Tensegrity - Elastically Bent Tensegrity Structures

Elastically Bent Tensegrity Structures

by Pablo Antu単a Molina

Table of Content Introduction

Tensegrity and Active Bending Concepts - Analogies History

Tensegrity Nowadays

Definitions

Historical Examples

Historical Ways of Experimentation

Pros and Cons of Standard Tensegrity

Pros and Cons of E-Bent Tensegrity Regular Polyhedra Categorization

Cable Categorization Nomenclature

Form Categorization Parametric Modeling Insights Potentials Weaknesses

Possible Applications

Elastically Bent Tensegrity Structures

Introduction The main reason of this research is the complete understanding of tensegrity structures and their basic principles in order to keep developing this structural typology and pursue bigger and newer goals.

For this study we have introduced two new variables: the use of active bending principles for the compression elements and the substitution of the tensors for membranes.

Active bending is an elastic deformation that generates tension in the struts as they will try to get a non-deformed stable position. The deformation has three downsides: Firstly, as time passes by the material loses elasticity and the elastic deformation can turn plastic which might end in fatigue fracture. Secondly, the curved compression struts are unstable and will rotate, that is why another fixation point is needed. Preferably in the middle of the bent arch. Thirdly, bending creates a momentum that will be hard to control and that we will need to take into consideration since it can cause the structure’s collapse.

The materials used for the compression elements should have high yield strength and good fatigue resistance to ensure structural durability. Metal, glass fiber, wood and some plastics are good materials to work with.

The second variable is the use of membranes to support tension loads instead of tensors. This material bears the structural effort with its whole surface and enables the possibility of controlling the momentum by wrapping the bent compression elements.

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This research focuses mainly in the experimentation and categorisation of Elastically Bent Tensegrity Structures (EBTS), laying the foundations for further experimentation with membranes.

Tensegrity and Active Bending Concepts Analogies First of all, we need to understand what Tensegrity means, how it can be defined and which the basic principles are. This will help us differentiate between tensegrity structures and pre-stressed non-tensegrity structures.

Tensegrity is a term created by Buckminster Fuller in 1955 from the contraction of “Tensional Integrity” for referring to a basic structural principle in which isolated compression elements are floating in a tension welter. This means that compression components do not touch each other and are joined only by traction components which define the whole system spatially.

This, apparently vague, definition allows us to understand the Tensegrity concept through a great variety of different dayto-day case scenarios beyond building structures, whose boundaries are limited by the usual compression struts and metal tensors.

As I have said, the best way to understand the tensional integrity or floating compression is by examples and analogies.

The most immediate and direct simile was created by Kenneth Snelson and explained in the article “Tensegrity, Weaving and The Binary World” [4]: “Tensegrity structures

Fig. 2: Serpentine Gallery Pavilion 2006, Rem Koolhaas and Cecil Balmond. are endoskeletal, as are humans and other mammals whose tension “muscles” are external to the compression members’ bones”. Even though this first example is quite graphical, it is not precise enough due to the fact that in the mammal’s bones-muscles’ structure the compression elements’ bones have junctions between them, touching each other. (Fig. 3) Another good analogy to understand what Tensegrity means relates it with pneumatic structures. There is a theory that considers any inflatable construction as tensegrty, because, once it is completely inflated and its inner pressure is higher than the outer one, it will absorb any stress as a unit and the whole surface will deform itself, getting back to its equilibrium-state once it is released from every load – as stepping on an inflated ball. Pneumatic structures are self-balanced systems composed by a tensed external surface which encloses gas atoms that behave as isolated compression components. Both, tensegrity and pneumatic structures are compressible, expanding, elastic, light, self-balanced and able to distribute local stress.

The easiest way to understand the active bending concept is by comparing it with a bow. This device, in its simplest configuration, is composed by two parts: the rigid wooden part and the string. The force transmitted to the arrow derives from the come back to the non-tensioned equilibrium state of both parts. This deformation has been also used traditionally for construction.

(Fig. 1)

Fig. 1: Mudhif, traditional madan house.

