building structures in cooperation with architects

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MASTER OF SCIENCE THESIS

STOCKHOLM, SWEDEN 2016

building structures in

cooperation with architects

Usage and evaluation of structural

plug-ins in 3D visualisation software

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Parametric

design of building structures in

cooperation

with architects

-

Usage and evaluation of structural plug-ins in 3D

visualisation

software

Daniel

Wallin and Martin Wasberg

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c

Martin Wasberg and Daniel Wallin 2016 Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering Division of Concrete Structures

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Abstract

Architectural and structural design process are closely connected but traditionally done in two separate steps in the design process. This requires effective coordination between the two disciplines and without the right tools problems often arise.

The thesis was done by support from structural engineers at Tyréns and in collaboration with a student from the department of architecture. The aim of the thesis was to investigate if the use of parametric design tools from both architects and structural engineers could be a way of making the design process more effective. This thesis also include test the structural plug-ins of the parametric design tools and compare them with the outputs from traditional structural software and hand calculations.

The comparison was made for different cases followed by a collaboration project. The cases was targeting different structural features which in turn gave the knowledge needed to develop the collaboration project.

The case studies consists of five cases where the first two gives an introduction to parametric modelling. The third case is a steel beam with fully restrained supports loaded by two point loads. It will compare the displacement calculations between the different software. The next case is a concrete slab with different supports along edges loaded by a uniformly load. The analysis includes calculation and evaluation of section forces. The final case is a concrete dome. It is built up by arches and five supports. The analysis of this case includes calculation and evaluation of the displacement.

The collaboration project is a concrete structure built up by a curved surface lifted by curved columns. The architect worked with the structure in parallel to this this thesis and targeted the development process whilst the authors targeted the structural parts and at the same time gave structural insight to the architect.

The results show a difference between the parametric structural tools and the traditional FE software regarding deformation and moments. The hand calculations in the collaboration project show that the amount of reinforcement will not work with the given inputs and assumptions due to practical reasons regarding spacing.

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Sammanfattning

De Arkitektoniska och strukturella designprocesserna är nära sammankopplade men traditionellt gjort i två separata steg i byggprocessen. Detta kräver en effektiv samordning mellan de två disciplinerna och utan de rätta verktygen uppstår ofta problem.

Uppsatsen genomfördes med stöd från konstruktörer på Tyréns och i samarbete med en student från institutionen för arkitektur. Syftet med uppsatsen var att undersöka om användningen av parametriska designverktyg från både arkitekter och konstruktörer kan vara ett sätt att göra de-signprocessen mer effektiv. Denna uppsats testade också de strukturella plug-in till de parametriska designverktygen och jämförde resultatet med traditionella finita element- och handberäkningar. Jämförelsen gjordes för olika fall följt av ett samarbetsprojekt. Fallen riktade in sig på olika strukturella egenskaper som gav den kunskap som behövdes för att utveckla samarbetsprojektet. Fallstudierna består av fem fall där de två första ger en introduktion till parametrisk modellering. Det tredje fallet är en fast inspänd stålbalk belastad med två punktlaster. Fallet kommer att jämföra deformationsberäkningen mellan de olika programvarorna. Det fjärde fallet är en betong-platta med olika stödvillkor längs kanterna belastad av en jämt utbredd last. Analysen inkluderar beräkning och utvärdering av snittkrafter. Det sista fallet är en betongkupol. Den är uppbyggd av valv och fem stöd. Analysen av detta fall inkluderar beräkning och utvärdering av nedböjningen. Samarbetsprojektet är en betongkonstruktion uppbyggd av en krökt yta som bärs upp av välvda pelare. Arkitekten arbeta parallellt med strukturen tillsammans med författarna av denna uppsats och riktade in sig på utvecklingsprocessen. Författarna fokuserade på de strukturella delarna och gav samtidigt strukturell insikt till arkitekten.

Resultaten visar en skillnad mellan de parametriska strukturella verktygen och det traditionella FE programmet vad gäller deformation och moment. Handberäkningarna för armeringsmängden visar av praktiska skäl, gällande avstånd, att detta inte fungerar.

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Preface

This master thesis has been carried out at Tyréns AB and the Division of Concrete Structures, Department of Civil and Architectural Engineering at the Royal Institute of Technology (KTH) in Stockholm.

We express our deepest gratitude to our supervisor, Adjunct Prof. Mikael Hallgren for the opportunity to work with this thesis as well as his advice and guidance. We also thank our examiner Professor Anders Ansell for all he has learned us about con-crete structures during our years at the Royal institute of Technology. Furthermore, we thank Tyréns AB for their warm welcome and for an inspiring work environment. Last but not least, we thank Ulf Edgren at the Department of Architecture for the collaboration and sharing his knowledge of architecture and parametric design. Stockholm, June 2016

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Symbols

Latin upper case letters

As.min Minimum cross sectional area of reinforcement

B Strain-displacement matrix Ce Exposure factor

Ct Thermal coefficient

D Global displacement vector Dbar Diameter of rebar

E Modulus of elasticity ; Material properties matrix Ecm Secant modulus of elasticity of concrete

Es Modulus of elasticity of reinforcement steel

F In plane force FC Compressive force

Fs Tensile force

Gk Area weight of permanent load

I Moment of inertia K Global stiffness matrix Lbeam Length of beam

Lspan Length of span

M Bending moment

MEd Design value of the applied internal bending moment

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NEd Design value of the applied axial force (tension or compression)

P Point load

R Global load vector

Smax.slabs Maximum spacing of bars

Sr.max Maximum crack spacing

V Shear force

VEd Design value of the applied shear force

VRd.c Shear force resistance

Latin lower case letters

b Width

cnom Concrete cover

d Effective depth of cross-section dbars Distance between bars

e Eccentricity

fcd Design value of concrete compressive strength

fck Characteristic compressive strength of concrete

fcm Mean value of compressive strength of concrete

fctm Mean value of axial tensile strength of concrete

fyd Design yield strength of steel

fyk Characteristic yield strength of steel

g Gravitational constant

h Height

k Coefficient ; Factor ; Local stiffness matrix nbars Number of bars per meter

qd Design area load

sk Characteristic snow load

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sn Design snow load

t Thickness wk Crack width

y Height of the effective compressive zone x Height of the compressive zone

Greek lower case letters

α Ratio of modulus of elasticity γ General partial coefficient γd Safety factor

γg Partial factor for permanent load

γc Partial factor for concrete

γS Partial factor for reinforcing steel

γq Partial factor for variable load

δ Deformation

ε Strain

εc Compressive strain in the concrete

εu Ultimate compressive strain in the concrete

εs Strain in the reinforcing steel

η Effective strength parameter

θ Angle

λ Effective height parameter µi Form factor

ξ Reduction factor ρconc Density of concrete

ρp.eff Reinforcement ration

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σc Compressive stress

σs Tensile stress

σcp Compressive strength from axial load

τ Shear stress ψ Load factor

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Chapter 1 Introduction

Architectural and structural design processes are closely connected but are tradi-tionally done in two separate steps in the design process. The architect focuses on geometry and arrangements of the elements that compose the whole structure while the engineer focus on the mechanical properties and behaviour of the struc-ture. This requires effective coordination between the two disciplines for a successful design process. (Po-Han Chen et al, 2005)

The information exchange between CAD systems used in different design phases has always been a problem. The introduction of building information models (BIM) has been believed to be essential in the information exchange in the AEC industry. In resent years a lot of CAD systems have become BIM-based CAD systems but still use different specifications depending on software vendor. (Changfeng Fu et al, 2006)

Without good compatibility the structural engineers have to interpret the infor-mation given by architects and remodel the geometry with structural properties (Po-Han Chen et al, 2005). This creates a lot of problems which in turn is very time consuming followed by increased costs.

