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I would like to identify the correct term for a long, lateral cut within a floor joist.

Based upon this guide and several other sources that I have come across, a notch is a cut that is allowed to have a width up to $\frac{1}{3}$rd of the depth of the floor joist. In other words, a 9" joist can have a notch cut in it up to 3" wide.

In my particular case, I would like to make a cut several feet long in order to increase the available head room. I would like to know the correct term so I can research joist reinforcement requirements.

Bonus points for providing additional references regarding US building codes related to this type of a modification.

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In my experience, this is called a rip. I've seen it done a lot (on the top of a beam section) to accommodate flooring thicknesses/slopes. Your reference actually mentions this, although briefly, on the first page under When a Notch Becomes a Rip.

I've always interpreted the code (NDS, IBC) such that, and received plan check comments implying that, notching (and therefore ripping) in the center 1/3 of the member span is prohibited.

The way around these code requirements would be by re-qualifying the structural member using the (new) shorter depth or re-qualifying the floor system assuming a joist spacing of twice the actual spacing. The second method is extremely conservative.

If you're doing a retrofit and neither of those approaches work, you need to start considering how you can reinforce the member after ripping it down to the size you need. There's nothing special about the process that sets it apart from the same basic approach you would take when reinforcing a member of any other material. I have accomplished this in the past by attaching steel plates or steel channel sections to the wood member. I've also accomplished this by simply sistering an additional wood member to both sides of the modified member. You're basically just trying to increase the member stiffness (i.e. It's moment of inertia) back up to around what it was before ripping.

Take care to deal with/check beam shear as well. Sometimes this can get tricky in retrofits because the beam support location (where beam shear is likely to be highest) is tougher to get to. It's not usually a concern if you only rip locally and don't add any additional load though.

I can't think of a good reference, really. This is all mostly accomplished by basic analysis and design techniques covered in most introductory statics/mechanics books.

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