Tensegrity Structres

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CONTENTS
FIGURE LIST………………………………………………...…………………………….…….…..IV
1.INTRODUCTION……………………………………………………………………………………1
2,.TENSION………………………………………………………………………………………...…2
2.1 Definition Of Tension…………..…………………………………………………….……..…2
2.2 Tension Members……………………………………………………………………….…..……4
3. TENSEGRITY………………………………………………………………………………...……7
3.1 Definition And Principles Of Tensegrity …………………………….…………………….…7
3.2 History Of Tensegrity …………………………………………………….......………….……..9
3.2.1 Origins Of Tensegrity……………………………………………………....................….9
3.3 Applications Of Tensegrity…………………………………………………………………... 14
3.4 Characteristics Of Tensegrities…………………………………………...……………..…..15
3.5 Advantages Of Tensegrity Over Conventional (Continuous) Structures……………………………………………………………………………….……….… 16
3.6 Disadvantages Of Tensegrity Over Conventional (Continuous) Structures ………………………………………………………………………………………..…17
3.7 Topological Classification Of Elementary Cells Of Tensegrity Structures ………………………………………………………………………………..……….…18
3.7.1 Rhombic configuration…………………………………………………………….….…..18
3.7.2 Circuit configuration………………………………………………………….…………...19
3.7.3 'Z type' configuration…………………………………………………………….……....20
3.8 Types Of Tensegrities…………………………………………………………..………....….21
3.8.1 Tensegrity prism (T-prism)…………………………………………………….……...…..22
3.8.2 Diamond tensegrity…………………………………………………………….…...…...24
3.8.3 Zig-zag tensegrity……………………………………………………………………….27
4.WEAVING : MOTHER OF TENSEGRITY…………………………………………….....….28
4.1 Introduction …………………………………………………………………………………...28
4.2 Right Helix And Left Helix……………………………………………….…………………..30
4.3 Examples of Weaving in The Tensegrity System……………………………………...….31
5.TENSEGRITY EXAMPLES ………………………………………………….…….…….35
5.1 Suspension Bridge……………………………………………………………………………35
5.2 The Skylon Tower……………………………………………………………………...36
5.3 Roof Structures…………………………………………………………………………37
5.4 Child's Baloon………………………………………………………………………….38
5.5 Human Body……………………………………………………………….………...…39
REFERENCES…………………………………………………………………………………….40



LIST OF FIGURES
2.1 Tension and Compression…………………………… …………….….……..2
2.2 Types of Members………………………..……………………………….….…4
3.1 Comparison between the "Gleichgewichtkonstruktion" or "Structure-Sculpture"……………………………………………………………………...…......10
3.2 Simplex by Snelson, simplest tensegrity system ……………….................11
3.3 "X-column" by Snelson …………………………………………..……….…...12
3.4 (a) Assembly of rhombic configuration and (b) its pattern ………………...18
3.5 (a) Circuit configuration of Cuboctahedron tensegrity having 4 non- touching triangular circuits and (b) its pattern …………………………….……...19
3.6 (a) Truncated Tetrahedron (T-tetrahedron) using Z type configuration,
(b) its pattern ……………………………………………………….…....…….20
3.7 Wood X-module ………………………………………………………….……21
3.8 Kite frame shape in various proportions ………………………………...….22
3.9 Diamond Configuration of T-prism ……………………………………….....23
3.10 (a) T-icosahedron and (b) its corresponding transformation from and
back to doubled-up octahedron ……….................................................25
3.11 (a) Octahedral structure (b) planar view of corresponding 3-struts
single layer tensegrity system …………………………………..........….25
3.12 Planar view of 6-struts two layer tensegrity system ……………….…...26
3.13 Planar view of 9-struts three-layer tensegrity system …………..……...26
3.14 T-Tetrahedron (Z type configuration ……….………………………….…27
4.1 The Common Square Weave Heaxagon ……………………………..………28
4.2 Three Way Triangle Weave……………………………………………..…...…28
4.3 Cross two pencils ………………………………………………………………..29
4.4 Right and Left Helix ………………………………………………………..……30
4.5 Right and Left Helixes ………………………………………………..…………31

