How to frame a small void in the floor system to support subfloor

floor

After tearing out the old subfloor in my half bathroom (due to water damage) I am replacing it with 3/4 in (2 cm) plywood. One side overhangs from the last joist about 9 in (22 cm) to the wall, and 11.5 in (29 cm) to the next joist. I plan to tile the floor with ceramics, so I need to brace that side somehow.

Subfloor

My plan was to attach three or four 11.5-inch 2x4s perpendicular to the joists and screw the plywood into those via 3-inch deck screws. My father also had a couple ideas:

  • Cut a couple 4x4s and set them on the concrete below pillar-wise (unfastened), and screw the plywood into them
  • Set a 2×10 on the concrete parallel to the joists, then screw the plywood into it — this would require more of that black stuff underneath to raise it up to the other joists' level

If it's not obvious, neither of us are carpenters. I already have the 2x4s cut, but is my method unsatisfactory?

EDIT: Here are the three ideas in picture form (my idea, then my father's ideas):

2x4s

4x4s

2x10s

Best Answer

Because of the short span I'd do the job with 2x4. 3" gold construction screws will do nicely.

  1. Run a horizontal ledger along the far joist 3-1/2" down from the top. Fasten it with pairs of screws every 12" or so.
  2. Run blocks across from the near joist on 16" centers (with one near the right end), end-screwing them twice at the near end and resting them on the ledger. Toenail them just once above the ledger to keep them in place. Your subfloor fasteners will do the lion's share of the work after that.

Consider using construction adhesive on everything to eliminate the chance of squeaks. 

___________________________________________
                 far joist
__________o______________________________o_
      | o|                           | o|    <-- new ledger, down 3-1/2"
_____o|  |__________________________o|  |__
      |  |                           |  |
      |  |                           |  |
      |  |                           |  |
      |  |                           |  | <-- cross blocks
      |  |                           |  |
      |  |                           |  |
      |  |                           |  |
      |x |                           |x |
__________________________________________
       x         near joist           x
__________________________________________

  o = toenailed screws
  x = end-fastened screws
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