In-plane vibrations of a circular arch of discontinuously varying cross-section

This paper deals with the exact solution of free in-plane vibration of a circular arch of discontinuously varying cross-section by considering axial extension, transverse shear and rotatory inertia effects. The governing equations of motion are six simultaneous linear homogeneous differential equations. It is not generally possible to find an exact solution to the equations. The exact solution can be obtained only for a circular arch with uniform cross-section. By using this exact solution, it is possible to find the exact solutions to arches with discontinuously varying cross-sections. Exact solutions are obtained also by considering each effect alone and by ignoring all effects as in the classical theory. The arches, studied in this paper, have three regions with uniform cross-sections. The numerical results are obtained for five different boundary conditions. The effects of step ratios, slenderness ratios and opening angles on the natural frequencies are given in diagrams. The asymmetric problems are studied and the results are given in diagrams.