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Fig. 3: Tensegrity Rabbit.

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History

Tensegrity Nowadays

Once the Tensegrity concept is understood in a visual and intuitive way is time to explain the history of its discovery and investigation and the controversy generated over its authorship. There are three names that need to be mentioned, if not as discoverers, as principal promoters of this structural typology: Richard Buckminster Fuller (1895-1983), David Georges Emmerich (1925-1996) and Kenneth D. Snelson (1927).

Nowadays Tensegrity research has taken three different paths:

Even though all of them claimed the authorship of the Tensegrity discovery, it was Emmerich who found the first tensegrity prototype, named “Gleichgewichtkonstruktion” (Fig. 4), designed by Karl Ioganson (1895-1929) – a Latvian constructivist artist from the USSR – in 1920. After this discovery, Emmerich started investigating new structural typologies based on tightened prisms and more complex tensegrity systems that he would name “structures tendues et autotendants” or tensioned and prestressed structures. As a result, he defined and patented the “self-pre-stressed nets”.

- The first one is focused on the active control and computational calculation of the structures, trying to predict its behaviour in a great variety of situations and conditions.

- The second one studies the behaviour of the structural typology while moving, submitted to dynamic loads and vibrations.

This has further applications in robotics and spatial engineering, being tested right now by associations like the NASA. (Fig. 5) - The third and the most spread one searches different ways of form-finding strategies varying the materials, introducing new ones and trying to optimize the typology. It is difficult to know when this way of developing these structures is useful or just purely formal and the sense lies on satisfying the creator’s intellect. (Fig. 6) This research belongs to this last way of developing tensegrities and the results of it will be analysed in the conclusion.

Fig. 6: Parametric Tensegrity Structure from students of Ball State, Indiana (USA)

Meanwhile, on the other side of the Atlantic Ocean, Snelson and Fuller would discover the same type of structures thanks to the combination of Fuller’s theoretical ideas and knowledge in physics and math and Snelson’s experimentation. The authorship of the discoveries was never clear, leading to a great dispute that was solved in 1959 when Fuller recognized Snelson’s contribution in the development of tensegrity structures. Both would patent many discoveries in Tensegrity during their lives. [1]

Fig. 4: “Gleichgewichtkonstruktion”, Simplex

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Fig. 5: NASA tensegrity landing device

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Definitions As we have seen, it is quite difficult to define Tensegrity and its boundaries. That is why David Emmerich tried to describe it in his multiple patents in a restrictive way, giving examples where Tensegrity appears and including drawings of some of them to explain it. The definition of Tensegrity appeared later and would change many times, while discoveries were made, before getting to the actual definition.

The first author that tried to put into words what Tensegrity means was Buckminster Fuller in his book “Synergetics” (1975): “Tensegrity defines a structural-relation principle in which the form of the structure is ensured by the continuous and finitely closed behaviour of the tensioned elements in the system and not by the discontinuous and localised behaviour of its compression elements.”

Fig. 7: Buckminster Fuller and his Tensegrity Geodesic Sphere.

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patents” and the “extended” definition. The last one is quite similar to Pugh’s but with some nuances: the compression elements are within the tension net and the system is self-balanced with no need of external loads. As a result in 2003 Motro gave this definition: “A tensegrity system is a balanced system, self-stable, that contains a discontinuous set of compression components within a continuous set of traction components.” [1] This last definition is the one used in this research and that will be adapted in order to introduce the two variables mentioned: active bending and membranes.

Fig. 8: Bead Chain Tower, Kenneth Snelson

This futuristic-looking, slender, vertical art structure was located by the Thames and the goal was to create a “vertical feature” that apparently floated in the air. The idea was designed by Hidalgo Moya and Phillip Powell and the duty to make it possible was given to Felix Samuely. The tower was made of six cables, three pillars and a large steel piece that “flew” over the ground.

This project was an important predecessor of floating-compression structures that encouraged new ways of experimentation.

Fig. 9: Skylon, Festival of Britain, 1951

It is quite complicated to define and get a complete understanding of tensegrity.

