To hold permanently a tenon that fits into a mortise in a chair (where there is much flexing) use a peg that goes through both.
A square piece of wood that is the same size (across flats) as the diameter of the hole is rounded off on one end - one end cylindrical, one end rectangular.
The round part slips into the round hole. A mallet is used to drive it in the rest of the way.
It will NEVER loosen up.