The tire () has a stiffening structure () comprising a stiffening element () anchored in or around radially inner reinforcing structures (A) and a radially outer reinforcing structure (A, B). Each stiffening element () extending from one of the radially inner reinforcing structures (A) comprises a portion () extending in the toroidal cavity () between first anchor points. Each stiffening element () extending from the other of the radially inner reinforcing structures (B) comprises a portion () extending in the toroidal cavity () between second anchor points. The portions () do not cross over in the toroidal cavity (). The first anchor points are axially on the other side of the same side of the median plane (M) of the tire () as the second anchor points.