4.6 Kagome Three-Way Basket Weave.…………………………. ………....……31

4.7 Kagome Three-Way Basket Weave…………………………………......…….32
4.8 Examples of Weaving in Tensegrity …………………………………….…..…33
5.1 Golden Gate Bridge, San Francisco, California …………………...…………35
5.2 Skylon Tower – London, England ………………………………………..……36
5.3 A dome designed by Robert Le Ricolais …………………………….…..……37
5.4 Child's Baloon …….……………………………………………………..……....38
5.4 Human body tensegrity model …………………………………………….…...39





1. INTRODUCTION

The technique used in constructions by architects around the world is based on holding the structure with the help of its weight and continuity of stresses induced basically in compression. For example, each component of a stone dome or bridge is pulled by tension acting downward through the structure, but the compressive continuity is still in charge of sustaining most of the load. On the other hand, tensegrity architecture represents a totally different principle with tension continuity and compression discontinuity. Tensegrities represent a system of 'equilibrated omnidirectional stresses' and they don't need any support and are self-equilibrated as well as pre-stressed. The theory of tensegrity structures is well known in civil engineering or statuary while in other engineering branches the potential of these structures is not yet fully explored.









2. TENSION

2.1 Definitions Of Tension
Tension force is a force that is exerted equally on both ends of a cable, chain, rope, wire or other continuous object and is transmitted between the ends by that object. On a microscopic level, objects under tension have a separation between molecules that creates potential energy in their bonds.

It requires force to put an object under tension, but technically, tension itself is not a force for as long as the situation is stable. Nonetheless, physicists measure the tension in a system in units of force. Once some part of the system begins moving, such as when one end of a rope under tension is released, the potential energy between the molecules of the rope is converted to mechanical energy, and the tension is converted to force. As long as an object under tension is straight, the tension is constant for its entire length.
A force is any push or pull acting upon an object as a result of its interaction with another object. Tension forces are one of the contact forces, which are the forces present when two objects are in contact with one another. There are also non-contact forces, such as gravity, magnetic force and electrical force, which act at a distance. [1]
Compression, decrease in volume of any object or substance resulting from applied stress. Compression may be undergone by solids, liquids, and gases and by living systems. In the latter, compression is measured against the system's volume at the standard pressure to which an organism is subjected—e.g., the pressure of the atmosphere at sea level is the standard, or reference, for most land animals, but the standard for deep-sea fishes and similar specialized forms is the normal pressure of their environment. Tension is the opposite of compression. (Fig.2.1)



Figure 2.1 Tension and Compression














2.2 Tension Members

The tension member considered for the design is a linear member which carries an axial pull. The members undergo extension due to this axial pull. This is one of the common types of force transmitted in the structural system. Tension members are very efficient since the entire cross section carries uniform stress unlike flexural members. The tension members do not buckle even when stressed beyond the elastic limit. Hence the design is not effected by the type of section used i.e., Plastic, Compact or Semi-compact. Some of the common examples of tension members in structures are; Bottom chord of pin jointed roof trusses, bridges, transmission line and communication towers, wind bracing system in multi-storey buildings, etc. The objective of this exercise is to determine the tensile strength of a given member having a specified end connection. The strength of these members is influenced by several factors such as the length of connection, type of connection (by bolts or welds), connection eccentricity, size and shape of fasteners, net area of cross-section and shear lag at the end connection. [2]

2.2.1 Types of Tension Members
Figure 2.2 Types of Members

The tension members may be made of single structural shapes. The standard structural shapes of typical tension members are( Fig.2.2);
(a) Rods and Bars (b) Cables and Wires (c)Single and double angles
(d)Rolled W-and S-section (e) Structural Tee (f) Buit-up Box Sections


Rods and Bars:The square and round bars are shown in Figure-2 are quite often used for small tension members. The round bars with threaded ends are used with pin-connections at the ends instead of threads.
The ends of rectangular bars or plates are enlarged by forging and bored to form eye bars. The eye bars are used with pin connections. The rods and bars have the disadvantage of inadequate stiffness resulting in noticeable sag under the self weight.

Wires and Cables:The wire types are used for hoists, derricks, rigging slings, guy wires and hangers for suspension bridges.

Single Structural Shapes and Plates:The single structural shapes, i.e. angle sections and tee-sections as shown in Figure-2 are used as tension members. The angle sections are considerably more rigid than the wire ropes, rods and bars. When the length of tension member is too ling, then the single angle section also becomes flexible.
The single angle sections have the disadvantage of eccentricity in both planes in a riveted connection.
The channel section has eccentricity in one axis only. Single channel sections have high rigidity in the direction of web and low rigidity in the direction of flange.
Occasionally, I-sections are sued as tension members. The I-sections have more rigidity, and single I-sections are more economical than built up sections.

