Disclosed is a tire building drum and a method for building a tire, wherein the tire building drum includes a crown section and a bead-lock section, wherein the bead-lock section defines a bead-lock position, wherein the crown section includes crown segments movable with respect to the base from a crown-down position into a crown-up position in a crown-up direction, wherein the crown segments together form a crown surface, a leading edge and a shoulder, wherein the tire building drum further includes a sleeve that extends around the crown segments in the circumferential direction, wherein the sleeve includes a first circumferential edge that is fixed to the crown segments at a side of the shoulder facing away from the bead-lock section, a second circumferential edge that is located between the shoulder and the bead-lock position, wherein the second circumferential edge is unfixed at least when the crown segments are in the crown-down position.