The article (website) with the title published on Trains.com is a complete error.
It is based on the same principle as “Build an adjustable guide to draw easement curves” published in the June 2010 issue of MR magazine, but unfortunately the way of thinking is wrong.
The method using the bar is perfectly correct based on the principle of cubic functions, but the direction of the straight line and the curve is reversed. MOROP, the European model railway standard NEM113 (pdf file), is the correct way of thinking.
Please ask someone knowledgeable in civil engineering.
I don’t know… I used pretty much the same principal for my easements when I put my track down in 1995 and today, thirty years later, I’m perfectly happy with my easements.
I think I surprised everyone a little too much. What I wrote is purely theoretical. In reality, all you need to do is connect a straight line and a curve with a gentle curve of about 24 degrees, and it will have no effect on train operation.
In the following diagram, point D, where the cubic function has a minimum curve radius, is approximately 24.1 degrees. On an actual railway, point A has an infinite curve radius, and it gradually gets smaller until it reaches point D. With a model railway, you don’t need to go through all that trouble; you can simply connect A and D with a uniform, large curve radius.
Thanks for posting this. I remember seeing that in Model Railroader and it didn’t dawn on me at the time that it was incorrect. I’m not sure what method I’ll use when I get to laying flex track on my current layout, but this is something to keep in mind.
The funny thing is, they seem to have used the correct physics when creating the drawing, because the spline does appear to have a sharper curve at the anchor point than along it’s path to the curve.
Well, if you’re happy with them, then I wouldn’t say you did anything “wrong”; however, if you used the method published in Model Railroader, you didn’t create a true easement.
As BN7150 pointed out, the spline is anchored at the wrong end. If you anchor a beam (or spline) at one end and apply a force to the other end to make it deflect, the beam will bend sharper at the anchored end and the radius will increase the further you go from the anchor point. For a true easement, you want the radius of the spline’s curve to increase from the curve to the tangent, so you should anchor it at the curve end.
doesn’t this depend on where the constant radius curve is fixed and the length of the curve between it and the straight track, the length of the A-D line is diagram?
how do you guarantee that the end of the spline is tangent to the straight track it is attached to?
The value Xc can be found by calculating a cubic function. Please see NEM113 of MOROP. The coordinate point (X1, Y1) and (X2, Y2) are intermediate points that are conveniently determined to draw the curve. Of course, these depend on the radius R.
I learned about the existence of easement curves in 1971, and laid an extremely accurate curve based on strict calculations. However, even when running trains, it is not clear at all. I was disappointed.
One day, my boss ordered me to check the safety of an amusement park ride that is like a small version of a roller coaster that moves around on a track. I was surprised that it did not have an easement curve. This is because as soon as it goes from a straight line to a curve, a centrifugal force is applied in the lateral direction. I was worried that the passengers’ necks would bend. However, I went to check the same facility and was convinced. That’s what people enjoy.
Simply applying a force to the opposite end will always create a sharper curve at the anchored end. In the above method, they overcome that somewhat by forcing the free end into a curve, but you still get the initially decreasing radius curve traveling away from the tangent, which ideally is not what you want.
By having the length from the curve to the tangent long enough based on the radius of the curve and the offset. The article above gives estimates, and according to the pdf BN7150 linked to, if I’m understanding correctly, the formula is L=sqrt(f x 24 x R), where f=offset and R=radius.
This is basically what I do when working with sectional track. I’ve done several temporary layouts with Unitrack and I always try to start my curves with a 718 mm curved piece to simulate an easement. It’s only 15°, but it’s enough to make a huge difference in the appearance of trains going into a curve vs going directly from straight track to a 15" or less curve.
When I have laid out the track route on my layout, always a plywood surface, I use Ribbonrail Metal Track Alignment Gauges to draw the curve (e.g., 32" radius). Then, when I lay flex track, I broaden the curve a bit. That’s my easement. Geometry and I don’t get along, so I don’t use it.
Cool, I never thought of that. By bending the straight “in” and the curve “out” where you able to even out the rail offsets so they wouldn’t have to be trimmed? How much did you notch the roadbed?
One thing I like about “springy” fleck track, like Atlas, as opposed to stiff flex track, like Micro Engineering, is that it naturally tends to form easements. One good point about it that this conversation brings up, however, is that like the “bent stick” method it’s better to anchor the curved portion first and let the natural bend of the track form the easement, which it sounds like you are doing by using the track alignment gauges in the curves.
