My garage ceiling framing consists of manufactured 2x4 trusses over a 20.5' span and on 24" centers. There is currently some plywood flooring for light storage. The sheetrock that was on the ceiling was bowing between those 24" spans. The 2x4 trusses are NOT sagging. I have removed all the sheetrock and intend to replace it with plywood. I would also like to add a joist between each of the trusses which will provide joists on 12" centers instead of 24". I need to use 2x4s so I don't have to raise the existing light storage flooring to accommodate 2x6s or 2x8s. I am thinking of putting 2x4 blocking between the existing trusses (offset from the center of the span to avoid the trusses' nail plates at the center) and using two 12' 2x4s (cut to length, of course) to make the span, nailing at the wall plate and but-joining to the 2x4 block with added blocking for support between the trusses and joists at a half or third the length of the new joists. Is this the best way to accomplish what I am trying to do?
This question really begs to have the analysis done by a structural engineer. That being said I can venture to suggest two ideas that may be suitable to your situation.
A) Probably the best way to provide adequate support for the drywall on the ceiling would be to screw flat 2x4's at right angles to the existing rafters onto the bottom of the lower rafter stringer. Place these on 16" centers so that you get the right spacing for the new ceiling materials.
B) If you try to add in 2x4 stringers in between the existing rafters you would definately not want to have those stringers have an unsupported span that was less than the segments on the lower side of the existing rafters. This likely means that a single cross header near the center is not going to be suitable. The rafter construction often has the support design of the rafter webbing in the shape of a W that divides the lower stringer into three sections. Your cross headers should be at points that the rafter webbing intersects the lower stringer of the rafters. Personally this method is so much more work than method A above I would try to avoid it.
You should check with your local building code administration to see if you can even use plywood to cover a garage ceiling of your style. If your garage is attached to your house there are often stringent regulations that specify the use of drywall at least 5/8" thick.