The free body diagram is
The moment of inertia of the cross section
is 
and c
= a/2. So all that's necessary to find the dimension a is the maximum moment.
Both the shear and the moment are zero at B'. (B' is the point at which the load ends; the remainder of the beam is of no consequence, since its weight makes no difference.) Both the shear and the moment are largest at A. Thus,
