ABSTRACT

This thesis is built around the conception, development and optimization of a helicoidal

skyscraper envelope in order to define a suitable structural choice. The shape is

conceived under geometrical consideration to define a fitting panelization. The aim of

this thesis is to achieve the reduction of costs and the fabricability optimization.

The envelope and the structural system are modelled entirely on Grasshopper, a

Rhino3D plug-in which allows to parametrically design objects. Particular attention was

focused on the studies of the skyscraper’s base shape in order to achieve different

envelopes to examine. Different approaches of geometrical panelization are applied on

defined shells: from the research of the same tangent on curve up to the principal

directions on a surface. This analytical study of the shell is concluded using the software

Evolute Tools PRO, an other Rhino plug-in that allows a complex and advanced analysis

of the considered geometry.

The internal structure is built after the chosen envelope and, because of iterative

optimization process, the best structural performance is found at constant weight.

RIASSUNTO

Questo lavoro di tesi si sviluppa intorno all’ideazione, sviluppo e ottimizzazione

dell’involucro esterno di un grattacielo di forma elicoidale per poi individuare una scelta

strutturale consona al progetto. La forma è stata concepita sulla base di considerazioni

geometriche mirate alla pannellizzazione dell’involucro, riducendo costi e ottimizzando

la fabbricabilità.

L’involucro e la struttura portante sono stati interamente modellati con Grasshopper, un

plugin di Rhino 3D che permette di generare delle geometrie in maniera parametrica.

Particolare attenzione è stata dedicata allo studio del piano di base del grattacielo a

partire dal quale sono state raggiunte diverse forme dell’involucro. A queste ultime sono

stati applicati diversi approcci di pannellizzazione geometrica partendo dalla ricerca

della planarità mediante l’individuazione di tangenti alle curve di piano fino all’ideazione

di algoritmi che mostrano le direzioni principali di una superficie. Lo studio analitico

dell’involucro si conclude con l’utilizzo del software Evolute Tools PRO, altro plugin di

Rhino 3D, che permette una definizione avanzata della geometria.

Successivamente alla scelta dell’involucro è stata concepita la struttura interna e, grazie

a processi iterativi di ottimizzazione, si riesce ad avere la maggior performance

strutturale a parità di carichi propri.

ACKNOWLEDGEMENTS

This thesis has been developed during my internship at the Technical University in

Vienna.

I would like to express my sincere appreciation to my Supervisors, Prof. M. Froli and

Prof. H. Pottmann, for their support and constructive suggestions during the

development of this research work.

A special thank goes to my whole family, especially my parents, my sister and my

grandmother for being such a huge support through my experiences.

I am deeply grateful to all the people known during my experience in Vienna. I found

amazing colleagues and great friends.

I would like to thank all the people who contribute the development and the support of

this thesis. Every suggestion, critique and help has been useful.

I want to acknowledge all the great people that I met this years of my studies, all of my

old and new friends. Everyone has been essencial for my personal growth.

An especially thank to you, who were present in the last three years of my life. I am here

because of you.

Thanks to the people that I met afterwards, thanks for the support and for all the smiles

during the final period before my graduation.

1

SUMMARY

1 Architectural Geometry ................................................................................ 9

1.1 Surface Discretization ............................................................................................. 9

1.1.1 Triangle Meshes .............................................................................................. 10

1.1.2 Quadrilateral Meshes ..................................................................................... 12

1.1.2.1 PQ meshes ............................................................................................... 12

1.1.3 Hexagonal Meshes ......................................................................................... 13

1.1.3.1 P-Hex meshes .......................................................................................... 13

2 Differential Geometry .................................................................................. 15

2.1 Parametric search .................................................................................................. 15

2.2 Definitions ................................................................................................................ 16

2.3 Curves ..................................................................................................................... 20

2.4 Surfaces.................................................................................................................... 21

2.4.1 Surfaces of Revolution .................................................................................. 22

2.4.1.1 Ruled Surfaces ........................................................................................ 22

2.4.1.2 Developable surface ............................................................................ 25

2.4.1.3 Developable surfaces with a NURBS .................................................. 26

2.4.2 Principal curvature directions ...................................................................... 28

2.4.2.1 Classification of points on a surface................................................... 29

2.4.2.2 Lines of curvature .................................................................................. 30

2.4.3 Minimal surfaces ............................................................................................ 30

3 Case study: twisting skyscraper ................................................................... 33

3.1 Design of high-rise buildings ................................................................................ 33

3.2 Softwares ................................................................................................................ 33

3.3 Check design ......................................................................................................... 36

3.3.1 Twisting and rotating forms .......................................................................... 37

3.3.1.1 My twisting form ..................................................................................... 38

3.3.1.2 Base shapes ............................................................................................ 39

3.4 Panelization ............................................................................................................ 40

3.4.1 Search for the same tangent on curve ..................................................... 40

2

3.4.2 Developable surface .................................................................................... 45

3.4.3 Panelization with diamonds and triangles ................................................ 52

3.4.3.1 Which triangles can be converted in flat diamonds with cold bending ......................................................................................................... 55

3.4.3.1.1 Panels deviation ................................................................................. 59

3.4.3.1.2 Min. Cold Bending radius ................................................................... 61

3.4.4 Corner modifications of base shapes ........................................................ 68

3.4.5 Principal curvature directions ...................................................................... 70

3.4.5.1 Shape with smooth corners ................................................................. 70

3.4.5.2 General B-Spline base with degree 3 ................................................ 74

3.4.6 Paneling Architectural Freeform Surfaces ................................................. 79

3.4.6.1 Case studies ............................................................................................ 83

3.4.6.2 Conclusions: Skyscrapers that can be built ....................................... 88

4 Site ........................................................................................................... 89

5 Actions ...................................................................................................... 91

5.1 Wind load ................................................................................................................ 91

5.2 Floor system ............................................................................................................ 96

5.3 Exterior walls ........................................................................................................... 96

6 Structural Systems ....................................................................................... 97

6.1 Frame system ......................................................................................................... 98

6.2 Shear wall system .................................................................................................. 99

6.3 Shear wall and frame system ............................................................................ 100

6.4 Framed tube system ............................................................................................ 101

6.5 Tube in tube system ............................................................................................ 102

6.6 Bundled – tube system ....................................................................................... 103

6.7 Braced – tube system ......................................................................................... 104

6.8 Outrigger – braced system ................................................................................ 105

6.9 Structural system choice .................................................................................... 106

6.9.1 Development of option A .......................................................................... 109

6.9.2 Development of option B ............................................................................ 111

6.9.2.1 2D Model ................................................................................................ 111

6.9.2.2 3D Model ................................................................................................ 118

6.9.2.3 The Project ............................................................................................ 126

7 Conclusions ............................................................................................. 129

3

LIST OF FIGURES

Figure 1: (left) British Museum Great Court Roof, London, completed in 2000 (right)

DG Bank Court Roof, Berlin, completed in 1998 .............................................................. 11

Figure 2: (left) Blob, Eindhoven, completed, (right) Vela Fiera Milano-Rho, Milano,

completed in 2005 .............................................................................................................. 11

Figure 3: (left) a node without an axis. Image of Waagner-Biro Stahlbau AG. (right)

Geometric Torsion in a Node ............................................................................................. 11

Figure 4: (left) Rotational PQ mesh, (right) Geometry of a conjugate curve network

[PAH07] ............................................................................................................................. 12

Figure 5: (left) Mannheim Grid Shell, Mannheim, completed in 1974, (right) Hamburg

History Museum Court Roof, Hamburg, completed in 1989 .......................................... 13

Figure 6: (left) Honeycomb subdivision, (right) Regular triangular tiling [PAH07] ..... 13

Figure 7: P Hex mesh computed using the progressive conjugation method [WLY08] 14

Figure 8: Tangent on a curve [PAH07]............................................................................ 16

Figure 9: Inflection point on a curve ................................................................................ 17

Figure 10: Osculating circles [PAH07] ............................................................................ 17

Figure 11: Bézier curve [PAH07] ...................................................................................... 18

Figure 12: Hyperboloid ..................................................................................................... 23

Figure 13: Hyperbolic paraboloid ..................................................................................... 23

Figure 14: Plücker's conoid ............................................................................................... 24

Figure 15: Möbius strip ..................................................................................................... 24

Figure 16: Cylinders, cones, and tangent surfaces of space curves [PAH07] ................. 25

Figure 17: Nurbs curve generated by control points polygon [PAH07] .......................... 27

Figure 18: Normal curvatures of a surface S at a point p are the curvatures of the

intersection curves with planes R through the surface normal [PAH07] ....................... 28

Figure 19: The osculating circle varies depending on curves [Jau11].............................. 28

Figure 20: Points change based on the position on torus: elliptic, hyperbolic, parabolic

[PAH07] ............................................................................................................................. 30

Figure 21: Star, lemon and monstar lines of curvature [WIKI] ....................................... 30

Figure 22: Catenary and catenoid ..................................................................................... 31

4

Figure 23: Helic and Helicoid ........................................................................................... 31

Figure 24: from the right 1. Schwartz minimal surface, 2. Riemann's minimal surface, 3.

Enneper surface, 4. Bour's minimal surface, 5. Gyroid, 6. Chen-Gackstatter surface.... 32

Figure 25: Development of tall buildings ......................................................................... 36

Figure 26: (from the left) Burj Kalifa, Millennium Tower, Shard, Sears Tower ............. 37

Figure 27: Twisting geometry process [PAH07] ............................................................ 38

Figure 28: (left) Turning Torso in Malmö, Sweden (right) 30 St. Mary Axe in London,

UK ...................................................................................................................................... 38

Figure 29: Base shapes case study and few stories that define the final building ......... 40

Figure 30: Generic curve with tangents in random points ............................................. 40

Figure 31: 74 computed points on curve with a distance of 1.5 m each. ......................... 41

Figure 32: Visualization of the adopted method to find planar meshes between two

consecutive curves. ............................................................................................................ 41

Figure 33: Algorithm generated to define planar panels with the same tangent on

curves ................................................................................................................................. 42

Figure 34: Research points with same tangents in the squared curve with smooth

verteces .............................................................................................................................. 43

Figure 35: Research points with same tangents in the convex curve .............................. 43

Figure 36: Research points with same tangents in the curve with inflection points ...... 44

Figure 37: Generic developable surface ............................................................................ 45

Figure 38: (left) Starting control points (right) Generic NURBS curve of degree 2 ....... 45

Figure 39: (left) Control point polygon (right) Control point polygon that intersect the

NURBS curve in inflection points ..................................................................................... 46

Figure 40: (left) Curvature graph for a generic NURBS curve (right) curvature graphs

for two generic NURBS curves .......................................................................................... 46

Figure 41: Groups of control points .................................................................................. 46

Figure 42: Algorithm to generate NURBS curves ............................................................ 47

Figure 43 Conic sections as special NURBS [PAH07] .................................................... 48

Figure 44: (left) intersection points of different NURBS curve in the same control

polygon (right) zoom of vectors tangent to this two curves ............................................. 48

Figure 45: Planes parallel to vectors tangent on every point selected for one curve and

points projected on the following curve ........................................................................... 49

Figure 46: Flat panels connecting two consecutive floors ............................................... 49

Figure 47: Algorithm to find random weights .................................................................. 49

Figure 48: Looping algorithm to create random skyscraper with developable surfaces 50

Figure 49 Developable skyscrapers with flat panels created by random NURBS curves 51

5

Figure 50: Developable skyscrapers with flat panels obtained by a scale alghoritm for

NURBS curves ................................................................................................................... 51

Figure 51: Initial steps to reach flat diamonds ................................................................. 52

Figure 52: Plane quite tangent to the curves .................................................................... 52

Figure 53: The intersection of two consecutive planes is a line. Picking the middle point

of every line and connecting these points with points previously found on curves we

found flat diamonds .......................................................................................................... 53

Figure 54: Steps to achieve planarity with diamonds and triangles ............................... 53

Figure 55: Planarity analysis with Evolute Tools Pro ...................................................... 54

Figure 56: The algorithm works well for the first floors, then the approximation

becames unacceptable ....................................................................................................... 54

Figure 57: Mesh with planar diamonds and triangles optimized by Evolute Tools Pro . 54

Figure 58: Panels deviation [EPR] ................................................................................... 59

Figure 59: Grasshopper definition for evaluating planarity ............................................ 59

Figure 60: Blue flat panels, yellow panels flat with cold bent, red panels with double

curvature ........................................................................................................................... 60

Figure 61: Screenshot of grasshopper definition for the analysis of principal curvatures

............................................................................................................................................ 61

Figure 62: Results of mesh analysis and cold bent .......................................................... 62

Figure 63: Analysis of cold bending ................................................................................. 62

Figure 64: Geometry of a laminated glass ........................................................................ 63

Figure 65: Model of a cold bent panel .............................................................................. 65

Figure 66: Table from CNR-DT 210/2012 for the analysis of displacement .................. 66

Figure 67: Base shapes [Bor13] ......................................................................................... 68

Figure 68: Evaluation of best shape for planarity check ................................................. 69

Figure 69: Principal curvature lines in the straight part and in the smooth part ........... 70

Figure 70: Starting mesh for the approximation of principal curvature directions of

smooth corners .................................................................................................................. 71

Figure 71: Planar panels obtained with meshes not weld ................................................ 71

Figure 72: Grasshopper definition of principal curvature directions of smooth corners

............................................................................................................................................ 72

Figure 73: (left) shape non-optimized (right) shape optimized with EVOLUTE Tools .. 72

Figure 74: Panelization method B .................................................................................... 73

Figure 75: (left) zoom of panelization alghoritm with method B (left) zoom of the node

with valence 5 .................................................................................................................... 73

Figure 76: Control points for a B-Spline of degree 3 ....................................................... 74

6

Figure 77: (left) Base shape, (right) Panels achieved from principal curvature directions

............................................................................................................................................ 76

Figure 78: (left) Base shape, (right) Panels achieved from principal curvature directions

............................................................................................................................................ 76

Figure 79: (left) Base shape, (right) Panels achieved from principal curvature directions

