My house was built in the 80s and in my basement, it appears that one of the ceiling joists was cut to accommodate venting. Haven't seen any problems so far in the room above but I worry about movement. How do I fix this and ensure it's properly supported?
Well that definitely is not safe to leave it. I'd highly recommend supporting that span with a beam going across the other way with posts. You'll probably need to add footings in. Other scenario Would be to re route the ducting and sistering that joist back. Not sure if this is an area in basement you wanted to finish or not so that will determine which route you will take.
If you are intending on framing a closet or a wall below the cut joist, that wi give you stability. Using an angle/i beam spaning 2 jousts on each side of the cut joist, lag screw to each joist. Then bulkhead everything in if finishing the basement.
That joist should never have been cut as it is structural. Prior to cutting and installing any pipe or ductwork, the floor should be framed properly and the HVAC company should realize this issue and discuss it over with the builder or homeowner.
This is why layouts are very important for every trade.
Also, adding a header frkm joist to joist is helpful.
Averax offered only really good advice, (you don't need another beam), just knock the little piece of bridging out on the side of cut joist and "sister in" another joist same size and about 4 feet long at least past the cut side and then between the 2 pipes and onto the top of the beam. Nail really good or screw to the existing joist every 12"- 3 in a row This is why guys like "holmes on homes" got rich, idiots doing crappy construction work. It actually isn't going to hurt anything for that little spot to be missing but it is "against code to leave as is" so for $20-$30 bucks why not fix and never worry about it again
Sistering a partial floor joist beside this one is a possible solution provided you can maintain a minimum 1" clearance from the furnace B-vent; however at kinda looks like this aint gonna work. If there's a perpendicular wall on the main floor (above) and it is securely attached where the floor joist is cut off (preferably with screws) this might be a possible engineered solution (not recognized by Part 9 of the Code).
Blocking across to the two neighboring joists isn't a possibility as the maximum allowable hole size would be around 2". Also as previously suggested if it works to frame a supporting wall below (preferably without compromising your desired layout) that would also work. Because the loads are so minimal it wouldn't be necessary to have a footing beneath however the basement floor should not be prone to measurable settling or seasonal displacement.
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