CURVES, BY DEGREE
By Nick Pull

The information provided here is related to railroad curves and will give you a method to measure your own Large Scale curves, just as a railroad would. Data provided, will show you how your curves compare to the real thing.

American railroads designate curves by degrees rather than by their radius. The degree of a curve is defined as the central angle created by two lines from the center of the circle to points on the circle segment separated by a chain of 100 feet. In England, the commonly used separation length is a chord (66 feet). Where the metric system is used, the common separation is 20 meters. The original, actual location of the track usually followed the lay of the land, but even then, it was required that the track be checked to see if it followed the railroads standards and to update the maps. When work was done in the future, the workers had accurate maps and exact directions to follow. This is important because track that was outside the railroad’s standards was difficult to maintain, wore badly, and rode rough.

American railroads have historically used 5734 feet as the radius of a 1 degree curve, based on a chord of 100 feet. This figure was derived in the mid 1800s. By 1900 it was refined to 5730 feet. With today’s tools the figure calculates to 5729.432506 feet. The improvements in accuracy are due to the advent of adding machines, then calculators and now, computers. The improved equipment allowed the use of more and more accurate trig. functions.

Railroads found that by using a chord of 62 feet instead of 100 feet, the versine or offset would by 1 inch per degree. A 7 degree curve would have a versine of 7 inches. This was a lot easier to use.

The formula for calculating a curve’s radius is: A) R =( L x L + (4 x V x V)) / (8 x V)

(all figures in feet)

This has been simplified to: B) R = (3 x L x L) / (2x V)

(R & L in feet, V in inches)

Or: C) R = (125 x L x L) / V

(R & L in meters, V in millimeters)

These formulas give slightly different answers. The difference between A) and B) is 4/100 of a percent.
Even if the formulas and theory seem a little esoteric to you, using the theory is quite easy. All that is needed is a straight board of the required length, a couple of nails , a ruler and a saw. The Chord length will change depending upon the scale use.

Ratio Chord length (feet) Chord length (inches)

If only one or two curves are to be checked, a yard stick and ruler can be used. If more use is expected, a simple set off tool should be made.

On a 1/2 x 4 inch board slightly longer than your scale’s cord, mark two holes the chord lengths apart, about 3/8" from the edge. Draw a line perpendicular to the edge exactly half way between them. Drive nails part way into the two chord end holes. With a straight edge against the nails (on the side nearest edge) mark where the straight edge crosses the midpoint line. Now drive the nails through the end holes and cut them so only ¼ to 3/8 inch shows.
Railroads measure their curves using the "high" rail. On a curve, they "super elevated" (raised) the outside rail for the same reasons race tracks are banked. It reduced the effects of centrifugal force and allowed higher, safer speeds. That is why it is called the high rail. We will do the same. Place the board’s nails against the inside edge of the outside rail. Measure the distance from the boards cross point to the near edge of the rail. Every 1/32 the rail is away from the cross indicates 1 degree of curvature. If you know the actual radius, the chart below will convert them to curvature degree.

Radius In Curvature of various large scale gauges (in degrees)

Standard gauge railroads built their fixed plant (track, roadbed, etc.)as broad and level as possible. Increased construction costs were outweighed by reduced operating costs. Narrow gauge roads were built to minimize construction costs and therefore used much sharper curves. Logging, plantation and other industrial roads spent as little on track construction as was possible and had curves as sharp as the rolling stock would allow. The chart below lists different railroads and their sharpest curves.

*these figures are from published charts and some radius do not compute to the degrees given.

As you can see by comparing both charts, very few large scale railroads (including mine) can afford to have scale curves.

Sources:
Field Manual for Railroad Engineers, Nagle, J., Braunworth & Co., Brooklyn, N.Y., 1917
Narrow Gauge Railroads in America, Flemming, H., Graham & Hardy, Oakland, Ca.
Hilton, George W., American Narrow Gauge Railroads, Stanford, University Press, Palo Alto, Ca. 1990