Built-up Box Sections:Two or more than two members are used to form built up members. When the single rolled steel section can not furnish the required area, then built-up sections are used.
The double angle sections of unequal legs shown in the figure are extensively used as tension members in the roof trusses. The angle sections are placed back to back on two sides of a gusset plate. When both the angle sections are attached on the same side of the gusset, then built-up section has eccentricity in one plane and is subjected to tension and bending simultaneously. The two angle sections may be arranged in the star shape (i.e. the angles are placed diagonally opposite to each other with leg on outer sides).

The star shape angle sections may be connected by batten plates. The batten plates are alternatively placed in two perpendicular directions.
A built-up section may be made of two channels placed back to back with a gusset in between them. Such sections are used for medium loads in a single plane-truss. In two-plane trusses, two channels are arranged at a distance with their flange turned inward. It simplifies the transverse connections and also minimizes lacing. The flanges of two channels are kept outwards, as in the case of chord members or long span girders, in order to have greater lateral rigidity.
The heavy built-up tension members in the bridge girder trusses are made of angles and plates. Such members can resist compression in reversal of stress takes place.









3. TENSEGRITY


3.1 Definition And Principles Of Tensegrity

Anthony Pugh gives the following definition of tensegrity: A tensegrity system is established when a set of discontinuous compressive components interacts with a set of continuous tensile components to define a stable volume in space. Tensegrity structures are distinguished by the way forces are distributed within them. The members of a tensegrity structure are either always in tension or always in compression. In the structures discussed in this book, the tensile members are usually cables or rods, while the compression members are sections of tubing. The tensile members form a continuous network. Thus tensile forces are transmitted throughout the structure. The compression members are discontinuous, so they only do their work very locally. Since the compression members do not have to transmit loads over long distances, they are not subject to the great buckling loads they would be otherwise, and thus they can be made more slender without sacrificing structural integrity. In the realm of human creation, pneumatic structures are tensegrities. For instance, in a balloon, the skin is the tensile component, while the atoms of air inside the balloon supply the compressive components. The skin of the balloon consists of atoms which are continuously linked to each other, while the atoms of air are highly discontinuous. If the balloon is pushed on with a finger, it doesn't crack; the continuous, flexible netting formed by the balloon's skin distributes this force throughout the structure. And when the external load is removed, the balloon returns to its original shape. This resilience is another distinguishing characteristic of tensegrity structures. Another human artifact which exhibits tensegrity qualities is prestressed concrete. A prestressed concrete beam has internal steel tendons which, even without the presence of an external load,

are strongly in tension while the concrete is correspondingly in compression. These tendons are located in areas so that, when the beam is subjected to a load, they absorb tensile forces, and the concrete, which is not effective in tension, remains in compression and resists heavy compressive forces elsewhere in the beam. This quality of prestressed concrete, that forces are present in its components even when no external load is present, is also very characteristic of tensegrity structures. In the natural realm, the structural framework of non-woody plants relies completely on tensegrity principles. A young plant is completely composed of cells of water which behave much like the balloon described above. The skin of the cell is a flexible inter-linkage of atoms held in tension by the force of the water in the contained cell. As the plant is stretched and bent by the wind, rain and other natural forces, the forces are distributed throughout the plant without a disturbance to its structural integrity. It can spring back to its usual shape even when, in the course of the natural upheavals it undergoes, it finds itself distorted far from that shape. The essential structural use the plant makes of water is especially seen when the plant dries out and therefore wilts. [3]

















3.2 History Of Tensegrity

Tensegrity is a developing and relatively new system (barely more than 50 years old) which creates amazing, lightweight and adaptable figures, giving the impression of a cluster of struts floating in the air. As it is explained in Gómez Jáuregui [1], it is not a commonly known type of structure, so knowledge of its mechanism and physical principles is not very widespread among architects and engineers. However, one of the most curious and peculiar aspects of tensegrity is its origin; controversy and polemic will always be present when arguing about its discovery. [4]

