Zigzag-Strut Tensegrities Spencer Hunter, 2000 This document, unaltered, is in the public domain. Links to accompanying public-domain images are on my web page at http://www.u.arizona.edu/~shunter/cads.html under the "Zigzag-Strut Tensegrities" directory, or one level up from this document. Abstract: A deployable triangular tensegrity truss to support a panel dome is outlined, tracing the development of the idea to current research and future studies. I do not claim that the truss is necessarily novel, only that I have developed the idea independently. 1. Derivation of the idea: flexible hubs for geodesic grid domes Various proposals have been made for flexible hubs to build geodesic domes. These hubs allow the dome builder to be concerned only with strut lengths, since the hubs will automatically adjust to the proper angles regardless of where they are used in the dome. Some examples may be seen at: http://reality.sculptors.com/~salsbury/House/details.html#Hub_Designs http://www.gardendome.com/sn1.JPG These proposals rely on a certain amount of tooling up for mass production. I had an idea of using rolls of flexible material instead, such as rolls of paper for my soda straw models or rolls of chicken wire for large projects involving metal or PVC pipe struts. These rolls may be squashed, bent, drilled, and loosely bolted to produce flexible hubs. Why wait for someone else to manufacture them when you can literally "roll your own!" 2. The basic unit: tensegrity tetrahedron I began to test my idea with two pairs of soda straws, each pair being connected by a roll of 1"x2" paper. I noticed that if the straw pairs were oriented at a ninety-degree angle to each other and each pair bent at an angle of approximately 109.5 degrees (with the pairs bent in opposite directions), the straws formed the radials of a tetrahedron. Estimating the length of the edges of the tetrahedron by eye, I cut out six equal lengths of strapping tape and connected the loose ends of the straw pairs in a tetrahedral pattern. I then forced the straw pairs together at the center, and kept them in close proximity with a loop of twine. Since I had underestimated the edge lengths, the straw pairs do not actually touch each other, but remain in suspension near the center. The resulting tensegrity is very intriguing. It is not a tensegrity in the Snelson/Fuller sense of the word, that being "floating discontinuous compression, continuous tension," since each straw does connect with another, making the compression elements not entirely discontinuous. Also, the paper rolls and the loop of twine at the center comprise the sides of a dual tetrahedron, counterbalancing the tensional forces of the strapping tape tetrahedron surrounding it, so the tension elements are not entirely continuous either. However, it does fit Ariel Hanaor's more precise definition in the book edited by Francois Gabriel, _Beyond the Cube_ (New York ; Chichester, [England] : John Wiley, c1997), that being an "internally prestressed cable [or fabric] network." This definition excludes bicycle tires, most tents, cable domes, and the Solar System; but it includes many of Kenneth Snelson's sculptures, Fuller's tensegrity spheres, and most rigid kite designs. One of the first applications I thought of for this structure was, in fact, a light and strong cell for a tetrahedral kite, where two of the sides are replaced with fabric. The individual cells could be stacked and tied to each other to form the larger structure (I would hesitate to call the entire kite a tensegrity since the cable/fabric networks do not overlap). A reinforced slit could be made through the fabric of the top cell and the kite string tied to the center, which is the strongest part of the cell. Another application is to prestress panels of a geodesic or parabolic fabric dome, where alternating triangular panels are equipped with struts and tensioning tendons. The tops of the tetrahedral elements are connected by cables to each other, forming a kind of octet truss that would be highly resistant to imploding forces on the dome, though relatively non-resistant to exploding forces. How (or even if) such a structure would hold up in a gale would make a worthwhile experiment. 3. Square tensegrity truss I was most interested, though, in somehow turning this structure into some kind of tensegrity truss. Shortening the lengths of four selected tendons distorts the tetrahedron and lets the rolled strut joints intersect the remaining two edges, bisecting them into four half-length tendons while preserving the 109.5 degree angle between the struts. The loop of twine at the center becomes a separate vertical tendon that forces the structure into rigidity. Such a distortion allows tetrahedra and dual tetrahedra to overlap side by side, with their struts forming a zigzag pattern very much like a basket weave. Because the 90 degree angle between the two strut pairs is preserved in each distorted tetrahedral element, the overlapping tetras may be used to build square or rectangular trusses. 4. Triangular tensegrity truss I did not build a model of the square truss because I was anxious to try the idea out with a triangular version, obtained by altering the 90 degree angle between strut pairs in each tetra element to 60 degrees and adjusting the distorted tendon lengths accordingly. The length of the bisected edges and the 109.5 degree angle between struts are still preserved. The pattern of zigzagging struts here is identical to that of a triangular basket weave. I built a sample model twice using three overlapping tetras with the same disappointing result, as some of the tendons always remained slack. A curious figure popped up unexpectedly in the middle of the model, however: a five-sided triangular prism with struts comprising the diagonals of the rectangular sides, as though a tensegrity octahedron had been twisted to the point where six of its sides had collapsed into three. I had discovered (or more probably rediscovered) a tensegrity version of the octet truss. 5. Current research I quickly built the pseudo-octahedron out of three straws and strapping tape and discovered that it was indeed a tensegrity in its own right. Curiously, the "octa" tensegrity is very good at resisting horizontal stress but poor at resisting vertical stress, just the opposite of how the distorted tetra tensegrities perform; designing with both in mind should make them complementary. The existence of the "octa" also explains why some of the tendons in my models always remained slack--they simply weren't needed. Eliminating the unnecessary tendons produces an added benefit of allowing the entire truss to fold up like an accordion when the separate vertical tensioning tendons are removed. 6. Future studies I will next build a truss that supports six triangular panels. The strongest points of the panels would be connected to each other at the corners, whereas the truss would support the panels near their weakest points at the middle of each side. Some solid engineering data on how this truss performs, including its strength-to-weight ratio, is desired. Ultimately, a multi-tetra truss would be used to support a geodesic or parabolic panel dome. Although it would have roughly the same amount of strut material as a grid dome, it should have superior strength and prevent local problems like panel "pop-in" by reason of its three-dimensional geometry. The dream is to have a rapidly deployable structure that can be delivered and erected on-site by relatively unskilled labor who unfold the zigzag-strut tensegrity, install and tighten the vertical tendons, and attach the panels to complete the dome within hours.