The "Degrees of Curvature" tabulated in the reference are specified per unit of length of trackage. A longer radius has a lower Degree of Curvature per unit length, as demonstrated by the table. The table does not define the unit length for the stated Degrees of Curvature, but the unit length will be constant for all entries. I could calculate the unit length from the table, but I'm too lazy, and it's not particularly germane to the argument. Sectional track curves are specified in degrees of span, and radius of curvature, which are two different yet independent quantities. Degrees of span define what part of a circle (360 degrees) the section spans. Degrees of Curvature should not be confused with degrees of span, but are directly related to radius of curvature, as demonstrated in the referenced table.. The reason for using Degrees of Curvature, rather than radius of curvature, on 1:1 railroads was that it was not always practical to locate (or measure to/from) the center reference point for that radius, for laying out the railway (the center point could be inside a mountain, in the middle of a river, or high in the air, etc.). By specifying curvature in degrees per unit length, the path of the railroad could be laid out without having to go far outside the bounds of the track itself. Using Degrees of Curvature in the field, every so many feet or meters of track length, it should bend left (or right) by so many degrees. When you consider how curved rails were bent during installation (they did not manufacture pre-bent rails for various Degrees of Curvature, they bent them in place,) it makes even more sense to use D of C, rather than Radius of Curvature and degrees of span. In model railroading, it is much easier and simpler to use radius of curvature and degrees of span for a curve when planning and building a model railroad (all you need is a protractor and a compass.) Laying out the easements gets a little more difficult, but not significantly. Sectional curved track is sold in increments of degrees of span. So, regardless of radius of curvature, a given degree of span always creates a given fraction of a complete circle. In other words, a 45 degree section spans 45 degrees out of a circle of 360 degrees. In other words, it takes eight 45 degree sections to make a circle, or 24 pieces of 15 degree sections to make a circle. Sectional track is also specified in different radius of curvature, which is the radius of a complete circle made out of the same radius sections. Therefore, if I have 180 degrees of a curve (not curvature), in any combination of sections of various radii, I have still turned the track 180 degrees. Granted, conventional easements incorporate a continuous rate of change in radius of curvature from straight to the final radius, and the sectional technique uses discrete steps in radius to accomplish the same goal, they both have the same effect of reducing the severity of change of curvature, which is what we modelers (and the 1:1 railroads) were seeking. Not everyone wants to use flex track (or hand-lay their track rails). But that does not mean they have to abandon the principle goal of easements!
To completely specify a curved piece of sectional track requires (by commercial convention) BOTH radius of curvature and degree of span. If so many total degrees of span are substituted with different radius pieces, the result will still form the same fraction of a complete 360 degree turn, but without specifying the radius of curvature, the new circle may be larger or smaller than the original it is replacing. And if you don't replace different radii sections symmetrically across the circle, then the result will not form a closed loop (the "ends" will not meet). If trying to ease an existing 180 degree curve by increasing the radius of the two end sections, then the result will be similar to half of an ellipse, and it will be slightly wider than the original half-circle (measured across the two ends). So, you will also need to substitute shorter radius section(s) in the middle of the curve to get the same width across the ends of the 180 degree curve. Easing a 90 degree curve works similarly: longer radius pieces at the ends, and narrower radius piece(s) in the middle. The shorter radius pieces in the middle of the curve do not usually present a big problem for equipment rolling over them, since they are not adjacent to straight track. And judicious placement of terrain and/or scenery can hide the disjointed appearance of long cars passing through the short radius section of the curve. It's a lot easier (and cheaper!) to experiment with easing sectional curves using layout planning software to see what works.
Here is the problem. Take a piece of 9.75 inch radius sectional track. Let's say that six pieces of such track will produce a half circle (180 degrees) 19.5 inches wide. Now take a piece of 11 inch radius sectional track. Six pieces will also produce a half circle (180 degrees) but now the width is 22 inches. Now take four pieces of the 9.75 inch radius track and 2 pieces of the 11 inch radius track (for the 'easement') and construct a half circle. Does the 'half circle' equal 180 degrees? You said in the quote above, "A longer radius has a lower Degree of Curvature per unit length, as demonstrated by the table." So by using the 2 pieces of 11 inch radius sectional track with their lower Degree of Curvature as compared to the 9.75 inch radius track will result in a 'half circle' less than 180 degrees which will require some form of a kink in the straight sectional track or use of flex track. One would better off not attempting an 'easement' with sectional track because, operationally, easements add nothing to a typical N scale layout
No, because the sections of 9.75" radius aren't as long. If they were as long, the same number of them couldn't possibly form a smaller circle.
Circumference is proportional to radius (and therefore, to diameter.) Proportional simply means that as one property gets larger/smaller by a given percentage, the other will also, by the same percentage. So a smaller radius curve of equal subtended angle (e.g. 180 degrees) will be shorter in length than a larger radius curve. And a longer radius one will be longer in length. I strongly disagree that easements do not add operationally to a typical N scale layout (whatever typical means). If your N scale layout is large-room-filling, your statement may be true, but such a layout is hardly typical by any measure, regardless of what books and magazines would have us believe. For every one of them, there are likely dozens 4x8, HCD or smaller layouts. And even if easements cannot add to the operability of a layout, they can certainly improve it visually. No matter what scale one operates at, there is always a limit to what can be accomplished in the available space. It may not be the same limit across scales, but there's always a limit. And when pushing the lower limits of radius of curvature in any scale, easements (whether true or sectionally-approximated) can make the difference between a marginal piece of equipment (too long, too much overhang, etc.) working or not on that layout. Easements can make a big difference by allowing more railcars to use body mounted couplers, which make a huge difference when backing up a cut of cars, especially uphill. And as new models of modern equipment continue to push the operational envelope on existing small layouts, this problem will get worse, not better. Adding easements can make the difference between minor track adjustments and a complete scrap-and-start-over.
