A Practical Guide to Tensegrity Design

Table of Contents

Appendix A

Other Double-Layer Technologies

Two other approaches to planar tensegrity truss
design similar to the approach outlined in
Chapter 5
have been independently developed by Kenneth
Snelson^{1}
and a group of authors consisting of
David Georges Emmerich^{2},
Ariel Hanaor^{3}
and René Motro.^{4}
For the most part, the tendon network for the outer
and inner layers of these structures is identical with the double-layer
truss described in Chapter 5.
However, the way members are connected between
the layers is different, and neither of these alternative approaches
have been elaborated for use in a spherical context by their authors.

A planar truss from Kenneth Snelson's work is the most similar to the trusses exhibited in Chapter 5. It only differs in the number and connection of the interlayer tendons. Figure A.1 exhibits a rendering of Snelson's truss as it would appear in an infinite planar context. For each strut, there is a corresponding interlayer tendon which makes a fairly direct connection between the two layers and skirts the strut closely. The close approach of the strut and the interlayer tendon makes the use of the truss in a spherical context difficult due to interference problems. A more usable approach for spherical contexts might be obtainable by threading the interlayer tendon through the middle of the corresponding strut.

Figure A.1: Snelson's Planar Truss |

The starting point of Emmerich *et al.*'s system is a planar assembly
of t-prisms.^{5}
Figure A.2
shows how such a planar assembly would
appear.^{6}
As is evident in comparing Figures
2.1 and
5.1, the topological
difference between a t-prism and
a t-tripod is not great: the t-prism is topologically equivalent
to a t-tripod with a tendon
triangle connecting the struts at the t-tripod's
base.^{7}
While the manner of Emmerich *et al.*'s
assembly retains triangulation in the
outer and inner tendon layers, the interlayer triangulation
exhibited by individual prisms is broken up by the arrangement.
In this situation a shallow dome can be induced by introducing
curvature in the planar assembly of prisms by transforming the prisms into
truncated pyramids.

Figure A.2: Planar Assembly of T-Prisms |

Figure A.3
shows how a truss based on t-tripods looks in a planar context.
Its appearance is most similar to Snelson's truss, the only
thing differentiating them being that there are two interlayer
tendons corresponding to each strut in the truss based on
t-tripods whereas Snelson's truss just has one. In the
t-tripod truss, the way
the tendons are connected in relation to the strut avoids
interference problems with that strut; however, interference
with other struts is always a possibility which may have
to be designed around.
As seen in Chapter 5,
the outer layer has been completed by interconnecting the t-tripod
apexes with tendon triangles to yield an outer layer which
is also identical with that obtained in Emmerich *et al.*'s arrangement.

Figure A.3: Planar Assembly of T-Tripods |

Each inner tendon triangle where
the struts of three t-tripods converge
is viewed as the apex of an inward-pointing t-tripod.
Adding the corresponding interlayer tendons
to complete these t-tripods (the secondary interlayer tendons
of Chapter 5)
provides more interlayer triangulation,
and thus more reinforcement of the structure.
In this configuration, each strut is secured by 12 tendons.
It is worth noting that this is precisely the minimum number
of tendons Fuller has experimentally found to be necessary to
rigidly fix one system in its relationship to
a surrounding system.^{8}

Hanaor's articles also present computations and models for
several structures. Computations for both member lengths and
forces are presented. All computations are based on
a methodology presented in
*Argyris72*.
An important qualification to these results is to notice that
only one layer of the truss is secured to a rigid support whereas,
for best performance, both layers should be. This is difficult
unless the truss has been elaborated into a hemisphere
of some kind. The structural behavior of Snelson's approach has not been
explored with civil engineering tools. The relative performance
of these two methodologies and the one presented in this work is
unknown.

^{1}
See photos in *Lalvani96*,
p. 48. A realization of his truss
described here was embodied as his *Triangle Planar Piece Model*
of 1961. It also appears in Figures 7 and 8 of some
1962 drawings which are part of an abandoned patent application.

^{2}
*Emmerich88*,
"reseaux antiprismatiques", p. 281.

^{5}
This is a typical application of the
technique. It has various ways it can be applied, and
is not limited to t-prisms.

^{6}
cf. *Hanaor87*, Figure 9.

^{7}
Hanaor might call the t-tripod a truncated pyramid with
its larger end triangle removed.
See *Hanaor92*, Fig. 3(f).

^{8}
*Fuller75*, pp. 105-107.