Yes, thank you for this point, my misunderstanding. The derivation is part of a solution for a moving neutral plane such as in the case for a curved tube; there should be a simpler derivation available for the case where the ring is not subjected to such a moment.
Another reference is Advanced Strength of Materials by JP Den Hartog where on page 238 he summarizes a von Karma derivation that shows the strain for a ring being compressed in its own plane by a point load as being
e_1 = - (3u_o/r_o^2) * cos 2 (theta) y
with the thin-walled tube assumption, as...