............................................................................................................................................ 76

Figure 80: B-Spline with inflection points ....................................................................... 77

Figure 81: Principal curvature lines through in two different points of the shape ......... 77

Figure 82: Steps to achieve planar panels ........................................................................ 77

Figure 83: Shape that follows only one principal direction in every floor ...................... 78

Figure 84: (left) panels completely planar that follow both principal directions (right)

in the red part we have to use non-planar panels ............................................................ 78

Figure 85: How principal direction lines change the side of the shape .......................... 78

Figure 86: Kink angle and divergence between panels [Evo12] ...................................... 81

Figure 87: Panels type used and costs .............................................................................. 83

Figure 88: Planarity analysis case study 1 ........................................................................ 85

Figure 89: Results of panel types for case study 1 ............................................................ 85

Figure 90: Planarity analysis case study 2 ....................................................................... 86

Figure 91: Results of panel types for case study 2 ............................................................ 86

Figure 92: Planarity analysis case study 3 ........................................................................ 87

Figure 93: Results of panel types for case study 3 ........................................................... 87

Figure 94 Analysis with 1 cm of gap, colors define different clusters for panels ........... 88

Figure 95: Analysis of 4 cm of gap and different clusters obtained ............................... 88

Figure 96: Location of the building .................................................................................. 89

Figure 97: Donau City (Vienna International Center) ..................................................... 89

Figure 98: DC Tower 1 ..................................................................................................... 90

Figure 99: Site .................................................................................................................. 90

Figure 100: External and internal pressure of the wind .................................................. 96

Figure 101: Structural systems .......................................................................................... 97

Figure 102: Frame system ................................................................................................. 98

Figure 103: Shear wall system .......................................................................................... 99

Figure 104: Shear wall and frame system ...................................................................... 100

Figure 105: Deflection profile ......................................................................................... 100

Figure 106: Framed tube system .................................................................................... 101

Figure 107: Tube in tube system ..................................................................................... 102

Figure 108: Bundled-Tube system .................................................................................. 103

7

Figure 109: Braced-Tube system .................................................................................... 104

Figure 110: Outrigger-braced system ............................................................................. 105

Figure 111: Structural systems ........................................................................................ 107

Figure 112: Utilization importance of structural elements ............................................ 109

Figure 113: Utilization of beams and slabs for the structural option A ......................... 110

Figure 114: Left: Top view of the skyscraper with the structural system and the external

shell. Right: System of the structure of one floor ............................................................ 111

Figure 115: Left: 2D model created with Karamba3D. Right: zoom of the model ......... 112

Figure 116: Detail of one floor in the 2D model .............................................................. 112

Figure 117: Load acting on floor ....................................................................................... 112

Figure 118: Grasshopper example of definition for the element that is indicated as

chord inf. ........................................................................................................................... 113

Figure 119: Natural vibration: Modal 1, modal 2, modal 3 ............................................. 117

Figure 120: 2D model's utilization factor of option B ..................................................... 117

Figure 121: Elements for the 3D model .......................................................................... 118

Figure 122: Method to find an equivalent adequate profil ............................................ 118

Figure 123: Structural system B with analysis result ..................................................... 122

Figure 124: Foundamental eigenmodes with SAP2000 ................................................ 125

Figure 125: Vienna International Centre ........................................................................ 126

Figure 126: Skyscraper floors.......................................................................................... 126

Figure 127: Skyscraper located in the site of construction ............................................ 127

Figure 128 Maquettes of the Skyscraper Envelope ........................................................ 128

8

LIST OF TABLES

Table 1: Comparing grid shells' topologies main properties. ............................................. 9

Table 2: Relevant material properties of basic soda lime silicate glass according to CEN

EN 572-1 2004 [BIV07] .................................................................................................... 55

Table 3: Values of coefficient Ψ for laminated glass beams under different boundary and

load condition .................................................................................................................... 64

Table 4: The left-hand number represent the floor number, the right-hand number

represent the wind force applied to a specific floor ( / 2 . .......................................... 95

Table 5: Modal analysis results with SAP2000 .............................................................. 125

9

1 ARCHITECTURAL GEOMETRY

Geometry is the core of the architectural design process and it is present from the initial

form finding to the construction. Free form surfaces represent the emblematic

expression of contemporary architecture, where the façade and the roof tend to merge

into a single element: the skin of the building.

Finding a proper shape by using geometric knowledge helps to ensure a good fabrication.

The complete design and construction process involves many aspects as form finding,

feasible segmentation into panels, functionality, materials, statics and cost. Geometry

alone is not able to provide solutions for the entire process, but a solid geometric

understanding is an important step toward a successful realization of such a project.

1.1 SURFACE DISCRETIZATION

There is a current trend toward architectural freeform shapes based on discrete surfaces,

largely realized as steel/glass structure. Topology is probably the most important

variable when dealing with free forms, and the most common topologies adopted are the

triangular, the quadrilateral one and seldom also the hexagonal one.

Table 1: Comparing grid shells' topologies main properties.

We can introduce some definitions to clarify the Table 1:

A node (vertex) is a point where more edges converge,

The valence of a node is the number of edges incident to the node,

Triangular Optimal Intrinsically flat 6 Yes High HighQuadrangular Good Quite easy 4 No Low LowHexagonal Quite good Not trivial 3 No Low Very Low

Sensitivity to Imperfections

Surface Approx.

Face Planariz. Complexity

Valence of Reg. Nodes

Torsion of Nodes

Overall Stiffness

10

The torsion of nodes consists in the twisting of sides of the meshes adopted due

to an applied torque: the tendence of a force to rotate an object about an axis.

This concept will be explain in chapter 1.1.1.

1.1.1 Triangle Meshes

Most of the basic tasks in geometric computing deal with the adaption of triangle meshes

to freeform surfaces. A triangle mesh M can approximate a surface in an aesthetic and

well fitting way, but it has to be noted that we obtain a valence of six using such meshes.

The valence or degree of a vertex is the number of edges incident to a vertex, this means

that in every node of a triangle mesh six edges merge.

To manufacture the mesh at the best possible cost, it is necessary to meet rather tight

constraints on the edge length and the angles in the triangular faces. Designing meshes

with large faces reduces the cost. Triangle meshes are easy to deal with from the

prospective of representing a given surface with the desired accuracy. To achieve

aesthetic aims as well as the proper requirements to statics, we use flat panels, which

provide overall high stiffness.

Howevere there are some disadvantages that we have to consider:

In a steel/glass or other construction based on a triangle mesh, six beams

meet in a node; this inplies a higher node complexity.

The cost of triangular glass panels are higher per-area than the cost of

quadrilateral panels.

More nodes imply more steel and glass, and as a consequence more weight.

Apart from simple cases triangle meshes do not possess offsets at constant

face-face or edge-edge distance.

Triangle meshes have high valence as geometric torsion on the nodes.

Excellent examples of triangular grid-shells are shown in the Figure 1 and Figure 2.

11

Figure 1: (left) British Museum Great Court Roof, London, completed in 2000 (right) DG Bank Court Roof, Berlin, completed in 1998

Figure 2: (left) Blob, Eindhoven, completed, (right) Vela Fiera Milano-Rho, Milano, completed in 2005

A geometric support structure of a connected triangle mesh with torsion-free nodes can

be simply realised if the shape is optimized; instead of a general free form triangle mesh

there is no chance to construct a practically useful support structure with torsion free

nodes. Essentially for the complexity of the nodes, nowadays the triangular topology is

decreasingly used. Instead one uses quadrilateral meshes in most applications.

Figure 3: (left) a node without an axis. Image of Waagner-Biro Stahlbau AG. (right) Geometric Torsion in a Node

12

1.1.2 Quadrilateral Meshes

Quadrilateral meshes exhibit two remarkable disadvantages: on the one hand their

stiffness is lower and on the other hand we have to consider non-planar panels in general.

Flat panels are of course cheap to produce, but also single curvature panels can be

obtained at little cost through the cold bending technique. In a quad mesh, an interior

vertex of valence four is called a regular vertex. If the valence is different from four, we

talk about an irregular vertex.

1.1.2.1 PQ meshes

Planar quad meshes also known as PQ meshes, can be easily used to represent

translational surfaces which are obtained by traslating a polygon along another polygon.

Also rotational surfaces can be generated by PQ meshes.

In a rotational PQ mesh, the mesh polygons are aligned along parallel circles and

meridian curves. Adjacent mesh polygons of the same family form PQ strips, which can

be seen as discrete versions of developable surfaces tangent to a rotational surface S

along the rotational circles and meridian curves. The network of parallel circles and

meridian curves is an instance of a conjugate curve network, and the PQ mesh can be

seen as a discrete version of it. [PAH07]

The tangents to the curves of one family of a conjugate curve form a developable ruled

surface that can be always represented by PQ meshes.

Figure 4: (left) Rotational PQ mesh, (right) Geometry of a conjugate curve network [PAH07]

13

Examples of structures with quad meshes are the following in Figure 5.

Figure 5: (left) Mannheim Grid Shell, Mannheim, completed in 1974, (right) Hamburg History Museum Court Roof, Hamburg, completed in 1989

1.1.3 Hexagonal Meshes

Hexagonal meshes might have non-planar panels and exhibit a low overall stiffness

compared to an equivalent triangular grid. Furthermore they are aesthetically pleasing,

most of the times they even resemble organic forms and additionally they have a very low

valence of the nodes which makes their production much easier.

The Honeycomb subdivision algorithm is a remeshing operator which translates a

triangular mesh into an hex-dominant one.

Figure 6: (left) Honeycomb subdivision, (right) Regular triangular tiling [PAH07]

1.1.3.1 P-Hex meshes Free form meshes with planar hexagonal faces, which are called P-Hex meshes, provide

a useful surface representation in discrete differential geometry and are demanded in

architectural design for representing surfaces built with planar glass/metal panels.

According to Liu’s algorithm [WLY08] the progressive conjugation method is used to

obtain a hexagonal mesh with planar panels. This method ensures that the resulting P-

Hex faces are nearly affine regular or quasi-regular hexagons, since ideal triangles are

14

computed within discretization error. A problem with this approach is that the widths

and orientations of the triangle layers cannot easily be predicted or controlled.

Figure 7: P Hex mesh computed using the progressive conjugation method [WLY08]

15

2 DIFFERENTIAL GEOMETRY

This chapter contains a brief summary of some important concepts and definitions that

will be useful in the remainder of this thesis.

The differential geometry of curves and surfaces has two aspects. One, which may be

called classical differential geometry, is connected with the beginnings of calculus.

Roughly speaking, classical differential geometry is the study of local properties of curves

and surfaces. By local properties we mean those properties which depend only on the

behavior of the curve or surface in the neighborhood of a point. In this thesis curves and

surfaces will be defined by functions which can be differentiated a certain number of

times.

The other aspect is the so-called global differential geometry. Here one studies the

influence of the local properties on the behavior of the entire curve or surface.

2.1 PARAMETRIC SEARCH

Parametric search is a technique that can sometimes be used to solve an optimization

problem when there is an efficient algorithm for the related decision problem.

The parametric search technique was invented by Megiddo as a technique to solve certain

optimization problems. It is particulary effective if the optimization problem can be

phrased as a monotonic root-finding problem and if an efficient algorithm for the

corresponding fixpoint problem can be constructed.

More specificall a root- finding problem consists of finding the largest value ∗ of with

the property that ∗ 0. Let be a monotonic function with a root and let be

an algorithm that computes , written in the form of a binary decision tree whose

nodes correspond to inequalities 0. The parametric search technique evaluates ∗ , and in the process discovers ∗, by evaluating the sign of at some of the roots

of .

16

Suppose that the optimization problem has inputs. Then the decision problem has

1 inputs where the additional input is for the parameter . [dC76]

2.2 DEFINITIONS

A Curve indicates any path, whether actually curved or straight, closed or open. A curve

can be on a plane or in three-dimensional space. Lines, circles, arcs, parabolas, polygons,

and helices are all types of curves

A Curve tangent is a line that touches a curve at a point without crossing over.

Formally, it is a line which intersects a differentiable curve at a point where the slope of

the curve equals the slope of the line.

Figure 8: Tangent on a curve [PAH07]

Curvature is the amount by which a geometric object deviates from being flat,

or straight in the case of a line, but this is defined in different ways depending on the

context. There is a key distinction between extrinsic curvature, which is defined for

objects embedded in another space (usually a Euclidean space) in a way that relates to

the radius of curvature of circles that touch the object, and intrinsic curvature.

A vertex V is a point with locally extremal curvature. At a generic vertex, the osculating

circle remains locally on the same side of the curve.

The inflection point is a point on a curve at which the sign of the curvature (i.e.,

the concavity) changes. Inflection points may be stationary points, but are not local

maxima or local minima.

17

Figure 9: Inflection point on a curve

The osculating circle o of a curve at a given point is the circle that has the

same tangent and curvature as the curve at point . Similar, as the tangent is the best

linear approximation of a curve at a point , the osculating circle is the best circle that

approximates the curve at . Let , , denote the circle passing through three

points on a curve , with . Then the osculating circle is given by

lim⟶

, , .

Figure 10: Osculating circles [PAH07]

An osculating paraboloid p is the counterpart of an osculating parabola that

approximates a surface at a point . We use a special coordinate system at to get the

equation:

∶ 2 2

(1)

Here the -plane is the tangent plane and the z-axis is the surface normal of at . Then

the -plane and the -plane are symmetry planes of and has locally the same

curvature behaviour as the surface . The two numbers and are called principal

curvatures of , whereas the -axis and -axis are called principal curvature directions.

Given a set of 1 control points P ,… , P the corresponding Bézier curve (or

Bernstein-Bézier curve) is given by

18

c t B , t P , (2)

where B , t 1 and t ∈ 0,1 .

A "rational" Bézier curve [MWW] is defined by

C t B , t w P / B , t w , (3)

where is the order, B , is defined as in (2), are control points, and the

weight of is the last ordinate of the homogeneous point. These curves

are closed under perspective transformations, and can represent conic sections exactly.