In order to experiment we need to set ourselves new rules that derive from previous experimentation or, at the beginning, from common sense.

In tensegrity history there are three different ways of studying and experimenting: - Use the platonic and archimedean solids to create geometrical forms that approach to the principal platonic solid, the sphere. This will lead to new rules for complex curved tensegrity structures.

(Fig. 10)

A year later, Anthony Pugh described Tensegrity in a really clear and precise way that was accepted almost worldwide: “A tensegrity system is established when a discontinuous set of compression components interacts with a continuous set of traction components, defining a closed volume in the space.” (1976)

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Skylon – Festival of Britain 1951, London (Fig. 9)

Historical ways of experimentation

Kenneth Snelson, who prefers calling it as “floating compression”, says: “Tensegrity describes a closed structural system formed by a set of three or more compressed struts within a net of stressed tensors (elements), both parts are mutually combined so that the struts do not touch each other but push to the outside against the nodes of the tensioned net for forming a firm, triangulated and pre-stressed compressionand-tension unit.”

René Motro, a French engineer that was introduced to Tensegrity by reading Emmerich’s work, made a distinction between a definition “based on the different

Historical examples

- Understand tensegrity modules as part of something bigger and work with them to create new systems and greater units. This satisfies the architectural idea of filling a great space with small repeatable pieces.

Fig. 10: Tensegrity Geodesic Sphere

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- It also exists a chaotic-artistic way of working in which, with a large knowledge in Tensegrity, small heterogeneous structures can be designed. The complexity lies in the lack of structurally-reasonable rules and the asymmetry of the “small” pieces of art, these lead to structural instability for bigger constructions. (Fig. 11) Buckminster Fuller could be considered as the greatest representative of the first mentioned way of studying tensegrity structures. He created the geodesic sphere/dome and, after studying and defining the Tensegrity, he designed the tensegrity geodesic sphere/dome that was a cardinal achievement in curved tensegrity structures. It has a multi-layered external surface that allows the correct load-bearing.

For the second and third possibilities of studying tensegrities the main author is Kenneth Snelson. He started studying the most primitive and ancient example of prestressed structures, the kite. This simple two-dimensional construction made by two struts forming a cross held together and stiffened by a tensioned string cannot be considered as a real tensegrity because the compression members touch each other, but it was the basis of his X-module (Fig. 13). With this module he found a way to build towers and planes filling the three-dimensional space (Fig. 12). After its exhaustive study, he designed the simplest real tensegrity structure, the “Simplex”, whose further developments would give birth to a whole new set of tensegrity modules that followed the Platonic Solids.

There is also a great variety of art work created by Snelson that applies the Tensegrity rules in an artistic way. Some of these pieces of art are stable and could have a further development and others are structurally unreasonable and impossibly developable.

Fig. 11: B-tree II, Kenneth Snelson

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Pros and Cons of Standard Tensegrity Pros

Cons

- The system bears local stress as a unit.

The structure is deformed globally.

- There might appear strut congestion when tensegrities grow in scale and the elements could disturb each other.

- Because the compression elements are isolated and the point loads are placed in the tips of the struts, there exists no possibility of unintended local torsion or bending stress, unless the system collapses.

- They are economic structures because the whole unit bears the loads. It has a great resistance with small amount of material and little weight.

- The self-equilibrium is a key point for this structural typology. Its performance does not depend on gravity or any external forces. They are stable in any position with no anchoring needed, unless they have to be fixed. In this case vertical supports are enough.

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Self-found equilibrium of the structure.

- “High structural deformation and little material efficiency in comparison with conventional geometrically-rigid structures”–Ariel Hanaor (specialist in tensegrity structures). [1] - Complicated and difficult manufacturing and assembling.

- There is a need of machinery for supplying the pre-stressed forces required in the structure’s construction. The bigger the structure gets, the more pre-stressed forces needed, the more machinery.

- Structural calculation is quite difficult due to geometrical complexity. Also, the shape is dependant on the structural performance. It hinders the design of complex structures.

