Zigzag-Strut Tensegrities

by 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.

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



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 tetrahedrons and dual tetrahedrons
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.

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