In the plane every Bézier curve passes through the first and last control point and lies

within the convex hull of the control points. The curve is tangent to and

at the endpoints. The "variation diminishing property" of these curves tells that no

line can have more intersections with a Bézier curve than with the curve obtained by

joining consecutive points with straight line segments. An other desirable property of

these curves is that the curve can be translated and rotated by performing these

operations on the control points only.

Figure 11: Bézier curve [PAH07]

A B-spline is a generalization of a Bézier curve. Therefore we define the knot vector

T t , t , … , t , (4)

where T is a nondecreasing sequence with t ∈ 0,1 . Furthermore we define control

points P , … , P and the degree as

19

p ≡ m n 1. (5)

The "knots"t , … , t are called internal knots.

We define the basis spline functions via

N , t1, if t t and t t0, otherwise

(6)

N , tt tt t

N ,t t

t tN , t , (7)

where j 0,1, … , p. Then the curve

c t N , t P , (8)

is a so-called B-spline.

Specific types include the nonperiodic B-spline ( 1 knots equal 0 and where the

last 1knots equal to 1) and the uniform B-spline (all internal knots are equally

spaced). A B-spline with no internal knots is a Bézier curve.

A B-spline curve is p k times differentiable at a point, where k duplicate knot values

occur. The knot values determine the extent of the control of the control points.

A nonuniform rational B-spline curve (NURBS) is defined by

, / , , (9)

where is the order, , are the B-spline basis functions, are control points, and

the weight of is the last ordinate of the homogeneous point. These curves are

again closed under perspect ive transformations and can represent conic sections

exactly.

20

2.3 CURVES

The world of Euclidean geometry is inhabited by lines and planes. If we wish to go beyond

this flat world to a universe of curvature, we need to understand more general types of

curves and surfaces.

A curve [Opr07] in 3-space is a continuous mapping :I → where Ι is some type

of interval on the real line . Because the range of is , ’s output has three

coordinates. We then write, for ∈ , a parametrization for ,

, , (10)

where the are themselves functions :I → . We say is differentiable if each

coordinate function is differentiable as an ordinary real-valued function of .

In order to define curvature and torsion, we will need each to be at least 3-times

differentiable.

The velocity vector of at is defined to be

| , | , | (11)

Where / is the ordinary derivative and | denotes the evaluation of the

derivative at .

Parametric search could be concretized in parametric representation of a parametric

curve that is expressed as functions of a variable . This means that the spatial curve

can be represented by , , , where is some parameter assuming all

values in an interval . We could consider a curve as the result of a continuous mapping

of an interval into a plane or three-dimensional space. Thereby, every parameter is

mapped onto a curve point . The functions , and are called the

coordinate functions and is a parametrization of .

Helics in parametric representation: Given the center , and the radius

, the points , of the circle are described as

(12)

21

2.4 SURFACES

Surfaces are 2-dimensional objects and should be describable by two coordinates. We

should try to spread part of the plane around a surface and, in terms of the required

twisting and stretching, understand how the surface curves in space.

Let denote an open set in the plane . The open set will typically be an open disk or

an open rectangle. Let:

:D →

, → , , , , , , (13)

denote a mapping of into 3-space. The , are the component functions of the

mapping . We can perform calculus on time depending variables by partial

differentiation. Fix and let vary. Then , depends on one parameter and

is, therefore, a curve. It is called a u-parameter curve. Simillary, if we fix then the

curve is , is a v-parameter curve. Both curves pass through , in .

Tangent vectors for the u-parameter and v-parameter curves are given by differentiating

the component functions of with respect to and respectively.

We write

, , (14)

, , (15)

We can evaluate this partial derivatives at , to obtain the tangent, or velocity,

vectors of the parameter curves at that point, , and , . [Opr07]

22

2.4.1 Surfaces of Revolution

Suppose is a curve in the -plane and is parametrized by , ,0 .

Revolve about the -axis. The coordinates of a typical point may be found as follows.

As it is mentioned in [Opr07], the -coordinate is that of the curve itself since we rotate

about the -axes. If denotes the angle of rotation from the -plane, then the -

coordinate is shortened to cos v and the -coordinate is given by

sin v . The function may be defined by:

, , cos v , h u sin v (16)

Examples of surfaces of revolution are:

Catenoid: obtained by revolving the catenary cosh about the -axis.

Torus: obtained by revolving the circle of radius about the -axis.

, cos , sin , (17)

Torus has been analised better in the chapter 2.4.2.1 and Catenoind in chapter

2.4.3.

2.4.1.1 Ruled Surfaces

A surface is ruled if it has a parametrization

, (18)

where and are space curves. The entire surface is covered by this one patch, which

consists of lines emanating from a curve going in the direction . The curve

is called the directrix of the surface and a line having as direction vector is called a

ruling.

Examples of ruling surfaces are:

Cones: , where is a fixed point.

Cylinders: , where is a fixed direction vector.

Helicoid: take a helix acos , asin , and draw a line through

0,0, and acos , asin , . The surface sweept out by this rising and

rotating line is a helicoid. A patch for the helicoid is given by

23

, , , (19)

Hyperboloid (a doubly ruled surface):

Figure 12: Hyperboloid

cos ∓ sinsin cos

cossin0

sincos (20)

Hyperbolic paraboloid (a doubly ruled surface):

Figure 13: Hyperbolic paraboloid

2

02

(21)

24

Plücker's conoid:

Figure 14: Plücker's conoid

cossin

2 cos sin

00

2 cos sin

cossin0

(22)

Möbius strip:

Figure 15: Möbius strip

cos cos12

cos

sin cos12

sin

sin12

cossin0

cos12

cos

cos12

sin

sin12

(23)

25

2.4.1.2 Developable surface

A developable surface is a ruled surface with global Gaussian curvature 0. Gaussian

curvature will be introduced in section 2.4.2. Developable surfaces include

the cone, cylinder, and plane.

A developable surface has the property that it can be made out of a sheet, since such a

surface must be obtainable by transformation from a plane (which has Gaussian

curvature zero) and every point on such a surface lies on at least one straight line. There

are three basic types of developable surfaces: cylinders, cones, and tangent surfaces of

space curves.

Figure 16: Cylinders, cones, and tangent surfaces of space curves [PAH07]

26

2.4.1.3 Developable surfaces with a NURBS

One of the advantages of NURBS curves [PT97] is that they offer a way to represent

arbitrary shapes while maintaining mathematical exactness and resolution

independence.

Among their useful properties are the following:

They can represent virtually any desired shape, from points, straight lines, and

polylines to conic sections (circles, ellipses, parabolas, and hyperbolas) to free-

form curves with arbitrary shapes.

They give great control over the shape of a curve. A set of control points and knots,

which determine the curve's shape, can be directly manipulated to control its

smoothness and curvature.

They can represent very complex shapes with remarkably little data. For instance,

approximating a circle three feet across with a sequence of line segments would

require tens of thousands of segments to make it look like a circle instead of a

polygon. Defining the same circle with a NURBS representation takes only seven

control points.

In addition to draw NURBS curves directly as graphical items; we can use them

as a tool to design and control the shapes of three-dimensional surfaces, for

purposes such as:

- surfaces of revolution (rotating a two-dimensional curve around an axis

in three-dimensional space)

- extruding (translating a curve along a curved path)

- trimming (cutting away part of a NURBS surface, using NURBS curves to

specify the cut)

One very important motivation for using NURBS curves is the ability to control

smoothness. The NURBS model allows you to define curves with no kinks or sudden

changes of direction or with precise control over where kinks and bends occur.

One of the key characteristics of NURBS curves is that their shape is determined by the

positions of a set of points called control points. As in the Figure 17, the control points

are often joined with connecting lines to make them easier to see and to clarify their

relationship to the curve. These connecting lines form is known as control polygon.

27

Figure 17: Nurbs curve generated by control points polygon [PAH07]

The second curve in Figure 17 is the same curve, but with the weight increased in one of

the control points. Notice that the curve's shape isn't changed throughout its entire

length, but only in a small neighborhood near the changed control point. This is a very

desirable property, since it allows us to make local changes by moving individual control

points, without affecting the overall shape of the curve. Each control point influences the

part of the curve nearest to it but has little or no effect on parts of the curve that are

farther away.

One way to think about this is to consider how much influence each of the control points

has over the path of our moving particle at each instant of time.

28

2.4.2 Principal curvature directions

Before we can introduce principal curvature directions we have to introduce the concept

of normal curvature [MWW]. Let be a surface in that is given by the graph of

a smooth function , . Assume that passes through the origin and its

tangent plane in is represented by the 0 plane. Let 0, 0, 1 be a unit

normal to at the origin.

Figure 18: Normal curvatures of a surface S at a point p are the curvatures of the intersection curves

with planes R through the surface normal [PAH07]

Furthermore we denote , , 0 a unit vector in . Let be the parameterized

curve given by slicing through the plane spanned by and . We obtain,

, , , (24)

For a plane curve we introduce the concept of signed curvature at with respect to

the unit normal : is the reciprocal of the radius of the osculating circle to at , taken

with sign as in the examples below:

Figure 19: The osculating circle varies depending on curves [Jau11]

More rigorously, is defined by the formula:

| (25)

29

where represents arc length with 0 0 (i.e., | . For

instance, one can readily verify that a circle of radius has signed curvature 1/ at each

point with respect to the inward-pointing unit normal.

[Jau11] The normal curvature has a maximum value and a minimum value . These

two quantities are called the principal curvatures and the corresponding directions are

orthogonal and are called principal directions. This was shown by Euler in 1760.

The quantity

(26)

is called the Gaussian curvature and the quantity

2 (27)

is the so-called the Mean curvature, which both play a very important role in the

theory of surfaces.

2.4.2.1 Classification of points on a surface

A point of a surface is called:

Elliptical, both principal curvatures , have the same sign, and the surface is locally

convex.

0. (28)

Umbilic, the principal curvatures , are equal and every tangent vector is a principal

direction. These typically occurs in isolated points.

Hyperbolic, the principal curvatures , have opposite signs, and the surface will be

locally saddle shaped.

0. (29)

Parabolic, one of the principal curvatures is zero. Parabolic points generally lie in a

curve separating elliptical and hyperbolic regions.

Flat umbilic, both principal curvatures are zero. A generic surface will not contain flat

umbilic points.

30

Torus:

Figure 20: Points change based on the position on torus: elliptic, hyperbolic, parabolic [PAH07]

2.4.2.2 Lines of curvature

A line of curvature of a regular surface is a regular connected curve ⊂ , such that for

all the tangent line of is a principal curvature direction at .

Typically forms of lines of curvature near umbilics are are star, lemon and monstar

(derived from lemon-star).

Figure 21: Star, lemon and monstar lines of curvature [WIKI]

2.4.3 Minimal surfaces

A surface that locally minimized its area is called a minimal surface. Equivalently one

can define a minimal surface as a surface whose mean curvature vanishes. We may

observe minimal surfaces as the shape of a soap membrane through a closed wire .

Neglecting gravity, surface tension implies that the soap membrane attains the shape of

the surface with minimal surface area. A minimal surface has vanishing mean curvature

in each of its points.

31

Examples of minimal surfaces are:

Catenoid: [Opr07]

Figure 22: Catenary and catenoid

A catenoid is a surface in 3-dimensional Euclidean space arising by rotating

a catenary curve about its directrix. Not counting the plane, it is the first minimal

surface to be discovered. It was found and proved to be minimal by Leonhard Euler in

1744. Apart from the plane, the catenoid is the only minimal surface of revolution.

The catenoid may be defined by the following parametric equations:

cos ,

sin ,

,

(30)

where ∈ , and ∈ and is a non-zero real constant.

In cylindrical coordinates:

, (31)

where is a real constant.

Helicoid: [Opr07]

Figure 23: Helic and Helicoid

32

Its name derives from its similarity to the helix: for every point on the helicoid, there is

a helix contained in the helicoid which passes through that point. It can be described by

the following parametric equations in Cartesian coordinates:

cos ,

sin ,

,

(32)

where and range from negative infinity to positive infinity, while is a constant.

If is positive, then the helicoid is right-handed, if negative then left-handed.

The helicoid has principal curvatures 1/ 1 .

Other minimal surfaces from the 19th century are: Schwartz minimal surfaces, Riemann's

minimal surface, Enneper surface, Bour's minimal surface, Gyroid, Chen–Gackstatter

surface.

Figure 24: from the right 1. Schwartz minimal surface, 2. Riemann's minimal surface, 3. Enneper surface, 4. Bour's minimal surface, 5. Gyroid, 6. Chen-Gackstatter surface

33

3 CASE STUDY: TWISTING SKYSCRAPER

After this short introduction on architectural geometry and differential geometry, we

want to consider a particular case study: a skyscraper. The goal of this research is to find

a good shape that can be easily built under the constraint of optimising materials and

costs. The work starts with different design processes in order to be aware of a good

choice.

The choice of the shape is furthermore influenced by:

the geometrical aspect,

the secondary structure exposed by wind loads.

3.1 DESIGN OF HIGH-RISE BUILDINGS

Technology and engineering of high-rise buildings have become far better and much

more sophisticated, but most, if not all of the skyscrapers constructed today remain

fundamentally the same in built configuration: in particular the basic planning remains

the same.

Whether built of concrete or steel, most are still nothing more than a series of stacked

trays piled homogeneously and vertically one on top of the other. [Yea02]

The shape of a skyscraper is mostly influenced by wind loads, which contribute to

aerodynamic modifications of the shape and different structural reinforcement. Thus

aerodynamics of the tower’s shape need to be considered as a critical parameter from the

first stage of design.

3.2 SOFTWARES

Complex surfaces, freeform geometry and relative structures are difficult to draw and

commonly used tools, such as AutoCAD or Revit, are not suited enough. Furthermore, it

is well known that projects change hundreds of times during their conception and their

realization: updating it with CAD and subsequenteky with all the softwares involved (for

structural analysis, energetics, fabrication, costs, etc…) it is unfeasible.

34

One of the software that allow to satisfy this demand is Grasshopper, a plug-in for the

highly advanced 3D modeller Rhinoceros. This allows the user to see and exploit all the

possible solutions inside the domain space (the imput parameters vary in a defined

domain).

