Graph: Google.com/trends/GH+Revit+ArchiCAD+Rhino+AutoCAD ..........................................................................13 Figure 7: David Rutten’s subjective understanding of terms commonly related to the notion of parametric design. Illustration: David Rutten ...............................................................................................................15 Figure 8: Charles Darwin. Illustration: simplecapacity.com ..........................................................................16 Figure 9: Seamless search progress illustrated for a two dimensional fitness landscape. Illustration: (Sarkisian et al. 2009) .................................................................................................................................17 Figure 10: Principles of crossover and mutation of GAs. Illustration: (Sarkisian et al. 2009) .........................17 Figure 11: GA procedural flow chart. Illustration: (Sarkisian et al. 2009) ......................................................18 Figure 12: GA procedural flow chart with equivalent gene dynamics depicted. Illustration: (Scheurer & Stehling 2011).............................................................................................................................................19 Figure 13: Meridian cross-sectional drawing of the 200-inch Hale Telescope and dome. Drawing: Russell W. Porter .........................................................................................................................................................20 Figure 14: Thee three only point symmetric grid structures; here shown with equal distances between neighbouring grid points. ............................................................................................................................21 Figure 15: The hexagonal bee honeycomb allocates the best ratio between storage space and cell wall material. Photo: homefurn.com/blog/decorate-hexagons/..........................................................................22 Figure 16: The Groningen Stadsbalkon. Photo: tunnelvariant.nu/stationsgebied.html .................................23 Figure 17: Tight circle packing. ....................................................................................................................24 Figure 18: Column habitat functional constraints. Illustration: (Scheurer 2003) ..........................................24 Figure 19: Agent dynamics. Illustration: (Scheurer 2003) ............................................................................24 Figure 20: Viewport examples of the Groningen Twister software. Illustration: (Scheurer 2003)..................25 Figure 21: The Stadsbalkon taking form. Photo: designtoproduction.com ....................................................25 Figure 22: The bicycle parking area. Photo: designtoproduction.com...........................................................25 Figure 23: LCD panel. Photo: news.softpedia.com/news/Korea-Leads-the-Way-in-LCD-Panel-Construction 26 Figure 24: Optimised support layout for beam structure using the L2 norm and two distinct support points as target. Illustration: (Jang et al. 2009) .....................................................................................................26 Figure 25: Beam solutions found for the two objective functions; L2 norm vs mean compliance. Table: (Jang et al. 2009) .................................................................................................................................................27 Figure 26: Plate solutions found for the two objective functions; L2 norm vs mean compliance. Table: (Jang et al. 2009) .................................................................................................................................................27

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Figure 27: Support convergence history of five-support locations for the G-shaped plate problem after (a) 35 iterations, (b) 70 iterations, (c) 100 iterations, (d) 130 iterations and (e) 150 iterations. .........................28 Figure 28: Hollow core slabs lined up after production. Photo: Concrete Technology L.L.C..........................31 Figure 29: Guiding load carrying capacity of HCS. Illustration: (Betonelement-Foreningen 1995) .................31 Figure 30: The Jespersen System. Illustration: Modul og Montagebyggeri, Nissen 1973 .............................32 Figure 31: Irregular opening in HCS element using slab hanger. Illustration: (Betonelement-Foreningen 1995) ...................................................................................................................................................................33 Figure 32: Piled TT-slabs. Photo: Moldtech S.L. ............................................................................................34 Figure 33: Guiding load carrying capacity of TTS. Illustration: (Betonelement-Foreningen 1995) ..................34 Figure 34: SL-Deck anatomy. Illustration: Abeo ...........................................................................................35 Figure 35: SL-Deck cross-section. Illustration: (Hertz et al. 2014) Photo: Abeo .............................................35 Figure 36: Longitudinal section in continuous, cantilevered SL-Deck. ..........................................................36 Figure 37: Moment transferring connection between adjoining SL-Decks. Illustration & Photo: Abeo .........36 Figure 38: Openings and recess in SL-Deck. Illustration: Abeo .....................................................................36 Figure 39: SL-Deck with curved ends. Illustrations & Photo: Abeo................................................................37 Figure 40: SL-Deck vs. Hollow Core Slab. Table: Abeo ..................................................................................37 Figure 41: First realised application of the Pearl-Chain Bridge system [Skjern, Denmark – March 2015]. Photo: Innovationsfonden ...........................................................................................................................38 Figure 42: Conceptual principles of the Pearl-Chain technology. Illustrations: Abeo ....................................38 Figure 43: Minimum slab heights for αfm ≤ 0,2. (Wight & MacGregor 2012) .................................................44 Figure 44: Standard Dome Dimensions and Other Data. Table: (Concrete Reinforcing Steel Institute 2008) .45 Figure 45: Estimate of waffle slab load carrying capacity. Illustration: (Concrete Reinforcing Steel Institute 2008) ..........................................................................................................................................................46 Figure 46: Moment strength and span of BubbleDeck. Illustrations: BubbleDeck.........................................47 Figure 47: Execution sequences for BubbleDeck. Illustrations: BubbleDeck .................................................47 Figure 48: Rolex Learning Centre. Photo: archicentral.com..........................................................................48 Figure 49: Planar, singly-curved and doubly-curved form symbols...............................................................50 Figure 50: Locked grid layout. .....................................................................................................................52 Figure 51: Free grid layout. .........................................................................................................................52 Figure 52: Free layout. ................................................................................................................................52 Figure 53: The FSC-Diagram ........................................................................................................................53 Figure 54: Free grid configuration layout.....................................................................................................54 Figure 55: Part of FSC-diagram (see Figure 53) which can be covered by script............................................55 Figure 56: Schematic flow chart of developed script. ..................................................................................57 Figure 57: Script in Grasshopper. ................................................................................................................57 Figure 58: Design Domain Input. .................................................................................................................58 Figure 59: Grid Generation..........................................................................................................................60 Figure 60: Generation of Free Point Support Positions. ...............................................................................61 Figure 61: Gathering Support Points. ..........................................................................................................63 Figure 62: FEA Input. ...................................................................................................................................65 Figure 63: FEA Routine. ...............................................................................................................................66 Figure 64: FEA Results. ................................................................................................................................67 Figure 65: Optimisation Routines. ...............................................................................................................69

