Aircraft tyre comprising: - two beads respectively intended to come into contact with a rim via at least one radially interior bead face, - a carcass reinforcement comprising at least one carcass layer connecting the two beads, - the or each carcass layer, being wrapped, in each bead, around a circumferential reinforcing element having a meridian section of centre and diameter, referred to as a bead wire, - in each bead, a portion clamped under the bead wire extending radially on the inside of a carcass reinforcement portion radially on the inside of the bead wire as far as the radially interior bead face, - the portion clamped under the bead wire comprising a surface layer intended to come into contact with the rim via the radially interior bead face, characterized in that the surface layer is made of an elastomeric material having an elastic modulus M 1 at 10% elongation at least equal to 5 MPa and at most equal to 9 MPa, and having a shear stiffness K 1, in that the portion clamped under the bead wire comprises a rigid layer, radially on the outside of and adjacent to the surface layer, which is made of a fabric comprising reinforcers that are mutually parallel and coated in an elastomeric material, and having a shear stiffness K 2 at least equal to five times the shear stiffness K 1 of the surface layer and in that the portion clamped under the bead wire comprises a deformable layer, radially on the outside of and adjacent to the rigid layer and radially on the inside of and adjacent to the carcass reinforcement portion radially on the inside of the bead wire, which is made of an elastomeric material having an elastic modulus M 3 at 10% elongation at most equal to the elastic modulus M 1 at 10% elongation of the surface layer, and having a shear stiffness K 3 at most equal to 0.3 times the shear stiffness K 1 of the surface layer. - two beads respectively intended to come into contact with a rim via at least one radially interior bead face, - a carcass reinforcement comprising at least one carcass layer connecting the two beads, - the or each carcass layer, being wrapped, in each bead, around a circumferential reinforcing element having a meridian section of centre and diameter, referred to as a bead wire, - in each bead, a portion clamped under the bead wire extending radially on the inside of a carcass reinforcement portion radially on the inside of the bead wire as far as the radially interior bead face, - the portion clamped under the bead wire comprising a surface layer intended to come into contact with the rim via the radially interior bead face, characterized in that the surface layer is made of an elastomeric material having an elastic modulus M 1 at 10% elongation at least equal to 5 MPa and at most equal to 9 MPa, and having a shear stiffness K 1, in that the portion clamped under the bead wire comprises a rigid layer, radially on the outside of and adjacent to the surface layer, which is made of a fabric comprising reinforcers that are mutually parallel and coated in an elastomeric material, and having a shear stiffness K 2 at least equal to five times the shear stiffness K 1 of the surface layer and in that the portion clamped under the bead wire comprises a deformable layer, radially on the outside of and adjacent to the rigid layer and radially on the inside of and adjacent to the carcass reinforcement portion radially on the inside of the bead wire, which is made of an elastomeric material having an elastic modulus M 3 at 10% elongation at most equal to the elastic modulus M 1 at 10% elongation of the surface layer, and having a shear stiffness K 3 at most equal to 0.3 times the shear stiffness K 1 of the surface layer.