New tile bathroom floor

I am gutting a 100 year old upstairs bathroom, and installing ceramic floor tile.
After removing the vinyl floor, 1/8" ply, 3/4" ply, and 3/4" plank sub floor, I am left with some saggy joists that are holding up the downstairs living room plaster ceiling. They have been notched and drilled quite a bit over the years for plumbing. The room is 6' x 10' with the joists running the 10' length.

The first question is for leveling and strengthening the joists.
I was thinking about sistering one LVL to each joist.
There is no place to land the ends of the LVL onto to transfer the load to the walls, so is construction adhesive and screws in shear enough to hold the floor up?
I need to keep the bottom of the LVLs up and away from the plaster keys of the ceiling below.

Is there a recommended height & thickness chart for sizing the LVL for that?
Is there a fastener spacing chart for that?

Second question is about sub floor. I thought I would put down 3/4" tongue & groove plywood Sturdi-floor, and then screw down 1/2" Durock on top of that in a mortar base. Just wondering if that 1-1/4" is thick enough for the tile, since I removed 1-5/8" of sub floor that was there.

I was going to use a roll-on liquid membrane on the Durock before tiling.

Thanks

 

Question, will the tops of new and old joists be at the same level?

 

You've told us the span but you haven't said the height of the floor joists. Also, what is their spacing?

Glue & screws or nails is acceptable for attaching an LVL or dimensional lumber beam. The instructions for a LVL will describe the fastening schedule.

Getting the ends of the sistering joists onto something bearing would be best. If you sister only in the center of the span you are still relying on the old sagging beams to still be the sole carrier of the load at the ends of the joist. It's better but not the best.

For a bathroom I prefer to use a premium floor sheeting like Advantech. It is stronger and more moisture resistant than plywood. The thickness required depends on the spacing of the floor joists.

I would only apply a waterproofing membrane inside a wet area like a shower. I would not apply it to the general floor area. If you have that much water on the floor you have bigger problems plus the edges won't be sealed.

 

The old joists are 2×8s, pretty badly notched by 100 years of remodels. There is about a 1" sag in the middle of the room, so the tops of the LVLs will not be level with the joists all the way across.
Do I need felt under the Durock? It's going to get punctured by all the screws. Is it okay to hit the joists with the Durock screws? They won't hold it off from the ply, the Durock is to brittle for that.