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Figure 66: IFC Model Export. .......................................................................................................................71 Figure 67: Equivalent Karamba Grid for LCD Panel. .....................................................................................72 Figure 68: Best support configuration generated for mean compliance, IE=2,00·10-4kNm. ..........................73 Figure 69: Best support configuration generated for L2 norm, |L2|=0,94m. .................................................73 Figure 70: Best support configuration generated for mean compliance when not constrained by symmetry, IE=1,64·10-4kNm. ........................................................................................................................................74 Figure 71: Best support configuration generated for L2 norm when not constrained by symmetry, |L2|=0,642m. ..............................................................................................................................................74 Figure 72: Mirrored version of best support configuration generated for L2 norm when not constrained by symmetry, |L2|=0,644m..............................................................................................................................75 Figure 73: Best support configuration generated for mean compliance, IE=3,90·10-5kNm. ..........................76 Figure 74: Best support configuration generated for L2 norm, |L2|=0,150m. ...............................................76 Figure 75: Best support configuration generated for mean compliance when not constrained by symmetry, IE=3,50·10-5kNm. ........................................................................................................................................77 Figure 76: Best support configuration generated for L2 norm when not constrained by symmetry, |L2|=0,136m. ..............................................................................................................................................77 Figure 77: FE mesh for a resolution of 0,14 yielding 1238 elements. ...........................................................79 Figure 78: FE mesh for a resolution of 0,06 yielding 6990 elements. ...........................................................79 Figure 79: The amoeba mesh, support reactions and peak deflection found in GH environment. ................80 Figure 80: The amoeba mesh, support reactions and peak deflection found in Robot. ................................81 Figure 81: Specimen 1 – found using Galapagos optimisation with IE as the fitness measure. Same configuration as in Figure 79 .......................................................................................................................82 Figure 82: Specimen 2 – found using Galapagos optimisation with wmax as fitness measure. .......................83 Figure 83: Specimen 3 – found using Galapagos optimisation with wmax as fitness measure. .......................83 Figure 84: Rendering of the Flying Carpet. Illustration: Polyform Architects .................................................84 Figure 85: Architect’s model of the Flying Carpet in Rhino...........................................................................84 Figure 86: Early structural principles developed by Søren Jensen. Drawing: Duncan Horswill ......................85 Figure 87: The Flying Carpet plan with configuration colour legends. ..........................................................85 Figure 88: Left surface part configuration using 35 columns. .......................................................................86 Figure 89: Søren Jensen’s initial column layout using 35 columns imposed in GH script. .............................87 Figure 90: FEM deflection results computed for initial column layout, wmax=0.005m. ..................................87 Figure 91: Vertical reactions in Robot (FZ) ...................................................................................................88 Figure 92: Initial peak and field distribution of 1st principal moments. .........................................................89 Figure 93: Smearing out of moment peak. Illustration: (Blaauwendraad 2010) ...........................................89 Figure 94: Reduced and smoothened 1st principal moment values. .............................................................90 Figure 95: 1st principal stresses calculated at top surface layer. ...................................................................91 Figure 96: Three representative stress check points have been chosen. ......................................................91 Figure 97: 1st principal forces. .....................................................................................................................92 Figure 98: Calculation of transformed moment of inertia for cracked cross-section. ...................................93 Figure 99: FEM peak deflection computed for initial column layout with cracked cross-section, wmax=0.014m. .............................................................................................................................................94 Figure 100: FEM peak deflection computed for initial column layout with cracked cross-section and h=400mm, wmax=0,006m. ............................................................................................................................94

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Figure 101: Candidate solution using 32 columns, wmax =0.009m. ................................................................95 Figure 102: Candidate solution using 32 columns with free grid point layout of 2,5m density. ....................95 Figure 103: Reduced and smoothened 1st principal moment values. ...........................................................96 Figure 104: Reduced and smoothened 2nd principal moment values. ..........................................................96 Figure 105: Cracked cross-section of concrete slab. ....................................................................................97 Figure 106: Updated candidate solution using 32 columns, wmax=0,005m ...................................................98 Figure 107: Simple force projection with axes representing orthogonal reinforcement net. ........................99 Figure 108: Left surface part with dimensions in GH. ..................................................................................99 Figure 109: Left surface part imported to Revit. ........................................................................................100 Figure 110: Circle packing by Kangaroo. ....................................................................................................100 Figure 111: Visual result of circle packing. .................................................................................................101 Figure 112: Cracked cross-section displacement of circle packing, wmax=0,004m .......................................101 Figure 113: Screenshot of youtube playlist. ...............................................................................................111

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List of Videos

Figure 113: Screenshot of youtube playlist.

Script Core Functionality: https://youtu.be/AF-565YbvyE Brute Force Search for LCD Panel w. 4 Supports: https://youtu.be/ZysIgKFwxEQ Galapagos Optimisation for L2 Norm of LCD Panel w. 4 Supports: https://youtu.be/r7ExqHHK6hc Octopus Optimisation for Multiple Fitness Functions (part 1): https://youtu.be/euOkFmzUPDY Octopus Optimisation for Multiple Fitness Functions (part 2): https://youtu.be/WniZpvbMRkY

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