A tire having a radial carcass reinforcement, anchored in each bead to an anchoring bead wire to form an upturn, characterized in that a profiled member of vulcanized rubber mix is arranged axially to the outside of the upturn of the carcass reinforcement, the secant modulus of elasticity in tension of which, measured for a relative elongation of 10%, is very much greater than the moduli of the other mixes forming the bead, the main part of the carcass reinforcement having a substantially rectilinear meridian profile between its point of tangency to the anchoring bead wire and a point A located at a radial distance HA from the base D of the beads of between 35% and 65% of the radial distance HE between the points of maximum axial width of the carcass reinforcement and said base D.