- Complex structures can be created from easy modules, enabled partly by the fact that the structure can work as scaffolding for itself.

- Foldable systems can be design due to the small energy needed, once they are built, for making them change their form.

- The structure deforms itself as a whole and returns to the initial position when released from the loads.

- If there is a vibration (earthquakes, volcanic eruptions, landings…), structural thrust is quickly distributed. [1]

Fig. 12: DCCT Towers, Kenneth Snelson

Fig. 13: WoodX-Column, Kenneth Snelson

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Pros and Cons of E-Bent Tensegrity

Regular Polyhedra

Pros

As I said before, historically there have been many ways of experimenting with tensegrities. One of them was to relate tensegrity structures with the different platonic solids and the different groups of polyhedra described during history. These latter groups of polyhedra are the dual solids, the archimedean solids and the geodesic structures.

- Because of active bending, tensors are immediately and automatically pre-stressed.

- The possible combination with membranes would control easily the momentum, creating a very efficient hybrid system.

Cons

The platonic solids group is composed by five polyhedrons: tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron, which have respectively four, six, eight, twelve and twenty sides.

- Momentum of the bent struts can collapse the system. This problem is solved by adding extra cables or extra nodes.

If we take the central points of each side and we unite them with the rest of the central points we create the dual solids.

These solids are also platonic, but there is a geometrical relationship between both figures in which one is the dual solid of the other and vice versa. The dual solid of a tetrahedron is itself, the dual solid of a cube is an octahedron and the dual solid of a dodecahedron is an icosahedron.

- The solution to the first problem makes the structure geometrically more complex.

In tensegrity, there is a way to create a structure that evolves from the main platonic solid to its dual solid. This method

Fig. 14: Platonic solids and their Dual solids

does not follow strictly the tensegrity rules and is based on the ideas of the geodesic spheres, in which the figures are made by triangulation.

The third group, the archimedean solids, is obtained by truncating the vertices of the platonic solids and also of the first grade archimedean solids. This creates high symmetrical semi-regular convex polyhedrons composed by more than one type of regular polygons that meet in identical vertices. The archimedean solids have also a great inner geometrical complexity with interlacing polygons.

This allows the creation of more complex tensegrity systems.

Finally we get to the geodesic solids.

The goal for this type is to create a figure as spherical as possible using triangles or rhombic polygons. For achieving this purpose the frequency property is key. It is a subdivision of the different polygonal faces in smaller pieces with the same shape.

All these types of polyhedrons can be built in tensegrity following the ideas explained in the categorization chapter that follows. In this research we focus on the exploration of platonic and simple archimedean solids.

A

B

C

D

E

F

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Fig. 15: Archimedean solids and their possible tensegrity homonymous structures A - Truncated Tetrahedron B - Truncated Octahedron C - Truncated Cube D - Cuboctahedron E - Truncated Dodecahedron F - Rhombicuboctahedron

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Categorization

Nomenclature

Cable Categorization

It is important to create a nomenclature that allows us to describe the geometry of a e-bent tensegrity in a direct proper written way. For this matter the author understands the nomenclature for tensegrities and adapts it to the topic. The nomenclature now used was created by René Motro in 2003 and has the following parameters:

This research is focused on setting the foundations for further experimentation with membranes instead of cables. Because of this, it is important to understand which are the main tasks of the different cables and see how they can be substituted for membranes. In other words, there is a need to categorize the cables and to know which ones and in which proportion they can be substituted.

A simple tensegrity module, as Kenneth Snelson discovered, needs three cables per tip to be stable. In the case of e-bent tensegrities appears the need of a junction in the middle of the compression element that joins it with a steady element and stops its momentum. Following these ideas tensors can be categorised in:

n: number of nodes.

S: number of struts.

C: cables or tensile components.

R: regular system or I: irregular system.

SS: spherical system, homeomorphic to a sphere.

a

Nomenclature example: [n6-S2-i-C8-R-SS] Edge tensors Momentum-stopping tensors Separating tensors Drawing tensors

Following this nomenclature a Simplex would be expressed as “n6-S3-C9-R-SS”.

