NY Signal drawn
At the curved end of each easement, it has a radius equal to the circle it adjoins. Zooming into the lower easement, we can see why it is also known as a Transition Curve (animated, 126K). It changes the radius from that of the curve to that of the tangent track, and the rate of change in radius is constant along the easement. By having the easement start farther along its length, any radius between infinity and the radius of the curve can be found.
Easements provide two valuable purposes in railroading. The first and most important for the prototype, is to gradually increase the side force felt as a train moves through a curve. For example, if you were driving your car and quickly snapped the steering wheel into the right position for a curve, you and your passengers would be thrown to the side. What we do while driving, and what the prototype railroads do, is "ease you into the curve". This familiar expression helps you understand this role of the easement.
The second purpose in railroading usually applies most to the small radius curves we use in model railroading. The two pictures below show this quite readily. Both of these image are the same example used above, with an SD-40 and a 60' flat car to demonstrate the second purpose of the easement. In both pictures, the SD-40 has been highlighted to show the position of its trucks.
In the top picture, we can see how much overhang results from track design without an easement. The truck of the SD-40 is aligned to the curve, just before its point of tangency with the straight track. The flat car is exactly horizontal, still on the tangent track. The end of the SD-40 and its coupler are well to the side of the track centerline. The resulting misalignment between the locomotive and flat car could easily cause a derailment.
One more railroading characteristic is associated with the easement, and that is Superelevation. On the prototype and on detailed models, the height of the outside rail is raised through the easement on then maintained through the curve, further counteracting the effects of centripetal force.
The Offset of an Easement
In the example, the circle has been extended and a second line has been drawn that is directly tangent to the circle. We can see that there is an Offset between track that is directly tangent to the circle, and track joined by an easement. The Offset is present to allow the radius to gradually increase to infinity as the easement leaves the circle. The Radius of the easement at any point along its length can be found using this equation:
Radius = (Length * FinalRadius) / DistanceAlongEasement
This equation states that the radius varies in proportion with distance along the easement. The varying separation between the original tangent track that achieves this rate of change can be found at any distance along the easement using this equation:
Separation = (Distance ** 3) / (6 * FinalRadius * Length)
This is the cubic equation referenced by John Armstrong is his excellent book Track Planning for Realistic Operation, and used by railroad surveyors to lay out track on the prototype. This equation is not quite what we need as model railroaders, as it is much easier for us to place the curve and then draw the easement from it. We need to know the offset from the tangent line of the circle to the tangent line from the easement. That is provided by this equation:
Offset = (Length * Length) / (24 * FinalRadius)
This is the distance between the two lines as shown in the diagram above. Using this, any easement can easily be laid out on your model railroad.
Laying out an Easement
To form an attractive, functional easement on your model railway, first locate the centerlines of the curve and tangent track to be connected by the easement. Mark the point of tangency with the circle as shown below. At this point, the easement is exactly halfway between the circle and the line.
Now take a yardstick and bend it so that it follows the circle at one end, is tangent to the tangent track at the other, and falls halfway between the two at the point of tangency, as shown above. This will form an excellent transition curve if you carefully align it with your eyes. Alternatively, you can print the transition curve from a computer program and use the output as a template.
It's really worth the time to plan easements into your layout. It will run better with long equipment, and it will look more like the prototype. Modelers using sectional track can substitute a larger radius section at each end of curves to simulate this easement effect.
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