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Problems Concerning Spherical Polyhedra and Structural Rigidity by Tibor Tarnai

Structural Topology #4,1980

Abstract

The intent of this paper is to present a short review of the author’s research on the subject of Structural Topology: polyhedra, juxtaposition and rigidity. This review deals with morphological problems of polyhedra inscribed into a sphere, bounded by triangular faces, and with the rigidity of a latticed cylinder. We treat them, first of all, from the aspect of engineering. Finally, future research and some open questions in rigidity of structures are mentioned.

Topologie Structurale #4, 1980

R&urn4

Cet article a pour but de presenter un bref compte rendu de recherche de I’auteur dans le domaine de la topologie structurale: polyedres, juxtaposition etrigidite. Ce rapport traite des pro- blemes morphologiques du polyedre inscrit dans une sphere, horde de faces triangulaires, et possedant la rigidite d’un cylindre a treillis. Nous les observons tout d’abord du point de .vue de I’ingenieur, puis, abordons la recherche a venir et certaines questions pendantes en matiere de rigidite des structures.

Spherical polyhedra with triangular faces

The general use of space frames and trusses called our attention to some geometrical problems concerning these structures. First we addressed ourselves to a problem in connection with constructing the network of single-layer grids, vertices of which lie on a sphere. The main problem was, how to subdivide the surface of the sphere into spherical triangles in order that

o even the longest one among the sides of the spherical triangles be shorter than a length given in ad- vance,

l the number of sides, of spherical triangles of dif- ferent lengths be small as possible,

l the lengths of the sides of the spherical triangles possibly differ only slightly from each other, i.e., the subdivision be approximately uniform, and

l the spherical network configuration produced by the subdivision determine a convex polyhedron.

This problem has been investigated by many resear- chers, e.g. R. B. Fuller, D. R. Stuart, J. D. Clinton, 0. Patzelt, W. Schonbach, H. C. Tiirkcii and others. We suggested a new method of subdivision of the spherical surface (Figure 1) in which the number of sides of dif- ferent lengths is equal to the frequency number of the subdivision (Tarnai 1974).

In (Makai 1975), the uniformity of the edge network of the polyhedron obtained by the subdivision is examined and, for different methods of subdivision, the

limit of the quotient of the length of the maximal edge and the length of the m;i nimal edge is determined as the frequency number of the subdivision increases. It is ascertained that this limit, for the method of subdivision sketched in Figure 1, is equal to 1.284 which is not much greater than the theoretically possible smallest value, 1.1756. A construction is given for a subdivision at which this limit is not greater than 1 .1780. The details of this construction and related problems can be found in (Makai 1973). (Recently J. D. Clinton discovered a new method of subdivision for which this limit is less than 1.1780.)

Other morphological problems on the sphere

In (Makai 1976), these problems are discussed inten- sively and some topological problems as well as the dual problem of subdivision (when the surface of the sphere is subdivided into pentagons and hexagons) are also mentioned. There is an unsolved topological problem. If a spherical polyhedron* is bounded by pentagons and hexagons, then it is known that the number of pentagons is always 12, but it is not known: How many hexagons may the polyhedron have? We only know that the number of the hexagons cannot be arbitrary. It cannot be, for example, equal to 1.

The packing problems are related to the tessellation and ’ juxtaposition problems. The problem of the densest spherical circle-packing, which is a well-known problem of discrete geometry, also belongs here. The problem is: how to give an arrangement of n non- overlapping congruent circles on a sphere in order that the area of the spherical surface outside the circles be a minimum. We only mention this problem now, because, in one of the next issues of the Bulletin, we in- tend to report on how this problem arose in a metal statue set up in Hungary, a statue which is essentially a sphere having a diameter 3m, composed of nearly 500 congruent truncated cone shells.

Figure 1. Refinement of the network

cases of frequencies 1 r&3, 4, 5. of the spherical icosa hed ron

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Rigidity of spatial structures

Nowadays, one of our basic problems is the rigidity of such bar and joint structures which, although satisfying the necessary condition of static and kinematic deter- minacy (3j-b-6 where j is the number of joints and b is the number of bars, or if the six bars connecting the structure to a foundation are included in b then 3j-b-0) are simultaneously statically and kinematically in- determinate*. The static indeterminacy means that the structure can be in a state of self-stress. The kinematic indeterminacy means that the structure is not rigid*. The question of rigidity arose in connection with the steel skeleton of a cooling tower whown in Figure 2. The reticulated cylinder consists of horizontal congruent n-gons and inclined bars connecting their joints in a symmetric arrangement. In (Tarnai 1980), it is ascertained that, if n is odd, then the structure is both statically and kinematically indeterminate, consequen- tly rigid. But when n is even then the structure is both statically and kinematically indeterminate and in this case the lowest ring is statically indeterminate (it can have a state of self-stress) and the uppermost ring is kinematically indeterminate (it is a mechanism* with a finite motion) (Figure 2). The degree of indeterminacy (the nullity) is always equal to 1 independently of the number of rings in the cylinder.

kinematically indeterminate ring

statically indeterminate ring

Figure 2. A reticulated cylinder having a finite motion.

