We explore an interesting three parametric family of essentially non-circular torus-like thin shells of revolution deforming under symmetrical loading without bending. It was found that these non-bending torus-like shells split naturally into two classes of shells lying either outside or inside of a given right circular cylinder. For each of these classes we obtain uniform parameterizations in terms of elliptic integrals. Explicit formulas for the aspect ratio of the non-bending toroids measuring their non-circularity are presented as well.
Digital Object Identifier: 10.7546/giq-21-2020-251-264