You should replace the old lead piping and drum trap but unless you move the tub drain location, it appears that the waste pipe will still be directly over the joist.

The important issue is how much load the cut joist is carrying and how much deflection is acceptable. If the tub is resting mostly on the adjacent joists, cutting the damaged joist off at the drain and installing a header to transfer the load to the adjacent joists might work. If that overloads the adjacent joists you could reinforce them with sistered joists.

Could the damaged joist be supported underneath? It's just my garage ceiling underneath - it can have an ugly patch in it. I'm kind of confused - I don't see why putting in a header would require cutting off the damaged joist. Suppose there were some 2x10 blocks nailed between the damaged joist and the joists to either side? Wouldn't that do the same thing as a header?
Someone else told me: "There need to be two double headers placed across the cut floor joist where the new headers connect to the two adjacent joists. This needs to be done in two places and is typical for any large opening in floor framing, i.e. a stairway."
Is he also talking about cutting the damaged joist completely?
The bathtub is supported on 3 other joists besides this damaged one. I don't want the floor to flex and maybe crack tiles or stone when somebody fills the bathtub. But I don't know whether the damaged joist makes the floor too flexible.
The waste pipe is under the tub overflow, not the tub drain. My plumber said he'd find out whether they can get drain plumbing that turns to avoid the joist. That was my original thought - that if the plumbing can go sideways, plywood plates could be attached to the sides of the damaged joist, with a hole in them for the drain pipe.

What other loads are on the joist?

Headering it off and adding a pair of new joists would restore the load bearing, but you need to decide if their IS any load to bear.

Bathtub, toilet, a short interior wall that runs parallel to it. The floor of my house in general.
Yes, that is certainly a question: it's been like that for 50 years, so perhaps it doesn't need any fixing. I do want to put in a stone or tile floor, though, so the floor needs to be rigid.

Blocks and/or plywood plates will not transfer the load . You must install a header or sister the damaged joist full length.

If the header is below the joists it would need to be lag screwed into the bottom of the good joists and it would not be as strong as a full height header even if you used a 4x6 on its side.

The notch occurs near a bearing wall so only one header is needed and perhaps some supplementary framing to minimize the floor opening size. The header does not need to be doubled for such a short span.

Since a new header would be so close to the ends of the good joists I would just install a single 2x10 header with joist hangers and replace or relocate the plumbing pipes using holes sized and located as specified by the building code.

Another possibility is to install a metal strap where the header would go. It would start at the top of a good joist, pass under the damaged joist and end on top of the other good joist. The strength would only be as good as the tension strength of the strap and the connections on top of the good joists so the straps might need be carried down the outer sides of the good joists so more fasteners could be used. It would need to be a heavy steel strap at least 45 inches long with drilled holes. Simpson makes some long 16 ga. straps with nailing holes: MSTA 49 & MSTC 52 or an 18 ga LSTI 49.

Replace the worthless plastic flex duct with a metal duct and consider relocating it if possible.

Based on your pictures, that's the end of the joist and it rests on a block wall with a wooden seat. If there were a significant bearing load then I expect the cut corners at the opening would have shown some signs of cracking.

Is that vertical stub next to the trap directly up to the tub drain and the water lines are for the tub faucets? It looks to me like the tub would be parallel to the cut joist, centered over it, and spanned over that connection. The tub loads would have been transferred to the adjacent joists through the tub (cast iron?). This would result in a lightly loaded section of this joist for approximately the 5 ft length of tub. The remaining span likely shared the load also throught he flooring.

To win some shear capacity at the end you could put some steel plates parallel to the cut opening and try to bring the load to the bearing pad, but unless your radiccally changing the structure above, past performance says you probably don't have to do anything.

The simplest solution is to put a post just before the cut section and support the end of the joist independent of the wall.

If you're going back with a stone floor be sure to nail the living stink out of substrate to create a very rigid diaphram (which would also cause any load to be shared by adjacent joists) to minimize deflections and associated cracking of tile, stone of mortar joints.

good luck.

I think what the other person meant was, instead of cutting the damaged joist completely to put in a header, split the header in half and put a half on either side of the damaged joist. Doubling the header to make up for the loss of strength from splitting it in half. I would have to drill holes in the header for the water supply pipes, but they wouldn't have to be very big.
That would avoid cutting the damaged joist completely. I don't know if the floor would sag a lot if the damaged joist were completely cut.
The steel strap idea sounds good but one of the adjacent joists is in the next room under a hardwood floor. It would be a lot of work to get a strap over that joist.

