Two analytical solutions of reduced rigidity plastic bifurcations, res
pectively for simply supported and clamped circular plates, are found
in the way of a post-confirmed presumption for the distribution of unl
oading area at the bifurcation. The J2 flow theory is used. While the
solutions, by their bifurcation modes and distributions of the unloadi
ng areas, recall many aspects of Engesser bifurcation, the correspondi
ng bifurcation loads he above the true Engesser loads.