(Fig. 4) These rules look simple and work perfectly for describing standard tensegrity geometry, but once the bending is introduced, two new variables appear. Firstly, the parameter C gets two new params described before with the classification of the cables and, secondly, the direction of the bent strut needs to be described in order to make a complete overview of the geometry. The authors proposal to this nomenclature would be:

- Edge tensors: this type makes each module stable by pulling the tips together in every direction and defines/closes its volume.

- Momentum-stopping tensors: their mission is simple, stop the momentum.

Edge and momentum tensors belong just to a single module, once two modules are joined two new types of tension cables appear:

n: number of nodes, including also the middle nodes that stop the momentum.

S: number of bent struts. t: the plane created by the bent strut remains in the perimeter of the figure, i: the curvature‘s direction of the compression elements points the centre of the figure or the direction crosses the centre, o: the direction of the curvature points outwards.

(Fig. 17) C: number of cables, all types of cables.

R: regular or I: irregular.

SS: spherical system.

(Fig. 16 a)

- Separating tensors: they create a physical boundary between the modules so that they do not touch each other, ensuring the Tensegrity principles. This typology replaces the edge tensors of the joined modules in the virtual intersection surface.

- Drawing tensors: they are always redundant, so in other words, not necessary. Nevertheless, sometimes the junction between two modules might be not as stable as it should. In this cases the drawing tensors are needed to link the modules with an extra force that pulls them together and equilibrates them.

(Fig. 16)

b Fig. 16 (a,b): Cable Categorization and Nomenclature example

o i t

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i

Fig. 17: Nomenclature explanation

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Elastically Bent Tensegrity Structures

Form Categorization After building a good number of models it is easy to realize that some of them follow similar rules for their formation, geometry and elements.

We cannot forget that the typology now studied is part of a huge family, the tensegrities, and because of this, it is important to understand the categorisation in which tensegrities are now classified so to locate the different e-bent tensegrities in the different pre-established groups. With the possibility of adding some nuances to this classification in order to fully embrace the new typology.

Although there have been some categorisations during the history of tensegrity structures, the first person that tried to make an exhaustive categorisation was Anthony Pugh. After some experimentation he described the three main patterns for creating a spherical or cylindrical tensegrity structure: the diamond pattern, the circuit pattern and the zigzag pattern. Even though this classification was made exclusively for polyhedra it is quite useful and if we combine it with Motro’s

Fig. 18: E-Bent Tensegrity Prisms a: eBT-prism vertices outside [n9-S3-o-C15-R-SS]

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book “Tensegrity: Structural Systems for the Future”, as Valentín Gómez Jáuregui does in his thesis “Tensegrity Structures and their application to Architecture” [2], a clear categorisation of tensegrities can be made. a)

Spherical Systems

These systems are homeomorphic to a sphere. In standard tensegrities, all cables can be mapped on a sphere without intersections between them and all struts are inside the cable net. In e-bent tensegrities, for now, this rule is also fulfilled but it is not sure that after further development it will.

Within this set we find three subsets: - Rhombic Configuration Systems This configuration derives from Pugh’s diamond pattern and it is based on a rhombus system composed by one strut and four tensors in which the strut represents the longest diagonal of the rhombus.

Tensegrity prisms (T-prisms) are included in this section. Prismatic tensegrities are generated from a straight prism in which

b: eBT-prism braid-like shape [n9-S3-i-C15-R]

cables form the edges of the shape and the struts are the diagonals that join the vertices from both polygonal bases and are located inside the volume.

In e-bent tensegrities, eBT-prisms do not necessarily follow this configuration of vertical/horizontal cables and diagonal struts. In fact, there are three ways that the author has found to create an eBT-prism.

Firstly, there is the possibility to follow the same rules that in T-prisms and take the vertices of the bent struts to the outside, needing an additional set of cables that would stop the momentum (Fig. 18 a).

