Tensegrity


Tensegrity, tensional integrity or floating compression is a structural principle based on a system of isolated components under compression inside a network of continuous tension, and arranged in such a way that the compressed members do not touch each other while the prestressed tensioned members delineate the system spatially.
The term was coined by Buckminster Fuller in the 1960s as a portmanteau of "tensional integrity". The other denomination of tensegrity, floating compression, was used mainly by the constructivist artist Kenneth Snelson.

Concept

Tensegrity structures are based on the combination of a few simple design patterns:
Because of these patterns, no structural member experiences a bending moment and there are no shear stresses within the system. This can produce exceptionally strong and rigid structures for their mass and for the cross section of the components. The loading of at least some tensegrity structures causes an auxetic response and negative Poisson ratio, e.g. the T3-prism and 6-strut tensegrity icosahedron.
, 1951
A conceptual building block of tensegrity is seen in the 1951 Skylon. Six cables, three at each end, hold the tower in position. The three cables connected to the bottom "define" its location. The other three cables are simply keeping it vertical.
A three-rod tensegrity structure builds on this simpler structure: the ends of each green rod look like the top and bottom of the Skylon. As long as the angle between any two cables is smaller than 180°, the position of the rod is well defined. While three cables are the minimum required for stability, additional cables can be attached to each node for aesthetic purposes or to build in additional stability. For example, Snelson's Needle Tower uses a repeated pattern built using nodes that are connected to 5 cables each.
Eleanor Heartney points out visual transparency as an important aesthetic quality of these structures. Korkmaz et al. has argued that lightweight tensegrity structures are suitable for adaptive architecture.

Applications

Tensegrities saw increased application in architecture beginning in the 1960s, when Maciej Gintowt and Maciej Krasiński designed Spodek arena complex, as one of the first major structures to employ the principle of tensegrity. The roof uses an inclined surface held in check by a system of cables holding up its circumference. Tensegrity principles were also used in David Geiger's Seoul Olympic Gymnastics Arena, and the Georgia Dome. Tropicana Field, home of the Tampa Bay Rays major league baseball team, also has a dome roof supported by a large tensegrity structure.
– Brisbane
On 4 October 2009, the Kurilpa Bridge opened across the Brisbane River in Queensland, Australia. A multiple-mast, cable-stay structure based on the principles of tensegrity, it is currently the world's largest tensegrity bridge.
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Since the early 2000s, Tensegrities have also attracted the interest of roboticists due to their potential to design lightweight and resilient robots. Numerous researches have investigated tensegrity rovers, bio-mimicking robots, and modular soft robots. The most famous Tensegrity robot is the Super Ball, a rover for space exploration currently under developments at NASA Ames.

Biology

Biotensegrity, a term coined by Dr. Stephen Levin, is the application of tensegrity principles to biological structures. Biological structures such as muscles, bones, fascia, ligaments and tendons, or rigid and elastic cell membranes, are made strong by the unison of tensioned and compressed parts. The musculoskeletal system maintains tension in a continuous network of muscles and connective tissues, while the bones provide discontinuous compressive support. Even the human spine, which seems at first glance like a stack of vertebrae resting on each other, is actually a tensegrity structure.
Donald E. Ingber has developed a theory of tensegrity to describe numerous phenomena observed in molecular biology. For instance, the expressed shapes of cells, whether it be their reactions to applied pressure, interactions with substrates, etc., all can be mathematically modeled by representing the cell's cytoskeleton as a tensegrity. Furthermore, geometric patterns found throughout nature may also be understood based on applying the principles of tensegrity to the spontaneous self-assembly of compounds, proteins, and even organs. This view is supported by how the tension-compression interactions of tensegrity minimize material needed to maintain stability and achieve structural resiliency. Therefore, natural selection pressures would likely favor biological systems organized in a tensegrity manner.
As Ingber explains:
In embryology, Richard Gordon proposed that Embryonic differentiation waves are propagated by an 'organelle of differentiation' where the cytoskeleton is assembled in a bistable tensegrity structure at the apical end of cells called the 'cell state splitter'.