3.2.1 Origins Of Tensegrity

Three men have been considered the inventors of tensegrity: Richard Buckminster Fuller, David Georges Emmerich and Kenneth D. Snelson. (As a precaution, these names have been mentioned in chronological order of their granted patents: Fuller-13 Nov 1962; Emmerich-28 Sep 1964; Snelson-16 Feb 1965). Although all of the three have claimed to be the first inventor, R. Motro mentions that Emmerich reported that the first proto-tensegrity system, called "Gleichgewichtkonstruktion", was created by a certain Karl Ioganson (some authors call him Johansen) in 1920 (Figure 3.1). As Emmerich explains: "Cette curieuse structure, assemblée de trois barres et de sept tirants, était manipulable à l'aide d'un huitième tirant detendu, l'ensemble étant déformable. Cette configuration labile est très proche de la protoforme autotendante à trois barres et neuf tirants de notre invention."


Figure 3.1 Comparison between the "Gleichgewichtkonstruktion" or "Structure-Sculpture"


This means it was a structure consisting of three bars, seven cords and an eighth cable without tension serving to change the configuration of the system, but maintaining its equilibrium. He adds that this configuration was very similar to the proto-system invented by him, the "Elementary Equilibrium", with three struts and nine cables (Figure 3.2).
All the same, the absence of pre-stress, which is one of the characteristics of tensegrity systems, does not allow Ioganson's "sculpture-structure" to be considered the first of this kind of structures.



Figure 3.2 Simplex by Snelson, simplest tensegrity system



The most controversial point has been the personal dispute, lasting more than thirty years, between R. B. Fuller (Massachusetts, 1895-1983) and K. D. Snelson (Oregon, 1927). As the latter explains in a letter to R. Motro, during the summer of 1948, Fuller was a new professor in the Black Mountain College (North Carolina, USA), in addition to being a charismatic and a nonconforming architect, engineer, mathematician, cosmologist, poet and inventor (registering 25 patents during his life). Snelson was an art student who attended his lectures on geometric models, and after that summer, influenced by what he had learnt from Fuller and other professors, he started to study some three-dimensional models, creating different sculptures (Figure 3.3).



Figure 3.3 "X-column" by Snelson, his first tensegrity art piece.
Illustration donated by the artist.



As the artist explains, he achieved a new kind of sculpture, which can be considered the first tensegrity structure ever designed. When he showed it to Fuller, asking for his opinion, the professor realized that it was the answer to a question that he had been looking for, for so many years. In Fuller's words:
"For twenty-one years, before meeting Kenneth Snelson, I had been ransacking the
Tensegrity concepts. (…) Despite my discovery, naming and development of both the multidimensional vectorial geometry and the three dimensional Tensegrity, I had been unable to integrate them, thus to discover multi-dimensional four, five and six axes symmetrical Tensegrity."

In contrast to other authors, and serving as an illustration of how important it was
considered, he always wrote "Tensegrity" starting with a capital T.
At the same time, but independently, David Georges Emmerich (Debrecen-Hungary, 1925-1996), probably inspired by Ioganson's structure, started to study different kinds ofstructures as tensile prisms and more complex tensegrity systems, which he called"structures tendues et autotendants", tensile and self-stressed structures. As a result, he defined and patented his "reseaux autotendants", which were exactly the same kind of structures that were being studied by Fuller and Snelson.




























3.3 Applications Of Tensegrity

The qualities of tensegrity structures which make the technology attractive for human use are their resilience and their ability to use materials in a very economical way. These structures very effectively capitalize on the ever increasing tensile performance modern engineering has been able to extract from construction materials. In tensegrity structures, the ethereal (yet strong) tensile members predominate, while the more material-intensive compression members are minimized. Thus, the construction of buildings, bridges and other structures using tensegrity principles could make them highly resilient and very economical at the same time. In a domical configuration, this technology could allow the fabrication of very large-scale structures. When constructed over cities, these structures could serve as frameworks for environmental control, energy transformation and food production. They could be useful in situations where large-scale electrical or electromagnetic shielding is necessary, or in extra-terrestrial situations where micrometeorite protection is necessary. And, they could provide for the exclusion or containment of flying animals over large areas, or contain debris from explosions. These domes could encompass very large areas with only minimal support at their perimeters. Suspending structures above the earth on such minimal foundations would allow the suspended structures to escape terrestrial confines in areas where this is useful. Examples of such areas are congested or dangerous areas, urban areas and delicate or rugged terrains. In a spherical configuration, tensegrity designs could be useful in an outer-space context as superstructures for space stations. Their extreme resilience make tensegrity structures able to withstand large structural shocks like earthquakes. Thus, they could be desirable in areas where earthquakes are a problem. [3]