Secondly, the vertices of the struts can be taken to the inside and, if desired, the new set of cables can be avoidided linking the struts to the tensors that create the bending in a braid-like shape (Fig. 18 b). Thirdly, the struts and tensors can be located in the vertical edges of the prism and the struts‘ vertices can be joined to the cables of the following edges (Fig.18 c). This last option does not belong to the Rhombic Configuration System. There is a possibility that it creates a new type, but it needs to be further investigated.

c: eBT-prism tangent [n9-S3-t-C12-R-SS]

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eBT Polyhedra – Icosahedron: A Standard Tensegrity Icosahedron is taken and modified pushing the vertices of the bent struts to the inside as seen in the categorization and creating an E-Bent Tensegrity Icosahedron.

The obtained figure, which is highly symmetrical, controls the problematic momentum of the struts in the inside appearing a central core in the structure.

Applying membranes to this configuration would be complicated as it is impossible to wrap the struts for controlling the momentum without disturbing other membranes. An option would be to use these for the external surfaces and cables for the internal core.

Nomenclature: [n18-S6-i-C36-R-SS]

Fig. 19, 20, 21 (left side, top to bottom): Different perspectives of the built model Fig. 22 (up): Floor plan of the structure generated with Rhinoceros and Nomenclature

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eBT Polyhedra – Icosahedron: In this case, for creating the E-Bent Tensegrity Icosahedron the vertices of the struts are pulled to the outside, obtaining a spherical-like figure that tries to cover a big volume with little external surface.

For this second option applying membranes would be a good idea since the struts can be easily wrapped for stopping the momentum.

Nomenclature: [n18-S6-o-C36-R-SS]

Fig. 23, 24, 25 (left side, top to bottom): Different perspectives of the built model Fig. 26 (up): Floor plan of the structure generated with Rhinoceros and Nomenclature

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- Circuit Configuration Systems In this case, compression components form circuits, which would mean that they are not “real” tensegrities as the struts touch each other. To this set belong some geodesic tensegrity spheres.

In e-bent tensegrities these circuit systems could be quite close to a sphere, remembering the platonic solids idea.

This configuration evolves from the rhombic configuration and is, with the same number of struts, much more rigid. eBT Polyhedra - Deployable Octahedron: The bending of the double struts is what allows the creation of the interlacing circuits and not the geometry of the figure. That is why the author prefers to name it as a particular case of the Circuit Configuration.

eBT Polyhedra - Cuboctahedron: Circuit Configuration eBT Cuboctahedron created with interlaced inner triangles. As known, the squares are less stable than the triangles. This, the uncontrolled momentum and the undesired torsion are the reasons why the model turns into an Icosahedron and loose tension in some cables.

In this case, due to the complex inner geometry of the figure, it would be difficult to use only membranes avoiding any cable.

Nomenclature: [n24-S12-o-C36-R] Fig. 30 (right): Explanation of structure’s inner geometry Fig. 31, 32 (below): Differente Perspectives of the built model

Nomenclature: [n12-S3(6)-o-C6/18-R-SS]

Fig. 27: Folded EBTS Octahedron

Fig. 28: Fixed EBTS Octahedron

Fig. 29: EBTS Octahedron, aerial view

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- Zigzag Configuration Systems The zigzag configuration, also called Type Z, depends on the position of the cables.

When having a rhombic configuration, for obtaining a zigzag one, some cables need to be changed so that three non-aligned tendons form a Z. The substitution of the cables has to preserve the stability of the system.

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In bent and e-bent tensegrity building a star system with a central node would underline the idea, together with the bent struts, of spherical geometry based structure.

Cylindrical systems can also be formed as an evolution of the eBT-prisms.

In e-bent tensegrities this configuration cannot be obtained only by changing the edge cables as the momentum nodes need to be also changed and previously meditated. b)

Star and Cylindrical Systems

Both are derivations of the previous class.

For the first type, if we introduce a strut in the centre of a rhombic configuration, for example, that follows the main symmetry axis and we link it to the rest of the cables a star system is obtained. This can also be created by introducing a spherical node in the centre of the structure.

