Your problem was the single layer cardboard. It is a bit flexible, and not at all intended to support the intense weight of a three tiered cake. I would use three of those cardboards, tape them together to make a drum, and then stack the cake on top. Your construction appears fine- but the fact that the bottom tier split, is indicative of the board flexing when lifted up, and set down repeatedly. Next time use a drum, or a home made drum (the three boards together).
10/6/13 at 3:04pm