Beams are sized based on 1) Deflection, (bending) and 2) Shear. Notches can affect a beam differently in those two situations differently.
1) A beam deflects most at the center of the span. This puts the fibers along the top and bottom of the beam in the most stress.
Basically, those fibers along the extreme edges are stressed the most at the center of ...
The lap joint is to position the top surface of the beams to the same level, and to provide a common surface for fasteners.
The ends of most beams can be reduced some without diminishing the maximum load capacity - usually found at the middle of a beam.
The problem you have, is that two 2x4 turned on side (as you propose) will be exactly 4" thick - which is the thickness of the other 2x4 studs - which means there is no space for the wiring. Given that you only need to create room for a 6" exhaust, I think you can go for a much simpler approach:
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The correct way is to frame the opening with a header, jack studs(aka trimmers) and king studs in the same manner that window and door openings are framed. See image below for reference.
The header is typically made of two pieces of 2x lumber face nailed together, sometimes with a piece of 1/2” plywood between the two to make the depth of the header match ...
The responses have been superb so far, however just to complete the breadth of understanding you should have prior to putting large openings into timber load bearing stud walls:
Indeed, bearing is one key issue to be aware of when removing studs. The suggestions for overcoming these in other comments are adequate, and following this should result in ...
Just a note that shoring the ceiling while you make the wall/stud modifications is non-trivial business in itself. From Litchfield's Renovation, you have a couple of options, using screw jacks or even an entire temporary stud wall, parallel to the one you're hacking. The latter is probably overkill for what you're doing, so I'm only illustrating the screw ...
The opening you want to create has to be framed like a window as shown*.
Notice the addition of a header to carry the load of the cut stud (cripple stud) to the sides, and the added jack studs which support the header. Because you're only supporting a light weight cabinet rather than a window, you probably don't need the double sill (single will do).
Can I safely remove one stud from a load bearing wall?
Yes, but you need to properly support the gap with a header. If you aren't willing to do this then don't remove the stud, period.
Will this damage the integrity of the bearing wall?
Without a header you are technically compromising the integrity of whatever the wall is holding up.
At best the floor ...
It is absolutely UNSAFE! - this will damage the structural integrity of the wall! The load in a load-bearing stud wall is carried by the studs.
Your options are:
Surface mount the cabinet
Flush mount the cabinet between studs
Cut the stud and insert a beam to carry the load from the stud to the two studs either side. You will need to reinforce each side ...