A cylindrical system is created by adding other levels of struts to the initial one, depending on the number of subsequent levels the tower would be more or less high.

Fig. 33: Tensegrity Cylindrical System

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Fig. 34: Bent Tensegrity Star System c)

Irregular Systems

Parametric Modeling At the beginning of the research, the models built were simple and based more on intuition than on precision. There was no need for having an absolute control on geometry and measurements.

Nevertheless, when the models’ complexity grew, the necessity of precise calculation appeared in order to predict the structural behaviour.

In the first stages of the research the software Rhinoceros was used for creating simple 3D models that outlined the desired geometry. At that point, the number of variables was low, having two or three struts and the tensors needed for keeping the system’s equilibrium. When the investigation with platonic solids and more complex systems started it became necessary the use of a tool that could give realism to the 3D Rhinoceros models. For this task the plug-in Kangaroo was used.

In this set we find normal tensegrity figures that have not been classified within the previous categorizations – for example, most of Kenneth Snelson tensegrity sculptures – and with the further development of e-bent tensegrities some of the structures will form part of this group.

Henceforth, the complexity in the relations between the different variables of the models grew, representing the reality in a more appropriate and accurate way in most of the cases.

There is also a further categorization for more complex systems based on simpler tensegrity modules that can be followed in the next steps of this investigation, but for now these fields are vast enough to experiment on.

Fig. 35, 36, 37 (below, top row): Modeling for different EBTS Prism possibilities Fig. 38, 39, 40 (below, bottom row): Modeling for different EBTS Polyhedra, Icosahedrons and Cuboctahedron

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Insights After developing this brief introduction on Elastically Bent Tensegrity Structures it is important to understand how it has broadened the boundaries of the Tensegrity family and which could be the potentials and weaknesses of this structural typology, as well as the possible future uses.

Potentials The major benefit of the active bending is how it solves the pre-stress problem of the structures. If all the struts are bent to a higher point than they would be in their final state, all the cable-nodes can be easily made and, just at the end, the over-compressed bars would be released.

At this point the whole structure would get pre-stressed acquiring the final shape and, more importantly, gaining its selfequilibrium.

Firstly, in the case of a tensegrity that would be designed for moving and rolling, the spherical-shape structures could help to stabilize the movement and make it more fluent. The possible downside would be the reduction of the friction coefficient since the curved edges would not grip so easily to the ground.

If we implement the use of membranes and the deployable structures principles to this typology, short-lived pavilions could be designed for fairs and one-time events.

This idea can be applied to a larger scale creating small shelter communities for human catastrophes – such as wars or epidemics – or natural calamities – the tensegrity structures, due to their stable self-equilibrium, can overcome any kind of earthquake or tremor. E-Bent tensegrity principles could also be used for lightweight roofs that would not need to bear important loads.

This means that there is no need for specialised machinery for tensing the cables. So this kind of structures can be built in a less sophisticated way, given the fact that bending has been used in construction for a long time. (Fig. 1) Another benefit lies in the geometry of the bent struts. The arc as the basic geometrical shape creates, if the curvature’s vertex tends to the outside, immersive inner spaces that have a lower shape coefficient than if straight elements are used. If we add the possibility of using membranes, this typology would be quite appropriate for creating temporal shelters, for example.

Weaknesses The biggest problem seen in this tensegrity typology, beyond the possible fatigue fracture and the deformation, is the momentum. It can be solved, but complicates the geometry of the structure and its calculation. This hinders the possible widespread use for solving a common problem and a personalized design for solving an isolated case.

Possible Applications As said at the beginning, it is not completely necessary to find a usage for this typology beyond the broadening of knowledge and the creation of possible sculptural and structural highlights. Anyhow, is the author’s belief that this typology is applicable to the current ways of experimentation in tensegrity and to problems that will always have the human beings.