The use of parametric design and its tools from both entities can be a way of making the collaboration more effective. This due to that both work from the same basis which decrease the errors developed from e.g. wrong interpretation. Furthermore, it gives possibilities of saving time by e.g. decreasing remodelling due to good compatibility between software’s.

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CHAPTER 1. INTRODUCTION

1.1 Background

The possibility to make changes without deleting or redrawing the model has for the designer been a much awaited opportunity. This has pushed parametric modeling to become a standard tool in traditional CAD programs which before 2005 was seen as a very sophisticated and expensive software made exclusively for manufacturing in shipping, aerospace and automobile industries. (Dominik Holzer, 2008)

One of the earliest examples of parametric design was when Antonio Guadi applied it in the design of Expiatory Temple of the Sagrada Familia, see figure 1.1. He created a model of strings weighted down with bird shot to create a complex vaulted ceilings and arches. This innovative methodology for parametric design is called Design Procedures (DP). The procedures are actions that generate parametric models where geometrical components are considered as variables. By adjusting the length of the strings or position of the weights Guadi could alter the shape of each component in the structure and also see how the changes influenced the components that worked together in the structure. (Dominik Holzer, 2008) (Wikipedia, Accessed: 2015-12-08)

Figure 1.1: Force model of the Colònia Güell, Sagrada Família Museum (Wikipedia, Accessed: 2015-12-08).

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1.1. BACKGROUND

Parametric design is a combination between mathematical thinking and new digital tools that with high efficiency can be used in the design process (White, 2015). It is done with the aid of parametric models which is a computer representation of a design constructed with geometrical entities that have attributes that can vary or are fixed. Parametric design therefore gives the architect free rein to design structures and enables creation and analysis of very complex structures that was not possible before, or required great craftsmanship. This is done just by changing the parameters in the parametric model. The process can include landscape or interiors, expressive facades, but also optimization of the structures, energy and natural lightning as a part of the design process. (Dominik Holzer, 2008) (White, Accessed: 2015-12-08)

The parametric models can be categorized into two kinds: those that generate new designs by combination of parametrized geometrical entities or those that perform variations. The model can also be a combination of both. (Hernandez, 2006) Different software where parametric design is applicable are CATIA, Autodesk 3DS Max, Autodesk Maya, Grasshopper 3D, Autodesk Revit, and GenerativeCompo-nents. (Wikipedia, Accessed: 2015-12-08)

A resent project where parametric design has been applied is the Louvre Abu Dhabi in Abu Dhabi, see figure 1.2. It is a museum that is designed as a "seemingly" floating dome structure. The web-patterned dome allows the sun to filter through. The overall effect is meant to represent "rays of sunlight" passing through date palm fronds in an oasis. The museum was planned to be finished in 2015 but the opening got pushed to 2016. (Wikipedia, Accessed: 2015-12-09)

Figure 1.2: The museum Louvre Abu Dhabi with features often used in parametric modelling (Rinaldi, 2015).

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early stages of architectural design. (Vincent Tourre, 2009)

Most of the software used by the architect for design and drafting are often limited to geometry while the structural engineer’s optimization and analysis software is adapted to specific structural tasks. Specialist often give feedback to the designers at discrete points in time with varying frequency which often leads to delays and dis-continuities in the workflow and consequently is responsible for coordination errors and the necessity for rework. The increased specialization within their individual domains has also led to a big gap in the understanding between the structural engi-neers and the architects. When working on a common project, both entities need to cooperate to reach a final result where both parts often have different theories and objectives. This in combination with the gap between understanding and limited feedback is very time consuming. By linking parametric design to the structural analysis and optimization, both entities can explore design in the conceptual de-sign phase through informed geometry alterations and therefore save a lot of time. (Dominik Holzer, 2008)

1.2 Aim

The aim of the thesis is to investigate if the use of parametric design tools from both architects and structural engineers could be a way of making the design process more effective. Mainly because the structural knowledge and optimization are brought in to the design in the early stage of the project process.

Furthermore, the structural outputs given from the used parametric design tools will be analysed and compared to the structural outputs given from a traditional structural software and hand calculations. The comparison will focus on differ-ences/similarities in deformations and forces. This is done to investigate the struc-tural evaluation capacity of the plug-ins.

1.3 Approach

The project will be done by support of structural engineers at Tyréns and in collab-oration with a student from the department of architecture who has already tested a method for structurally optimized generative geometry. He will write a thesis project parallel with the aim of exploring a structurally informed design process. The main parametric tool to be used in the project is Grasshopper (Scott Davidson, 2016) which is a visual programming editor. As a plug-in for Rhino3D (Robert McNeel & Associates, 2016), Grasshopper is integrated with the modelling environ-ment used in e.g. architecture and engineering. Grasshopper offers the opportunity to define precise parametric control over models, the capability to explore generative design work flows and a platform to develop higher-level programming logic by using an intuitive, graphical interface. (ModeLab, Accessed: 2015-12-09)

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1.3. APPROACH

For structural evaluation of structures modelled within the parametric design tools Karamba (Matthew Tam, 2016) will be used. Karamba is a plug-in for Grasshopper that makes it easy to combine geometric models, finite element calculations and optimization algorithms like Galapagos (Karamba, Accessed: 2015-12-09).

For evaluation of the structural outputs given from Karamba, SAP2000 (CSI, 2016a) will be used which also uses the finite element method (FEM).

The idea is to support the architectural student in the design process with structural insight as well as testing Karambas structural outputs to make an early optimization of different parameters and by comparing it to outputs obtained from SAP2000. This will be achieved by exporting the studied model/models from Grasshopper/Karamba to SAP2000 or by, depending on the complexity of the structure, remodelling it with SAP2000.

