Force and Displacement Influence Functions for the Circular Ring

If a complete circular ring is subjected to a known, selfequilibrating system of forces, the equations of equilibrium are not sufficient to determine the internal forces and moments. In certain cases, the defect can be made up by arguments from symmetry, but it is usually necessary to impose the condition of continuity of displacements around the ring. This condition is often greatly simplified by the use of Castigliano’s theorem, but the solution can be tedious in asymmetric practical problems, particularly if a number of concentrated forces and moments are applied to the ring. In the design field, it is usual to build up the solution to such a problem by superposition. Lists of elementary solutions appropriate to the complete circular ring are given by Roark (l)t (in-plane loading only) and Blake (2) (in-plane and transverse loading). However, neither of these authors gives a set of influence functions that is sufficiently general to cover all possible problems. In this paper, a set of functions is given that satisfies this requirement of generality. It is assumed throughout that the cross-section of the ring is uniform and small in both directions in comparison with the mean diameter. It is also assumed that extension of the centre-line and shear deflection can be neglected.