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Fig. 41: Structural Skin for Emergency Shelters, Abeer Seikaly

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Bibliography

List of Figures

[1] Gómez Jáuregui, Valentín: Tensegridad, Estructuras de compresión flotante, Madrid 2008, (ISBN 978-8489670-62-4)

The figures not included in this list have been created by the author.

[2] Gómez Jáuregui, Valentín: Tensegrity Structures and their Application to Architecture, School of Architecture, Queen‘s University, MSc in Architecture, Belfast 2004, http://www.tensegridad.es/Publications/MSc_Thesis-Tensegrity_Structures_and_their_Application_to_Architecture_by_GOMEZJAUREGUI.pdf [3] Paul, Chandana; Lipson, Hop; Valero Cuevas, Francisco, Evolutionary Form-Finding of Tensegrity Structures, Mechanicall and Aerospace Engineering, Cornell University, http:// creativemachines.cornell.edu/papers/ GECCO05_Paul.pdf [4] Snelson, Kenneth: Tensegrity, Weaving and the Binary World, http://kennethsnelson.net/tensegrity/ [5] Snelson, Kenneth: The Art of Tensegrity, reprinted from International Journal of space Structures, volume 27, number 2&3, Multi-Science Publishing Co. LTD., Brentwood (UK) 2012, http:// kennethsnelson.net/articles/TheArtOfTensegrityArticle.pdf

1 https://es.pinterest.com/ pin/294141419391768141/ from http:// pyramidbeach.com, P.2 2

http://www.dailyicon.net/2008/08/ serpentine-gallery-pavilion-by-remkoolhaas/, 2008, P. 2

3

https://ittcs.files.wordpress. com/2010/07/img_0265.jpg, P. 2

4

Manske, Magnus: Tensegrity, in: Daytar Group: Randform - Blog of maths, physics, art and design, 2004, P. 3

5

Sunspiral, https://commons.wikimedia. org/wiki/File:NASA_SUPERball_Tensegrity_Lander_Prototype.jpg, P. 3

6

Riether, Gernot: Students of Ball State Construct Parametric Tensegrity Structure, http://www.archdaily.com/553311/ students-of-ball-state-construct-parametric-tensegrity-structure-for-local-artfair/542db7bac07a809a0e00043e_students-of-ball-state-construct-parametric-tensegrity-structure-for-local-artfair_14_detail-jpg/, P. 3

14 Cundy and Rollett 1989, Table II following p. 144, http://mathworld.wolfram. com/DualPolyhedron.html, P. 8 15 Bardzik, Andrew: Experimental Tensegrites: Inscribed within Archimedean Solids, 2014, http://pr2014.aaschool. ac.uk/DIP-03/Andrew-Bardzik, P. 8 33 Gómez Jáuregui, Valentín: Tensegrity Structures and their Application to Architecture, 2004, http://www. tensegridad.es/Publications/MSc_ Thesis-Tensegrity_Structures_and_ their_Application_to_Architecture_by_ GOMEZ-JAUREGUI.pdf, P. 11 34 http://tensegrity.wikispaces. com/6+struts 41 http://www.plataformaarquitectura.cl/ cl/625200/en-detalle-tejido-estructural-para-refugios-de-emergencia

7 http://www.pbs.org/wnet/ americanmasters/r-buckminster-fullerabout-r-buckminster-fuller/599/, P. 4 8

Snelson, Kenneth: Bed Chain Tower, The Art of Tensegrity [5], P. 4

9 https://lightwater.wordpress. com/2014/03/11/aspects-on-thetowers-in-the-1951-festival-of-britain/, P. 4 10 http://www.artvalue.com/auctionresult-fuller-buckminster-20-usa-90-struttensegrity-geodesic-d-2524996.htm, P. 4 11 Hebert, William J., http://www.meijergardens.org/explore/b-tree-ii/, P. 5 12 Snelson, Kenneth: DCCT Towers, The Art of Tensegrity [5], P. 5 13 Snelson, Kenneth: WoodX-Column, http://republicofnowhere.tumblr.com/ post/2862355498/early-x-piece-madeof-wood-and-nylon-by-kenneth, P. 5

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Structural Research