Geometric approach to form finding of a spoke wheel system and findings in forms in light of graphic statics

The form finding of a spoke wheel structure is a highly nonlinear problem because it has to determine the geometry of cables, struts, and compression rings as well as its equilibrium under prestressing forces and the dead load, at one time. The affine spoke-wheel system, developed and realized by Schlaich Bergermann Partners for circular and ellipse-like plans, is a simple but structurally efficient practical solution, which does not require computation for the form finding. This paper extends the affine concept to apply to more general shapes in the plan with only graphical steps. It demonstrates that the extended method allows a designer to choose a more arbitrary compression ring shape for covering an asymmetrical plan shape with a spoke wheel system. Yet, the forces in the compression and tension rings are uniform. Those in the radial cables are also equal. Among the spoke wheel geometry generated by the extended affine concept, the author discovered two remarkable properties of the equilibrium geometry. 1) The equilibrium ring geometry that minimizes the sum of the squared side lengths forms the evolute of other equilibrium rings with different radii, minimizing the variance of the side lengths. 2) The limit curve of such evolute polylines is a cycloid. The two findings are investigated in light of graphic statics to give graphically comprehensive proofs of them. The proofs are rendered in the dual space represented by the Airy stress polyhedra. A regular polygon force diagram represents the above structurally optimal force distribution; it plays a vital role in rendering the proofs. The proof illustrates that, in the dual space, the necessary condition for the minimum sum of the squares represents a cyclogon, a polyline inscribed in the trajectory of a rotating regular n-polygon, and that for the minimum variance of the spoke wheel's segment lengths can straightforwardly express the evolute-and-involute relation. The potential application of this approach for form-finding is significant.