Free Vibration of Arches Flexible in Shear

An analysis of the in-plane, free vibration of uniform, circular arches is made, considering the effects of both shearing deformation and rotatory inertia. From a study of numerical solutions, it is concluded that the natural modes of such systems are of the following three types: (1) almost entirely inextensional, with the flexural and shearing actions coupled as in a straight beam; (2) primarily extensional, in which case rotatory inertia and shearing deformations have negligible effects; and (3) a mixture of those described under (1) and (2). It is shown that the diagram representing the variation of the natural frequencies as a function of the slenderness ratio of the arch may be approximated by a set of diagonal straight lines corresponding to the predominantly extensional modes, and a set of curved lines corresponding to the predominantly extensional modes, and a set of curved lines corresponding to the predominantly inextensional, flexure-shear modes. When the effects of rotatory inertia and shear are neglected, the latter curves become horizontal lines. The approach used to derive the approximate formulas may also be extended to arches of other profiles, boundaries, and distributions of mass and stiffness.