Analysis of Noncoplanar Elastic Rings

The six scalar equations required to determine the components of the indeterminate moments and forces in loaded or distorted elastic rings are derived using vector algebra. Simple sign conventions and easily remembered formulas for "elastic areas" result. The equations may be transformed to the column analogy or to the shear and torsion analogy for the analysis of coplanar rings, or to a corresponding analogy using a single elastic centroid in space for noncoplanar rings. The analysis of a superelevated helical beam is illustrated, using cylindrical rather than cartesian coordinates. The expressions are easily converted to any orthogonal coordinate system.