The work will be done with both reference models and fictitious models which satisfy some limited architectural core values such as e.g. proportion, mass/void relation-ship, light, material, etc. Furthermore, the models will focus on honest architecture, where structural and architectural concepts are tightly connected. This is likely where the focus on the transdisciplinary collaboration between architecture and en-gineering will take place. The collaboration process will be built up according to figure 1.3.

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The models will be divided into different cases depending on features. Some will be analysed with greater scope while others more superficially. This will be presented in chapter 4 and 5. A model acquired by the collaboration will be presented which includes the biggest part of the evaluation of the collaboration and investigation of the possibilities of making a more effective and smooth process bringing in the engineer’s in to the parametric design.

The cases presented in chapter 4 will also be used to increase knowledge about used parametric design tools. Here different tools within used software’s. By this the knowledge needed to evaluate the collaboration project is acquired.

1.4 Limitations and assumptions

• Only linear elastic analyse will be performed, i.e. the calculations are based on the following conditions, Ansell et al. (2012):

• Hook’s law σ = E ε is valid, i.e. the materials are ideally elastic.

• Bernoulli’s hypothesis that the linear distribution of strain in the cross-section

is retained during loading.

Furthermore, Karamba is also limited to linear elastic calculations.

• Only thin plate theory or Kirchhoff theory, i.e. a line that is straight and normal to the surface before loading remains straight and normal to the deformed mid-surface. Furthermore, this prohibits transverse shear deformations. (Cook et al., 2002)

• Only three-node triangular element. This is a limitation in Karamba, used FE software integrated with the parametric design tools, that only work with triangulate elements.

• The structures or parts of structures that are analysed have the same cross-sectional thickness everywhere, i.e. non varying cross-section.

• Buildability or construction method are not investigated in depth. A smaller presentation of this will although be presented for the collaboration project.

• Converting from the general FE software to used software for parametric design will not be performed, i.e. the structure will at first be modelled in used parametric design tool and after that exported to used general FE software.

• The core plug-ins but not every used plug-ins will be described in detail. This due to that a lot of plug-ins have been used and some of them to a small extent, i.e. small influence on the overall project.

• The most important components in different plug-ins but not all components will be described in detail. For further information about all components for every plug-in the reader is referred to the manual in question.

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1.5. USED SOFTWARE

• The main focus of the structural evaluation will be put on the behaviour of the concrete shell structures.

• The parametric design is limited to structural aspects such as quantity, position and structural element properties. The form finding aspects of parametric design will mostly be taken into consideration with regards to structural outputs.

1.5 Used software

The software used in the project are presented in chapter 3. The software within the parametric design have been chosen from discussion between the authors of this project and their co-worker, the architect, together with research and comparison. Here Rhino3D and Grasshopper were predetermined which, in this case, are the foundation of the parametric design tools and consequently the most important tools for e.g. defining geometries. Karamba is also one of the most essential tool in the thesis. It is a well known FE motor within the parametric design tools. Other tools presented have been chosen during the process to solve the problems at hand. This include Millipede, Galapagos, Octopus and Kangaroo.

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Chapter 2 Theory

2.1 Finite element method

The finite element method (FEM) is a numerical solution method for field problems, which are described by differential equations or by an integral expression. The field problem is used to formulate finite elements which can be seen as small pieces of a structure. In each finite element a field quantity is allowed to have only a simple spatial variation which is an approximation of the real variation in the region spanned by the element. The elements are connected to each other at nodes and their arrangement build up a so called mesh. The mesh is represented by a system of equations with unknowns at the nodes. The degrees of freedom (DOF), the ability to rotate or move, at the nodes as well as the stiffness of the structure governs the spatial variation of the field. The solution is aquired by interpolating between elements using shape functions, usually polynomials denoted Ni, and the number of

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CHAPTER 2. THEORY

Figure 2.1: Mesh of a dome with triangular elements.

The choice of element is dependent of the structures geometry and what kind of results is desired. Most elements are based on displacements and are classified by the number of nodes and their arrangement within the element. Usually the geometry of the elements in shell or plate models are rectangular or triangular which are easy to formulate. Rectangular elements are however impractical since it is difficult to mesh a complicated geometry.

The general procedure of a finite element analysis consists of assembling a global stiffness matrix [K] from the local stiffness matrices, [k], of each element, Determin-ing the gloabal load vector [R] and reducDetermin-ing the system due to boundary conditions. Equilibrium gives

[K][D] = [R]

where [D] is a vector containing the displacements. The stiffness matrix for a element [k], is given by

[k] = Z

[B]T[E][B] dV

where [E] is the material property matrix and [B] is the strain-displacement matrix containing the shape functions and is given by

[B] = [∂[N ]

The limitation of rectangular elements is why most software makes use of isopara-metric elements which allows quadrilateral elements to have non-rectangular shapes

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2.1. FINITE ELEMENT METHOD

and/or curved sides. The isoparametric element uses reference coordinates that map the element into a squared or cubed reference element.

Figure 2.2: Four-node plane element in physical space mapped into ξ η space (Cook et al., 2002).

This transformation produces algebraic forms that require numerical integration since they cant be integrated in closed form. The numerical integration is usually performed by Gauss quadrature which evaluates the function at specific points, called Gauss points, multiplies the resulting number with a weighting factor to minimize the integration error. The Gauss points are located symmetrically with respect to the interval centre and paired points share the same weight factor. The result is then extrapolated to the nodes. (Cook et al., 2002)

When dealing with shell structures one usually needs to consider two distinct, com-monly used theories. The first is membrane theory which usually applies to a large part of the entire shell. A membrane is identified as a body incapable of convey-ing moments or shear forces. The other is bendconvey-ing theory or general theory which includes the effects of bending which means it allows discontinuities in the stress distribution taking place close to where loads act and/or structural discontinuity is present. The bending theory usually includes a membrane solution but corrected in areas where discontinuity effects are present. (Ugural, 1981)

The membrane stiffness of a thin shell is much greater than the bending stiffness which could lead to ill-conditioned FE equations. That is when the solution vector is sensitive to small changes in the stiffness matrix, [K], or the vector of constants, [R]. (Cook et al., 2002)

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CHAPTER 2. THEORY

quality of the model which can be done e.g. by making easy hand calculations for comparison, mesh refinement to check for convergence and using different element types. (Cook et al., 2002)

2.2 Principal stress-lines

Any solid body can be broken down to infinitely small cubical elements to describe the state of stress for each point, see figure 2.3. Furthermore, by consideration of only one individual plane the state of stress can be represented by two normal stress components and one shear stress component. (Das and Sobhan, 2014)

The 2D principal stresses σ1 and σ2 are derived by rotating this infinity small

ele-ment, see figure 2.3, by the critical angle θp1 and θp2 calculated from the expression:

tan2θp =

τxy

(σx− σy)/2

given in Das and Sobhan (2014). This two principal stress components corresponds to either the maximum or minimum normal stress. Furthermore, for either orien-tations there will be two pairs of normal stresses with no shear stress. (Das and Sobhan, 2014)

Figure 2.3: Two elements taken from a solid body showing the normal stress together with shear stress and after rotation, principal stress.

