On inner constraints in plane circular arches

A one-dimensional model of plane circular arches with rigid sections is introduced. Suitable strain measures are defined as differences from rigid displacements. If the arch is thin, constitutive arguments make the shearing strain negligible. Hence, the shearing indeformability will be assumed as inner constraint. By means of a formal power series expansion of the exact measures of deformation, it is shown that the shearing indeformability implies some constraints on the axial strain. In particular, the first-order axial strain must vanish in the case of infinitesimal displacements. The same procedure is applied to the pure flexible arch, in order to compare the two sets of results. It is shown that the hypothesis of finite pure flexibility is not compatible with small deformations of the arch. An example is provided to evaluate the effects of the two constraints at the first non-linear step of the perturbation expansions.