A
BSTRACT
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.
R
IASSUNTO
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.
A
CKNOWLEDGEMENTS
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
S
UMMARY
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 ... 252.4.1.3 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
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
L
IST OF
F
IGURES
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
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
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
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
8
L
IST OF
T
ABLES
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 A
RCHITECTURAL
G
EOMETRY
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 S
URFACED
ISCRETIZATIONThere 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 High
Quadrangular Good Quite easy 4 No Low Low
Hexagonal 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.
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.
15
2 D
IFFERENTIAL
G
EOMETRY
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 P
ARAMETRIC SEARCHParametric 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
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 D
EFINITIONSA 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 byC 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, t 1,0, if totherwiset and t t (6)
N, t t t
t t N,
t t
t t N , 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 1 knots 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 C
URVESThe 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
21
2.4 S
URFACESSurfaces 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 ∓ sin sin cos cos sin 0 sin cos (20)
Hyperbolic paraboloid (a doubly ruled surface):
Figure 13: Hyperbolic paraboloid
2
0
24 Plücker's conoid:
Figure 14: Plücker's conoid
cos sin 2 cos sin 0 0 2 cos sin cos sin 0 (22) Möbius strip:
Figure 15: Möbius strip
cos cos 1 2 cos sin cos 1 2 sin sin 1 2 cos sin 0 cos 1 2 cos cos 1 2 sin sin 1 2 (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.
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:
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
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]
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 C
ASE 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 D
ESIGN OF HIGH-
RISE BUILDINGSTechnology 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 S
OFTWARESComplex 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 C
HECK DESIGNDevelopment 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:
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 P
ANELIZATIONIn 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).
42
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.
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.
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.
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 theweight 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).
47
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
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.
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.
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.
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.
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
