I want a makeup vanity but have only enough space for one 18" base. I have granite and a base of drawers 22" deep by 18" wide. If I affix the granite atop the base and extend it to reach the wall, leaving about 22" empty space for a chair, will it be stable (assuming the granite is secured to the wall)?
The granite should span this without any problems. But there are a lot of variables, including how thick the granite is, as well as what loads might be applied, and whether the granite is truly free of invisible cracks.
If the granite is easy and inexpensive to replace, I would "go for it". On the other hand, if you want to play it safe, a good underlayment or some support ribs would strengthen it. The most important thing is it needs to be evenly supported everywhere it is supported. Just a speck on a support that prevents it from evenly seating can cause it to crack when a load it placed on it. So it needs full contact on all supports to prevent this.
Granite is usually placed over plywood or particle board. Many people prefer plywood, claiming it is stronger. That is partially true, but the fact is particle board is stiffer, and makes a better underlayment. People will argue this all day long, but I have seen it demonstrated. MDF is puported to be stonger, but it is less stiff. The demonstration was a simple apparatus that measured how many pounds of force are required to flex a board to a small deflection, and the particle board was the most stiff. The particle board will eventually fail in ways plywood won't, but to support granite, the less flex the more effective the underlayment.
The problem with granite is that it's brittle. You didn't specify thickness, but I think even with relatively thick (eg 1 1/4" granite) it's still at risk of breaking if someone jumps onto the counter top or something.
Granite is normally installed overtop of 3/4" plywood, with the bullnose overhanging and hiding the edge of the plywood.
3/4" plywood will have no problem with this span, especially if secured to the back wall as well. Install the granite on top of the plywood and you should have no problems.
Any granite installer will tell you that 10" is the max without supports. It really doesn't matter the thickness either, as they will crack on fissure lines. Now you there are some really strong granite formations that can handle more than 10" but this is all based on load bear.
If you promise that no one will ever ever stand or sit or lay something heavy on that section then it won't break. Granite doesn't break for no reason. The same could be said with granite at a 50" span (unless the granite's own weight took it down).
To give you an idea of granite spans, if I am putting a "black" granite in a kitchen, installers will be fine going over a dishwasher without a "top". Dishwasher is 24"... so you say what about 10". Well it is supported on each side so the breach is only 2" not 14". If I were to get some exotic blue or red I might be putting plywood flush with cabinet tops.
About plywood - installing on plywood is just a poor method. If you bullnose it is OK, however it makes the counters too permanent in my opinion. I prefer directly on cabinets with corner block gluing.
As for what you should do with your bathroom... If this is a big piece of granite that you are paying a lot for I would use some metal bracing between the cabinets. You can by these stock or have a local welder do these for you. If you wanted to get really fancy your granite installer could router the metal bar sections out (they should have braces in hand or buy them themselves). If you have metal bracing installed times 2 you will never see or feel it unless you are lying on the floor.
i have had dozens of granite countertops done for me, and here are my comments:
1) plywood base is always necessary to space off the granite from the cabinets properly and two provide a base to glue to. i have seen many jobs where the granite is installed without the base plywood, only to have it fracture later (usually when someone sits on the counter or jumps up on it to get something high). its a cheapout that lots of shoddy corner-cutters do to save time and money. if you put no plywood under the granite, it will generally fail the first time a significant load or impact is put on the granite.
2) the normal proper way to do granite is plywood first, silicone adhesive to mount the granite to the ply. then fiberglass mesh epoxied to the bottom of the granite to strengthen the granite and mitigate cracks or fracture. avoid any of these steps and you are getting a sub par job. but like everything in life, you get what you pay for.
3) when you want to freespan granite over anything larger than 12", just use steel plate. 1/4" steel plate is easy to cut or have cut (if you don't have a plasma cutter) and easy to work with. just install it over 1/2" ply wherever you haf to freespan. i personally wouldn't go more than 24" with this approach, but thats up to you. the largest freespan we have ever done with literally no gussets was 60", but we put that on 3/4" aluuminum plate. it was completely hidden by the bullnose, and was really not that expensive. i think a piece of 3\4 x 24 5000 series aluminum was around $100/ft. not much if it solves the problem well.