My wife built a compost bin out of 2- 4x4's, cut in half; and 3 pieces of 4ft x 4ft heavy wooden lattice. The front is 5 pieces of 5/4 decking slid into a groove. The bin is inexpensive, free standing, and 4ft x 4ft x 4ft holds quite a bit of trimmings, scraps, grass, etc. We recently covered it with plastic deer netting to keep the chickens out of it. They could empty the bin in a days time.
You might find that the compost will loosen up the underlying soil unless it actually is rock or concrete.