© 1952 by Oxford University Press
THE STABILITY OF A COMPRESSED ELASTIC RING AND OF A FLEXIBLE HEAVY STRUCTURE SPREAD BY A SYSTEM OF ELASTIC RINGS
( Air Ministry London, S. W.1 )
The stability of a compressed elastic ring has been studied by a method which can be extended to solve the problem of the stability of a flexible heavy structure spread by a system of hoops as in a crinoline skirt. The original work by Levy, which was developed by Timoshenko and Love, cannot be generalized to problems in which the compressing forces are affected by the deformation of the ring.
It is shown that the load at which a ring will buckle depends not only upon the magnitude of the load but also upon its first derivative relative to the radial distance. A positive derivative causes the ring to buckle at a higher load. When this result is applied to a cone of heavy and loosely draped fabric spread by a rigid hoop of radius r2 and a larger and flexible hoop of radius r1 below it, both hoops being in horizontal planes, then various modes of buckling other than oval are possible according to the relative magnitudes of r1 and r2. It is found that oval buckling changes to three-wave buckling when r2/r1 = 13/24, three-wave changes to four-wave when r2/r1 = 97/120, and as r1 and r2 approach nearer to equality the buckled form progressively changes to more waves. When applied to a structure spread by many horizontal hoops of which the top one is rigid and oval, it is found that all other hoops, if each is designed to the criterion qr3 = 3EI, will have the same absolute deviation from circularity as the rigid hoop. If any one hoop is designed so that qr3 > 3EI, then the oval shape of the rigid hoop is magnified on all flexible hoops.