Grasshopper hosts a lot of different plug-ins, each one specialized on some aspect of

design workflow. Numerous plug-ins have been used throughout the entire project

development. In the following, a list of them is presented along with a short introduction.

Kangaroo is probably the most known and used plug-in of all. What it does is to add a

physical engine and number of physical forces and interactions. A physical engine may

have a lot of useful uses. In this thesis it has been used to find planar quad mesh of the

case-studies.

Hoopsnake came to solve one of the biggest flaw of Grasshopper: the impossibility to

perform a so called for loop. Since Grasshopper has a linear workflow, meaning that the

flow of data goes in one and one only direction. Hoopsnake comes to change this habit.

There one can imput the initial data, at step 0, run them through the script, and re-input

the updated data from a different input plug.

Python Script is a plug-in that include Python inside Grasshopper. Python is a modern

programming language that is used to automate a repetitive task in Rhino much faster

than a manually way. Perform task that are not accessible in the standard set of Rhino

commands or Grasshopper components are available in Python.

WeaverBird is a powerful mesh editing tool. It can perform various mesh subdivisions

(e.g. Catmull-Clark, Sierpinsky, midedge, Loop).

LunchBox is a plug-in that include unusual geometry, panels and structures.

Furthermore with this component it is possible to write/read a .xls file.

Karamba3D is a commercial plug-in by Bollinger+Grohmann ingenieur, a German

structural engineering firm. It allows to perform structural analysis, right inside

grasshopper. Although it is not as powerful as other stand-alone well-known softares,

first of all SAP2000 and GSA, its being directly connected to the design workflow

increase the time performance.

35

Moreover, since it is inside grasshopper, it can perform various operations at any step of

the workflow, something which would be quite difficult to achieve with a stand-alone

software. Here’s a list of some interesting operations Karamba is able to do:

Actively operate on the design geometry (topology);

Modify any set of data, according to certain structural output;

Allowing the creation of an optimization loop, with the aim of minimize

(maximize) of some structural output.

Geometry Gym is another plugin for anyone interested in structural

analysis/optimization, keeping linked with Grasshopper. This plug-in enables

reading/writing of files from/to any structural software (e.g. SAP2000, GSA, Robot

structural Analysis). Projects realized inside Grasshopper can be exported, analysed and

re-imported into it, giving the designer an important edge over the workflow process.

Evolue Tools PRO is a plug-in for Rhino, not for Grasshopper, differently from the

others. This software is an advanced geometry optimization tool for freeform surfaces

with a user friendly interface. Established computational tools from Evolute's core

software library as well as ground breaking technology from our cutting edge research

results provide you with optimization functionality not offered by any other CAD system.

This software offers:

multi-resolution mesh modelling;

global and local subdivision rules;

mesh editing tools and mesh optimization for various goals (closeness, planarity,

fairness, coplanarity, edge length repetition, ballpacking);

specification of vertices as anchor/corner points, constraints (floor slabs, generai

co-planarity constraints, reference curves, fine grained fairing);

specific analysis such as closeness, planarity, edge length, principal curvature and

asymptotic line analysis;

pattern mapping;

NURBS fitting.

Evolute Tools is used at the end of the research process based on knowledges of

differential geometry.

36

3.3 CHECK DESIGN

Development of tall buildings has been changing from year to year starting with the

Home Insurance Building in Chicago (1885) up to the tallest building the Burj Khalifa

realised in Dubai in 2010. In the Figure 26 the evolution of the building design of

skyscrapers is depicted; examples include the WTC, the Sears (Willis) Tower built in the

1907 and Taipei 101 in 2004.

Figure 25: Development of tall buildings

Figure 25 shows how reducing the section area gradually toward the top is a good

strategy to enhance lateral performance of a tall building.

Examples of tapering buildings are the Burj Kalifa in Dubai (828 m height), the

Millennium Tower in Tokyo with his 840 meters (not yet built), the Shard in London

(319 m height) and the Sears Tower in Chicago (527 m), where tapering is very often

associated with the changing of the cross section.

37

Figure 26: (from the left) Burj Kalifa, Millennium Tower, Shard, Sears Tower

3.3.1 Twisting and rotating forms

An interesting approach in contemporary tall building design is a twisted form. In

general, twisting and rotating forms are effective in reducing vortex-shedding induced

dynamic response of tall buildings by disturbing vortex creation.

The twisting of buildings minimises the wind loads from prevailing directions and avoids

the simultaneous vortex shedding along the height of the building. Rotating the building

can also be very effective because its least favourable aspect does not coincide with the

strongest wind direction.

To define a twist deformation, we introduce a fixed bottom plane and a straight line ,

which is called the twist axis, orthogonal to the plane . The layers of the object in the

planes orthogonal to the axes are rotated about as follows (Figure 27). The bottom

plane remains fixed and the rotational angle of the top plane is prescribed. The

distance between the bottom and top planes is , the height of the object to be deformed.

The rotational angle / .

This is a linear variation of the rotational angle with respect to the distance. For the

bottom plane, we have 0 and thus 0 0 which means that the bottom slice

remains fixed. As desired, the top plane is rotated by an angle . The plane

at bottom distance /2 is rotated by /2, and so on. [PAH07]

38

Figure 27: Twisting geometry process [PAH07]

Examples of twisting towers are:

The Turning Torso (190 m height) designed by Calatrava, which has a twist of 90

degrees from the bottom plane. It is composed by a central concrete core that is

able to take wind loads even without a secondary structure in the façade.

30 st. Mary Axe in London (180 m) with a triangulated perimeter steel structure

to eliminate extra reinforcement.

Figure 28: (left) Turning Torso in Malmö, Sweden (right) 30 St. Mary Axe in London, UK

3.3.1.1 My twisting form

In this thesis we study the design of a twisted 74 high building, where each story is

assumed to have an heigh of four meters. We use a twisting algorithm for the rotational

angle based on an exponential function ∗ 1 where 0.011 and ∑ . The

goals of the project are the following:

To optimize the panels for the secondary structure in a geometrical way

39

To give a substance to the secondary structure and analyse the actions of wind

loads.

Modern architecture employs different kinds of geometric primitives when segmenting

a freeform shape into simpler parts for the purpose of building construction. For most

materials used (glass panels, wooden panels, metal sheets,…), it is very expensive to

produce general double-curved shapes. A popular way aims to use approximation by flat

panels, which most of the time are triangular. A third way, less expensive than the first

and capable of better approximation than the second, is segmentation into single curved

panels. The decision for a certain type of segmentation depends on the costs, but also on

aesthetics. The visual appearance of an architectural design formed by curved panels is

different from a design represented as a polyhedral surface.

The planarity constraint on the faces of a quad mesh however is not so easy to fulfill, and

infact there is only little computational work on this topic. So far, architecture has been

mainly concentrating on shapes of simple genesis, where planarity of faces is

automatically achieved. For example, translational meshes, generated by the translation

of a polygon along another polygon, have this property: all faces are parallelograms and

therefore planar.

3.3.1.2 Base shapes

After fixing the twisting form, the next step deals with the analysis of different kind of

basis shapes in order to find the best solution for a given task.

Three different solutions have been proposed whose initial shape is a square in each case.

The analysis gives differents results depending on the shape analysed.

The first shape is simply a square with smooth vertices; the second shape is completely

curved and convex with no inflection points; the third one is composed of eight inflection

points and it is a NURBS curve. As mentioned in Section 3.3.1.1 the task consists of

constructing a twisted building with 74 stories. We will perform this task for the three

basic shapes described above.

40

Figure 29: Base shapes case study and few stories that define the final building

3.4 PANELIZATION

In the next chapters there will be different approaches to check the best base shape and

the best pannelization type. The target is to achieve flat panels and reduce costs of

fabrication with clusters of panels.

3.4.1 Search for the same tangent on curve

Figure 30: Generic curve with tangents in random points

One way to find planar panels aims to have the same tangent from one floor to the next

floor.

For finding planar trapezoids, a given number of points are fixed on the first curve in

order to achieve 74 panels of a length of around 1.5 meters each (Figure 31).

41

Figure 31: 74 computed points on curve with a distance of 1.5 m each.

For every point on the curves the ortogonal plane tangent has been identified using a

parametric algorithm (Figure 33). To define a strictly planar mesh, the point projected

to the curve above has the same tangent of the point to the curve below (Figure 32).

Figure 32: Visualization of the adopted method to find planar meshes between two consecutive curves.

42

Figure 33: Algorithm generated to define planar panels with the same tangent on curves

43

The behaviour of the base shapes is completely differente for each of the studied cases:

1. The shape with straight edges and smooth vertices is the worst shape for this

algorithm. In the straight area, one can never find a corresponding point

projected from the floor below that has the same tangent on the curve: lines are

always oblique and all the points in the curve below converge in the same point

on the consecutive curve.

Only in the convex part of the shape, which corresponds to the smooth vertices,

it is possible to find points with the same tangent for two consecutive curves and

thus obtain complete planarity of the panels.

Figure 34: Research points with same tangents in the squared curve with smooth verteces

2. For a basis curve of the second type we obtain much better results. Here, it is easy

to find the same tangent from two consecutive floors and the result is satisfying;

every panel is completely planar.

Figure 35: Research points with same tangents in the convex curve

44

3. For the third type of basis curve we obtained mixed results. The results are not

satisfying near the inflection points but the results in the other regions are

acceptable.

Figure 36: Research points with same tangents in the curve with inflection points

45

3.4.2 Developable surface

Another way to find planar panels is to use a developable surface. In mathematics, a

developable surface is a surface with zero Gaussian curvature. Such a surface can be

flattened onto a plane without distortion. Therefore, it is always possible to find planar

panels for a developable surface.

Figure 37: Generic developable surface

It is however not possible to find a developable surface for a twisting shape. Instead, we

will consider a slightly simpler problem namely a simple translation in the z-direction

for every curve. For basis curves of the first and the second type, it is easy to see that this

construction always yields a developable surface. Therefore, we will perform the analysis

in this section only for curves of the third basis type, i.e., for a NURBS curve of degree 2

with 8 control points.

For a curve with degree 2 it is easy to find the inflection points since they are the

intersection points between the spline and the control point polygon.

Figure 38: (left) Starting control points (right) Generic NURBS curve of degree 2

46

Figure 39: (left) Control point polygon (right) Control point polygon that intersect the NURBS curve in inflection points

Figure 40: (left) Curvature graph for a generic NURBS curve (right) curvature graphs for two generic NURBS curves

In order to create a free form skyscraper, it has been decided to assign different random

weights to the control points based on a algorithm that considers two groups of control

points: these external and these internal to the curve (Figure 41). Every group of control

points has the same weight for every floor (this means that there are two different values

for the weight of every floor, for example a value of 1,2 for external control points and a

value of 0.1 for internal control points) (Figure 42).

Figure 41: Groups of control points

47

Figure 42: Algorithm to generate NURBS curves

48

The values of the weights are connected from one curve to the consecutive with a function

sin where 0.5 1 for internal control points and 1 2 for

external control points. Figure 49 explains the results of this choice: we obtain a shape

with a sinusoidal motion in z-direction.

We can obtain also different special NURBS curves by changing the weights as arcs of a

parabola, hyperbola, ellipse or circle according to the following table.

Figure 43 Conic sections as special NURBS [PAH07]

By changing the weights of the control points of a NURB curve of degree 2 we obtain a

different shape. However, the curve intersect the polygon in the inflection points.

Therefore, one can compute easily tangent vectors at the inflection points, which are

parallel to every curve.

Figure 44: (left) intersection points of different NURBS curve in the same control polygon (right) zoom of vectors tangent to this two curves

The curve is divided into 8 segments, where the separation points are choosen to be the

8 inflection points. Note that the tangents in the inflection points are parallel to the curve.

74 panels are created again for each floor and moreover found the pairs of corresponding

points and segments within the 74 points. Every segment has its own curvature and if we

49

look at the convex part of each segment we can find easily the associated point in the

curves above with the same tangent vector.

For finding the associated point from one floor to the next we create a vertical plane with

inclination based on the tangent vector and find where this plane intersect the plane of

the consecutive curve. Making intersections between planes, we project the point on this

line and find the closest point in the curve above. With this procedure, two points are

found in two different curves with the same tangent vector. Planar panels can be achieve

for every floor (Figure 46) and we are able to find different envelopes changing the

weights of control points. To make an example, a sinusoidal function algorithm has been

created to modify every NURBS curve (Figure 47).

Figure 45: Planes parallel to vectors tangent on every point selected for one curve and points projected on the following curve

Figure 46: Flat panels connecting two consecutive floors

Figure 47: Algorithm to find random weights

50

With a looping algorithm, as shown in Figure 48, it is possible to find different

skyscraper’s shells with planar panels changing the weight of the control points.

Figure 48: Looping algorithm to create random skyscraper with developable surfaces

51

Figure 49 Developable skyscrapers with flat panels created by random NURBS curves

In addition, a simple way to find a developable surface consist in scaling the curve:

parallel tangent vectors can be easily obtained from one curve to the consecutive curve

and thus all panels are completely planar. Examples are in Figure 50.

Figure 50: Developable skyscrapers with flat panels obtained by a scale alghoritm for NURBS curves

52

3.4.3 Panelization with diamonds and triangles

Another way of pannelization is to use planar quads and triangles for every floor. The

algorithm that has been created works for all the three types of curves. As an example we

present the algorithm for the convex curve. This consists in finding a mesh, which is

completely planar. We started with 74 arbitrary points in the first curve and we projected

every point to the curve above; the corresponding points have been connected with a line

and we pick the middle point as an additional point. Now we have 3 points allineated and

a plane can be approximated throught this three points. Obviously, this input does not

uniquely define the plane (Figure 51).

Figure 51: Initial steps to reach flat diamonds

To construct the desired plane, an additional point is add on the above curve, namely we

consider a point very close to the original point (the approximation is 10-13). This

construction yields planes that are quite tangent to this two curves (Figure 52).