@sdello One corner where the joist was cut does have a crack. It's separated about 1/8".
The vertical stub goes straight up to the tub overflow. The cast-iron tub was laid perpendicular to the joists, so the tub drain is over the space between joists.

@brickeyee I like the idea of fastening angle to the damaged joist where it was cut. It would be less work and a cleaner fix.
By "angle", you mean L-shaped steel, which can be cut to whatever length is needed? What gauge steel?

However, the place where the joist cracked, at the cut corner, is over the sill plate. So I couldn't put L-shaped steel under the joist there, and that's where the stress seems to be. So reinforcing it with a steel angle may not work either.
I've read of bolting plywood plates onto the damaged section of joist. If the plumbing can be rearranged to avoid the joist, maybe that would work.

You're at the end of the span so any real load is vertical (shear) and not tension (bending). An 1/8 in crack in 50 years doesn't demonstrate any real problem and the shear load is finding its way to the adjacent framing through the flooring and has been for years. Otherwise that end piece would have completely failed.

What you want to do is add vertical support the end of the cut joist where it has a full section before the cut.

If it means that much to you, I'd go with a post in the garage below. It looks like you can get one right next to the vertical sewer pipe.

Your blocking is fine to bridge sheathing over the cut section for the floor installation above,

"So I couldn't put L-shaped steel under the joist there, and that's where the stress seems to be. So reinforcing it with a steel angle may not work either. "

The steel does not need to be under the joist if it is fastened well enough.
One side of the 90 degree angle goes flat against the side of the joist, the other sticks out away from the joist and about 1/8 inch above the bottom of the joist (putting the angle further up reduces the stiffness being added to the joist).

If the sides of the joist are clear at the bearing point you could sister them for at least twice the joist height to help spread the shear load at the end.

It would be better to take the weight off the joist with a jack column, then attach the sisters (16d nails are adequate in a pattern like the number 5 on a die) and then put the load back on the bearing point by lowering the jack.

@brickeyee "The steel does not need to be under the joist if it is fastened well enough.
One side of the 90 degree angle goes flat against the side of the joist, the other sticks out away from the joist and about 1/8 inch above the bottom of the joist"

I see! That sounds like much the best solution, if it's strong enough. It wouldn't be in the way of water supply lines. I could put 2-inch "L" steel in there.

How thick should the steel be?

Would it be a good idea to support the joist before putting on the steel angles, to get any sag out of it? Somebody suggested to me: get a 2x4, cut to 1/4" longer than the 8' distance between joist and my garage floor, hammer into place. Would this work, or would it be better to use an actual jack of some kind?

"If the sides of the joist are clear at the bearing point you could sister them for at least twice the joist height to help spread the shear load at the end."

There isn't room to sister the joist where it rests on the sill plate. The drain plumbing is in the way, and also water supply pipes.

thanks!
Laura

"How thick should the steel be?"

I would use at least 2 in x 2 in x 1/8 thick on each side with 3/8 fasteners about every 4 inches.

The joist is sagging about 1/4" at the LH side of the cut.
It seems like a good idea to take some of that sag out before attaching steel to it. Is that a good idea and do-able?
Would hammering a 2x4 under the joist, about 1/4" longer than the distance between the floor and the joist, work? Would it have to be pushed up gradually?
thanks
Laura

If you need to raise the joist gradual is a lot better.

Screw type jack columns are not very expensive and allow for a lot more control.

Quick lifts using brute force can result in all sorts of cracking occurring.

I wonder - since a short section is bent a lot right at the cut - maybe that wouldn't straighten out if I tried to jack up the joist. Maybe the wood has just re-shaped itself, and I should just leave the floor a bit bent there.
It wouldn't cause problems except for perhaps making it more difficult to attach the steel angle.
How about using 2" x 1/2" steel plate? Would that work as well as the angle?
Laura

I wonder - since a short section is bent a lot right at the cut - maybe that wouldn't straighten out if I tried to jack up the joist. Maybe the wood has just re-shaped itself, and I should just leave the floor a bit bent there.
It wouldn't cause problems except for perhaps making it more difficult to attach the steel angle.
How about using 2" x 1/2" steel plate? Would that work as well as the angle?
Laura

I just realized I probably *can* sister the joist, by putting a 2x10 behind it. I'm pretty sure the plumber can redo the plumbing so it doesn't interfere with a sister.
:)
Better than metal angles, I think.

