Potential solutions could range from nothing, to armoring the notch, to new post beneath, to sistering, or to infinity and beyond.

Nobody here can say much without more information! If you provide the following we would be in much better shape to provide guidance:

1. What is the approximate size of the notch (area, perimeter, shape)
2. Where is the notch located relative to the support, and load on the joist (e.g. right underneath a line load above, middle of the room, etc). What are the loads on the floor generally?
3. What's below the joist and in the room (e.g. are we dealing with a finished area, mechanical room, etc). Basically, what access are we looking at?
4. When you say joist that could be anything. What are the specifications of the joist (e.g. 2x8, TGI, etc)
5. What is the spacing of the joists for the floor we are looking at
6. Pictures and/or sketches would also be superb

Those should get us rolling. Also, are you sure the notch is in the tension part of the joist? Most residential joists are single span so a notch above the neutral axis would be in compression in that case. If it's continuous over multiple supports it would be in tension or if for some odd reason you had fixed connections. Could you clarify on that?

Thanks for the reply- I apologize for the lack of clarity and information in my original post.

Firstly, you are correct that this notched would be in compression, not tension- shows how long it's been since I've thought about beam deflection. Regarding the rest of your questions:

• The joists, being extremely old, are 2-1/8" x 7-3/4". Unsure of the species, house is ~100 years old in Ontario Canada.
• The notch is 9" long and 5" deep, starting about 6" off the support.
• Joist spacing is 16" OC
• Finished kitchen is below this joist.
• Room will be a bathroom (notch was done by previous owners to fit plumbing). Currently room is demolished and cleaned- just joists and studs (and old plumbing, but that will probably go).

I had a structural engineer come in to take a look and he told me to replace the joist- he would not elaborate on what calculations he performed to determine the joist must be replaced (I see this is a common point of contention). Completely understandable from a liability perspective, and while it is my most likely course of action, I would still like to know exactly how to determine that this joist is not acceptable.

For removal of over 60% of the depth, I agree with the engineer's assessment. From an ease of calculation standpoint more than anything. Is there something preventing you from just sliding in a new joist? If there's existing plumbing or electrical in the way, there are methods of reinforcing, but they are generally both costlier and harder to perform than just installing a new joist.

Is it possible to add a steel strap to the bottom and secure it with screws and adhesive. How important is it to repair the joist? There is provision under Part 11 of the OBC, essentially, to provide a repair that isn't necessarily code compliant as long as the long as the 'end result is better than what was' (there's a little more to it, but that's the gist). I've use the provision often.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Do you feel any better?

-Dik

Again, I agree with the engineer's assessment- I'm looking for the calculations to determine HOW the assessment (beyond code and intuition).

There is nothing preventing me from replacing this joist, would just like to do the math myself.

If OK for shear, then can the moment be resisted by treating the remaining rectangular 'block' of wood in compression (for old lumber a value of about 1500 psi might be appropriate) and the steel strap taking the tension (stressed to maybe 25ksi) and use the screws and adhesive (generally adhesive) to resist the moment? If shear is an issue, can you add some diagonal bridging to help transver the load to adjacent joists? You may not get it to work, exactly (it's a serious notch), but you may get it to work.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Do you feel any better?

-Dik

Admittedly I don't know if shear will be an issue- would you be able to recommend what topics I should research to learn more on the subject? Similarly, where did you get the 1500 psi value for compressive stress?

1500 is a fairly typical allowable stress design value for old DFir lumber. You can find it in many old wood design manuals. Dik has forgotten more about engineering than many of us will ever learn so that's why he's able to just recall stuff like that without reference.

Depending on the age of the residence, there were a few books from the early 1900s for wood design. But really, 1500psi is a conservative allowable bending stress value for any lumber from that era. I can dig up a reference for you, but not until likely next week as I'm working from home and all of our old reference books from that era are in the office library.

I really feel that the shear calculation of the remaining section will not calculate out. I can't imagine a 2 3/4" x 2 1/8" piece of lumber having almost any shear capacity.

For design of the replacement joist, you could essentially just pick one out of the part 9 span tables as long as there's no bearing walls in the middle of the joists or anything like that. Alternately, ForteWeb is a free program that will also design commodity lumber. Lastly, if you really want to know the ins and outs of the design, you'll want to look at CSA O86 and the CWC Wood Design Manual chapter 2. Should have everything you need in there.

The value is likely OK for most timbers that are 100 years old. Almost anything that old is extremely strong. Also the quality of lumber was much better. We slavaged a bunch of lumber from a building that was being demolished that had 100 year old Doug Fir lumber. It was so tough you couldn't drive a nail into it, or pull one out. There were many 2x12 x 20' that hardly had a knot in them.

As far as shear... If necessary, I'd put a couple of small blocks or steel angles in place and put diagonal comapression members to adjacent members and use a steel strap to take care of the force. It's not likely a 'real' design, but will likely be suitable for residential construction.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Do you feel any better?

-Dik

The post from enable has a good hint about sistering which on my part would be on both sides of the area of concern or by adding a new joist along the length of the unacceptable joist.