Figure 52: Plane quite tangent to the curves

A family of lines is obtained from the intersection of consecutive planes. Afterwards we

pick the middle points of these lines to find flat diamonds (Figure 53).

53

Figure 53: The intersection of two consecutive planes is a line. Picking the middle point of every line and connecting these points with points previously found on curves we found flat diamonds

Reorganizing the points and connecting them, the algorithm achieves flat quad diamond

panels with triangles. In Figure 54, the whole process to achieve planar diamonds and

triangles is depicted.

Figure 54: Steps to achieve planarity with diamonds and triangles

54

Figure 55: Planarity analysis with Evolute Tools Pro

Repeating the same algorithm for all 74 floors, this works sufficiently well for the first

loops, but the approximation error for the construction of the planes grows from floor to

floor and at a certain point becomes unacceptable. (Figure 56)

Figure 56: The algorithm works well for the first floors, then the approximation becames unacceptable

One way to find good planar quad diamonds associated to triangles is to use the software

Evolute Tools, which achieves the task without an approximation error. Using triangles

and diamonds and the Evolute Tools optimization, all the meshes are strictly planar as it

is shown in Figure 57.

Figure 57: Mesh with planar diamonds and triangles optimized by Evolute Tools Pro

55

3.4.3.1 Which triangles can be converted in flat diamonds with cold bending

We have to think that every facette in a mesh is a glass panel that has its own property as

a real material. For architectural applications, glass is generally considered to be a

homogeneous and isotropic material. At temperatures below the deformation point

(which is 520° C for basic soda lime silicate glass), it is generally accepted that glass can

be assumed to be a linear elastic material. This behaviour abruptly endes when the failure

strength is reached: glass is brittle. Glass is usually employed as a shelter, or envelop for

the building. It guarantees solar lighting, whilst at the same time protection for external

adverse conditions. Due to recent technological advancements, its mechanical properties

can be exploited. Glass panels can be colored, multi-layered with films in between panels,

so as to five protection from UV rays, or as to change transmissivity with heat.

Table 2: Relevant material properties of basic soda lime silicate glass according to CEN EN 572-1 2004 [BIV07]

Using well-controlled residual stress, a toughened glass, which can be very useful for

structural applications, can be obtained. In this way, one can cause an overall

prestressing effect on the glass element, which increases its resistance against tensile

(bending) stresses: it virtually becomes stronger. Most prestressed glass is made by

means of a temperature treatment, but also chemical processes exist. Depending on the

level of prestress, the glass is called toughened (fully tempered) or heatstrengthened. The

strength of glass is a very complex characteristic which depends on external factors like

humidity (corrosion), ageing, surface flaws and scratches, loading history, loading speed,

and so on. The strength value corresponding to a fully tempered glass, according to CEN

EN 572-1 2004, is 120 / . [BIV07]

Curved glass can be applied in an interesting way in e.g. facades and canopies.

Traditionally, curved glass is manufactured from float glass that is heated above the

weakening point and formed in a heavy curving mould. However, this technique is time-

and energy consuming and consequently relatively expensive. For this reason, a more

affordable alternative has been developed. The technique is called a “cold bending

56

process” because it is used to bend glass plates on the building site at room temperature.

In this process, toughened float glass laminates are gradually bent on a curved frame.

Finally, the newly curved panel is mechanically fixed to the frame, which implies that the

glass is continuously subjected to bending stresses during its lifetime. In this

contribution, time dependent loading-deformation interaction during the bending

process as well as relaxation after the bending process are closely examined [BIV07].

Only FLOAT or tempered glass is acceptable for structural use. Nowadays, the fail-safe

method is employed, which consists in using a multilayer panel, so that in case of a layer

failure, the others will support the load.

Float Glass Uses common glass-making raw materials, tipically consisting of sand,

soda ash (sodium carbonate), dolomite, limestone, and salt cake (sodium sulfate) etc.

Other materials may be used as colorants, refining agents or do adjust the physical and

chemical properties of the glass.

1. The raw materials are mixed in a batch mixing process, then fed together with

suitable cullet (waste glass), in a controlled ratio. The mixture is wet, so that it

will not realise dust. The whole process is computer controlled.

2. The production phase is subdivided into three main parts:

a. The mixture is brought into a furnace where it is heated to approximately

1500 °C. Common flat glass furnaces are 9 m wide, 45 m long, and contain

more than 1200 tons of glass.

b. The homogenization process, where gas bubbles are eliminated.

c. The cooling process at low viscosity, where the temperature of the glass is

stabilized to approximately 1200 °C to ensure a homogeneous specific

gravity.

3. The molten glass is fed into a “tin bath”, a bath of molten tin (about 3-4 m wide,

50 m long, 4 cm deep) at about 1100°C, from a delivery canal and is poured into

the tin bath by a ceramic lip known as the spout lip. The amount of glass allowed

to pour onto the molten tin is controlled by a gate called Tweel. Tin is suitable

immiscible into the molten glass. Tin, however, oxidixes in a natural atmosphere

to form Tin dioxide (SnO2). Known in the production process as dross, the tin

dioxide adheres to the glass. To prevent oxidation, the tin bath is provided with a

positive pressure protective atmosphere consisting of a mixture of nitrogen and

hydrogen. The glass flows onto the tin surface forming a floating ribbon with

perfectly smooth surface on both sides and an even thickness. As the glass flows

57

along the tin bath, the temperature is gradually reduced from 1100 °C until the

sheet can be lifted from the tin onto rollers at approximately 600 °C. The glass

ribbon is pulled of the bath by rollers at a controlled speed. Variation in the flow

speed and roller speed enables glass sheets if varying thickness to be formed. Top

rollers positioned above the molten tin may be used to control both thickness and

the width of the glass ribbon.

4. A pyrolitic layer may be added, to add extra properties to the glass, such as low

emissivity, different transparency, etc.

5. Glass can now be heated again (heat tempering), to increase mechanical

properties.

6. Once off the bath, the glass sheet passes throught a so called étenderie, a cooling

tunnel, for approximately 100 m, where it is further cooled gradually from the

change in temperature. A slow air cooling comes right after, in order to eliminate

internal stress. On exting the “cold end” of the kiln, the glass is cut by machines.

Tempered glass With this method, the external layer is precompressed, so that

when bended, the external part won’t fissure. Moreover, when the glass breaks, only

small fragments with no sharp edges are generated.

For glass to be considered toughened, this compressive stress on the surface of the glass

should be a minimum of 69 MPa. For it to be considered safety glass, the surface

compressive stress should exceed 100 MPa. The greater the surface stress, the smaller

the glass particles will be when broken. It is this compressive stress that gives the

toughtened glass increased strength. This is because any surface flaws tend to be pressed

closed by the retained compressive forces, while the core layer remains relatively free of

the defects which could cause a crack to begin.

Any cutting or grinding must be done prior to tempering. Cutting, grinding, sharp

impacts and sometimes even scratches after tempering will cause the glass to fracture.

The glass solidified by dropping into water, know as “Prince Rupert’s Drops”, which will

shatter when their “tails” are broken, are extreme examples of the effects of internal

tension. The strain pattern resulting from tempering can be observed with polarized light

or by using a pair of polarizing sunglasses.

Toughtened glass must be cut to size or pressed to shape before toughening and cannot

be re-worked once toughening. Polishing the edges or drilling holes in the glass is carried

out before the toughening process starts. Because of the balanced stresses in the glass,

damnage to the glass will eventually result in the glass shattering into thumbnail-sized

pieces. The glass is most susceptible to breakage due to damnage to the edge of the glass

58

where the tensile stress is the greatest, but shattering can also occur in the event of a hard

impact in the middle of the glass pane or if the impact is concentrated. Using toughened

glass can pose a security risk in some situations because of the tendency of the glass to

shatter completely upon hard impact rather than leaving shards in the window frame.

The surface of tempered glass does exhibit surface waves caused by contact with

flattening rollers, if it has been formed using this process. This waviness is a significant

problem in manufacturing of thin film solar cells.

Laminated glass is a type of safety glass that hold together when shattered. In the

event of breaking it is held in place by an interlayer, typically of polyvinyl butyral (PBV),

between its two or more layers of glass. The interlayer keeps the layers of glass bonded

even wen broken, ant its high strength prevents the glass from breaking up into the large

sharp pieces. This produces a characteristic “spider web” cracking pattern when the

impact is not enough to completely pierce the glass. Laminated glass is normally used

when there is a possibility of human impact or where the glass could fall if shattered. The

PVB interlayer gives the glass a much higher sound insulation rating, due to the damping

effect, and also blocks 99% of incoming UV radiation.

A typical laminated glass is composed as as follows:

Glass

Transparent thermoplastic material like TPU, PVB or EVA

LED (led emitting diodes) on transparent conductive Polymer

Transparent thermoplastic material like TPU, PVB or EVA

Glass

59

3.4.3.1.1 Panels deviation

Panel’s deviation is the distance between diagonals of a mesh (Figure 58) and the

planarity of a mesh face is measured by the shortest distance between its diagonals.

Figure 58: Panels deviation [EPR]

This distance is measurable and quantifiable. The function for the total planarity, ,

of a mesh with some number of faces, , can be represented as the sum of this distance

for each face in the mesh. [EPR]

… (33)

If triangle meshes are replaced with diamonds, it is possible to analyse with simple initial

geometrical considerations which panels are planar and which are not planar:

6008 meshes are strictly planar,

2847 meshes have a deviation smaller than 0,001 (maximum deviation should be

1/250 0,004 that for now is reduced to 0,001 for safety reason)

2391meshes have a deviation higher than 0,001.

Figure 59: Grasshopper definition for evaluating planarity

60

In the following pictures we can see in blue planar panels, in yellow panels with cold

bending and in red panels with higher curvature.

Figure 60: Blue flat panels, yellow panels flat with cold bent, red panels with double curvature

61

3.4.3.1.2 Min. Cold Bending radius

The minimum cold bend radius has been calculated from some companies depending on

the structure of the glass and other properties. In this thesis we chose the numbers

provided by SEDAK, a company specialized in structural glass and cold bending.

The minimum radius depends on the thickness of the glass panel:

R 1500 ∗ s (34)

Where:

The thickness adopted for the panel to verify is composed by three layers:

20

8

0.38

With this thickness the minimum radius is R 42.57m .

In a next step we found the radius of the principal curvature in the point ; in

the main direction of every panel to analyse which of the panels respect this limits.

Figure 61: Screenshot of grasshopper definition for the analysis of principal curvatures

With this analysis the results are:

6008 meshes are strictly planar (results from the first analysis)

3486 meshes have a radius higher than 42.57

1752 meshes have a radius smaller than 42.57

62

Figure 62: Results of mesh analysis and cold bent

In the following pictures we can see in blue planar panels, in yellow panels with cold

bending and in red panels with higher curvature.

Figure 63: Analysis of cold bending

In Figure 63, it is shown how panels on the top of the skyscraper could be planar with

cold bending. The deviation for panels approximated planar with cold bending goes from

0.025 to 5.6 ; therefore the most high deviation is 5.6 .

The other panels have a deviation between 11.6 and 67.5 .

To verify the structure of the panel under the given thickness, one glass panel has been

created as a model using the software SAP2000 and it has been used to confirm that the

choosen thickness satisfies the wind loads.

63

The thickness adopted is evaluated with the Enhanced Effective Thickness method

[EET] that is extended to the case of laminated glass beams composed by three layers of

glass of arbitrary thickness.

Figure 64: Geometry of a laminated glass

The deflection-effective thickness is:

1

121 (35)

And stress-effective thickness is:

,1

2 ,

12

(36)

,1

2 ,

12

(37)

Where:

, thickness of the glass

, , , , , ,

∙ ∙

is the effective thickness of the laminated glass beam.

where:

is PVB thickness

64

Young’s module

Shear’s module

Poisson’s module

Bending stiffness of the glass where 1,2

the value of Ψ depends upon the geometry, boundary and loading

conditions of the beam, and it is reported in Table 3 for the most common

cases.

Table 3: Values of coefficient Ψ for laminated glass beams under different boundary and load condition

In this specific case, the thickness evaluated with this method is:

Method EET 

Enhaunced Effective Thickness 

L 1diamond  1400  mm 

L 2diamond  3740  mm 

hint  0.38  mm  d1  10.38  mm 

h1  20  mm  d2  4.38  mm 

h2  8  mm  dTOT  28.38  mm 

D1  666.66  mm3  d  14.76  mm 

D2  42.66  mm3  hs,1  10.54  mm 

DTOT  1829.33  mm3  hs,2  4.22  mm 

E  70000  MPa  A1  28000  mm2 

ν  0.22    A2  11200  mm2 

ψ  0.068  mm‐2  A*  8000  mm2 

Gint  0.44  MPa  Is  1244.9   

η  6.02831E‐08   

65

hw  20.42  mm 

h1,σ  20.63  mm 

h2,σ  32.62  mm 

Using this thinkness, two models have been created in the software SAP2000; one with

the deflection effective thickness, the other with the stress effective thickness. The model

is created using the high deviation for the cold bending (5,2 mm).

Figure 65: Model of a cold bent panel

The panel is analysed using the following loads:

Distributed wind on the shell (wind load is evaluate at the top of the

skyscraper with h = 300 m).

Tangential wind dir.x, dir.y.

Crowd loads :

‐ Distributed

‐ Distributed on a line at h = 1.20 m

‐ Concentrated load.

The action force use different coefficients for the analysis of ultimate limit state and

serviceability limit state. The results obtained from the analysis with the action force

used for ultimate limit state is compared with the project resistance. Meanwhile, with the

action force calculated for the serviceability limit state, we compare the displacement.

The resistance at the ultimate limit state using the CNR-DT 210/2012 is calculated as:

66

,∙ ∙ ∙ ∙ ∙

(38)

Where every coefficient is explained in the chapter 7.4 of the CNR-DT 210/2012.

In the specific case:

Results from the analysis with SAP2000 are:

The displacement is verified with the values shown in the Table 7.12 of the CNR-DT 210/2012 (Figure 65).

