Drywall Joists – Code Compliance for Notching a Joist Block

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.

You don't have joists or rafters. You have engineered roof trusses. The bottom chords are 2x4 because that's all that's required for your scenario. They don't span the ceiling themselves. They're part of a rigid structure that's supported by both the top chords and the diagonal truss members. This is typical and has been standard practice since the 1970s with virtually no static failures on record.

Some trusses, in the case of very large spans or girders (which are sometimes doubled or tripled), are designed with 2x6 or 2x8 bottom chords. But for common trusses in modestly-sized homes 2x4 is standard.

The blocking was probably added to support a particular ceiling finish--tiles or wood panels, for example. Unless you have a particular need for more weight-bearing capacity, leave things alone. If you do, consult a licensed engineer who will examine things much more closely than we're able to from one shady photo.