Keep in mind, 31" tires is a bit subjective. Actual 31x10.5 tires aren't 31" tall. They are pretty close in diameter to 245/75R16, but wider. Their diameter is influenced by tread type, and wheel width as well. 265/75R15(note 15" wheel in the size) is about identical to a 31x10.5r15. My 30x9.5 tires are pretty meaty, and are barely smaller in diameter to a mild 31" tire. I've just thrown a bunch of numbers up and added to confusion,lol. Are you wanting to run "31 inch tires" just because that's what you hear people run, or are you lifted and you have a set need? If you are lifted, then 245/75R16 is all you'll need, and will for all intents and purposes be the same diameter as a 31x10.5 tire, just not the width. If you want to run a tire that says 31x10.5r15 on it, then you need 4" offset 15" wheels, of only a couple of brands that will fit, plenty of lift, flattened pinch welds, etc.
245/75-16s are pretty close to a 31x10 tire. Most go with that size anyways. Could always go standard and find a "legit" 31" tall tire, but I'm not too sure if they make 31x(10/10.5)x16 and if they do it won't be common. A common size would be 31x10.5x15, but as said it will need the right 15" rims(only some steel rims with the right backspacing, I believe it's about 4"), and about 2.5-3" of lift.
2011 Deep Cherry Red KK - Skyslider - OME 927/731 - Bilstein Shocks - 1/4" Spacer Plate - 3/8" Clevis - Al's A-arms - 245/75-16 Duratracs
I'm running Ironman springs front and rear, about 22" from fender to hub. I had 15" alloy wheels, 235/75r15 tires on it when I bought it. I now have 30x9.5r15 Cooper STTs on it now. Its about as wide as 245mm tires, just a hair smaller diameter. Due to the 4" offset, I do get rubbing of the fenders at the pinchweld. I had slight rubbing at stock sagged height with the smaller tires at the pinchwelds. 10.5" wide tires are about 265mm wide, to help with comparisons.
Tire Rack has a specs sheet for every tire they have in every size.... it lets you compare on a chart side by side all the different tire sizes, weights, tread width, section width, diameter, revolutions per mile, max psi, max load, etc etc...just click on whatever tire you are interested in and scroll down to see the specs sheet.