folding + bending
TRIANGLES
B O O T C A M P 2 0 1 3 / GR OU P 7 ra d h i k a a mi n
re b e c c a b ra d l e y
j o s e g a rc i a
a mro k a bba r a
CONTENTS
02 INTRODUCTION rapid description of aim of content
COMPONENTS OF THE SYSTEM 04 Component Geometry
primitive geometry - variation
06 Local Component
component - geometric structure, properties
08 Regional Component
assembly of the basic component
EXPERIMENTATION 10 Investigations a. Compression
strength under external load
b. Materiality
deflection comparison between different materials
c. Flexibility
range of deflection under prestressing
THE SYSTEM 14 Regional Interaction/ Structural Behaviour
prestressing in the structure
16 Global System
organisation of the global structure
18 Fabrication
construction from local to global
22 CONCLUSION
Analysing the global geometry for its strength and weaknesses and a way forward
FORWARD before we begin... This was an exploration to develop a material system T through a series of physical and some digital tests in an effort to achieve a hierarchal form composed of local and regional components. Through collective actions produced by the local and regional components a global form was defined. The definition of the final explored form was through building a system designed to learn from ground level, in which the macro-formation was derived from the micro, or local geometry.
DEVELOPING THE SYSTEM The design is based on three hierarchal modules: The local, regional, and global. The aim of the design was to create a system with each unit informing the other. The local is developed upon a basic geometry; the triangle. Several local components join to create the regional component, which after several tests and explorations, multiplied into a global system.
THE DESIGN The design acts as a material system that responded according to the applied post-tension wires. The regional component was used as a flexibility form that could be manipulated when tension was applied. Each local geometry, through its flexibility, informed the regionals’ flexibility, which in turn, created a material system that was able to produce a double curvature.
DEVELOPMENTAL TIMELINE MANY GEOMETRY
DEFINING THE GEOMETRY Defining the geometry as the triangle: the minimum number of points needed for a surface.
1 GEOMETRY
PAPER
DEFINING THE LOCAL 4 GEOMETRY
DEFINING THE MATERIAL
2 GEOMETRY
3 GEOMETRY
Two folded triangles create a structural yet flexible geometry.
PAPER
POLYPROPYLENE
1 LOCAL MANY LOCAL A series of re-testing the material in its regional module
POLYPROPYLENE
PAPER
6 LOCAL
RE-DEFINING THE MATERIAL
DEFINING THE REGIONAL
5 LOCAL
2 LOCAL 3 LOCAL
4 LOCAL
ANALYSE FOCAL POINT
BRANCHING CENTRIFUGAL
2
DEFINING THE GLOBAL
LINEAR
PREDEFINED FORM
EQUALIZED PARALLEL
Introduction
Bootcamp 2013
Introduction
3
COMPONENT GEOMETRY
DIGITAL EXPLORATION OF THE GEOMETRY
The triangle gives the possibility to generate many geometric forms and this begins to act as the building block for our system. Using a triangular grid system, there began an exploration to find the shape of our module. Using the equilateral triangle as a structural component, a digital and physical exploration in geometry was instigated with the aim to create a structural and efficient local component. The geometry was explored from the basic to complex through three main approaches by using one, two, or three triangles. (fig. 6.1)
These two approaches were multiplied digitally to explore the different connections and systems that could be derived from these two approaches. (fig 6.2) Complexity, however, led to the loss of simplicity in structural stability that was achieved through simply connecting two triangles together as connections became limited and what was successful with the basic folding of a triangle were also lost.
