The problem with using a king-size bed frame without box springs is that the box springs provide the structural support for the bed. Much like your body's skeleton, without box springs beneath it, the mattress pad will sag in the middle. In many cases, it will sag far enough to slip through the frame and wind up on the floor. The solution is simple: Create structural support via a project that takes less than half an hour.
Cut the plywood to match the dimensions of your mattress. For a standard king mattress, cut each sheet to 76 inches by 40 inches. For a California king, cut each to 72 inches by 42 inches.
Lay the plywood in the rails of your bed frame. The long sides of the plywood should run perpendicular to the long sides of the bed.
Roll out egg crate foam to cover the plywood, stapling it in place with the staple gun. This will help accommodate for the fact that plywood is harder and less pliable than box springs.
Place your mattress pad on top of the egg crate foam.