Tyre for a motor vehicle, comprising: - two beads intended to come into contact with a mounting rim, each bead comprising at least one annular reinforcing structure, each annular reinforcing structure having a radially innermost point; - two sidewalls extending the beads radially outwards, the two sidewalls meeting at a crown comprising a crown reinforcement having at least one ply (80; 90) provided with crown reinforcing elements, the crown reinforcement being furthermore surmounted by a tread; - at least one carcass reinforcement extending from the beads through the sidewalls as far as the crown, the carcass reinforcement having a plurality of carcass reinforcing elements and being anchored in the two beads by a turn-up around the annular reinforcing structure, so as to form in each bead a main strand and a turn-up strand, - a stiffening reinforcement arranged in at least one sidewall of the tyre, said stiffening reinforcement having a radially inner end and a radially outer end, the stiffening reinforcement being formed of a plurality of stiffening elements oriented at an angle less than or equal to 10° relative to a circumferential direction of the tyre, wherein - each turn-up strand of the carcass reinforcement extends radially outwards as far as an end situated at a radial distance DRE from the radially innermost point of the annular reinforcing structure of the bead, the radial distance DRE being greater than 20% and less than or equal to 95% of a radial height H of the tyre; and - said stiffening reinforcement is arranged such that the radial distance DRR between the radially inner end and the radially outer end is at least equal to 10% of the radial distance DRE from the turn-up strand of the corresponding sidewall; characterized in that at least a portion of the carcass reinforcement comprises crossed reinforcing elements oriented at an angle of between 70° and 80° relative to the circumferential direction, the crossed reinforcing elements of the carcass reinforcement being radially inside the radially outer end of the stiffening reinforcement. - two beads intended to come into contact with a mounting rim, each bead comprising at least one annular reinforcing structure, each annular reinforcing structure having a radially innermost point; - two sidewalls extending the beads radially outwards, the two sidewalls meeting at a crown comprising a crown reinforcement having at least one ply (80; 90) provided with crown reinforcing elements, the crown reinforcement being furthermore surmounted by a tread; - at least one carcass reinforcement extending from the beads through the sidewalls as far as the crown, the carcass reinforcement having a plurality of carcass reinforcing elements and being anchored in the two beads by a turn-up around the annular reinforcing structure, so as to form in each bead a main strand and a turn-up strand, - a stiffening reinforcement arranged in at least one sidewall of the tyre, said stiffening reinforcement having a radially inner end and a radially outer end, the stiffening reinforcement being formed of a plurality of stiffening elements oriented at an angle less than or equal to 10° relative to a circumferential direction of the tyre, wherein - each turn-up strand of the carcass reinforcement extends radially outwards as far as an end situated at a radial distance DRE from the radially innermost point of the annular reinforcing structure of the bead, the radial distance DRE being greater than 20% and less than or equal to 95% of a radial height H of the tyre; and - said stiffening reinforcement is arranged such that the radial distance DRR between the radially inner end and the radially outer end is at least equal to 10% of the radial distance DRE from the turn-up strand of the corresponding sidewall;