elastic-ideally plastic material

Dear iMechanica,

suppose you have a beam with a square cross-section, manufactured from an elastic-ideally plastic material.

Now apply a load that rises linearly in time, but is locally constant along the beam length. Upon sagging, the beam will develop a plastic zone beginning in the top and surface regions at mid-length.
This is the "straight" problem solved in Prager, Hodge: Theory of perfectly plastic solids, publisher: Springer