EXPLORATIONS OF GEOMETRY
02
03
Physical
D i g i ta l
01
fig. 6.1
fig. 6.2 fig. 6.2 4
Components of the System
Bootcamp 2013
Components of the System
5
LOCAL COMPONENT A. GEOMETRY - ANGLE With the chosen geometry, two adjacent equilateral triangles were used to form a ‘rhombus’, creating one folding edge and four free edges. With various experimentations, the equilateral triangle was best exploited when its dimensions was divided into thirds (See Fig 8.1); At the folding edge, smaller equilateral triangles were created at the corners that bend inwards. These triangular flaps measure 1/3 the folding edge length, leaving 1/3 of the bigger triangle length for the body core and 2/3 of the length for the trailing edge. Bending was achieved by scoring or hash-doting the material at the smaller triangular corners (fig 8.2). fig. 8.1
B. PROCESS OF FOLDING
fig. 8.2
One way we have found to take the 2 dimensional triangle and create a structural three-dimensional component was to begin to fold two triangles together so they overlap. This increases the moments of connections and sturdiness as it grows into 3-dimensions. C. PROPERTIES FROM FOLDING Using different materials in building the local component, paper was too foregiving and cardboard was too rigid a material to achieve the characteristics sought for in the global surface. The choice of POLYPROPYLENE relayed in three aspects: 1. Hand crafted material. 2. Bending is possible just by scoring or hash-doting the material. 3. Given the component structure, by folding and overlapping the material in specific areas provides the rigidity needed to withstand load-bearing yet still flexible in the material itself where needed. Having this three traits we can generate a component to achieve our goals in an efficient way: Flexible without loosing Stiffness (fig 8.3).
JOINTS Of all the methods tested for joining the module, eyelets worked the best for polypropelene
fig. 8.3 6
Components of the System
Bootcamp 2013
Components of the System
7
REGIONAL ASSEMBLY After compiling and forming the local module, we started experimenting in joining the individual components together to form a Regional module. Joining the modules at corners and edges allowed for multiple orientations and structure formations. However, connecting at the edges provided a weak joint and increased the chances of failure. One configuration that allowed for a more organic growth keeping the local components intact while maintaining regional rigidity and still allowing for material flexibility was achieved by joining the local components from the side triangles. This contributed for further overlapping of material, thus achieving more stiffness for load-bearing and lateral compressions. Joining the modules in the same orientation allowed for a natural ring form, or ‘crown’ shape, as shown in fig. 10.1.
We started stacking the regional module vertically, by joining the trailing edge flaps together which created columns.These columns showed interesting characteristics. 1. Very strong in sustaining vertical compressions 2. The flexibility of the lattice structure can be deflected by applying tension on the side flaps, making the component even stronger. This allows us to control the direction of growth and achieve the curvature needed. 3. The column structure is stable with a maximum of 7 rings. Nevertheless, more testing is required to confirm this, covered in later sections of this document.
EXPLORATIONS OF REGIONAL
1
2
3
15 local modules in 700micron Polypropelene
6 local modules in 100gsm Paper
5 local modules in 100gsm Paper
4
5
6
5 local modules in 700micron Polypropelene
2 rings (10 local modules) in 700micron Polypropelene
3 rings (15 local modules) in 700micron Polypropelene
7 fig. 10.1 fig. 10.2
We tested the column structure and growth patterns in 2 orientations: vertically and horizontally. While both maintain the same growth pattern, it was evident through various testing that the column structure is more robust and controllable when the global surface was realized with horizontal orientation. When joining lattice structures of different length and applying tension in various connection points in the column structure, we were able to control the shape and define the curvature. However, looking at the heirarchical component logic, it was evident that the ring becomes the new local component making the column structure the new regional component, out of which the global surface emerges.
4 columns composed of 5 rings each (total 100 local modules) in 700micron Polypropelene
8
fig. 10.3 8
8 9
5 rings (25 local modules) in 100gsm Paper
5 rings (25 local modules) in 700micron Polypropelene
10 4 columns in 700micron Polypropelene
fig 11.1 Explorations of the Regional component
fig. 10.4 Components of the System
8
Bootcamp 2013
Components of the System
9
EXPERIMENTATION a. COMPRESSION TESTING UNIFORM EXTERNAL LOAD
Aim: To test the compressive strength under uniform external load. We conducted the test on a series of 5 regional modules joined consecutively on the sides.
1250 gms L1
A comparitive study analysing: Material strength between paper, polypropelene and cardboard. Structural strength by changing the size of the module for the same thickness of material. The tests gave an understanding of the structural and material resistance under external load. Conclusion: Both paper and polypropelene show an equal range of flexibility. Though as compared to paper, polypropelene can resist double the load. Polypropelene modules of 90x90x90, .7mm thickness provides the maximum resistive force under compression. Increasing the module size for the same thickness decreases the resistive forces.
