An assembly (1) for a tire comprises: a first structure (10) formed by first cord elements (15), a second structure (12) formed by second cord elements (16), a bearing structure (14) comprising bearing cord elements (17) linking the first cord elements (15) to the first structure (10) and the second cord elements (16) of the second structure (12), and at least one cord securing element (18) fixed to the first cord elements (15) and to the second cord elements (16), said securing element (18) exhibiting an elongation at rupture at least equal to a minimum elongation greater than or equal to the ratio (A.sub.1) between a conformation height (h) of the assembly (1) and the sum of a thickness (e.sub.1) of the first structure (10), of a thickness (e.sub.2) of the second structure (12) and of a lock length (E).