Joist reinforcement question. Engineer types?

[jslo]

There are basically 3 types of basement floors being used in construction these days, especially here. Your typical floating slab which is in most homes. It is a 4" slab, put in after foundation, directly on grade and not attached to foundation. Meant to be able to move independently of your foundation (which shouldn't move). The second is a structural wood floor which is attached to the foundation and above finished grade (think of a crawl space below). 3rd is a structural concrete floor. It is framed hanging from the foundation walls also but uses concrete panels for the floor.

The structural floors will require much more work for you.

You can normally tell if its a floating slab by looking around the perimeter. A 1/2" fiber expansion joint is normally used to keep it independent of the foundation. If you have a radon system that joint has probably been caulked.

What I would do (and I'm not an engineer):
Inspect the floor for signs of movement (large/uneven cracks, dropped or heaved areas)
If no signs of large movements I'd bolt the steel column directly to slab (steel shims under to plumb) and monitor from time to time for movement
If signs of anything other than minor cracks in concrete, cut concrete and add caison under post

If you could dislodge it later with a hammer, I'd consider that unstable for holding up something structural.