L2
1250 gms 120 mm 170 mm
1000 gms
fig 12.1: For a unit of 5 modules L1 - Length in normal condition L2 - Length in compression under external load
MM 350
120 mm
LEGEND
300
PAPER 100gsm
L1 L2
250 200
POLYPROPELENE .75mm
L1 L2
150 100
2350 gms
CARDBOARD 1.5mm
L1 L2
50 0
170 mm
1000 gms
90X90X90 90X90X90 90X90X90 120X120X120 120X120X120 MODULE SIZE 1000
1250
2800
2350
750
2800 gms
WEIGHT (GMS)
200 mm
fig 12.2: Change in length under maximum external load to the point of buckling. % 100
100
100
100
100
80
80
80
80
80
60
60
60
60
60
40
40
40
40
40
20
20
20
20
20
0
90X90X90 1000
0 120X120X120 1250
0
90X90X90 2800
0
120X120X120 2350
0
140 mm
9X9X9 750
MODULE SIZE WEIGHT (GMS)
fig 12.3: Decrease in length in percentage 10
Experimentation
Bootcamp 2013
Experimentation
11
b. MATERIALITY DEFLECTION COMPARISON
PAPER 100 GSM
POLYPROPELENE .7 MM THK 90 mm EDGE WITH SCORING
POLYPROPELENE .7 MM
Aim: Comparitive study of deflection between paper and polypropelene. By studying the the displacement in length under varying degrees of prestressing.
POLYPROPELENE .7 MM THK 90 mm EDGE WITHOUT SCORING
L=340
L=350
L=385
L=395
L=415
L=415
CENTRE
A comparitive study analysing: Change in deflection by applying a prestressing load to 1,3 and 5 modules.
90X90X90 MM
CENTRE
Comparing the results between polypropelene and paper.
c. FLEXIBILITY DEFLECTION ANGLES Aim: To understand the deflection induced due to bending in polypropelene. Comparing the deflections between modules with and without scoring as well as by varying the edge length. A comparative study analysing: The deflection by measuring the degree of deflection with respect to the centre and the resulting curved arc length.
CENTRE THREE
CENTRE THREE
ALL
POLYPROPELENE .7 MM THK 120 mm EDGE WITH SCORING
ALL
L=460 ALL
L = 495
CENTRE
CONCLUSION
CENTRE
The degree of deflection achieved is the maximum in the modules with 90mm edge length, unscored. Also, if the size of this module increases further the strength achieved in pre-stress condition decreases due to its inherent flexibility.
CONCLUSION For a series of 5 modules, for polypropelene the relative increase in deflection is more when the prestressing is increased from a single module to 3 modules as compared to the relative increase during prestressing from 3 to 5 modules. For paper, there is a periodic increase in deflection length as the prestressing increases from 1 to 3 to 5. 12
CENTRE THREE
120X120X120 MM
L=550
CENTRE THREE
L=570
ALL
fig 15.1: Deflection measurements under prestressing. L= deflected arc Bootcamp 2013
After testing the module by scoring and not scoring the edges, we realised that transfer of forces occurs more smoothly in the module without scoring. Taking into consideration the above factors, we maintained the thickness: edge length ratio of 0.7 mm : 90mm.
ALL
fig 14.1 : Deflection comparison Experimentation
fig 15.2: Load transfer in modules with scoring and without scoring
Experimentation
13
REGIONAL INTERACTION - STRUCTURAL BEHAVIOUR
external load
The tests gave a better understanding of the geometry in terms of the strengths and failing in the structure.
external load
Joining 5 basic geometry modules formed a ring. Due to the closed geometry formed, the ring in itself became stable in one direction yet maintained a flexibility in the other. In order to exploit this flexibility, a series of modules were joined from the sides along this axis. The interaction of regional modules resulted into flexible structure with a bouncing effect. Using prestressing cables on one of the sides, induced a curvature and at the same time stabilised the geometry.
stiffness
fig 16.1: Basic geometry interaction: Stiffness v/s Flexibility
r
flexibility
flexibility
stiffness due to pre-stress
pre-stress r = rise of the arc formed due to prestressing
fig 16.2 Regional geometry interaction: Stiffness v/s Flexibility
Locally, each piece acts independently however the resultant deflection affects the global geometry. Thus the resultant global interaction is dependent on the interaction at the level of a single basic module.
