A Fourier Integral Solution for the Plane-Stress Problem of a Circular Ring With Concentrated Radial Loads

A Fourier integral solution for the stresses in a straight bar of uniform cross section loaded by various combinations of loads applied normally to the edges of the bar was published by L. N. G. Filon in 1903 (3). Solutions for the stresses in circular rings, loaded on one or both boundaries by radial loads, have been limited to Fourier-series solutions for closed circular rings (1, 12, 13, 14, 15), except that solutions in closed form have been obtained for the limiting cases which occur either when the inner radius becomes very small or when the outer radius becomes very large. This paper presents a Fourier integral solution for the plane-stress problem of a curved bar bounded by two concentric circles and loaded by radial loads on the circular boundaries. It treats only the particular case of a curved bar in equilibrium under the action of two equal and opposite radial forces, one on each boundary. However, the method can be extended so as to deal with other combinations of loads. Sufficient numerical results are given to show that the Fourier integral method permits the calculation of numerical values of the stresses in the particular case considered. It is the purpose of this paper to show that the Fourier integral method can be used successfully in what is probably the simplest problem of concentrated loads acting on a curved bar and to furnish a background of material for use in less simple problems such as bending of curved bars due to concentrated loads.