The principal stress-lines follow the direction of the principal stresses and are there-fore a good way of indicating the trajectories of internal forces where no shear forces act. Furthermore, they are pairs of orthogonal curves that idealize paths of material continuity. (Li and Chen, 2010-09-23)

In (Li and Chen, 2010-09-23) a method where principal stress-lines are used for de-signing beam structures, including those of size, shape/geometry and topology, is derived and presented. By an example they show that the topology design, derived from the principal stress-lines, is not affected by load size and material proper-ties. With this in mind together with what was mentioned in preceding paragraphs

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2.2. PRINCIPAL STRESS-LINES

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Chapter 3 Description of used software

3.1 Rhinoceros 5.0

Rhinoceros or Rhino3D, is a computer-aided design (CAD) and a 3-D modeller application software developed by Robert McNeel & Associates. The first version, Rhino 1.0, was released in October 1998. (Wiki, Accessed: 2016-03-06)

The geometry is based on NURBS (Non-Uniform Rational B-Splines) which focuses on producing 3-D free-form surfaces or solid and mathematically precise represen-tation of curves. (McNeel and Associates, 2014-08-01)

NURBS has a good flexibility and accuracy and can therefore be used in any process from illustration and animations to manufacturing. The geometry is an industry standard for designers who work in 3-D where both form and function is important. (McNeel and Associates, 2014-08-01)

The models can be rendered at any resolution and a mesh can be crated from the model at any resolution. (McNeel and Associates, 2014-08-01)

3.2 Grasshopper

The precursor to Grasshopper was born in 2008 and was then titled Explicit History. Later the same year it was re-branded to Grasshopper. It is devoloped by David Rutten at Robert McNeel & Associates. (ModeLab, 2016-02-08)

It uses a visual programming language (VPL) which, by manipulating logic elements graphically rather than by specifying them textually, lets users create programs. (ModeLab, 2016-02-08)

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CHAPTER 3. DESCRIPTION OF USED SOFTWARE

Grasshopper is a platform for development of higher-level programming logic by using an intuitive, graphical interface and the capability to explore generative design work flows. (ModeLab, 2016-02-08)

The logic elements is functional blocks, so called components, that is added to a canvas. The components are connected by "wires" where the only syntax required is that the inputs of the blocks receive the data of the appropriate type. One can ei-ther design a geometry in Rhino3D and add it to the components in Grasshopper, or one can define the geometry in Grasshopper which in turn is shown and updated in Rhino3D’s viewport wile changing parameters/components in Grasshopper. (Mod-eLab, 2016-02-08)

In figure 3.1 one can see a code for a dome in Grasshopper (a) that automatically updates in Rhino3D (b) wile changing parameters.

(a) Interface of Grasshopper. (b) Interface of Rhino3D.

Figure 3.1: Grasshopper and Rhino3D.

The algorithms are step by step procedures designed to perform an operation and when using Grasshopper the user designs these algorithms that then automate tasks in Rhino3D. (ModeLab, 2016-02-08)

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3.3. KARAMBA

3.3 Karamba

Karamba is a Finite Element program which predicts how structures behave when subjected to external loads. It is fully embedded in the Grasshopper environment which makes it easy to combine geometric models, finite element calculations and op-timization algorithms (Preisinger, 2015). Karamba takes full advantage of Grasshop-pers visual computing environment and enables an instant update of the structural response when certain parameters are altered. The ability to get instant feedback on the structural performance, without additional software, gives a faster understand-ing of the structural mechanisms and reduces time in the design phase (Preisunderstand-inger, 2013). One reason for the speed of Karambas calculations are the deliberate limi-tations of the software e.g. instead of isoparametric finite beam elements Karamba uses hermitian elements which are confined to linear elastic calculations of elements with straight axes. The calculation of the element stiffness matrix can be done with-out the need for numeric integration which greatly reduces computation time (Mira et al., 2012).

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Figure 3.2: Karamba workflow.

Karamba offers several different ways of evaluating a structural model. The nu-merical evaluation options consists of second order theory, large deflections, eigen-modes, natural vibration eigen-modes, evolutionary structural optimization, cross section optimization and iterative elimination of tension or compression elements. For each calculation option there is a component which takes a model as input, calculates it and adds the results to the model data (Preisinger, 2015).

Karamba provides truss-, beam- and shell-elements (Preisinger, 2015). The shell element formulation used in Karamba is based on the TRIC element with six DOFs per node, constant strain state assumed for each layer, no in-plane rotational stiffness added but contrary to the TRIC element it is based on Kirchhoff theory (Clemens, Accessed: 2016-03-01).

The Finite Element Analysis (FEA) is performed with the assumption that deflec-tions are small as compared to the size of the structure. There is however a com-ponent that enables calculations with large deflections which increases the load in several steps and updates the deformed geometry but this approach leads to a solu-tion which drifts away from the exact solusolu-tion. Another assumpsolu-tion is linear elastic behaviour of the materials which suits the purpose of an initial design. (Preisinger, 2015).

It is possible to do analysis with both first and second order theory in Karamba.

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3.4. GEOMETRY GYM

The big difference between the two is that the latter count for the influence of axial forces in beams and in-plane forces in shells. These two influence the structures stiffness. Also, compared to higher order theory, the first do not count for geometric non-linearity.

Second order theory is considered with the use of the "AnalyzeThII"-component. It accounts for axial forces via the element’s geometric stiffness matrix and is based on small displacements. (Preisinger, 2015)

Utilization of a analysed structure or part in Karamba is calculated as Von Mises stress divided by the yield strength of the material. The results are possible for both beams and shells given as contour plots where the utilization is calculated in every element.

Karamba only has components giving the maximum displacement of the analysed structure which means that one has to define a Grasshopper/Karamba definition if one wants to find the deformation in specific nodes.

