Method for manufacturing a bonded reinforcing textile filamentary element, the bonded reinforcing textile filamentary element comprises: - a core made up of a first strand comprising at least one monofilament, and - a layer comprising at least two second strands, each second strand of the layer comprising at least one monofilament, each second strand of the layer being wound in a helix around the core and each second strand of the layer is wound around the core without being wound around the or the other second strands of the layer, wherein, in said method: - the first strand of the core and the second strands of the layer are assembled so as to form a reinforcing textile filamentary element in the natural state, - the reinforcing filamentary element in the natural state or a pre-bonded reinforcing filamentary element is obtained, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element is coated with an external layer of at least one heat-crosslinkable adhesive composition, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element that is coated with the external layer is thermally treated so as to crosslink the adhesive composition in order to obtain the bonded reinforcing textile filamentary element, wherein, in said method, the steps of coating with and of thermally treating the external layer of the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element are carried out such that, for an elongation equal to 30% of the elongation at break of the reinforcing textile filamentary element in the natural state, the tangent modulus of each bonded reinforcing textile filamentary element is increased compared with the tangent modulus of the reinforcing textile filamentary element in the natural state, with the tangent modulus is calculated on the basis of a force-elongation curve obtained by applying the standard ASTM D 885/D 885M - 10a of 2014 and characterized in that the sum of the stiffnesses in tension at 1% elongation of the second strands Sc, is greater than the stiffness in tension at 1% elongation of the first strand Sa. - a core made up of a first strand comprising at least one monofilament, and - a layer comprising at least two second strands, each second strand of the layer comprising at least one monofilament, each second strand of the layer being wound in a helix around the core and each second strand of the layer is wound around the core without being wound around the or the other second strands of the layer, wherein, in said method: - the first strand of the core and the second strands of the layer are assembled so as to form a reinforcing textile filamentary element in the natural state, - the reinforcing filamentary element in the natural state or a pre-bonded reinforcing filamentary element is obtained, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element is coated with an external layer of at least one heat-crosslinkable adhesive composition, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element that is coated with the external layer is thermally treated so as to crosslink the adhesive composition in order to obtain the bonded reinforcing textile filamentary element, - the first strand of the core and the second strands of the layer are assembled so as to form a reinforcing textile filamentary element in the natural state, - the reinforcing filamentary element in the natural state or a pre-bonded reinforcing filamentary element is obtained, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element is coated with an external layer of at least one heat-crosslinkable adhesive composition, - the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element that is coated with the external layer is thermally treated so as to crosslink the adhesive composition in order to obtain the bonded reinforcing textile filamentary element, wherein, in said method, the steps of coating with and of thermally treating the external layer of the reinforcing textile filamentary element in the natural state or pre-bonded reinforcing textile filamentary element are carried out such that, for an elongation equal to 30% of the elongation at break of the reinforcing textile filamentary element in the natural state, the tangent modulus of each bonded reinforcing textile filamentary element is increased compared with the tangent modulus of the reinforcing textile filamentary element in the natural state, with the tangent modulus is calculated on the basis of a force-elongation curve obtained by applying the standard ASTM D 885/D 885M - 10a of 2014 and characterized in that the sum of the stiffnesses in tension at 1% elongation of the second strands Sc, is greater than the stiffness in tension at 1% elongation of the first strand Sa.