3.4 Characteristics Of Tensegrities

Characteristics of tensegrities can be summarized as follows :
3.4.a) They have a higher load-bearing capacity with similar weight.
3.4.b) They are light weight in comparison to other structures with similar resistance.
3.4.c) They don't need to be anchored or have to lean any surface as they don't depend on their weight or gravity. They are stabilized in any position by equilibrium of compressive forces in struts with tensional forces in prestressed cables. Prestrain in the cables can be transformed into prestress only if the structure is statically indeterminate.
3.4.d) They are enantiomorphic i.e. exist as right and left-handed mirror pairs.
3.4.e) Elementary tensegrity modules can be used (such as masts, grids, ropes, rings etc.) to make more complex tensegrity structures.
3.4.f) Higher the pre-stress, stiffer the structure would be, i.e. its load bearing capacity increases with the increasing pre-stress. The degree of tension of the pre-stressed components is directly proportional to the amount of space they occupy.
3.4.g) In a tensegrity structure the compressive members are short and discontinuous, hence they do not undergo buckling easily and no torque is generated in them.
3.4.h) The resilience depends on the structure assembly and material used.
3.4.i) They work synergically i.e. their behaviour cannot be predicted by considering the behaviour of any of their components separately.
3.4.j) They are sensitive to vibrations under dynamic loading. Slight change in load causes the stress to redistribute in the whole structure within no time and thus, they have the ability to respond as a whole. k) Kenner introduced a term 'Elastic Multiplication' for the tensegrity structures. It is a property of tensegrity structure which depends on the distance between two struts. If two struts are separated by a certain distance the elongation of tendons (tensile members) attached to them is much less compared to this distance.
3.4.l) The deformation response of entire tensegrity structure to load is non-linear as its stiffness increases rapidly with increasing load, like at a suspension bridge.
3.4.m) The tensegrities are commonly modelled with frictionless joints, and the self-weight of cables and struts is neglected. [6]


3.5 Advantages Of Tensegrity Over Conventional (Continuous) Structures

3.5.a) As the load is distributed in whole structure there are no critical points of weakness.
3.5.b) They don't suffer any kind of torsion and buckling due to space arrangement and short length of compression members.
3.5.c) Forces are transferred naturally and consequently, the members position themselves precisely by aligning with the lines of forces transmitted in the shortest path to withstand the induced stress.
3.5.d) They are able to vibrate and transfer loads very rapidly and hence, absorb shocks and seismic vibrations which makes them applicable as sensors or actuators.
3.5.e) They can be extended endlessly through adding elementary structures.
3.5.f) Construction of structures using tensegrity principle makes it highly resilient and, at the same time, very economical. [5]


















3.6 Disadvantages Of Tensegrity Over Conventional (Continuous) Structures

3.6.a) If the structure becomes too large it faces a problem of bar congestion (i.e. the struts start running into or touching each other).
3.6.b) They show relatively high deflections and low material efficiency as compared to with conventional continuous structures.
3.6.c) Fabrication complexity is a major barrier in developing floating compression structures.
3.6.d) Adequate design tools are not available for their design, software 'Tensegrite 2000' (developed by R. Motro et al.) is the most advanced tool available to design tensegrity structures.
3.6.e) At large constructions the structure cannot withstand loads higher than the critical, related to its dimensions and prestress. [6]




























3.7 Topological Classification Of Elementary Cells Of Tensegrity Structures

3.7.1 Rhombic configuration

The name of tensegrity patterns is based on the way they are constructed (tendon patterns). In Fig. 3.4, each strut of a system represents the longest diagonal of
a rhombus formed by four corresponding tendons and can be folded following
these diagonal. Generally, this configuration corresponds to the diamond tensegrity.
T-prism (section 3.8.1) and T-icosahedron (section 3.8.2) tensegrities are well known examples of the rhombic configuration where rhombus represents a non-planar quadrilateral formed by tendons. [5]