3.4 Geometry Gym

Geometry Gym develops utilities and plugins for Rhino3d and Grasshopper3d among others that enable BIM generation and exchange. The exchange is provided by di-rect API interaction or OpenBIM formats such as IFC, Industri Foundation Classes. (Mirtschin, 2014) IFC is represented in EXPRESS language. This enables users to model information. The IFC model defines a integrated scheme to depict the main physical and logical building objects, their characteristics and their inter-relationships in the form of a class hierarchy. The IFC hierarchy covers the core project information such as building elements, the geometric and material proper-ties of a building, project costs, schedules and organizations. Moreover IFCs enable inter-operability of among AEC/FM software applications and this means the end-users in the AEC/FM area can effectively share the model data through IFC files. (Fu et al., 2006)

3.5 Millipede

Millipede is a plug-in for Grasshopper which allows optimization and structural anal-ysis. This contains analysis of shell elements in 3D and frame, 2D plate elements for in plane forces and 3D volumetric elements. Compared to Karamba this plug-in is more user-friendly for varying cross-sections. It uses structural analysis algorithms for linear elastic systems. The algorithms used for optimization are based on topol-ogy optimization. This can be used in combination with e.g the plug-in Galapagos for solving generic form finding problems. (Panagiotis, 2014)

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of reinforcement patterns and principal stress aligned grid shells. (Michalatos, Ac-cessed: 2016-03-22)

3.6 Galapagos

Galapagos is a evolutionary solver using genetic algorithms which enable optimiza-tion of problems. The optimizaoptimiza-tion consist of random testing of chosen "lists" of inputs which aim towards a better solution with respect to specified outputs during the process. The process of an evolutionary solver run consists of both advantages and disadvantages. The advantages includes e.g. that the evolutionary algorithms are flexible, meaning in this case that they can be used on a wide variety of prob-lems. Furthermore, the algorithms are quite "forgiving", meaning that they can work with poorly formulated problems. It is also, due to that the run-time process is progressive, possible to get answers at any time, which in many cases can be of big importance. One example of this is for structure optimization when the design is of big importance. A good structural solution is not always the neatest. The disadvantage consists of e.g that the evolutionary algorithms are slow. For prob-lems where a single iteration consume a lot of time other types of solvers may work better. Secondly, the evolutionary algorithms do not guarantee a solution. Unless the user predefine a interval where the solution fit. (Rutten, Accessed: 2016-04-01) The input to the Galapagos component consists of variables, so called genes, that one wants the solver to change while running. When changing one or many variables the state of the model changes where the outcome either gets better or worse. The user defines if the component should strive for a higher or lower value. One example is obtaining the lowest possible displacement in a concrete slab standing on columns where the thickness of the concrete and placement of the columns are the inputs (genes). In this case the solver starts by randomly testing values or "genomes" which are specific values for each gene. In other words, it randomly places the columns under the slab and picks a random thickness from the given interval. After evaluation of these, the solver learns a pattern how to choose the next set of random values. It continues until it reaches or comes very close to a defined value or if the user stops it. (Rutten, Accessed: 2016-04-01)

The process of Galapagos is presented in figure 3.3.

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3.6. GALAPAGOS

(a) Random popolation of points in landscape. (b) Best output chosen from (a).

(c) Random population of area by outputs in (b).

(d) Best outputs chosen from (c).

Figure 3.3: (a), (b), (c) and (d) shows the process of Galapagos, optimization of finding the highest point in a landscape Rutten (Accessed: 2016-04-01).

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Figure 3.4: Process of Galapagos.

3.7 SAP2000

SAP2000 is a finite element software which features a graphical user interface that is suitable in terms of ease-of-use and productivity linked with design capabilities. The capabilities consists of linear and non-linear analyses, fast equations solvers, many different constrain options, force and displacement loading and highly accurate layered shell elements to name a few. (CSI, 2009)

The frame element in SAP2000 uses a general, three-dimensional, beam-column formulation which includes the effects of biaxial bending, torsion, axial deformation and biaxial shear deformations. SAP2000 includes several database files for section properties with the option of creating your own database files using a software, PROPER, available by request from the developers. (CSI, 2009)

The shell element formulation consists of either a three- or four-node element which combines membrane and plate-bending behaviour with the option of thin- or thick-plate theory. Each shell element has the option to be modelled as pure membrane, pure plate or full shell behaviour. Stresses and internal forces and moments are evaluated at the 2-by-2 Gauss integration points. (CSI, 2009)

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3.7. SAP2000

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Chapter 4 Case studies

To get an understanding for parametric modelling the first two cases are created, see section 4.1 and 4.2. This is a way of learning the basics of the used software and plug-ins which in turn build the foundation for the development of the collaboration project presented in chapter 5.

The next three cases, section 4.3, 4.4 and 4.5, are used to evaluate the structural out-puts from Karamba with outout-puts given from SAP2000. The reason why these case studies are chosen is because they contain one or several features of the collaboration project.

Furthermore, the model codes and specific codes are presented in appendix B.

4.1 Ribbed concrete shell dome

The Palazzetto dello sport is used as a reference object. The structure is a ribbed concrete shell dome, braced by concrete flying buttresses. It has a simple main geometry built by more complex geometries. (Jérémy Rinaldi, 2016)

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CHAPTER 4. CASE STUDIES

(a) Configuration 1 (b) Configuration 2

(c) Configuration 3

Figure 4.1: Different configurations. The colours represents stresses and are shown to give a more clear view of the configurations.

The geometry is adjusted based on the reference object and outputs obtained from Karamba.

4.2 Atrium roof

It is a beam structure acting as a roof which is constructed as a grid system of steel beams, see figure 4.2. The structure comes from a reference object located in Stockholm, Sweden. It is a small part of the main building and is an example of how a open part through the building could be covered.

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4.3. FULLY RESTRAINED BEAM

Figure 4.2: Atrium roof built up by beams. Surface corresponds to glass. The sup-port condition, in this case, for the beams are fully restrained.

The idea is that the "main" beams (following the y-direction in figure 4.2) in the grid should follow the shape and columns under the roof/grid-system. e.g. the same inclination as the shape and amount of columns. Between the beams, glass will be used which in this case has no structural capacity. To get the shape of the grid a surface is defined in Grasshopper where the lines that creates the grid are projected onto the surface. The lines are then converted to beams with the "LineToBeam"-component in Karamba.

This structure with its quite complicated shape and appearance is made to increase the understanding and knowledge of how one can develop/create and work with surfaces in Grasshopper.

The shape/measurements of the bottom line is locked, but the shape of the surface is fully parametrized including e.g. curvature and height. Figures showing the appearance, sections and measurements can be seen in appendix B.2. These are made to investigate the presentation capabilities of Rhino3D.

4.3 Fully restrained beam

A fully restrained beam is modelled and analysed, see figure 4.3. The idea of analysing this structure is to investigate the basics with a simple structural member and if there is any difference in the structural outputs given from SAP2000 and Karamba. And to see how the displacement is given in Karamba, e.g. when using the analysing-component for second order theory for small deflections.

The beam is also analysed with simple hand calculations where the results are com-pared with the results given from the software.

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235 MPa. The weld between web and flanges will not be taken into consideration. The condition for the supports is fully restrain, i.e. no translation or rotation in any direction. The self weight is not taken into consideration in the analysis.

Figure 4.3: Steel beam in 3D view from the interface of Rhino3D with load values (in kN) and visible supports (fully restrained).

Used equations, section inputs and material inputs for hand calculation are taken from Byggformler och tabeller (Johannesson and Vretblad, 2013). The calculations are presented in appendix A. Furthermore, the calculations are divided into two calculations, i.e. load in z-direction for displacement in z-axis, and load in y-direction for displacement in y-axis.

