Radial tyre for a heavy vehicle of the construction plant type comprising two beads (2) joined together by a carcass reinforcement (4) comprising at least one layer (5), this layer comprising a main part (6) turned up within each bead (2), axially from the inside to the outside of the tyre, around a bead wire (7) to form a turn up (8) having a free end (E). The distance (l) between the turn up (8) and the main part (6) decreases continuously from the bead wire (7) to a first minimum distance (a) reached at a first point (A) of the turn up (8), then increases continuously as far as a first maximum distance (b) reached at a second point (B) of the turn up (8). The distance (l) decreases continuously from the second point (B) as far as a second minimum distance (c) reached at a third point (C) of the turn up (8), and then increases continuously from the third point (C) as far as a second maximum distance (d) reached at a fourth point (D) of the turn up (8), which is radially on the inside of the free end (E) of the turn up (8).