Figure 66: Table from CNR-DT 210/2012 for the analysis of displacement

kmod 0.88 k'ed 1

ked 1 kv 0.95

ksf 1 RM;v 0.9

fg;k 45 γM;v 1.35

RM 0.7 λgl 1

γM 2.55 k 0.145

fb;k 150 λgA 0.85

fg;d 100.92 MPa

Stress

Smax Top Smax Bot

h1,σ 26.53 40.26

h2,σ  10.78 16.36

h1,σ 7.47 9.26

h2,σ  4.27 5.74

h1,σ TOT 34.00 49.52 < fg;d 100.9

h2,σ TOT 15.05 22.10 < fg;d 100.9

Stress Dead + Wind Pressure + Tangential Wind Pressure + Crowd loads

Stress cold bending 

67

Conclusion One panel stressed with cold bending is verified at the top of the

skyscraper with the thickness supposed.

1/60*Lmin < 30 mm

wmax 23.33 mm w 56 mm

1/200*Lmin < 12 mm

wmax 7.00 mm wwind 0.689 mm

Displacement

Cold Bending

68

3.4.4 Corner modifications of base shapes

Corner modifications of the shapes, as we have already seen in some of the approaches

analysed before, are really relevant to check planarity. This aspect is important also for

the structural part and the analysis of wind loads.

Investigations have established that corner modifications such as slotted corners,

chamfered corners/corner cut, corner recession are in general effective in causing

significant reductions in both the along-wind and across-wind responses compared to

basic building plan shape. [Bor13]

Figure 67: Base shapes [Bor13]

The most common cross-sectional shape modifications are shown for a rectangular

shape, such as slotting, chamfering, rounding corners and corner cutting.

The modification of windward corners is very effective to reduce the drag and fluctuating

lift through changing the characteristics of the separated shear layers to promote their

reattachment and narrow the width of wake.

Experiments have proved that chamfers of the order of 10% of the building width

produce up to 40% reduction in the along-wind response and 30% reduction in the

across-wind response. [Bor13]

After these considerations, it is interesting to evaluate which form is geometrically more

efficient using EVOLUTE Tools.

For the analysis, the three different shapes, already mentioned in Figure 29, have been

evaluated.

Using as an example the same algorithm of panelization (diamonds and triangles) we can

analyse the planarity of three different skyscrapers with the curves illustrate in Figure 29

as base shapes. The results achieve with Evolute Tools show that the shape with round

corners is the best of the three (

Figure 68) . That is, as we already quoted, also the best solution for decrease wind loads.

69

Figure 68: Evaluation of best shape for planarity check

70

3.4.5 Principal curvature directions

To obtain an overview of the principal directions, one can use principal curvature lines.

A principal curvature line is a curve on a surface whose tangents are in principal

direction. Thus, through each general point of a surface there are two principal curvature

lines that intersect at a right angle and touch the principal directions.

Different alghoritms has been created depending on the behaviour of the initial shape.

In particular, the principal curvature line of the shape with round corner has been

analysed because that results the more efficient in many ways. Afterwards, two general

alghoritm have been created in order to find planar panels that follows principal

curvature lines according to convex shapes or shapes with inflection points.

3.4.5.1 Shape with smooth corners

As we mentioned before, there are two principal curvature lines for every point computed

in a surface. The shape with smooth corners presents two different principal directions

depending on which part of the skyscraper we want to analyse: the linear part or the

curve part.

Figure 69: Principal curvature lines in the straight part and in the smooth part

As it is shown in the Figure 69, principal curve directions are completely different. We can present two different methods:

Method A In order to have a unique direction for panels, the directions present in the surface

of the corners have been chosen as principal directions for the whole shape because

71

this is the area with more bending. The other parts can be easily optimized with cold

bending because have low curvature.

The starting mesh could be the following in Figure 70.

Figure 70: Starting mesh for the approximation of principal curvature directions of smooth corners

One way to find planar panels consist in find flat meshes not entirely weld as it is

shown in Figure 71. The verteces have a gap from one to the other.

Figure 71: Planar panels obtained with meshes not weld

72

Welting verteces, panels planarity get enhance, and conditions for cold bending

are satisfied. After the optimizations with Evolute Tools, the results are even more

satisfied.

Figure 72: Grasshopper definition of principal curvature directions of smooth corners

Figure 73: (left) shape non-optimized (right) shape optimized with EVOLUTE Tools

Figure 73 shows on the left the shape before optimization where panels deviation

is between 1.63 and 4.13 . After optimization steps, the Figure 73 shows

panel deviations between 0 and 3.22 .

73

Method B

Another solution aims to keep going both principal directions. This takes more

planarity but cause problems in the fabrication of the panels in the corners surface

where the principal directions change (Figure 74).

Figure 74: Panelization method B

After this optimization steps, the planarity scale invariant is between 0 and

2.42 , the best solution we could find so far but the intersection knot is not

easy to fabricate (zoom in Figure 75).

Figure 75: (left) zoom of panelization alghoritm with method B (left) zoom of the node with

valence 5

74

3.4.5.2 General B-Spline base with degree 3

The goal of the next analysis aim to find a general algorithm for a generic B-Spline that

follow principal curvature directions. We have to work differently if the shape is convex

or presents inflection points.

Defined 8 control points, different B-Spline curves have been created starting with

convex shapes up to shapes with inflection points. As we can notice with the study

completed so far, if the B-spline has a low convexity or has inflection points, principal

curvature lines converge in the same points and it is impossible to find a good

panelization.

Figure 76: Control points for a B-Spline of degree 3

The algorithm that shows principal directions in every point evaluated in a surface is

created with a script:

Dim p As Point3d = uv Dim U As Interval = srf.Domain(0) Dim V As Interval = srf.Domain(1) Dim samples As New List(Of Point3d) Do 'Abort if we've added more than 10,000 samples. If (samples.Count > 10000) Then Exit Do 'Get the curvature at the current point. Dim crv As SurfaceCurvature = srf.CurvatureAt(p.X, p.Y) If (crv Is Nothing) Then Exit Do 'Add the current point. samples.Add(srf.PointAt(p.X, p.Y)) 'Get the maximum principal direction. Dim dir As Vector3d If (crv.Kappa(0) > crv.Kappa(1)) Then If (max) Then dir = crv.Direction(0) Else

75

dir = crv.Direction(1) End If Else If (max) Then dir = crv.Direction(1) Else dir = crv.Direction(0) End If End If dir.Rotate(angle, crv.Normal) If (Not dir.IsValid) Then Exit Do If (Not dir.Unitize()) Then Exit Do 'Scale the direction vector to match our accuracy. dir *= accuracy 'Flip the direction 180 degrees if it seems to be going backwards Dim N As Integer = samples.Count If (N > 1) Then If (dir.IsParallelTo(samples(N - 1) - samples(N - 2), 0.5 * Math.PI) < 0) Then dir.Reverse() End If End If 'Move the last point in the list along the curvature direction. Dim pt As Point3d = samples(samples.Count - 1) + dir Dim s, t As Double If (Not srf.ClosestPoint(pt, s, t)) Then Exit Do 'Abort if we've wandered beyond the surface edge. If (Not U.IncludesParameter(s, True)) Then Exit Do If (Not V.IncludesParameter(t, True)) Then Exit Do 'Abort if the new point is basically the same as the old point. If (Math.Abs(p.X - s) < 1e-12) AndAlso (Math.Abs(p.Y - t) < 1e-Then Exit Do p.X = s p.Y = t Loop Return samples End Function.

We select a certain number of points on the surface in order to show principal directions.

Afterwards meshes that follow these directions have been created. As usual, the number

of subdivision is 74 and in the Figure 77-78-79, we can understand how principal

curvature directions change. All the panels obtained are strictly planar.

76

1. Convexity extreme : Circle

Figure 77: (left) Base shape, (right) Panels achieved from principal curvature directions

2. Convexity

Figure 78: (left) Base shape, (right) Panels achieved from principal curvature directions

3. Convexity

Figure 79: (left) Base shape, (right) Panels achieved from principal curvature directions

77

4. Inflection points

Figure 80: B-Spline with inflection points

From the studies carried out it is hard to obtain good planar panels from not-convex

shapes following the principal curvature lines.

Figure 81: Principal curvature lines through in two different points of the shape

Principal curvature lines converge in the inflection point proximity and it is possible to

simplify the shape obtained by this curve connecting principal curvature lines for every

floor and interpolating the curve near the inflection points (Figure 82).

Figure 82: Steps to achieve planar panels

78

We divide every curve in a given number of points and connecting this we obtain a shape

that follow one principal directon. Panels are quite planar. The planarity scale invariant

gives a deviation of 3.6mm from planarity (Figure 83).

Figure 83: Shape that follows only one principal direction in every floor

If we want to construct panels that follow both principal directions, we obtain panels that

are completely planar (left picture in Figure 84). Near the inflection points it is not

possible to find panels constructed by principal curvature lines as it is shown in the red

area of the right picture in Figure 84 because principal curvature lines change direction

close to the inflection points (Figure 85). In this part of the surface, we have to use non-

planar panels.

Figure 84: (left) panels completely planar that follow both principal directions (right) in the red part we

have to use non-planar panels

Figure 85: How principal direction lines change the side of the shape

79

3.4.6 Paneling Architectural Freeform Surfaces With the emergence of large-scale architectural freeform surfaces, the essential question

arises of how to proceed from a geometrically complex design towards a feasible and

affordable way of production.

The goal is to find a paneling solution for a given freeform design that achieves prescribed

quality requirements, while minimizing production cost and respecting application-

specific constraints. This method is shown in the publication [EKS10]. The quality of the

paneling is mainly determined by the geometric closeness to the input surface, the

positional and normal continuity between neighboring panels, and the fairness of

corresponding panel boundary curves. The cost mostly depends on the size and number

of panels, the complexity of the panel geometry, and the degree of reuse of molds that

need to be custom-built to fabricate the panels. A key objective is to solve instances of

the paneling problem on large-scale architectural freeform designs that often consist of

thousands of panels. Due to the high complexity and global coupling of optimization

objectives and constraints, manual layout of panels for these freeform surfaces is

infeasible, mandating the use of advanced computational tools.

The Evolute research group in [EKS10] developed an algorithm based on costs of panels

and molds in order to achieve the best solution for fabricating panels.

Let be the given input freeform surface describing the shape of the design. Our goal is

to find a collection , . . . , of panels , such that their union approximates .

The quality of the approximation strongly depends on the position and tangent

continuity across panel boundaries: Divergence quantifies the spatial gap between

adjacent panels, while the kink angle measures the jump in normal vectors across the

panel intersection curves.

Curved panels are commonly produced using a manufacturing mold . We call the

collection , . . . , with the mold depot.

To specify which mold is used to produce which panel(s), we define a panel-mold

assignment function:

∶ 1, → 1,

that assigns to each panel index the corresponding mold index. The arrangement of

panels in world coordinates is established by rigid transformations that align each

panel to the reference surface . Panels produced from the same mold are sub-patches

of the mold surface and need not be congruent.

80

Let be the fabrication cost of mold and , the cost of producing panel

using mold.

The total cost of panel production can then be written as:

, , , , (39)

Ideally, the same mold will be used for the fabrication of multiple panels to reduce cost.

The choice of panel types depends on the desired material and on the available

manufacturing technology. Materials may be transparent or opaque, and include glass,

glass-fibre reinforced concrete or gypsum, various types of metal, and wood.

Currently Evolute tools support five panel types:

planes,

cylinders,

paraboloids,

torus patches,

general cubic patches.

Planar panels are easiest to produce, but result in a faceted appearance when

approximating curved freeform surfaces, which may not satisfy the aesthetic criteria of

the design. A simple class of curved panels are cylinders, a special case of single-curved

(developable) panels. Naturally, such panels can lead to a smooth appearance only if the

given reference surface exhibits one low principal curvature. General freeform surfaces

often require double-curved panels to achieve the desired tolerances in fitting error,

divergence, and kink angles.

Approximate a given freeform surface by a collection of panels of preferred types such

that the total production cost is minimized, while the paneling respects pre-defined

thresholds on divergence and kink angle between adjacent panels, and reproduces the

initial curve network as well as possible. A closer look at this specification reveals that

any solution of the paneling problem has to address the following central aspects:

• Mold depot: determine the number and types of molds that should be fabricated.

• Assignments: find the optimal assignment function to establish which panel is

best produced by which mold.

• Registration: compute the optimal shape parameters for each mold and the

optimal alignment of each panel such that the reference surface is faithfully

81

approximated, thresholds on kink angles and divergence are met, and the panel

intersections curves respect the design intent.

Mold depot and assignment function determine the total cost of fabrication, while

registration affects the quality of the rationalization. Minimizing fabrication cost calls for

a maximum amount of mold reuse and the wider use of panels that are geometrically

simple and thus cheaper to manufacture.

The quality of a panelized surface is mainly determined by the gaps (“divergences”) and

the angles (“kink angles”) between neighboring panels [Evo12]. Depending on the panel

shapes, panels might not smoothly align, which will strongly influence the visual quality

of the surface and the reflections on the surface. Using more complex panel shapes

results in a smoother panelized surface. This usually stands in direct conflict to the

objective to minimize cost.

Figure 86: Kink angle and divergence between panels [Evo12]

Cost is often directly related to the geometric complexity of the panels. Minimizing cost

means preferring simple panel shapes like planes or cylinders as well as exploiting the

repetition of the same panel shapes at different parts of the surface. The cost of glass

panels is directly linked to the type of curvature. Therefore, we organize the case studies

into the categories planar, single curved, and double curved panels. We show how our

general framework ensures high quality results for various panel types and budgets.

Planar panels [Evo12]

The most cost-efficient way to realize a freeform surface with glass is to use planar

panels. For triangulated curve networks, paneling with planar glass is trivial,

since the three corner points of a triangle always lie in a plane. For 4 corner points

or more, the curve network does not automatically define a planar paneling.

Single curved panels [Evo12]

A nicer surface reflection and refraction pattern, lower kink angles, can be

achieved with single curved panels. The divergences achievable with single

82

curved panels are also depending on the geometric properties of the curve

network but there is more flexibility for the curve network layout due to the

higher flexibility of the panel shapes.