History

The origins of tensegrity are controversial. Many traditional structures, such as skin-on-frame kayaks and shōji, use tension and compression elements in a similar fashion.
In 1948, artist Kenneth Snelson produced his innovative "X-Piece" after artistic explorations at Black Mountain College and elsewhere. Some years later, the term "tensegrity" was coined by Fuller, who is best known for his geodesic domes. Throughout his career, Fuller had experimented with incorporating tensile components in his work, such as in the framing of his dymaxion houses.
Snelson's 1948 innovation spurred Fuller to immediately commission a mast from Snelson. In 1949, Fuller developed a tensegrity-icosahedron based on the technology, and he and his students quickly developed further structures and applied the technology to building domes. After a hiatus, Snelson also went on to produce a plethora of sculptures based on tensegrity concepts. His main body of work began in 1959 when a pivotal exhibition at the Museum of Modern Art took place. At the MOMA exhibition, Fuller had shown the mast and some of his other work. At this exhibition, Snelson, after a discussion with Fuller and the exhibition organizers regarding credit for the mast, also displayed some work in a vitrine.
Snelson's best known piece is his 18-meter-high Needle Tower of 1968.
Russian artist Viatcheslav Koleichuk claimed that the idea of tensegrity was invented first by Kārlis Johansons, a Soviet avant-garde artist of Latvian descent, who contributed some works to the main exhibition of Russian constructivism in 1921. Koleichuk's claim was backed up by Maria Gough for one of the works at the 1921 constructivist exhibition. Snelson has acknowledged the constructivists as an influence for his work. French engineer David Georges Emmerich has also noted how Kārlis Johansons's work seemed to foresee tensegrity concepts.

Stability

Tensegrity prisms

The three-rod tensegrity structure has the property that, for a given length of compression member “rod” and a given length of tension cable “tendon” connecting the rod ends together, there is a particular value for the length of the tendon connecting the rod tops with the neighboring rod bottoms that causes the structure to hold a stable shape. For such a structure, it is straightforward to prove that the triangle formed by the rod tops and that formed by the rod bottoms are rotated with respect to each other by an angle of 5π/6.
The stability of several 2-stage tensegrity structures are analyzed by Sultan, et al.

Tensegrity icosahedra

The polyhedron which corresponds directly to the geometry of the tensegrity icosahedron is called the Jessen's icosahedron. Its spherical dynamics were of special interest to Buckminster Fuller, who referred to its expansion-contraction transformations around a stable equilibrium as jitterbug motion.
The following is a mathematical model for figures related to the tensegrity icosahedron, explaining why it is a stable construction, albeit with infinitesimal mobility.
Consider a cube of side length 2d, centered at the origin. Place a strut of length 2l in the plane of each cube face, such that each strut is parallel to one edge of the face and is centered on the face. Moreover, each strut should be parallel to the strut on the opposite face of the cube, but orthogonal to all other struts. If the Cartesian coordinates of one strut are and, those of its parallel strut will be, respectively, and. The coordinates of the other strut ends are obtained by permuting the coordinates, e.g., → → .
The distance s between any two neighboring vertices and is
Imagine this figure built from struts of given length 2l and tendons of given length s, with. The relation tells us there are two possible values for d: one realized by pushing the struts together, the other by pulling them apart. For example, for the minimal figure is a regular octahedron and the maximal figure is a quasiregular cubeoctahedron. In the case we have s = 2d, so the convex hull of the golden ratio figure is a regular icosahedron. Since no article on the kinematics of polytopes would be complete without a Coxeter reference, it is appropriate to note here that by 1940 Coxeter had already shown how the twelve vertices of the icosahedron can be obtained by dividing the twelve edges of an octahedron according to the golden ratio, as one of the continuous series of icosahedra with faces consisting of eight equilateral triangles and twelve isosceles triangles, ranging from cuboctahedron to octahedron, that can be produced by such a process of division.
In the particular case the two extremes coincide, and, therefore the figure is the stable tensegrity icosahedron.
Since the tensegrity icosahedron represents an extremal point of the above relation, it has infinitesimal mobility: a small change in the length s of the tendon results in a much larger change of the distance 2d of the struts.

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