I have a 12' span with a floor joist that is currently 2 2x12's. It was originally a 16'+ span with an attic opening that was 3' wide thus the reason for the double 2x12's. I'm installing an L staircase to the attic (live load, but no bathroom or kitchen will be up there) and the headroom at the point of the double floor joist is not tall enough to meet code. If I was able to cut the double 2x12's down to 2x6 size, which I think is roughly comparable to a single 2x8 (a 2x8 spanning 12' meets code for a floor joist 16 OC where I am), then the headroom would meet code.
You can see in the attached drawing what I'm trying to do. Can I notch a double 2x12 like this where they're still 2x12 depth at the bearing point, reduced to 2x6 depth for the 3 feet of stairs and then back to 2x12 for the remainder? Again, I only need a single 2x8 for this span so my thinking is double 2x6 depth for that 3 feet of 2x12 will suffice.