Inverse problem in beam bending, 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

I am curious about the inverse problem: how do I have to load the beam in time and space so as to achieve a circular arc along the length ?
Beneficial (though not required) is a circular arc for the entire duration of loading. But it might be sufficient to achieve a circular arc at the end of the loading process.

I'd need this for both the square cross-section and for a round cross-section.

Any advice is gratefully acknowledged.

Kind regards

Frank Richter
Bochum, Germany