A cutting segment for a False Drum (100,200,300) having a cylindrical outer surface (202,306) comprises a slot (204,224,330) extending generally axially from end-to-end of the cylindrical surface, a door panel (322) having substantially the same size and shape as the slot; and a mechanism (320) for opening and closing the door panel. The mechanism for selectively opening and closing the door panel includes an elongate lever (324), one end of which supports the door panel, the other end of which is pivotally attached to a stationary portion (point) of the drum, a spring (326) extending between an inner surface of the door panel and another stationary portion of the drum; and an inflatable pneumatic tube (328) disposed between the lever and yet another stationary portion of the drum. There is thereby provided a method of supporting material overlying the cutting slot of the False Drum, by opening the slot when cutting the material, and closing the slot when not cutting the material. Importantly, after the material is cut, the door panel closes the slot so that the cut end of the material does not hang in the slot, but rather is supported, thereby ensuring reliable transfer of the material to a tire building drum.