Exact solution of free in-plane vibration of a stepped circular arch

This paper investigates the free in-plane vibration of stepped circular arches. The effects of axial extension, transverse shear deformation and rotatory inertia are included in the governing equations. The solution is obtained exactly by using the initial value method. The solution procedure is also applied to cases in which only one effect is considered in order to determine its contribution to the results. The results show that the aforementioned effects must be taken into account to get a better approximation to the actual behaviour of an arch, even if it is slender. The effects of step ratio, location of the step, boundary condition, opening angle and slenderness ratio on the frequency coefficients are studied. The mode transition phenomenon is also investigated and the mode shapes are given in figures. The examples in the literature are solved and comparisons are presented in the tables. Numerical solutions are obtained by using ANSYS which is a commercial finite element program. Experiments are also performed to verify the theoretical and numerical results.