Curve calculations for 15" gauge railroads

The surveying and layout of prototype railroad curves in the U.S. involves the concept of "degrees of curvature". This is a convention which states that the curvature of an arc is stated as the angle subtended by a chord of 100 feet. The radius (R) of a curve of A degrees is given by:

         R  =  50  / sin(A/2)  ft

 What makes such a scheme useful for railroad surveying is that this angle is also the deflection of successive chords along the curve. Merely placing a transit at the end of a 100 foot tape sighting back to its beginning, and then forward to the end of the next 100 foot interval at the curve angle will produce a succession of points along the required circle. Notice that when entering the curve from its tangent that the first deflection is half the curve angle.

Hobby railroads are usually built on a much less extensive basis than the real thing and a similar scheme using 20 foot chords seems more appropriate. Remember that a 50 foot radius curve would be a 180 degree curve with a 100 foot chord and that sharper curves could not even be described.

Tthe table below relates the angular deflections of chords of 20 feet to various curve radii. Often the terrain allows more than one station on the curve to be sighted. Therefore, additional columns are shown giving deflections and distances for additional stations along the curve. To use this table, the transit would be positioned at Station 0, at the end of the tangent. Sight backward parallel to the tangent track and rotate the transit 180 degrees. With a 100 foot tape fastened at Station 0, the successive Stations may be quickly laid out by placing markers at the angles and distances given. Remember that this table assumes that the first deflection is from the tangent leading into the curve to the first deflected point on the curve. If the point preceding Station 0 is already on the curve, as when a curve already started is to be continued, the deflection angle shown for Station 0 should be doubled.

The column in this table labeled "chordal offset" is the distance from the center of any 20 foot chord to its circle. This means, for instance, that if a 20 foot string were stretched between two points on the inside of a rail curved with a 50 foot radius, that the distance from the string at its middle to the rail would be 12.1 inches.  Since  much 12 lb rail (normally used for 15" gauge track) comes in 20 ft lengths,  you would bend  a rail to this offset before laying.

Good railroad practice does not allow a tangent track to immediately enter a constant radius curve. Rather, the curvature begins to increase along the path of a spiral until it matches the desired radius. Although precise mathematical treatments of such transition spirals are available, they are too complicated to be of practical use on the small railroad where the differences in curve shapes amount to only a few inches when speaking of 100 foot radius curves. The natural spring of steel rail and the short distances usually involved means that satisfactory transitions can be accomplished with a simple table of offsets.

What the table shows is the case where the circular part of the curve is displaced from the actual end of the tangent track. This allows the first section of rail from the end of track to the circular part of the curve to be of larger radius than the curve itself. The rail in this transition may be easily bent to provide a suitable transition.