I'm looking to correct the sagging, bouncing floors beneath the dining room in my 150 yr old house. My plan for this is to install a new beam in the basement supported by lally columns. Not to sure how much of the sag I can fix, but if I can get rid of the bounce I'll be content.

The problem is this: The bottom of the existing joists are not even (off by as much as ~3"), due both to differential sagging of the beams and the fact that it seems at least two of them had been replaced prior. How can I properly support uneven joists with the even top of the new beam? I assume notching the beam to fit is a no-no. Should I notch what already appear to be defficient joists? Can I shim the difference? Any other ideas?

Thanks in advance.