Table 4.1: Node global axis and resultant translation displacement in middle (x = 5 m) according to SAP2000, Karamba and hand calculations.

Given by x-dir. (m) y-dir. (m) z-dir. (m) Translation disp. (m) SAP2000 0 −3.71 · 10−2 _{−1.3 · 10}−2 _{3.93 · 10}−2 Karamba 0 −3.71 · 10−2 _{−1.3 · 10}−2 _{3.93 · 10}−2 Hand calc. 0 −3.7 · 10−2 −1.19 · 10−2 3.91 · 10−2

By looking in table 4.1 one can see that the resultant translation displacement calculated by SAP2000 and Karamba is the same if comparing to the second decimal. If comparing this to the hand calculation, there is a difference. Hand calculation gives approximately 0.5 percent smaller resultant translation displacement.

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4.3. FULLY RESTRAINED BEAM

(a) Moment diagram for Mz given from Karamba, all valus given in kNm or kN.

(b) Shear force diagram for Vy given from Karamba, all valus given in kN.

(c) Moment diagram for My given from Karamba, all valus given in kNm or kN.

(d) Shear force diagram for Vz given from Karamba, all valus given in kN.

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(a) Moment diagram for Mz given from SAP2000, all valus given in kNm.

(b) Shear force diagram for Vy given from SAP2000, all valus given in kN.

(c) Moment diagram for My given from SAP2000, all valus given in kNm.

(d) Shear force diagram for Vz given from SAP2000, all valus given in kN.

Figure 4.5: Moment and shear force diagrams for beam presented in 4.3 given from SAP2000.

4.4 Slab with different support condition along edges

The structure is a concrete slab with two clamped and two simply supported edges, see figure 4.6. The measurements are 5.1×3.9×0.2 m2 _{and the concrete is of class}

C20/25, i.e. fy = 13.3 MPa. It is loaded with an uniform load of 8.30 kN/m2.

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4.4. SLAB WITH DIFFERENT SUPPORT CONDITION ALONG EDGES

Figure 4.6: Structure in question - two clamped and two simply supported edges showing supports, mesh and global directions. Showed in the interface of Rhino3D.

The idea with the structure is to investigate the difference between moments calcu-lated by Karamba and SAP2000. Furthermore, the structure is selected because it is fairly simple and easy to evaluate. It is taken from an example in Concrete Slabs Theory and design methods see (Nilsson et al., 2012) example 4.2. This will give the geometrical inputs and external load where the latter is determined according to EC (2002). The evaluation will be performed where the critical moment appear, see figure 4.7. Furthermore, the analysis will be made for second order theory.

(a) (b)

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The section forces will be needed to calculate the reinforcement area for the concrete shell in the collaboration project presented in chapter 5. In that case the forces and moments are taken from SAP2000 due to the limitations in Karamba (for shell structures, only possible to get the forces and moments in the center of each element for the principal directions in Karamba). SAP2000 also gives it in the global axis-directions and in all points in the element. When the forces and moments are available in the same axes they are easier to combine when calculating the amount of reinforcement along a determined distance, e.g for a one meter strip.

Furthermore, from forces and moments calculated by Karamba the possibilities of determining the reinforcement area in general and for this structure will be presented and discussed.

The structure is first defined in Grashopper/Karamba and then exported to SAP2000, using Geometrygym, see chapter 3.4.

The values for comparison are chosen from the center of each element. This due to that the forces and moments only are available in that specific point in Karamba. Figure 4.8, 4.9 and 4.10 show the center point of each element where the critical moment appear creating tension at the top of the slab. Furthermore, when taking values from Karamba one has to be careful with signs. Karamba gives values in the local axes for each element in the principal direction.

Figure 4.8: Points from where the principal moments are taken from in Karamba.

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4.4. SLAB WITH DIFFERENT SUPPORT CONDITION ALONG EDGES

Figure 4.9: Contour plot from SAP2000 showing points from where the moment M_maxis chosen, values in legend shown in kNm.

Figure 4.10: Contour plot from SAP2000 showing points from where the moment M_min is chosen, values in legend shown in kNm.

Table 4.2: Principal moments calculated by Karamba and SAP2000 in points pre-sented in figure 4.8, 4.9 and 4.10.

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Table 4.3: Moment along a 0.4 meter strip along direction 22 in critical area, calcu-lated by SAP2000. Element Joint M11 (kNm) M22 (kNm) M12 (kNm) MMax (kNm) MMin (kNm) ggGHa3934 Node 2058 2.86 12 0.2 12 2.85 ggGHa3932 Node 2057 2.87 11.98 0.21 11.99 2.87 ggGHa3936 Node 2059 2.83 11.98 0.19 11.98 2.83 ggGHa3930 Node 2056 2.88 11.92 0.22 11.93 2.87 ggGHa3938 Node 2060 2.8 11.92 0.18 11.92 2.79

By looking at table 4.2 one can see that the difference between the mean values of bending moment according to Karamba and SAP2000 for Mmax is 0.39 percent

(higher for Karamba compared to SAP2000) and for Mmin 2.7 percent (lower for

Karamba compared to SAP2000). Both software give zero shear force in all points of the structure.

Table 4.3 and table 4.2 present nodal values along the strip in figure 4.11. Node numbers in table 4.2 can be visualized in figure 4.9 or 4.10. The vertical and hor-izontal distance between the nodes is 0.1 meter. One can see that the moments acting in the global axes 11 and 22 corresponds well to the principal moments.

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4.5. COMPLEX DOME/SHELL STRUCTURE

Figure 4.11: Contour plot for M22 calculated in SAP2000 showing a 0.4 meter strip

(the red line) along direction 22.

4.5 Complex dome/shell structure

This structure was the starting point in the collaboration project between the au-thors and the architectural student. The idea was to be able to change the amount of supports and vaults, size of the structure and other desired parameters by a total or partly parametrized model.

After further work the collaboration evolved passed this structure, see chapter 5. However, the authors believe this structure have great value in terms of structural appearance and will therefore be one of the cases used for the comparison between SAP2000 and used parametric software.

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Figure 4.12: Complex dome/shell structure.

The structure is only analysed with five supports/vaults. After testing different inputs the ones that give the best structural outcome is chosen. I.e. inputs that give the smallest amount of tensile stresses among tested inputs. The total hight of the structure is 18 meters.

A less extensive analysis with second order theory is performed where the only structural outputs evaluated are displacement and mass of the structure exposed to only gravity load. The class for the concrete is C50/60, i.e. fc = 33 MPa and the

hight of the concrete section/shell is 220 mm.

Figures including measurements and codes in Karamba and Grasshopper are shown in appendix B.5.

