Drywall – How to prevent sagging ceilings when joists are spaced 24″

This will likely be a location specific issue, so it doesn't hurt to check your local codes.

All the fire rated work that I've done has been between units (both to other units and common areas) and any load bearing structures of a multi-unit dwelling. When we do that, fire blocking is installed in the ceiling space as well to prevent a fire from traveling through the ceiling to another unit. For everything else within a single unit, and in a single family home, we don't use any fire rated drywall.

Edit: A few reasons I'd disagree with Eric on this one. First, I'm presuming you already have a certificate of occupancy (or someone did) to move in with an unfinished utility room. That required an inspection that presumably allowed the space as is. Next, most codes for combustion devices (furnace, gas hot water) require that you have ventilation. Installing a vented door is perfectly acceptable and won't provide any fire stopping. And, finally, most HVAC's (if the furnace is forced air) will be connected to vents that run to every room in the house. From the little I saw, the furnaces and utility rooms in searches likely referred to multi-unit dwellings, and that does need to be fire stopped. You also have to use fire caulk around every hole. Our multi-unit buildings also have sprinklers throughout and spring loaded vents that snap closed in event of a fire.

That all being said, there's absolutely nothing bad about installing fire rated material around the furnace, other than the extra cost and that it's a bit more difficult to cut. If you want to do it, then by all means, do so.

Edit 2: One last note, realize that fire rated drywall increases the resistance from 30 minutes that you'd typically see with 1/2" drywall to 1 hour. Fire rated doesn't mean fire proof.

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.