14
The System
Bootcamp 2013
GLOBAL SYSTEM
GLOBAL FORM EXPLORATION
The global system is achieved through repetition of the regional component. The local geometry of the folded triangle is modulated in order to begin to create a hierarchical system. (fig 18.1) Depending how tight or how apart each regional component was connected to one another resulted in how flexible or how static the form was as investigated in the two investigations of organization: branching vs linear.
00 1 Local 2 Regional
3 Global
fig. 18.1
01
A. LINEAR ORGANIZATION
B. BRANCHING ORGANIZATION
The regional modular components are placed in linear parallel modules.
The regional components are placed in rows that extend away from a central axis and are connected by an exponentially increasing number of regional components. Post-tension was applied along the axis of each row to allow that allowed for maximum flexibility in directional curvature. (fig 18.3)
Post-tension was applied parallel to the main axis of the composed regional modules that resulted in a parallel controlled curvature (fig 18.2)
Exploration 00: A branching global form allowed for maximum flexibility and gave opportunity for a global double curvature with a minimal amount of modules. There was the required three points of support, however, with minimal stability.
Exploration 01: A Linear global form allowed for maximum structure and stability with the maximum amount of modules: however, little flexibility with a single curvature configuration.
Axis Post Tension
02
2
Exploration 02: Three repeated structures connected to a singular point was found weak at the center and needed much more support through the addition of many more regional modules.
1 1
2
Axis Post -Tension fig. 18.3
fig. 18.2
DIFFERENTIATION IN APPLIED TENSION Differentiation in shape of the global form was achieved through posttentioning specific rows of the universal linear organization module. (fig 18.4)
Post-Tension Cables
03
Exploration 03: Formed linear global form of a particular shape gave structure and stability by increasing the size of the modules in proportion to thickness: however, in need of more stability and greater control.
Post-Tension Cables
The form then stabilized itself under its own weight as a result of the posttensioned strips. This gave the global form structure and variation in its overall curvature and reorganizes its local and regional components in order to accomplish a predefined global double curvature. fig. 18.4 16
The System
Bootcamp 2013
The System
17
FABRICATION
FORM ANALISYS
A. CONSTRUCTION
A. ORIENTATION LINES
00 Local We used a laser cutter to cut polypropylene to get the flatten geometry (two sided triangles). With the laser cutter we had the possibility to scored de material to fold it. Holes were done in the two extremes of the triangles (future lateral connectors). The module remained just folded in order to assemble it latter with another component. Eyelet puncher was used.
01
02
Regional Each Local module has 4 triangle connectors (wings), 2 in each side of the principal triangle. Attaching modules together using eyelets gave us the chance to get stiffness not only in the Local component but also in the Regional, which is generated by joining 5 modules. Forming a resistant structural shape around a pentagon’s geometry.
Strips Once we obtained the crown we started to connect them one next to each other forming linear strips. The holes in the extreme part of the principal triangles Enable us to use eyelets to connect the resulting components among them. We used eyelets because the holes which were used latter to connect and pre-stressing strips.
03 Global Strips are organised linearly, using the eyelets connector, attaching 4 layers of polypropylane. For effective joining we made a sandwich between the principal triangle’s layers of the Regional components. The lateral holes are connected by passing a nylon string to prestress and post-stress the strips to get stiffness and curvature.
The aggregation of the Regional module generate stripes that have two different structural behaviour. The pentagon organisation creates an axis in which the flexibility of the composition gets a 100%. The bouncing behaviour was going to be useful to achieve multifunctional orientation. Transversal to the Regional axis we found that the configuration of the crown component acquired the best structural stiffness.
Pentagon axis
Pre-stressing
Lateral eyelets
Learning from the exploration we realise that the efficient way to start aggregating the stripes was parallel to the axis of the pentagon’s organisation. It was not only positive for the connectivity among the stripes but also for the rotation of the stripes in order to give us the possibility of create the form of the global structure. Even though the eyelet was an additional connector, it not only had the possibility of joining 4 layers of polypropylene, but also was able to let us have a hole where a nylon string had passed through to prestress the stripe.