Table 4.4: Node global axis and resultant translation displacement according to Karamba, nodes are visualized in figure 4.14 and 4.15.

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Table 4.5: Node global axis and resultant translation displacement according to SAP2000, nodes are visualized in figure 4.14 and 4.15.

Name SAP x-dir. (m) y-dir. (m) z-dir. (m) Translation disp. (m) S1 4.07 · 10−3 1.4 · 10−2 −3.3 · 10−2 _{3.61 · 10}−2 S2 3.97 · 10−3 1.39 · 10−2 −3.29 · 10−2 _{3.59 · 10}−2 S3 1.7 · 10−2 −5.57 · 10−3 _{−3.07 · 10}−2 _{3.55 · 10}−2 S4 4.06 · 10−3 1.36 · 10−2 −3.25 · 10−2 _{3.54 · 10}−2 S5 1.69 · 10−2 −5.59 · 10−3 _{−3.06 · 10}−2 _{3.54 · 10}−2

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Figure 4.14: Node K1, K2 and K4 for Karamba and node S1, S2 and S4 for SAP2000 in table 4.4 and 4.5 given from Area 1 in figure 4.13.

Figure 4.15: Node K3 and K5 for Karamba and node S3 and S5 for SAP2000 in table 4.4 and 4.5 given from Area 2 in figure 4.13.

Table 4.6: Maximum resultant translation displacement and total self mass of struc-ture according to Karamba and SAP2000. Maximum displacement occur in Area 1 in figure 4.13 for both software.

Karamba disp. (m) Karamba mass (kg) SAP2000 disp. (m) SAP2000 mass (kg) 3.3 · 10−2 1.45 · 106 _{3.61 · 10}−2 _{1.48 · 10}6

Table 4.6 shows that the displacement computed by SAP2000 is about 9 percent

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larger than that computed by Karamba. The difference in mass is smaller, about 2 percent larger computed by SAP2000 compared to Karamba.

To get a better solution a mesh convergence study is performed by mesh refinement. The amount of elements is quadrupled and the same analysis is performed again. The area for the elements before mesh refinement is approximately 0.39 m2 _{and 0.64}

m2 _{in area 1 and area 2 respectively and after mesh refinement, consequently, four}

times smaller.

Contour plots for displacement after mesh refinement is presented in figure 4.16 and 4.17.

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Figure 4.17: Contour plots showing deformation in SAP2000, values given in [cm], shown from above.

By looking at figure 4.16 and 4.17 one can see that the deformation corresponds well in all points between the software with exception that SAP2000 shows a slightly higher deformation. This can be seen by comparing values in the legends.

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Table 4.7: Node global axis and resultant translation displacement according to Karamba after mesh refinement, nodes are visualized in figure 4.19 and 4.20. Name Karamba x-dir. (m) y-dir. (m) z-dir. (m) Translation disp. (m) K1 −1.61 · 10−2 _{6.35 · 10}−3 _{2.88 · 10}−2 _{3.36 · 10}−2 K2 −1.61 · 10−2 _{6.38 · 10}−3 _{2.87 · 10}−2 _{3.36 · 10}−2 K3 −1.61 · 10−2 _{6.31 · 10}−3 _{2.87 · 10}−2 _{3.35 · 10}−2 K4 −1.6 · 10−2 _{6.39 · 10}−3 _{2.87 · 10}−2 _{3.35 · 10}−2 K5 −3.97 · 10−3 _{−1.29 · 10}−2 _{3.05 · 10}−2 _{3.34 · 10}−2

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Figure 4.18: Mesh of model after mesh refinement in Karamba and SAP2000, max-imum nodal displacement in Area 1 and 2.

Figure 4.19: Node K5 in table 4.7 given from Karamba from Area 1 in figure 4.18.

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Figure 4.20: Node K1-K4 for Karamba and S1-S5 for SAP2000 in table 4.7 and 4.8 given from Area 2 in figure 4.18.

Table 4.9: Resultant translation displacement and total self mass of structure ac-cording to Karamba and SAP2000 after mesh refinement.

Karamba disp. (m) Karamba mass (kg) SAP2000 disp. (m) SAP2000 mass (kg) 3.36 · 10−2 1.45 · 106 3.6 · 10−2 1.48 · 106

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Chapter 5 Collaboration project

The main task was to assist the architect with structural knowledge in the design process as well as support with choices regarding architectural core values. The initial thought was to create a structurally true design where the geometry somewhat follows the flow of forces.

The location of the structure was decided to Kungsträdgården which is a park in central Stockholm. The goal of the structure is to both offer space for commercial and recreational activities. The idea was initially a dome structure with large open-ing arches as mentioned earlier. Duropen-ing the work process it evolved into a curved roof-slab supported by hollow pillars with a curvature that that smoothly connected to the roof, for pictures of how the model evolved see figure B.14 in appendix B. The material of choice fell upon reinforced concrete since it is a versatile material which is ideal for structures with a complex geometry.

The work was done together and separately by using the parametric design tools. Since the architectural part of the collaboration will continue to the very end of the thesis project the decision was made for the authors to settle on a present design, with focus on structural performance, that would be analysed more comprehensive, see figure 5.1.

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CHAPTER 5. COLLABORATION PROJECT

5.1 Structural evaluation

5.1.1 Limitations

• The structural analysis is limited to only checking sectional forces and moments in the critical area for the critical direction against requirements in ultimate limit state (ULS) and serviceability limit states (SLS) with hand calculations according to Eurocode 2.

• The deformation will be checked against limits stated in EC 2 (2004).

• Only the roof of the structure is analysed and only the section with the largest stress is considered.

• Creep effects and instability will not be considered. • No dynamic analysis will be performed.

5.1.2 Method

The geometry of the structure was adjusted to the topology of the site where con-sideration was taken to the trees and the fountain/pool. The number of pillars was decided by balancing the visibility and the structural outputs from Karamba since the ability to see the park from the opposite of the structure is of great importance. The purpose of the structure is after all to complement and further enhance the recreational capabilities of the park and not the opposite.

Karamba was used as an early optimization tool regarding cross section, material properties and geometry. The pillars geometry and location are parametrised which enabled an early optimization of the inclination and position with respect to defor-mation and utilization without compromising the visibility through the structure too much. The curvature of the pillars were adjusted to smoothly transition into the curved roof both from an aesthetic and structural point of view. The structural reason is to avoid any abrupt changes in the curvature which could lead to local stress concentrations.

The loads considered to act on the roof of the structure is the self weight of the roof and a snow load which corresponds to the location of the structure according to Boverket (2015). The loads are combined for ULS and SLS according to EC (2002), see appendix A

The concrete class C30/37 is used, which is recommended by (EC 2, 2004) based on the environmental exposure condition of cyclic wet and dry. The reinforcement is B500BT with 12 mm i diameter. The concrete cover is calculated with respect to environmental exposure and with a life span of 50 years, see appendix A.

