INFLUENCE OF CENTERLINE EXTENSIBILITY ON THE INPLANE FREE-VIBRATIONS OF LOADED CIRCULAR ARCHES

Free vibrations of circular arches about a prestressed static equilibrium state are investigated. The equations of motion for the small amplitude free vibrations about the initial equilibrium state attained under distributed tangential and normal loading derived in a previous paper, and specialized to constant curvature, are used. Particular attention is given to the influence of centerline stretching during the vibratory motion. The vanishing of the loaded free vibration frequencies signals static buckling, and the load intensities at with no initial loading is treated as a special case of the problem considered. It is shown that exact numerical values for the frequencies can be obtained in this case. In general, the presence of distributed initial loading precludes the possibility of obtaining closed form analytical solutions, barring a few simple loading situations such as uniform, constant directional pressure, as the governing differential equations have variable coefficients. In the present work, numerical results are obtained for uniform arches subjected to gravity loading. The Galerkin method, with polynomial trial functions that satisfy the geometric boundary conditions, is employed in obtaining accurate values of free vibration frequencies. Convergence of the method, with increasing number of terms in the assumed expansion representing the true solution, is demonstrated. Arches with pinned and clamped end support conditions are considered. Considerably different are found, depending upon whether one includes or ignores the influence of centerline stretching during the vibratory motion.