Single curved panels can be smoothly aligned along strips producing to a nice

“semi discrete” pattern. This is a good solution if general developable glass panels

can be manufactured. Unfortunately, general single curved glass is still very

expensive to produce.

A much more relevant glass paneling in practice is a paneling with cylinder

panels, because cylindrical glass can be produced cost-efficiently through hot or

cold bending, allowing for thermal tempering.

Double curved panels [Evo12]

The highest surface quality can be achieved using double curved panels. Creating

a custom shape for each panel can produce a perfectly smooth solution. This

perfect solution, however, is usually also the most expensive one. Between the

solution with only custom panels and the solution with only planar panels lies a

whole range of solutions defining a tradeoff between approximation quality and

cost.

83

3.4.6.1 Case studies In practice, two important cost factors are the panel types and the mold reuse.

Geometrically simpler panels are usually cheaper than more complex ones. We

implement the following 5 panel types: plane and cylinder, because they are the most

relevant for many materials including glass; parabolic and torus patches, because they

carry families of congruent profiles which can reduce cost in the production process; and

cubic patches, because they have a high geometric flexibility and can approximate even

complex shapes with high accuracy. The optimization framework is very flexible and

further panel types have already been added in the meanwhile, like cones, general

cylinders, and ruled surfaces. Usually, planar panels are the cheapest to produce,

followed by cylindrical panels and then the more costly double curved panel types. A

molding process often produces curved panels: first, a mold is created and then this mold

is used to produce the panel. Since the cost of creating a mold is usually much higher

than the cost of creating a panel using this mold, significant cost savings can be achieved

when the same mold is reused to create several panels.

Figure 87: Panels type used and costs

As already mentioned, there is thus a strong link between divergences and kink angles

and the cost of a paneling solution. The larger the allowed divergences and kink angles,

the more cost can be saved by using only simple panels and reusing the same mould

many times.

We take three case studies where the divergence is constrained to lie below six

millimeters and the kink angle is respectively one, five, nine degrees in order to compare

different results; the cold bent setting is always one millimeter.

Panel types Panel types Cost mold Cost panel

plane plane 0 1

cold bent cold bent 0 1

cylinder cylinder 2 2

paraboloid paraboloid 18 5

torus torus 24 5

cubic cubic 30 5

84

The case studies take in exam for paneling are the most relevant within constructability:

1. The first case study is composed by panels created from the direction of the

principal curvature lines that are shown in the corner part of the surface;

2. The second case study is the same of the previous case except for the optimization

analysis carried out with Evolute Tools;

3. The third shape is composed by diamond panels.

In the next sections, the results achieved using the setting panel fitting of Evolute Tools

Pro are presented in order to evaluate which is the more efficient shape of the three.

Afterwards the chosen shape will be analysed for the structural part of this thesis.

85

Case study 1

Figure 88: Planarity analysis case study 1

Figure 89: Results of panel types for case study 1

kick

divergence

panel type

panels 8240 160 8240 160 8240 160

molds ‐ ‐ ‐ ‐ ‐ ‐

cluster 1 cm

cluster 4 cm 50|8400

1° 5° 9°

6 mm 6 mm 6 mm

50|8400 50|8400

31|8400 31|8400 31|8400

86

Case study 2

Figure 90: Planarity analysis case study 2

Figure 91: Results of panel types for case study 2

This shape has been previously optimized for planarity check. With the planarity scale

invariant this resulted better respect the case study 1; after panel fitting the first one is

more suitable.

kick

divergence

 Panel type

panels 4896 1024 4896 1024 4896 1024

molds ‐ ‐ ‐ ‐ ‐ ‐

cluster 1 cm

cluster 4 cm

1° 5°

6 mm 6 mm

9°

6 mm

1380|5920 1380/5920 1379/5920

460|5920 460/5920 459/5920

87

Case study 3

Figure 92: Planarity analysis case study 3

Figure 93: Results of panel types for case study 3

This case is the most difficult to achieve. Analysis are done for only one case because of

the complexity of the computatonal time of the Evolute Tools software.

kick

divergence

 Panel type

panels 8516 549 1656

molds ‐ ‐

1°

6 mm

88

3.4.6.2 Conclusions: Skyscrapers that can be built

The case study 1 results the best of the three for fabrication. With a gap of 1 between

every panel, we can define 50 different clusters for 8400 panels.

Figure 94 Analysis with 1 cm of gap, colors define different clusters for panels

Figure 95: Analysis of 4 cm of gap and different clusters obtained

A gap of 1 is perfect to define the fabrication of the secondary structure of a façade.

Only for testing the surface, the analysis has been computed also with a gap of4 . In

this case clusters becames only 10; as we could expect, increasing the gap, number of

clusters decreases. Figure 95 shows clusters and their position in the surface.

89

4 SITE

The primary site considerations for tower design include the effect of wind, seismic and

geotechnical conditions. For the purpose of this thesis, the site that has been chosen is

located in the International Center of Vienna, Austria (Figure 96, Figure 97) and wind

action and geotechnical conditions have been evaluated for the assessment of the tower’s

structural behaviour.

Figure 96: Location of the building

Figure 97: Donau City (Vienna International Center)

The project is thought to be located in front of the DC Tower 1 designed by the architect

Dominique Perrault and built in the 2014 (Figure 98, Figure 99).

90

Figure 98: DC Tower 1

Figure 99: Site

The DC Tower 1 is actually the highest office building throughout Austria featuring a

height of over 220 m, 60 floors and 86,500 m2 gross floorspace. In the site where the

case study is supposed to be located, the DC Tower 2 is planned to reach a height of about

160 m, 46 floors and 58,000 m2 gross floor space.

91

5 ACTIONS

The next step consists in identifying the size of building elements and choosing an

appropriate structural system. In addition to the self-weight of the structure (based on

density of material used), several dead and live loadings must be considered in the

design. In this thesis, compliances with structural design standards have been checked

using wind loads, distributed gravity loads and crowd live loads.

5.1 WIND LOAD

In the following part, the procedure described in EN1991-1-4:2005 ”Actions on

Structures, Part 1-4: General actions - Wind actions” is presented.

The wind action is represented by a simplified set of pressures or forces whose effects are

equivalent to the extreme effects of the turbulent wind.

The wind velocity and the velocity pressure are composed of a mean and a fluctuating

component. The mean wind velocity vm should be determined from the basic wind

velocity which depends on the wind climate as described in chapter 4.2 of EN1991-1-

4:2005, and the height variation of the wind determined from the terrain roughness and

orography as described in 4.3 of EN1991-1-4:2005.

Basic wind velocity

The fundamental value of the basic wind velocity, , , is the characteristic 10 minutes

mean wind velocity, irrespective of wind direction and time of year, at 10 m above ground

level in open country terrain with low vegetation such as grass and isolated obstacles with

separations of at least 20 obstacle heights.

The basic wind velocity shall be calculated from:

∙ ∙ , (35)

92

Where:

is the basic wind velocity, defined as a function of wind direction and time of year

at 10 m above ground of terrain category II.

, is the fundamental value of the basic wind velocity; this value may be found in the

National Annex. The recommended value is 1,0.

is the directional factor; this value may be found in the National Annex. The

recommended value is 1,0.

The 10 minutes mean wind velocity having the probability for an annual excedence is

determined by multiplying the basic wind velocity with the probability factor:

1 ∙ ln ln 11 ∙ ln ln 0.98

(40)

Where:

is the shape parameter depending on the coefficient of variation of the extreme-

value distribution. The reccomended values is 0.2.

is the exponent. The recommended value is 0.5.

Main wind velocity

The mean wind velocity at a height above the terrain depends on the terrain

roughness and orography and on the basic wind velocity, , and should be determined

using:

∙ ∙ (41)

Where:

is the roughness factor.

is the orography factor, taken as 1,0.

is calculated as:

∙ ln for (42)

for (43)

93

where:

is the roughness length that depends on the terrain category of the site takes in

exam. In the specific case is 0,3 for category III illustrated in the Table 4.1 of

EN1991-1-4:2005

is to be taken as 200 meters but in the specific case is taken as 300 meters, the

height of the building.

is the terrain factor depending on the roughness length calculated using:

0.19 ∙0.05

. (44)

In the specific case is calculated with (38).

Turbulence intensity

The turbulence intensity at height is defined as the standard deviation of the

turbulence divided by the mean wind velocity.

In the specific case it is calculated as:

∙ ln (45)

Where:

is the turbulence factor taken as 1.0

is the orography factor that is take as 1.0for this specific site.

Peack velocity pressure

The peak velocity pressure at height , should be determined:

1 7 ∙ ∙12∙ ∙ (46)

Where:

is the air density, which depends on the altitude, temperature and barometric

pressure to be expected in the region during wind storms. The value 1.251 /

is taken from the ÖNORM – Austrian standard regulations for constructions.

94

Wind force

The wind force acting on a structure or a structural component may be determined

directly by:

∙ ∙ ∙ (47)

Where:

is the structural factor that should be calculated with a detailed procedure

described in chapter 6.3.1 of EN1991-1-4:2005. In this specific case, the factor is

equal to 1 in order to simplify the procedure and evaluate a feasibility study of the

building take in exam.

is the force coefficient for the structure or structural element. The external and

the internal pressure coefficients are respectively equal to 1 and 0.65 as the

ÖNORM describes for this specific site in Vienna.

is the peak velocity pressure at reference height .

is the reference area of the structure or structural element.

The Table 4 shows the wind force ( / ) applied in the x-direction for every floor of the

skyscraper as the external pressure. The internal pressure is the same force multiply for

0.65.

95

Table 4: The left-hand number represent the floor number, the right-hand number represent the wind

force applied to a specific floor ( / .

One method for applying external and internal pressure acting on a twist shape is

identified with normal vectors of the shell’s faces. When the x-vector of wind load is

concordant with the external face vector, the force is to be considered for the internal

pressure, otherwise the force has been considered for the external pressure (Figure 100).

0. 0 20. 1.356787 40. 1.598262 60. 1.749732

1. 0.643471 21. 1.373025 41. 1.607278 61. 1.756059

2. 0.751437 22. 1.388619 42. 1.616103 62. 1.762295

3. 0.83184 23. 1.403619 43. 1.624745 63. 1.768442

4. 0.896378 24. 1.418071 44. 1.633213 64. 1.774503

5. 0.950518 25. 1.432017 45. 1.641513 65. 1.780481

6. 0.997282 26. 1.445491 46. 1.649652 66. 1.786378

7. 1.038522 27. 1.458527 47. 1.657638 67. 1.792195

8. 1.075463 28. 1.471153 48. 1.665474 68. 1.797936

9. 1.108955 29. 1.483396 49. 1.673169 69. 1.803602

10. 1.139618 30. 1.495279 50. 1.680726 70. 1.809195

11. 1.167912 31. 1.506823 51. 1.68815 71. 1.814718

12. 1.194196 32. 1.518049 52. 1.695448 72. 1.820172

13. 1.218748 33. 1.528974 53. 1.702622 73. 1.825559

14. 1.241794 34. 1.539615 54. 1.709678 74. 1.83088

15. 1.263516 35. 1.549987 55. 1.716619

16. 1.284066 36. 1.560104 56. 1.723449

17. 1.303569 37. 1.569978 57. 1.730172

18. 1.322132 38. 1.579622 58. 1.736791

19. 1.339845 39. 1.589047 59. 1.74331

96

Figure 100: External and internal pressure of the wind

5.2 FLOOR SYSTEM

Gravity loads are generally considered to be uniformly distributed over an occupied floor.

This loads vary based on building use and include dead load (self-weight) and live load

(in this specific case a crowd loading has been used).

When considering dead load, all components of the primary structure must be included,

typically floor slabs, floor framing beams and girders, and columns.

5.3 EXTERIOR WALLS

Exterior walls for tower structures produce specific loads that must be considered in the

structure design. These loads are small because of the glass panels. In this specific case

the exterior wall is supported on perimetral girders, in some other cases it is attached

directly to columns. For initial calculations, exterior wall loads may be considered to be

evenly distributed along perimeter spandrels, considering the exterior wall weight and

the floor-to-floor heights.

Exterior wall loads are often considered as distributed load over the face area of the

structure.

97

6 STRUCTURAL SYSTEMS

Considerations about geometrical design and economy of the shell are concluded;

afterwards there are several factors to be accounted for choose a structural system for

tall buildings. Safety, occupant comfort, and economy are the most important. But also

materials, available construction time and contractor expertise must also be considered.

The following structural systems are mentioned in [Sark11].

Concept of the structural design is conditioned by:

Height/Number of stories

Form of stability system

Key structural dimension as column, wall position, slab, beam depth

Form of floor structure

Based on this elements, eight different structural systems can be considered (Figure 101).

Figure 101: Structural systems

98

6.1 FRAME SYSTEM

This system is applicable for buildings of up to approximately 75 m in height. The

structural system is relatively simple, in which beams and columns are rigidly connected

to form moment-resisting frames, resisting lateral and gravity loads. Each frame resists

a proportion of the lateral load, determined on the basis of its relative stiffness compared

to the sum total stiffness of the frames. For increasing structure height, there is an

associated direct increase in the size of the frame elements to satisfy lateral drift and

deflection limits.

Figure 102: Frame system

99

6.2 SHEAR WALL SYSTEM

Shear walls within a tower are most commonly centrally located around service areas,

including elevators, mechanical spaces, restrooms. In some cases, shear walls are located

eccentrically in the floor plan and must resist significant torsion due to eccentrically

applied wind loads or seismic loads arising from the eccentric relationship between

center of mass and lateral stiffness.

General limit of number stories in a shear wall system is 35 for a height of 110 meters.

Figure 103: Shear wall system

100

6.3 SHEAR WALL AND FRAME SYSTEM

This is the combination of the first two systems. The combined lateral stiffness of rigid

frames and core shear walls allows designers to reach heights of around 160 m.

