The payload limit rating for your specific truck should be available in your owner documents or on-line. For example, my 2012 F-150 with 5.0L V8, 8-ft bed, and regular towing package has a maximum payload of 2,000 lbs. I also have the option of using a trailer, with a maximum GVWR of 7,000 lbs. So with a trailer, I could safely and legally haul about 2 yards per trip if I really wanted. The optional "max tow" package for my truck raised the payload limit to 3,000 lbs and the GVWR to 10,000 lbs. Your truck and options will vary.
There's some safety margin factored into these limits, so you can probably "get away" with a little more weight on a short drive. Frankly, it's not worth the risk. If the drive is short, then carry loads well below your limit and make several trips. It's not the leaf springs, ride height, engine, or transmission you must worry about. It's your brakes and how you'll stop when some yahoo dives in front of you and slams his brakes.