I should be more careful, but the span of an angle is also called the subtended angle. If you think of a typical piece of pie, the angle can be measured at the pointy end (assuming it hasn't been eaten yet) with a protractor. But when the piece of pie, except for the crust, is eaten, the crust resembles a curved piece of sectional track. It has the same span or subtended angle as you would have measured at the pointy end (before you ate it.) This is also assuming it was not a rum pie, and and that you had not eaten too many pieces before you cut the slice we are considering!
I should be more careful. A half of a pie is not considered a typical piece of a pie, for the purposes of the above discussion.
Ugh... For the purposes of the above discussion, a typical piece of pie should be less than 1/4 of the pie (lest the pointiest end would not be the end you would measure before eating it.) Regardless of whether anyone is watching when you slice and/or eat the piece of pie, or would notice afterward.
Maybe this example will help: Kato Unitrack. The top loop is just an oval made of six 248mm straight pieces and 381mm, 30° curves. For the bottom loop, I took one curved section out of each end and put in a total of four, one at each end of each curve, 718mm, 15° curved pieces. Complete loop with no kinks and simulated "easements" at each end. And yes, operationally the "easements" absolutely do make a difference. I have certain combinations of equipment that will not go around the first loop without derailing, but will the second one, and everything looks much better on the second one.
Oh, sure, a picture IS worth a thousand words! And to take that excellent example a little further, What happens if adding the easements made the oval taller (top to bottom) than it needed to be to fit the space? Just replace the middle 381R30 on each curve with a 348R30 (or maybe two 315R15's) to narrow the entire curve a little (top to bottom). Generally speaking: large, abrupt changes in radius (e.g. uneased transitions from straight to curve) cause more operational problems than even smaller radius turns, if properly eased. In my example solution, the sharper, curved sections in the middle are "eased" by the original radius curved sections on either side of them. Just like a race car on a track, the smoothest way (and easiest on the tires) through a curve is to start on the outside edge of the race track at the beginning of the curve, gradually work to the inside edge of the race track at the middle* of the curve, and back out to the outside edge of the curve at the end of the curve. Similarly, you can gradually reduce each curved section's radius from the entrance of the curve to the midpoint of the curve, then gradually increase each curved section's radius on the way out of the curve. *Since race cars decelerate (brake) more quickly than they accelerate, for maximum speed around the entire course, they shift the apex of their path through the curve towards the entrance of the curve, allowing them to accelerate more coming out of the curve, into the straight-away.
Humm? I thought I kept it simple enough. This is one of those subjects that can go on forever and etc. Enjoy your layout no matter what decisions you make. It's all about Fun!
Wait, you mean we're supposed to STOP planning, build the bloomin' layout, and actually RUN TRAINS on it? Preposterous!
Yep, it's all about the fun! And I find it a lot more fun when I have smooth, derailment free running, which easements and simulated sectional track easements can help with a lot.
I've had my layout running for 9 years. The width of the layout is 32 inches. Unitrack sectional. Double mains. NO easements. Trains run flawlessly. No derailments. No problems at all. I'm going to put this out there. Make sure your trackwork is flawless...easements or not...less problems all the way around.
That's great, if you can get your layout running flawlessly with no easements, then you don't neeed them (although in my opinion the trains still look a lot better with them). A big part of the equation is your equipment. As I mentioned before, some of mine would not run reliably on the curves I had without easements (and that was with perfectly laid Unitrack). The main culprit was a Kato SD80MAC, a very long engine with body mount couplers. With some cars its overhang would pull the car to the side hard enough to derail it. if I added one extra wide curve (481 or 718) before the straight section the problem was solved.
Yes; I've often used 481R15's to ease the entrance to a curve. Looks really nice in operation. But in practice, often anything with larger radius than what the rest of your curve is will help. And surprisingly, using smaller radius curves in the middle of the curve (themselves eased by the remaining original radius curved pieces) are still reliable, and can allow the eased curve to fit in the available space. This reinforces my opinion that it's not so much the minimum radius that determines un/reliability, it is the severity of transitions in radius, especially at the entrances and exits of the curve, when transitioning from/to straight track.
Man !! This guy is getting pummeled with answers haha! Glad to see you all are still at it!! This is all great info and accurate. maybe this is the worlds way of letting you give N scale a try. I wanted to do HO too but space and also the cost of HO products is double to what n is. So that is a plus. Not to mention what you can do in a small space. My layout is less then 3 ft wide and I run three turns at each end. Now I can’t run a big boy but most everything else works. another thing I don’t see here. Banking your turns. That is, unless your using ink track you can’t do that. If you are using flex track or any other track with no base you can set masking tape in thin strips on the outside edge of said track is going to lay. That way when your rolling with 100 cars at 75mph the train will lean into the turn a bit and help it not derail. just thinkNASCAR! Cheers
