Thank you for your replies,

The grey colour is heating duct and the span of the joist is 9ft with the chunk taken out in the middle of the span.

I discovered this afternoon that the plumber/person put a copper pipe through the joist 1 inch from the bottom and 2ft from where the big chunk was taken out.  So the idea of strapping the bottom side of the joist is out.

The joist is above two heating ducts and a furnace.

Just getting in any length of lumber will be a challenge.

Considering I want to put in a tile shower above I have a lot of decisions to make.