B. PRESTRESS LINES Assembling the stripes through the lateral connectors enable the system to find an structural solution. Our objective was to generate a surface with double curvature. Once the parallel organisation is done we started passing through the lateral eyelets nylon cords to prestress the stripes in order to form not only a curvature in one axis but also stiffness in the other, therefore the structure started to grow vertically and the module begun to communicate to each other. The forces of the cord create stiffness, the structure required another force to compensate the prestressing. In order to maintain the surface stable a post-stressing force was applied to the opposite side of the strips. Without this two antagonist forces the structure would not be able to have self supporting condition. Eye lets play a very important roll in this configuration. It permits to connect all the components, moreover it eases the building process. . 18
Introduction
Bootcamp 2013
Introduction
19
CONCLUSION
Z Y
A Brief Through this exploration of the simple geometry of the triangle in creating a material system there was further development to our understanding of the hierarchy, the fabrication, and the modular differentiation in the physical and digital application of this particular system. Disconnection Between Global and Local In this system, the local component information trickles up, however, the global system must also inform the the local. As a result, the global form was a direct representative of what the local and regional components are capable of. The local components held a strength, and when placed in their gathered system of the regional they gave a buoyancy and high levels of flexibility. The global form, however, was not a derivative of these qualities or systems and instead some of the qualities that were found in the local and regional components were lost in the global. The overall shape of the local component didn’t contribute to the global form. The direction of relationship between the local – regional – and global was in one direction rather than both. For success in a global form there must be a relationship in both directions. A constant understanding of each components’ relationship with each other and the global form; where the components are going and what they are doing. By observing the entire system at work, the global behavior become apparent. Complications with Fabrication The jointing of components was the most critical issue we encountered during the final fabrication phase of the global form. We changed the thickness of the polypropylene from 700 to 1000 micron maintaining the size-tothickness ratio, however we overlooked the size of the eyelets in relation to the combined thickness of superimposed polypropylene, specifically where 4 corner triangles overlapped. One way we tackled this challenge was to clench two eyelets ‘back-to-back’ on both sides of the triangular interconnection. Although the joints were fixed tightly, it still resulted in feeble joints with respect to the weight of the global surface, which relays as a single point of failure. One crucial improvement is to use specialized eyelets, like the ones used in sail fabrication, with lengthier height. These however need special order or customization. 22
The use of metallic eyelets introduced another material type to the structural system. Although a single brass eyelet is lightweight (0.015g), the collective mass of eyelets employed in the system sums up and is deemed unnecessary. An ideal solution to this is to utilize a jointing mechanism from within the geometry, which is what we initially explored (see page 7) but failed to make secure fasteners. An alternative solution is to employ thermo fusion technique using a heat-shrink sleeving gun to join polypropylene at the interconnecting triangles. Apart from using less material, the upside of this solution is that we can benefit from the tensile properties of polypropylene in forming the global surface. The downside of it however is the complexity of the procedure, cost and time it takes to fuse all the parts together. Nevertheless, at a much bigger scale it can be indemnified. Acknowledging that the initial regional module (the ‘crown’) became the new local component, it was clear that the latter became complex since it was built from 5 smaller modules. It originated two important weaknesses in the project: Firstly, the joints had 4 overlapping layers to connect, therefore too much redundancy in the system. Secondly, the process was time consuming and labor intensive to attach each of the eyelets. Further study in the joints is mandatory. We considered that the joints of the new module could be done by folding the polypropylene more effectivly into the triangle interconnections,consequently the eyelets would become unnecessary.
Differentiation of the modules One of the shortcomings of the form is that the shape of the local module does not contribute to the global form.
X
One of the methods to introduce a heirarchy in the global geometry would be to vary the scale of the module. This could allow for getting a double curvature without the introduction of tensile members in the system. There could be two approaches for such a system. 1. A gradual change in module size. In such a system, each module would be different in size. Digital fabrication makes the manufacturing feasible. We tried modelling this computationally using box morphing. 2. Limiting the variation in size of module. Box morphing approach A surface was populated with boxes with gradual change in dimensions along x,y and z axis. The basic module was morphed into these boxes. The change in module sizes results into a self generated double curvature.
Another weakness that contributed negatively to the global connectivity was the type of string used in pre-stressing. The Nylon string was too thin to handle the weight, too loose to align the regional components together, and has an elastic property which is disadvantageous for applying tension. During the exploration of structural behavior, we were connecting the regional columns with a regular candlewick (diameter:4mm), it worked better than the nylon string. We assumed that friction is the physical property that helped control each point of connection.
The Conclusion
fig 25.1 computational model using box morphing Bootcamp 2013
The Conclusion
23