There is approved hardware that can be used to strengthen joists that must be cut excessively for DWV lines. Metal plates designed to allow pipes to pass through while preserving strength.

It is only a few sizes (joist heights) but might give you some relief on sistering if you still need a DWV path through the joist(s).

It would not fix your existing joist, but a sister with one of these might solve the problem.

@brickeyee I don't think I would need any incursions into the sister. It would be behind the joist in the picture, not in front. It can extend a couple feet past the cut.
Do you think this wouldn't be enough support?
I asked a plumber whether the plumbing could be redone so the drum trap goes in the space between joists, and there's only a pipe passing through the joist. Last I heard the answer was no.

"I asked a plumber whether the plumbing could be redone so the drum trap goes in the space between joists, and there's only a pipe passing through the joist. Last I heard the answer was no. "

The odds of him knowing how to work lead DWV is vanishingly small.
It went out of use a long time ago.
About the only thing you are likely to be able to do is remove the old lead and replace it with newer material.
If you still need to pass a DWV line through the joist investigate the kits for strengthening joists that must be bored for DWV lines.

You must be in a very rural area to have lead DWV lines from 1950.
Galvanized and cast iron were pretty much the standard from shortly after WW2.

The house I owned that still had some lead DWV lines was built just after the turn of the century, in the early 1900s. It also had brass NPT threaded water lines.

@brickeyee "If you still need to pass a DWV line through the joist investigate the kits for strengthening joists that must be bored for DWV lines. "

Do you think such a thing is necessary even if the joist is sistered on the back side?

I don't think there's any way to avoid having plumbing in the place where the joist was cut, since the joist is right under the tub overflow pipe. But I think they can avoid cutting into a sister that would be attached behind the joist.

It's not all that rural here, it's near Cornell.

thanks
Laura

@brickeyee "There is approved hardware that can be used to strengthen joists that must be cut excessively for DWV lines. Metal plates designed to allow pipes to pass through while preserving strength."

I had my plumber over this morning, and he said yes, they can redo the drain line so it only has one pipe hole coming through a metal plate in front!

So if anyone can show me the metal plates to strengthen joists, please do (how far does it go past the cut in the joist, etc.)

I'm still planning to sister it with a 2x10 in the back, which would go about 3' past the cut part.

Here is one company.

There are some others.

Here is a link that might be useful: Metwood

Rather than buying a premade reinforcer for the joist, I need to buy a steel plate, and get a metal shop to drill holes in it and cut it as needed.
How thick should the steel plate be? It would be almost the same width as the joist, like 9" vs 9.25" for the joist. It could sit on the sill plate and go about 15" past where the joist is cut (farther if it's notched to make room for a 2x4 attached to the joist).
I also plan to sister the joist on the other side with a 2x10; sandwiching the joist between the 2x10 and the steel plate, and fastening it with say 3/8" bolts with big washers on each end.
I'm currently jacking up the joist :) Slowly. 16" past the sill plate, it's about 1/3" below horizontal. That's a lot of bend in a short distance, so I figure I'll jack it up over about a month.
thanks
Laura

"I need to buy a steel plate, and get a metal shop to drill holes in it and cut it as needed.
How thick should the steel plate be?"

Unless you want to do the engineering calculations, buy a pre-made repair plate.

The work has already been done.

The pre-made repair plate is quite thin and it's made for a small hole in the joist, not for most of it cut away. So no, they haven't done calculations for me.
So I need to get a metal plate that would certainly be adequate in strength. So I wonder what a reasonable thickness would be for that.

"The pre-made repair plate is quite thin and it's made for a small hole in the joist, not for most of it cut away. So no, they haven't done calculations for me. "

Did you read the listing?

It is designed for more than the maximum cutting, but yes, it still has limits.

If you need a PE sealed fix to satisfy an AHJ, open your wallet.

It should only be under $1,000 to examine the joist, determine the loads, and design a fix.
PE stamps come with liability, so they are not used for free.

Or you can just guess (but be careful, your comment about "repair plate is quite thin" indicates you may be in over your head.
Even thin metal has surprising strength when formed and applied correctly.
All the metal is fixing is the maximum shear load the joist is designed to carry so it does not need to be especially heavy if it is formed and fastened correctly.