Abstract: The method of rigidifying portions of flexible structures comprised in part or wholly of yarns comprised of man-made fibers of filaments wherein the fibers or filaments of the yarns become welded to each other, specifically to a method of welding the constituent yarns of shoelaces at the ends of the laces so as to form rigid tips of reduced cross section at the ends; to joining two or more yarns at predetermined intervals to form a compound structure; and to joining parallel or crossing portions of yarns arranged in a predetermined relation to each other to form a dimensionally stable structure, by means of sonic vibration supplemented by pressure; and to apparatus for carrying out the method.