Spiral bands having a plurality of convolutions of a high tensile strength composite material are formed by feeding a plurality of continuous high tensile strength fibers through an uncured resin to form a web, depositing the web on a release web having at least one substantially flat surface to form a laminate, winding the laminate around the mandrel to form a plurality of overlapping convolutions, curing the resin in the composite web to set the convolutions in a spiral configuration, and then removing the release web from the composite web. The convolutions of the spiral bands in accordance with this process have substantially flat surfaces and an elastic memory. These bands can be used to reinforce axially extending structures having internal forces directed radially outward therefrom by wrapping the band around the structure with a layer of adhesive applied to hold the convolutions in place.