**A pair of rubber rings**

This pair of models demonstrates that *a tied ring is much stiffer than a similar ring without a tie.*

Fig.9-8: Comparison of the deformations of two rubber rings

Fig.9-8 shows two rubber rings, one with and one without a wire tied across the diameter. The dimensions and material properties of the rings are described in Example 9-1 where the calculated vertical displacements of the rings are given . The same weight of 22.3 N is placed on the top of each of the two rings and the reduced deformation of the tied ring is apparent and its increased stiffness can be seen and felt. This may be explained since the force in the wire increases as the applied load increases, and produces a bending moment (Eq. 9-10a) in the ring in the opposite direction to the bending moment (Eq. 9-8a) caused by the external load (Fig. 9-2). In this way the force in the wire balances part of the bending moments in the ring, reducing the internal forces in the ring, thus making it stiffer.

Examples of this criterion in practice are tied arches and tied-pitched roofs. The ties help to balance horizontal forces and reduce horizontal displacements, thus effectively increasing the structural stiffness.