This section is from the book "A Treatise On Architecture And Building Construction Vol3: Stair Building, Ornamental Ironwork, Roofing, Sheet-Metal Work, Electric-Light Wiring And Bellwork", by The Colliery Engineer Co.. Also available from Amazon: A Treatise On Architecture And Building Construction.
At (c) and (e) in there are 4 winders, differing but little from the plan shown in Fig. 23; while in (b) there are 3 winders with slightly curved ends. The 5 risers so changed in their arrangement in (b) take 4 inches more run than the plan with winders as shown. The placing of the carriage timbers in such stairways is a matter requiring care, as the strength and stability of the stairway largely depend thereon.
Fig. 23 are shown the methods of developing the front stringer when each end curves around a cylinder. The plan (c) of the front stringer is the same as in (a). The distance a b in (c) is equal to the "layout" of the semicircular well, obtained as in (d), where d e is the outstretched, or developed, length of the curve a f b. The length of d e may be determined as follows: Make a b equal to the diameter of the cylinder; from c, with a radius equal to one-half of a b, describe an arc a f b; from a and b as centers, with a radius equal to a b, describe arcs intersecting in g. From g, through the points a and b, draw lines g d and g e; through f draw a line tangent to the arc and parallel to a b; the points where this line intersects with the lines drawn through g, will define the length of d e, or the developed length of the arc a f b. A C B in (c) is the front stringer, any outstretched step, as C, being obtained by projecting the line of the riser vertically from the plan, and the tread horizontally from the story rod. As the steps D, E, etc., in the cylinder, have their width marked on the plan as 3 1/4 inches, that width is therefore transferred to the development. The stringer in this case is 6 1/2 inches, measured from the angle between the treads and risers. This width is marked out with a compass from each angle, and a pleasing curve is drawn tangent to the arcs, as shown. When a cylinder is of this diameter, or smaller, at the bottom of the flight, the curve of the hand rail presents a better appearance, and is more convenient when the rail over the regular treads is continued on its pitch into the wreath, without ramping the straight rail. In this case there would be an easement introduced between the straight rail and the wreath piece.