Out-of-plane free vibration of a circular arch with uniform cross-section: Exact solution

In this study, free out-of-plane vibrations of a circular arch with uniform cross-section are investigated by taking into account the effects of transverse shear and rotatory inertia due to both flexural and torsional vibrations. The governing differential equations for out-of-plane vibration of uniform circular beams are solved exactly by using the initial value method. The solution does not depend on the boundary conditions. The same solution procedure is also used to obtain the results of other cases in which each effect is considered individually in order to assess its importance. The frequency coefficients are obtained for the first five modes of arches with various slenderness ratios and opening angles. The results show that the flexural and torsional rotatory inertia and shear deformation have very important effects on resonance frequencies, even if slender shallow arches are considered. It is concluded that the torsional rotatory inertia effect is the most significant effect to be included in the analysis. A phenomenon known as transition of modes from torsional into flexural is characterized by the sharp increment in resonance frequencies of modes that occurs at certain combinations of curvature and length of the arch. The mode transition phenomenon is shown in figures. Vibration problems for circular beams that have been analysed in the literature are solved and the results are compared in tables. The comparison shows good agreement between the results.