Some problems for future research:

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1. For a simultaneously statically and kinematically in- determinate bar and joint structure, knowing the coor- dinates of the joints and the topological character of the structure (what joints are connected by bars), how can it be ascertained by the help of matrix methods of struc- tural analysis, for instance by the help of the equation of state of the structure (Szabo 1978), whether the kinematic indeterminacy appears in the form of an ((in- finitesimal mechanism*, or a ((finite mechanism*))?

2. Let the elastic properties of material of the structures be taken into account. What is the criterion in order that the prestress give rigidity for a simultaneously statically and kinematically indeterminate bar and joint structure? Some results in this problem have been presented, e.g. in (Calladine 1978) and (Connelly 1980).

3. Finally, there is an open question which we address to mathematicians: Do there exist closed, flexible, smooth surfaces (topologically equivalent to a sphere) which can have curved edges, that is, do there exist closed membrane shells as ({finite mechanisms)), similar to Connelly’s flexible bar and joint structure? It is known that there exist closed shells forming ((infinitesimal mechanisms)) similar to certain closed bar and joint structures (Figure 3). It should be noted that a very similar question has been mentioned in (Connelly 1979).

Definitions

Spherical polyhedron. In the topological sense, a polyhedron which is topologically equivalent to a sphere. In the geometrical sense, it is the same as in the topological sense and moreover the polyhedron is an inscribed or circumscribed one of a sphere.

Statically indeterminate structure. A structure in which the inner forces arisen from any given external load can be determined by equilibrium relationships but the uniqueness for the inner forces does not hold.

Kinematically indeterminate structure. A structure in which the external motions arisen from any given inner (relative) motion can be determined by kinematic relationships but the uniqueness for the external motions does not hold.

Rigid. Statically rigid, not permitting even any in- finitesimal motion, save rigid body motions of the entire space.

Mechanism. A structure which has a motion not causing inner forces.

Infinitesimal mechanism. A mechanism having only an infinitesimal motion.

Finite mechanism. A mechanism having a finite motion.

aI .- c .- z .-

c 0

CJ, l z c o .- > E

0

space truss (polyhedron)

1963 v.2 .Vlasov

a=2b

R.Connelly 1978

membrane shell (closed curved surface smooth per section)

V. Z .Vlasov 1955

VV. Fltigge 1962

,,-a+ b - h

Figure 3. Just-non-rigid structures. Analogy between closed single-layer space trusses and closed membrane shells. 64

Bibliography

The code in the first block of each bibliographic etem consists of

three parts, separated by dashes. The first letter indicates whether

the item is a

B ook,

A rticle,

The middle letter(s) indicates whether the piece was intended

primarily for an audience of

M athematicians,

A rchitects, or

E ngineers.

The key words or other annotations in the third column are inten-

ded to show the relevance of the work to research in structural

topology, and do not necessarily reflect its overall contents, or the

intent of the author.

P reprint, or

C ourse notes. The final letter(s) indicates if the piece touches on one or more of

the principal themes of structural topology:

G eometry, in general,

P olyhedra,

J uxtaposition, or

R igidity.

Calladine 1978

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Buckminster Fuller’s *Tensegrityn Structures and Clerk Maxwell’s Rules for the Construction of Stiff Frames

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A-E-R International Journal of Solids and Structures 14 (1978), 161-l 72.

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A Flexible Sphere

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P-M-R

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Connelly 1980

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P-ME-R

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Fliigge 1962

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B-EM-R

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Makai 1973

Endre Makai Jr. and Tibor Tarnai

A-ME-P

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Manuscript, Math. Inst. of the Hungarian Academy of Sciences,

Budapest, 1973

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faces, uniformity of edge network.

Makai 1975

Endre Makai Jr. and Tibor Tarnai

On Some Geometrical Problems of Single-Layered Spherical Grids with Triangular Network

A-ME-P

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tures, University of Surrey, Guildford, England, September 1975, 675-682.

Inscribed polyhedra of sphere, subdivision of sphere surface,

uniformity of edge network

65

Makai 1976

Endre Makai Jr. and Tibor Tarnai

A-MAE-P

Morphology of Spherical Grids

Acta Technica Acad. Sci. Hung. 83 (1976), 247-283.

Inscribed and circumscribed polyhedra of sphere, subdivision of sphere surface, uniformity of edge network.

Szabc3 1978 Anwendung der Matrizenrechnung auf Stabwerke

janos Szabo and Bela Roller Akademiai Kiad6, Budapest 1978.

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Spherical Grids of Triangular Network

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A-AE-P . .

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(in Russian) Izdatel’stvo Akademii Nauk SSSR, Moscow 1963, 501-502.

Reticulated cylinder, rigidity. Tarnai 1980

Ti bor Tarnai

Simultaneous Static and Kinematic Indeterminacy of Space Trusses with Cyclic Symmetry

International Journal of Solids and Structures 16 (1980), 347-359. A-E-R

Vlasov 1955 To the Theory of Membrane Shells of Revolution * Kinematic indeterminacy of membrane shells.

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I

(in Russian) lzvestiya Akad. Nauk SSSR, Otdelenie Tekhn. Nauk, N05, 1955,55-84. I I

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66