A structural member that contains a solid lineal profile that is formed from a plurality of consolidated ribbons is provided. Each of the ribbons includes unidirectionally aligned continuous fibers embedded within a thermoplastic polymer matrix. The continuous fiber ribbons are laminated together during pultrusion to form an integral solid profile having very high tensile strength properties. Contrary to conventional wisdom, the present inventors have discovered that careful control over certain aspects of the pultrusion process can allow such high strength profiles to be readily formed without adversely impacting the pultrusion apparatus.