After the initial optimisation mentioned above the model will be exported with Geometrygym to SAP2000. The outputs extracted from SAP2000 are the sectional

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5.1. STRUCTURAL EVALUATION

forces and moments acting in the critical direction, regarding stress, for a one meter strip, see B.14(a). The moments and normal forces are extracted from the critical stress region and the shear force from the region where it is largest. Equilibrium conditions are set up for both ULS and SLS according to Ansell et al. (2012) with the simplification that the mean values are constant over the whole strip.

The hand calculations are made on the basis of equilibrium conditions in a static state with reinforcement only in the tension zone and checked against limits stated by EC 2 (2004).

For the ULS calculation several assumptions are made, a rectangular stress distribu-tion, the strain is assumed to be linear, the concrete cannot take any tensile stress, the compressive strength of the reinforcement is equal to its tensile strength and the reinforcement will yield before the concrete fails in the compressive bending zone (Ansell et al., 2012). The minimum flexural reinforcement is according to Swedish national annex accepted as the reinforcement acquired from the equilibrium condi-tions set up in the ULS (Boverket, 2015). The shear force is checked to see if shear reinforcement is required.

In the SLS the calculations are made with the assumption that the linear strain distribution over the cross-section is retained, Hook’s law is valid and the concrete cannot take up tensile stresses, i.e. the bending tensile cracks reach the neutral axis (Ansell et al., 2012). The required minimum reinforcement content is checked to control the cracks.

The required reinforcement content is calculated with respect to both ULS and SLS. The crack widths are also checked. The deflection is checked against simple conditions that are based on the length of the span according to EC 2 (2004). The length of the span is measured as the distance between the support of the columns on each side of the critical region. The orientation of the reinforcement is not designed in this thesis and the principal stress lines, calculated by Karamba, is presented only to show how one can use it for that purpose.

Furthermore, the structure is optimized regarding the placement of the base of the columns using Galapagos, see chapter 3.6. The result will not be analysed further then discussing the outcome.

5.1.3 Construction method

The consideration regarding the construction method is a combination of traditional methods with some experimental methods currently in development in a project called TailorCrete.

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The project is developing tree new formwork types. The multi-edge formwork con-sists of special cut plywood sheets placed on a flexible formworks system of beams and adjustable telescopic girders. The wax formwork is a waste-free fabrication method which is used for complex, double-curved large scale projects. The method consists of wax formwork elements created by pouring the wax on a flexible actuated mold and then assembling it on standard scaffolding after curing, see figure 5.2. The wax formwork elements can then be melted and reused again. (Danish Technological Institute, 2015)

Figure 5.2: Construction method with tailorcrete (Danish Technological Institute, 2015).

The third method is called Milled formwork. It consists of milled out shape-giving parts. Expanded polystyrene (EPS) is used as the shape defining structure. A robotic cell uses a, for the purpose, developed milling strategy to form the EPS into the desired shape and are coated with a rubber skin. The coating makes it possible to reuse or recycle the EPS. (Danish Technological Institute, 2015)

The current assessment from the developers regarding the curvature of the structure, defined by the bending radius, is that the multi-edge formwork method is the most cost effective for medium curvatures. The Wax formwork used less energy with low-and high curvatures. The milled formwork method gives good surface results low-and provides most possibilities regarding architectural usability. (Danish Technological Institute, 2015)

These methods are presented here only to show the possibilities and which method that would be most suitable for the collaboration project in this thesis would require more extensive research and consideration.

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5.1.4 Results from Karamba and SAP2000

The thickness of the concrete is 200 mm and Karamba calculated the resulting translation displacement to 34.9 mm and the utilization of the concrete compressive strength of the concrete according to figure 5.3 where red is compression and blue is tension. Figure 5.4 show contour plots of the stress calculated by SAP2000 for the top and bottom face of the structure. Figure 5.5 and 5.6 show the sectional forces and moments for ULS and SLS loads respectively in the critical region enhanced and with adjusted limits for better visualisation. The deformation in the critical area acquired from SAP2000 is 31.5 mm for the ULS load and 22.5 mm for the SLS load.

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(a) S11 bottom surface with critical section

marked with a red ring.

(b) V13 with critical section marked with a red

ring.

Figure 5.4: SAP2000 countour plots of the critical sections.

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(a) M11. (b) F11.

(c) V13.

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(a) M11. (b) F11.

(c) V13.

Figure 5.6: Forces and moments in SLS from SAP2000 in the critical regions with a one meter strip marked.

Table 5.1: Sectional forces and moments in the critical areas, calculated by SAP2000.

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5.2. OPTIMIZATION

According to the hand calculations the required reinforcement area is 1131 mm2/meter which gives a spacing of 55 mm and a crack width of 0.06 mm.

Figure 5.7: Principal stress lines given from Karamba.

5.2 Optimization

By the use of Galapagos solver the structure is optimized to give a minimal dis-placement with consideration taken to the dis-placement of the corner columns. The process is described below.

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Figure 5.8: Input structure showing area for which the columns are allowed to move.

The result gave columns with a slightly larger inclination towards the surrounding and the difference in displacement went from 34.8 mm to 32.6 mm, i.e a 6.3 percent smaller displacement.

The same process was also done for when the columns were allowed to move in a larger determined area where inclination in all direction was possible. This according to figure 5.9.

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5.3. FINAL ARCHITECTURAL DESIGN

Figure 5.9: Input structure showing area for which the columns are allowed to move.

This gave a solution visualized in figure B.18 in appendix B. With two columns inclined towards the middle, one inclined towards the surrounding and one straight. For this configuration the displacement went from 34.8 mm to 32.0 mm, i.e a 8.0 percent smaller displacement.

5.3 Final architectural design

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Chapter 6 Discussion

6.1 Discussion on the case studies presented in

chap-ter 4

6.1.1 Ribbed concrete shell dome

The visual presentation of the results makes it easy to establish parameters that are more suitable and to get an somewhat optimised structure is a fast and easy process. Cross-sections and quantity of different beams are established based on the deformation and utilization provided by Karamba. The structure is a good example of a project where the tools examined in this thesis could be used. By changing the parameters one gets an immediate understanding of the deformation and which beams that are critical with respect to stresses.

6.1.2 Atrium roof

The definition for the program code creating the surface was both hard to create and time consuming. The most difficult part was to make the code general so that one, by the use of parameters, could change the size in all directions. When it comes to the definition code creating the beams, a lot more parameters were fixed due to the complexity for the connection between the surface and the beams. Another factor that made it difficult to make the code for the beams general was the beam grid. This due to that it is built up by beams in two separate direction following different boundary lines.

building structures in cooperation with architects

MASTER OF SCIENCE THESIS

STOCKHOLM, SWEDEN 2016