Figure 104: Shear wall and frame system

An advantage of the combined system, besides the ability to build taller, is the reduction

of lateral drift at the tip of the building. At the top of the building the shear-core walls

are restrained by the frames while, lower down the building, the frames are restrained

by the shear walls (Figure 105).

Figure 105: Deflection profile

101

6.4 FRAMED TUBE SYSTEM

Conceptually, this system is based on a closed hollow tube, with the large distance

between the tension and compression elements in both directions serving to resist lateral

forces. Internal elements positioned as necessary to adequately support the gravity loads

from the floor framing.

Typically, columns placed at relatively close centres of 2-4 m, connected by beams to

create rigid frames around the perimeter.

The structural principle is based on the flange of the tube frame being perpendicular to

lateral wind forces, tied at each end by the webs of the framed tube which are oriented

parallel to the wind. This system is applicable for buildings of up to approximately 170 m

in height.

Figure 106: Framed tube system

102

6.5 TUBE IN TUBE SYSTEM

Combines the stiffness of the perimeter-framed tube (described in 6.4) with internal

reinforced concrete core walls. Structurally, this arrangement will act in a similar manner

to the shear-wall and-frame system but will be considerably more robust due to the

strong lateral strength of the outer tube. This system is suggested for height up to 200

m.

Figure 107: Tube in tube system

103

6.6 BUNDLED – TUBE SYSTEM

This type is suited for super tall buildings (up to 225 m). Performs in same manner as

the tube system, the number of ‘flange’ frames is increased by introducing inner frame

elements splitting the plan area into a series of modules.

The principal benefit of this system lies in the robust linkage of relatively smaller flange

sections by numerous web elements. This system also typically allows for greater space

between the outer columns.

Figure 108: Bundled-Tube system

104

6.7 BRACED – TUBE SYSTEM

Similarly to the bundled tube, this system uses diagonal bracing added to the perimeter

tube frames to increase the tube system’s lateral stiffness and accommodate increased

building height (with this system height is up to 300 m).

The external tube elements act as bracing frames by transferring lateral loads to the

foundations along the diagonal tension and compression lines, and also redistribute the

gravity loads from the highly stressed to less stressed columns, ensuring a high degree of

structural redundancy and many load paths.

Figure 109: Braced-Tube system

105

6.8 OUTRIGGER – BRACED SYSTEM

Perimeter columns form a fundamental part of the structural system. The introduction

of horizontal outrigger elements (often trusses) of one or two floors deep, connecting the

core with the outer columns at regular intervals up the building. At the same level as the

outriggers will be ‘belt trusses’ – up to two storeys deep, connecting the perimeter

columns to the outriggers and also serving to distribute vertical loads. This system is

applicable for buildings of up to approximately 350 m in height.

Figure 110: Outrigger-braced system

106

6.9 STRUCTURAL SYSTEM CHOICE

The case study identified as the best choice for the geometrical envelope is shown in

Chapter 3.4.6.2 .

The structural system chosen is a composite tube in tube system with belt outrigger that

is ideal for 75 stories and 300 meters of height. Columns spacing approximately matches

the floor-to-floor height (4.5 meters). The tubular frames are engineered to

approximately equalize the bending stiffness of the columns and the beams.

We can evaluate two different structural systems in order to select the most efficient

(Figure 111).

The choice is made taking into consideration the twisting shape of the shell. Both the

structural system options feature a central core designed to be 1/3 where is the length

of one of the sides, consisting of a composite steel-concrete scheme counting 20 steel

columns so that resulting on-center spacing of 4.5 m could be achieved.

Structural system A consists of:

- Eccentric steel columns that follow the same twist of the shell;

- Central core in concrete;

- Floor slabs;

- Belt outrigger situated at 1/3 , 2/3 and ; where is the height of the

skyscraper.

Structural system B consists of:

- Straight steel columns situated in the internal part of the skyscraper;

- Central core in concrete;

- Steel trusses connecting the core to the columns cantilevering out at every

floor;

- Braces;

- Corrugated steel-lightweight concrete composite slab;

- Belt outrigger located at 1/3 , 2/3 and ; where is the height of the

skyscraper.

107

Figure 111: Structural systems

108

Materials

Both of the options are analysed using the same materials:

S355 is a structural steel. Typically application of this steel are:

Power plants

Mining and earth-moving equipment

Load-handling equipment

Wind tower components

Structural steels are used in many ways and their application can be diverse. Thus

are particularly useful because they offer the unique combination of good welding

properties with guaranteed strengths. Structural steel is an extremely adaptable

product and helps maximize strength or structure while minimizing its weight.

The construction industry is the bigger consumer of structural steel, where it is used

for all manner of purposes and employed at diverse scales.

This steel has the following properties: :21000 / , : 8076 / ,

:78.5 / , α : 2.1 51/ ° :36 / .

C80/95 is an high strength concrete. Concrete has relatively high compressive

strength, but significantly lower tensile strength, and as such is usually reinforced

with materials that are strong in tension (often steel). The elasticity of concrete is

relatively constant at low stress levels but starts decreasing at higher stress levels as

matrix cracking develops. Concrete has a very low coefficient of thermal expansion,

and as it matures concrete shrinks. All concrete structures will crack to some extent,

due to shrinkage and tension. Concrete which is subjected to long-duration forces is

prone to creep.

This concrete has the following properties: :4200 / , : 1750 / ,

:25 / , α : 1.0 51/ ° :5.33 / .

109

6.9.1 Development of option A

Preliminar analysis have been made for the option A to identify the structural elements

and verify with adequate dimentions, the displacement at the top of the skyscraper:

1.2 with 300 (48)

Option A has been excluded because it resulted in having:

- Core of 10 10 and thickness of 2.5 ;

- Twenty eccentrical circular columns of 1.267 0.30 ;

- Braces composed by circular profiles of 0.711 0.30 ;

- Concrete slab with a thickness of 0.35 ;

And with such a layout the displacement evaluated at the top of the skyscraper does not

satisfy the condition in the equation (48). For the analysis, wind distributed loads that

increase for every floor and gravity loads have been identified, they have been applied to

the model and amplified respectively by 1.5 and 1.3 for safety reason. In the calculation,

crowd loads have been omitted.

In addition, the use of eccentrical columns create problems in the distribution of loads

which have a vector component that is not completely directed as the vertical axis.

Figure 112 shows how structural elements behave under wind and gravity loads;

compression in the core and in the columns reduces as the floor number increases; slabs

transfer their weight to the core and the columns stiffening the whole structure.

Figure 112: Utilization importance of structural elements

110

The utilization of beams and slabs is shown in the Figure 113. The elements take into

consideration are able to resist under wind loads and self weight but the displacement

results more than the maximum admitted.

Figure 113: Utilization of beams and slabs for the structural option A

111

6.9.2 Development of option B

Figure 114: Left: Top view of the skyscraper with the structural system and the external shell. Right: System of the structure of one floor

If we analyse the twisting shell from the top view, we can see that there is a gap of four

meters between the external and the internal circumference. Every column, in order to

be straight, is located on the internal circle with a radius of 14.5 meters from the center

of the skyscraper. We choose twenty columns because the distance between two

consecutive columns should be the height of one floor (four meters in this specific case).

The core should be 1/3 of the length of one side, in this specific building we choose a

circular core of 11 meters compared with the length of 30 meters for one side.

6.9.2.1 2D Model

A 2D model has been analysed to identify optimized cross sections for the structural

system B based on a limit displacement of 1.2 meters. This kind of optimization has been

made using the plug-in Karamba3D for Grasshopper and afterwards the model is

imported in the software SAP2000 to verify the results obtained with Karamba3D. The

model is exposed to wind loads, gravity, crowd and dead load from flooring structure

calculated according to the Eurocodes.

112

Figure 115: Left: 2D model created with Karamba3D. Right: zoom of the model

Figure 116: Detail of one floor in the 2D model

This model includes steel trusses for every floor, columns, one core and no cantilevers.

The model has the following properties:

Moment-resisting joints for columns and trusses chords

Hinged joints for braces

Meshed core

Fixed base supports

Figure 117: Load acting on floor

Karamba3D is a plug-in of grasshopper where every element is a geometrical element

that is converted into a beam or a shell made of real material (steel or concrete in this

case) and real cross section. In Figure 118 it is shown how a line can be modified in a real

113

beam with a tubular steel element of 114.3x3.6 mm. This process is made for all the

elements of the model.

Figure 118: Grasshopper example of definition for the element that is indicated as chord inf.

Afterwards the model is assembled with elements, materials, cross sections, loads and

supports. Loads are described in Chapter 5 while fixed supports are chosen for the model.

For the first analysis fixed cross sections are chosen for every element in the whole

model. Afterwards, elements are optimized for every floor in order to choose the best

combination for the model. The limitation parameter is the maximum displacement that

should be 1.2 meters. After an iterative process we chose different cross sections for every

floor.

The following list represent the cross sections that where optimized with Karamba. The

left-hand number represent the floor being considered, meanwhile the right-hand code

defines the structural profile used for a specific element in that specific floor. Finally the

model is imported in SAP2000 for a confirmation of the results obtained with Karamba.

114

Bottom chord

Top Chord

115

Diagonal beam

Vertical beam

116

Columns

Frame with X-braces

Core

10 ; 1.2 ; 300 .

With this sections:

The maximum displacement is . < 1.2 .

Natural periods:

1. . 2. 1.58 3. 0.63

117

The Figure 119 shows the natural eigenmode for the 2D model.

Figure 119: Natural vibration: Modal 1, modal 2, modal 3

The utilization factor is:

Figure 120: 2D model's utilization factor of option B

As it is shown in Figure 120, the 2D model is able to resist against wind loads with the optimized cross section previously mentioned.

118

6.9.2.2 3D Model

Cross sections obtained from the 2D model are used for the first analysis of the 3D model.

The process for the analysis is the same described for the 2D model. The differences are

the cantilevers, braces and floor mullions that are included in this model. Every floor is

composed by the elements illustrated in Figure 121.

Figure 121: Elements for the 3D model

An iterative process is used to optimize again the following cross sections with

Karamba3D. Every cross section is associated with the respective number on the left that

represent the floor being considered. Circular hollow sections have been used to define

size and a resistance of every element in the case study. For the trusses members an

adequate equivalent profile had to be chosen to simplify construction, so that the intertial

properties of the original section could have be keep costant.

Figure 122: Method to find an equivalent adequate profil

119

The following tables show cross sections optimized using Karamba.

Columns

*For the first twenty floor columns are composed with concrete and steel.

Top chord

120

Bottom chord

Diagonal beam

121

Vertical beam

Frame with X-Braces (Floor)

Frame with X-Braces (Structure)

122

Core

5.5 ; 1.5 ; 300 .

The utilization results obtained from Karamba 3D are shown in the Figure 123.

Figure 123: Structural system B with analysis result

123

External shell and internal system

Skyscraper floor type

124

Trusses

Façade spider

Details

125

The model of the structure has been exported in SAP2000. We obtained the following

results about displacement and modal vibrations:

The maximum displacement is . < 1.2 .

Modal:

Table 5: Modal analysis results with SAP2000

Figure 124: Foundamental eigenmodes with SAP2000

This results are acceptable compared with results of buildings of the same height and dimentions.

OutputCase StepType StepNum Period UX UY UZ SumUX SumUY SumUZ

MODAL Mode 1 7.121161 0.05715 0.52561 1.299E‐16 0.05715 0.52561 1.299E‐16

MODAL Mode 2 7.121099 0.5256 0.05715 2.219E‐15 0.58275 0.58276 2.349E‐15

MODAL Mode 3 1.617402 0.03085 0.17836 7.791E‐15 0.61361 0.76111 1.014E‐14

MODAL Mode 4 1.617375 0.17835 0.03085 5.46E‐13 0.79196 0.79197 5.562E‐13

MODAL Mode 5 0.737388 1.444E‐12 2.793E‐10 1.533E‐12 0.79196 0.79197 2.09E‐12

MODAL Mode 6 0.65839 0.04107 0.03141 4.622E‐15 0.83303 0.82338 2.094E‐12

MODAL Mode 7 0.658386 0.03142 0.04107 1.789E‐12 0.86445 0.86445 3.884E‐12

MODAL Mode 8 0.351887 0.0289 0.00673 1.394E‐12 0.89335 0.87118 5.277E‐12

MODAL Mode 9 0.351885 0.00673 0.0289 9.207E‐12 0.90008 0.90009 1.448E‐11

MODAL Mode 10 0.307497 1.14E‐11 1.016E‐10 4.687E‐10 0.90008 0.90009 4.831E‐10

MODAL Mode 11 0.299543 1.053E‐13 7.387E‐13 0.75763 0.90008 0.90009 0.75763

MODAL Mode 12 0.226935 0.02373 0.00091 5.228E‐12 0.92381 0.901 0.75763

126

6.9.2.3 The Project

The Vienna International Centre (VIC) is the campus and building complex hosting

the United Nations Office at Vienna. The twisting skyscraper is thought to be the tallest

building of the area and it is located on the DC towers site as it is mentioned in Chapter

4. The location of the building is important to collect real data for wind actions in order

to evaluate real loads for the analysis take in exam.

Figure 125: Vienna International Centre

Figure 126: Skyscraper floors

127

Figure 127: Skyscraper located in the site of construction

128

Figure 128 Maquettes of the Skyscraper Envelope

129

7 CONCLUSIONS

This work presents a parametric geometry definition, with two different approaches:

geometrical and structural. Standardization of elements in tall buildings is important, in

order to reduce costs and fabricability. This thesis aims at the optimization of the internal

structure and the external envelope.

The final shape of the external skyscraper shell presents 8400 flat panels clustered in 50

different molds. Iterative analysis of optimization are performed also for the cross

sections of the structural system. This result is achieved through an in-depth study in

mathematics and engineering. It was possible to define the whole workflow by scripts

and advanced softwares.

Differential geometry plays an important role in the design of free form shapes.

Knowledges in this field allow a rational and critical awareness of the advantages

applicable to the structural engineering. With a mathematical method, it is easy to reach

optimization settings that could be difficult or impossible to achieve with traditional

tools.

130

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