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Imagine now that we take several of the tied half arches in Fig. 5b and arrange them in a radial configuration as shown in the equilibrium network G (Fig. 6). Instead of having a tension tie for each pair of aligned arches, this same spatial layout of arches can be balanced by a continuous, polygonal tension ring (Fig. 7). The horizontal equilibrium, and hence the plan geometry, of this funicular tension ring is defined by the (equal) thrusts of the half-arches, and can be represented by the reciprocal force diagram Γ* in Fig. 7.

Such reciprocal diagrams of form and forces are used in Thrust Network Analysis (TNA) (Block & Ochsendorf 2007; Block 2009) to explain and control the horizontal equilibrium of compression-only vaults. To guarantee compression, Block (2009) explains that corresponding directed edges ei and ei* in both diagrams need to have the same directions, i.e. not only be parallel, but furthermore have the same orientation, resulting in positive force densities q, which are the ratios of the corresponding edge lengths of the force diagram Γ and the form diagram Γ*. Tension elements have negative force densities, and their corresponding edges in form and force diagram thus have opposite directions. For two reciprocal diagrams, the sign of axial forces in the elements can thus directly be obtained by using the normalised dot product of the corresponding edges, -1 for tension, and +1 for compression (Eq. 1).

Fig. 6 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration based on two half arches as shown in Fig. 5b (tension elements marked in blue; for illustration purposes overlapping edges are shown offset to each other)

Fig. 7 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration with a continuous tension tie based on the configuration shown in Fig. 6 (tension elements marked in blue)

Fig. 8 shows the same configuration in plan as in Fig. 7, but with inverted support conditions. While the inner node is free to move vertically, the nodes along the outer boundary are fixed in the vertical direction. Note that this results in a simple dome structure with a continuous tension tie resisting the tensile hoop force at the supports.

Fig. 8 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration with a continuous tension tie resisting the tensile hoop force at the supports (tension elements marked in blue)