The common curved beam theory covered is the Winkler method (e.g. in Johnston and Beer, or Boresi and Schmidt). As can be seen by reading any derivation, this method applies to beams with a common center of curvature. See, for example, here.
In essence, there is a single center and the radius goes from the inner radius to the outer radius, passing through the neutral axis radius and the centroidal radius somewhere. Those two will not coincide as they do in a straight beam.
The curved beam analysis has a lot of interesting applications, such as the stresses developed in parts shaped like chain links and like hooks. Of course, it is only first order accurate for these cases, as is always a caveat when combining together various mechanics of materials methods to form reasonably complex geometries. However, it is useful for things like initial sizing and rating of a chain.
Is anyone aware of a similar analysis for a curved beam where the centers do not coincide? I am interested in order to get a design equation for shapes like more complicated hooks or for D Rings. Please see a rough sketch of what I would mean below.
My initial thought was that something like the neutral axis or centroidal axis could be taken as being a radius at some sort of "averaged" center dependent on the inner and outer curves. Viewed as a circle at this center, the profile would then change along the arc, since the top and bottom are centered elsewhere.
Any help or reference is appreciated.