^^ That is a good read!
The ONLY answer to this question is "It depends". The answer is obtained by reading the above, using common sense, and possibly a little learning.
I routinely run the 285/16's on my TJ at 4.5psi (yes, four and a half) in really soft and dry sand. I don't do donuts, but I do run it pretty hard, and I've not lost a bead. At pressures that low though, the tire will tend to slip on the rim a little, which makes the balance go out.
On my 2500 GMC, I'm running 235/80R17's, and I haven't taken the fronts below 20. (They look SQUASHED at 20, so I think going lower is not really in my best interest...) The rears look about the same at 13-14psi.
You're on the right track... 15psi for a 31 on a little truck is still much higher than I'd run. You should be safe to 8psi without worry, particularly if they're on the stock 7" wheels. The wider the wheel, the easier it can be to roll the bead off if you're driving a little too boldly. (IE, turning sharp and hard and fast, donuts, jumps, etc.)
Have fun!!
Chris