Spiral Easement Curve Calculator
To compute the Tangent Offset, o, and the Spiral Angle, Δ, enter the radius of the circular curve, R, the desired length of the spiral, Ls, and click "Calculate."
The Spiral Easement Curve
On the prototype, curves are banked, or superelevated, to allow faster speeds through the curve and to reduce lateral forces, which could overturn the train. At the scale speeds of model railroads, superelevation is not required; however, many modelers use superelevation because it greatly adds to the realistic appearance of a train rounding a curve.
A superelevated curve cannot be directly connected to tangent track—superelevation must be gradually increased from zero to full elevation. On highways, superelevation is run in on the tangent adjacent to the curve; however, this method is not suitable for railroads, because in the absence of counter-balancing lateral forces, the train may overturn to the inside of the curve.
Prototype railroads use spiral easement curves on high-speed main lines to provide a constant superelevation run in along the length of the spiral. Spiral easements also provide a gradual increase to full lateral acceleration, resulting in a smoother ride for passengers and freight.
Layout of the spiral curve.
The spiral curve recommended by the American Railway Engineering Association (AREA) possesses these fundamental properties:- The degree of curve varies proportionately from 0° (tangent) to the degree of the circular curve along the length of the spiral.
- The length of the spiral is exactly twice the length of the circular curve for the same central angle, Δ.
- The tangent offset of any point on the spiral is approximately proportional to the cube of the distance along the spiral.
- The tangent offset of the point of curve, PC, is twice the offset of the spiral's midpoint.
For model purposes, the spiral easement curve can be laid out using the wooden "batten" method described in John Armstrong's Track Planning for Realistic Operation (Fig. 8-8 in the 2nd Ed., Fig. 9-8 in the 3rd Ed.). Some computer-aided drafting (CAD) programs, such as AutoCAD, have a "spline" object that can be used to lay out spiral easements.
For both the "batten" method and CAD method, the two parameters you need are the offset of the curve from the tangent, o, and the spiral angle, Δ. To lay out a paper template by hand, the Microsoft Excel worksheet calculator computes x and y coordinates at tenth-points along the spiral. To download the Excel worksheet, click the link below:
• Spiral Easement Curve Calculator
For more information on spiral easement curves, please read the JGL Technical Bulletin on the subject:
• Technical Bulletin 2: Spiral Easement Curves