Keywords: spherical truss, potential energy, second-order stiffness, spherical circle-packing, covering problem, retractable dome.

This paper discusses some questions on the modelling discrete structures of spherical layout. Large-span self-supporting structures such as a spherical dome are often built from trusses, but since the characteristic orientation of external loads and displacements does not follow the spherical surface, a three-dimensional space truss model with straight bars and ball joints should be used.

If a retractable roof is designed to be able to move on a spherical surface [1], it may be necessary to analyse its mobility just on the sphere. In such an analysis, it would be sufficient to describe the equilibrium and compatibility conditions of the assembly just in two coordinates that would also mean a reduction of the matrix dimensions by almost one third.

In this model, straight bars are replaced by curved elements that fit to great circles of the sphere and bear only normal forces in analogy to a net with a ball in it; the joints between curved elements supposed therefore to be radial hinges.

Mathematical description is made in spherical coordinates. In the first approach, equilibrium and compatibility conditions are formulated with the aid of simple geometrical arguments, but unlike in the space truss model, the derived equilibrium and compatibility matrices are not transpose to each other. For the sake of a more consistent description, the second formulation is based on energy principles and uses the Hellinger-Reissner functional [2], considering potential energy together with additional terms related to kinematical constraints (arc lengths) and their Lagrange multiplier.

The presented linear analysis is a spherical adaptation of the technique implemented earlier for space trusses [3], and it is suitable also for detection of states of self-stress and infinitesimal mechanisms. Since equilibrium and compatibility equations are derived from the first variation of the same functional, it can be seen that the equilibrium and compatibility matrices are necessarily transpose to each other, but instead of forces in tangent planes to each node, equilibrium conditions are formulated in terms of moments of those forces about a specific axis or point.

An infinitesimal mechanism is called to have second-order rigidity if can be stiffened by pre-stressing [4]. It is also possible to point out second-order rigidity by this model by checking the sign definiteness of the reduced complementary stiffness matrix of the structure, which matrix can be computed considering the second variation of the functional.

Theoretical description of the model is completed by simple numerical examples in order to show the applicability of the procedure presented: by slight modifications of a regular cube projected onto the unit sphere, different arc networks are obtained and analysed with the same topology. The results of computations show that some of these networks are (finite) mechanisms without any state of self-stress, but others are first-order infinitesimal.

In addition to the mobility analysis of hinged retractable roof structures moving on a sphere, the analysis can be used for convenience in the solution of spherical circle packing and covering problems: locally optimal arrangements of the spheres can also be represented by spherical graphs [5,6] whose states of self-stress should then be investigated in the way presented above.

1

F. Kovács, "Foldable Bar Structures on a Sphere", In: IUTAM-IASS Symposium on Deployable Structures: Theory and Applications (eds S. Pellegrino and S.D. Guest), Kluwer Academic Publishers, Dordrecht, 221-228, 2000.

2

K. Washizu, "Variational Methods in Elasticity and Plasticity", 3rd edn, Pergamon, Oxford, 1982.

3

T. Tarnai and J. Szabó, "Rigidity and stability of prestressed infinitesimal mechanisms", In: New Approaches to Structural Mechanics, Shells and Biological Structures (eds H.R. Drew and S. Pellegrino), Kluwer Academic Publishers, Dordrecht, 245-256, 2002.

4

R. Connelly, "The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces", Advances in Mathematics, 37, 272-299, 1980. doi:10.1016/0001-8708(80)90037-7

5

T. Tarnai and Zs. Gáspár, "Improved packing of equal circles on a sphere and rigidity of its graph", Mathematical Proceedings of the Cambridge Philosophical Society, 93, 191-218, 1983. doi:10.1017/S0305004100060485

6

T. Tarnai and Zs. Gáspár, "Covering the sphere by equal circles, and the rigidity of its graph", Mathematical Proceedings of the Cambridge Philosophical Society, 110, 71-89, 1991. doi:10.1017/S0305004100070134

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