Figure 3.4 (a) Assembly of rhombic configuration and (b) its pattern





3.7.2 Circuit configuration

In this system, the compressed members are formed by close circuits (see Fig. 3.5(a)) which do not comply with standard definition of tensegrity. This can be constructed by closing the rhombus generated by struts and tendons of the diamond pattern tensegrity, such as T-icosahedron (see Fig. 1.12). [5]



Figure 3.5 (a) Circuit configuration of Cuboctahedron tensegrity having 4 non- touching triangular circuits and (b) its pattern












3.7.3 'Z type' configuration

A 'zig-zag' configuration (also being an enantiomorphic) is obtained from the rhombic
configuration as the basic structure. Both ends of any strut should be connected by three non-aligned tendons arranged to form a 'Z' shape. Truncated tetrahedron (see Fig. 3.6) is a classic example of Z type configuration obtained from truncated icosahedron which belongs to the class of rhombic configurations. [5]


Figure 3.6 (a) Truncated Tetrahedron (T-tetrahedron) using
Z type configuration, (b) its pattern










3.8 Types Of Tensegrities

The tensegrity structures are widely classified as prestressed and geodesic structures. They are divided into three main categories: Tensegrity prism, Diamond tensegrity and Zig-zag tensegrity. The X-module (see Fig. 3.7) build by Snelson had given birth to tensegrity principle. This simple tensegrity structure consists of two X-shaped wooden struts suspended in air by stretched nylon cables. The simple kite frame is the basic prestressed tensioncompression cell of X-module tensegrity structure. It consists of two crossed struts firmly held together by a girth of four tension members, because the (vector) sum of compression forces pushing out equals to the sum of tension forces pulling in. As shown in Fig. 3.8, the
change in length of both struts and tendons i.e. change in proportion of frame is related to the variation in distribution of forces, both tension and compression. [5]




Figure 3.7 Wood X-module





Figure 3.8 Kite frame shape in various proportions

It is a quasi-tensegrity structure because it is planar (2D), and the struts touch each
other. If any one of the tendons is removed, then the blank side will work as a compressed component; this is called as 'strut effect'. This basic principle is required to design various elementary components of two layer and three layer tensegrity systems. The lengths of the four tendons and the lengths of the two struts determine the shape of the kite frame (see Fig. 3.8).

3.8.1 Tensegrity prism (T-prism)

Also known as 'Three struts T-prism' was invented by Karl Ioganson in Moscow in
1921. It is the simplest and therefore one of the most instructive members of the
tensegrity family. The T-prism has 9 tendons and 3 struts (see Fig. 3.9) and belongs to a subclass of prismatoids. It has been called tensegrity prism or T-prism as it can be considered as a twisted prism consisting of two triangular faces twisted with respect to each other. Generally, these tensegrity structures are designed by keeping the lengths of one set of tendons and struts constant, and determining the lengths of another set of tendons. When one end of the prism is twisted relative to the other, the rectangular sides of the prism become non-planar quadrilaterals. Thus, two opposite angles of each quadrilateral become obtuse and acute. For structure to be stable and

prestressed, the prism is twisted in such a way that the distance between the obtuse angles is least (an intermediate stage of twisting) and hence, a completely stable T-prism is formed.



Figure 3.9 Diamond Configuration of T-prism














3.8.2 Diamond tensegrity

The tensegrity icosahedron also known as T-icosahedron depicted in Fig. 3.10(a) is a classic example of diamond tensegrity. These tensegrities are characterized by the fact that each triangle of tendons is connected to the adjacent one via a strut and two interconnecting tendons. It was first exhibited by Buckminster Fuller in 1949 and is one of a few tensegrities which exhibit mirror symmetry. This tensegrity is classified as a 'diamond'
type because each of its struts is surrounded by a diamond form of four tendons which are supported by two adjacent struts making them distinct from a Zig-zag tensegrity. It has 6 struts and 24 tendons with tendon to strut lengths ratio of 0.613.
If the quadrilaterals nested with struts are changed to squares then the tendons form
a cuboctahedron network. Fig. 3.10(b) illustrates the change in system of tendons from an octahedral arrangement (with each strut doubled-up by presence of two struts in the identical position) to cuboctahedron and back. Small arrows indicate the direction of movement of the struts and of the corresponding pair of opposite points of the quadrilateral as the tendon system goes through transformations.
Opening the octahedral structure carefully from one end gives a single layer diamond
structure as depicted Fig. 3.11(a). New tensegrity structures with spherical symmetry can be generated by addition of new layers of struts and tendons and joining both ends of each layer. Planar views of tensegrity systems based on this approach are depicted in Fig. 3.12 and Fig. 3.13.


