Beam and Joist repair in balloon framed home
I have a sagging beam and joists in an old balloon framed house that I would like to remedy before I start any finish work. The beam is a 2 ply rough 2x7ish spanning 16' and supported by 3 posts (8' max span). The rough 2x6 joists are notched over the beam and over the sill at the other end of their 12' span. This is all basement ceiling framing. Both are sagging due to age and lack of size.
I was thinking of replacing the beam with an appropriately sized LVL(s) and either sistering or building a wall under the middle of the current joists. The 2" concrete slab in that area is weak and already cracked in places.
I want some opinions on this repair. Is cutting out and replacing the beam the best way or is reinforcing it a better idea? What about the joists. How would either of these be accomplished safely and effectively?
See the attached diagram.