Figure 3.10 (a) T-icosahedron and (b) its corresponding transformation
from and back to doubled-up octahedron



Figure 3.11 (a) Octahedral structure (b) planar view of corresponding
3-struts single layer tensegrity system



Figure 3.12 Planar view of 6-struts two layer tensegrity system




Figure 3.13 Planar view of 9-struts three-layer tensegrity system










3.8.3 Zig-zag tensegrity

The tensegrity tetrahedron also known as (T-tetrahedron) depicted in Fig. 3.14 is a classic example of diamond tensegrity developed by Francesco della Sala in 1953. The T-tetrahedron is the zig-zag counterpart of the diamond T-icosahedron (see Fig. 3.10). Although both structures have 6 struts, the major difference is that T-tetrahedron has four tendon triangles, whereas the T-icosahedron has eight of them.





Figure 3.14 T-Tetrahedron (Z type configuration)

In general, zig-zag structures with Z type configuration are simpler and less rigid due
to their lower number of tendons than their diamond counterparts with rhombic configuration.









4. WEAVING: MOTHER OF TENSEGRITY
4.1 Introduction
The ancient invention of weaving displays the basic properties of natural structure: modular-repetition, left and right helical symmetry and the close association between geometry and physical structure.
Two and only two fundamental fabric weave structures exist: the standard two-way plain weave made up of squares (Fig.4.1), and the three-way triangle/hexagon weave (Fig.4.2) used most often in basketry. Though there are many variations such as criss-crossing, doubling, etc. these two are the only primary forms.



Figure 4.1The Common Square Weave Figure 4.2Three-Way Triangle/ Heaxagon Weave(In Asia: It called KAGOME)




A single weaving event, two filaments crossing and in contact with one anothe (Fig.4.3), each warping the other where they press in contact is, in itself, an elementary structure. At the point of crossing the two threads create dual helical axes one clockwise, right-rotating and the other counterclockwise, left-rotating. [7]




Figure 4.3Cross two pencils. Place a thumb and index finger on the pencils and slide toward the center. Your hand will tend to rotate either clockwise or counterclockwise



4.2 Right Helix And Left Helix

Figure 4.4 Right and Left Helix
Just as the individual crossings of filaments have alternating helical axes so each square in a plain weave alternates with its neighbors like chess board squares. In order to prove whether a weave cell is right or left handed, imagine your fingers sliding in contact with the frame of a cell. Your hand will move down-hill" in a clockwise/ counterclockwise sense according to the cells "rotation".(Fig.4.4),(Fig.4.5)


Figure 4.5 Right and Left Helixes


Figure 4.6 Kagome Three-Way Basket Weave

In three-way, or Kagome weaving, hexagons alternate with triangles. If the hexagons have a clockwise helix the triangles are counterclockwise. If the hexagons are counterclockwise the triangles are clockwise.(Fig.4.6),(Fig.4.7)


Figure 4.7 Kagome Three-Way Basket Weave








4.3 Examples of Weaving in The Tensegrity System


Figure 4.8 Examples of Weaving in Tensegrity

Weaving and tensegrity share the principle of alternating helical directions, of left-to-right, of bypasses clockwise and counterclockwise.
In the top row above are five primary weave figures. Below them are the equivalent tensegrity modules. Individual tension lines strings, wires or rope are attached to the ends of the struts as shown so that each assembly is a closed system made of tension and compression parts. Each tension line connects individually to the ends of two struts. They do not thread through like a string of beads. The tension lines must be adjusted for tightness as with tuning a stringed instrument or inflating a car tire.
Tightening the tension system stores both tension and compression forces in equal amounts, a state that engineers call "prestressing." The energy remains stored inside the structure until it is disassembled.
In the figures above, only the 2-strut "x-module" and the 3-strut prism have tension networks with total triangulation. The networks of the square prism, the pentagon prism and the hexagon prism are not composed of triangles. In tensegrity structures triangulation in the tension network is significant because it determines if the structure will be firm or not. Tensegrity structures are endoskeletal, as are humans and other mammals whose tension "muscles" are external to the compression members' bones. Unique to tensegrity, the compression struts are separated one from another, non-touching within their tension envelope. The exception is the two-strut x-module, or traditional kite frame. This essentially flat figure' lacks a compression force in the "z" direction. In order to separate the crossed struts a third strut or else an additional X-module, must be added to pull the two struts apart.(Fig.4.8)


























5. TENSEGRITY EXAMPLES AND APPLICATIONS

5.1 Suspension Bridge

This is a non-self-sufficient tensegrity. It is anchorage dependent. When it is supported by pylons which are rigidly connected to the ground, the pylons should be considered part of the anchorage, and the bridge should be considered a composite structure, only the non-pylon part of which is a tensegrity. If the proper function of the bridge depends on the gravity field coming from one way rather than another, as seems likely given the adjective "suspension", but some local fastenings working with gravity, as might be desirable for wind and earthquake resistance, would firmly restore its tensegrity-hood; and a suspension bridge would not completely vulnerable to disintegration in the absence of gravity, as opposed to the case of the Roman arch below which would be.(Fig.5.1) [8]


Figure 5.1 Golden Gate Bridge, San Francisco, California







5.2 The Skylon Tower

In 1951, just three years after the official discovery of tensegrity, the Festival of Britain's South Bank Exhibition took place in London. In that occasion, a competition was organised to erect a "Vertical Feature", a staple of international exhibitions grounds. Philip Powell and Hidalgo Moya (helped and inspired by their former Felix Samuely) designed the Skylon , which was selected as the best proposal and built near the Dome of Discovery. Some authors (Cruickshank, 1995; Burstow, 1996) state that this needle like structure was a monument without any functional purpose, but it became a symbol for the festival, a beacon of technological and social potentialities and, finally, a reference for future engineers and architects. The 300 foot high spire was a cigar-shaped aluminium-clad body suspended almost invisibly by only three cables, and seemed to float 40 feet above the ground.(Fig.5.2) [8]



Figure 5.2 Skylon Tower – London, England




5.3 Roof Structures

An important example of Tensegrity being employed in roof structures is the stadia at La Plata (Argentina), based on a prize winning concept developed by architect Roberto Ferreira. The design adapts the patented Tenstar Tensegrity roof concept to the twin peak contour and the plan configuration, and consequently, it is more similar to a cable-dome structure than to a conventional roof structure. The first studies for the design of Tensegrity grids were carried out by Snelson, but its applications were limited. For the past few years, the main focus has been in the development of double-layer Tensegrity grids and foldable Tensegrity systems. This kind of grid has its most feasible possibilities in the field of walls, roofs and covering structures.(Fig.5.3) [8]



Figure 5.3 A dome designed by Robert Le Ricolais


5.4 Child's Baloon

Another common example of tensegrity is child's balon. The rubber skin of the balloon continuously pullswhile the individual molecules of air are discontinuously pushingagainst the inside of the balloon keeping it inflated. All external forces striking the external surface are immediately and continuously distributed over the entire system. This makes the balloon very strong.(Fig.5.4)
Molecules of air discontinuously pushing against the continuously pulling rubber skin of the balloon. In this example;Tensegrity is a balance of continuous pull and discontinuous push. [8]


Figure 5.4 Child's Baloon














5.5 Human Body

The muscles, tendons and ligaments are the purely tensile components which bind together the bones and cartilage(Fig. 5.5). These statements could probably use the attention of someone better versed in anatomy. Plants, fungi, single-cell creatures and other animals should be considered for inclusion in the tensegrity category on a case-by-case basis. [8]



Figure 5.4 Human body tensegrity model




REFERENCES
[1] http://www.ask.com/science/tension-force-5d179244f93b3d05
[2] http://theconstructor.org/structural-engg/types-of-tension-members/4800/
[3] A Practical Guide to Tensegrity Design 2nd edition Copyright 2004-2008 by Robert William Burkhardt
[4] Controversial Origins of Tensegrity by Valentín GÓMEZ JÁUREGUI
[5] Engineering MECHANICS, Vol. 21, 2014, No. 5, p. 355–367
[6] Overview of Tensegrity – I: Basic Structures by Bansod Y.D. et al.
[7] Tensegrity , Weaving And The Binary World by Kenneth Snelson

[8] http://bobwb.tripod.com/synergetics/tenseg