A number of years ago I read that 1:1 railroads do not use a fixed radius for their curves but calculate them from a formula. Does anyone model curves using that prototype formula?
Theses are called easements (there is another topic on this in the prototype section right now.) The idea is that the radius changes smoothly (ie linearly)
The formula is called an Euler spiral. Almost nobody uses the actual formula. There is actually no exact solution, so even the prototype is actually using an approximation.
However, many people use a method of offsetting the curve and using a stick to transition. This is described in John Armstrong's "Track Planning for Realistic Operations". It turns out the Euler spiral is also the solution of the bent stick. So while nobody does the math, the stick is a very effective way of active the solution.
There are other ways people achieve the easement, but they can describe it better then I can. I did go down the rabbit hole of the math of these easements so that I could get the offsets for any curve. (Armstrong just has a table for a few radio). However, I would not recommend that. I had a lot of time on my hands and didn't have space to start building my layout.
Prototype curves are not expressed in terms of radius but rather in terms of degree of curvature (which could be expressed as a radius but typically is not).
And unlike modelers who use sectional track (be it Lionel or Atlas or whatever), prototype railroads do not have a tangent meet a curve of a fixed degree of curvature/radius, but transition into it with what is often called an easement curve.
There are formulas for this easement curve, and entire texts (and thick ones at that!) have been published on the topic. The one in my library is Allen's Railroad Curves & Earthworks. There are others. Many factors go into the calculation, such as train speed. The formulas tend to make my brain hurt.
There are templates that provide a practial usable easement curve for modelers, published over the years by the NMRA as well as Model Railroader magazine, that are likely based on formulas. John Armstrong gives some suggestions and a formula in his book Track Planning for Realistic Operation. I am not sure it is a scaled down version of the prototype formula however, and moreover I am not sure we need to do so -- our curves are too tight perhaps.
But the advantages of easement curves in running our trains are real - it isn't just cosmetic (which super-elevated curves largely are). But having said that, track laid with generous easement curves can be beautiful to look at and run on with the smooth flow of the track; our British colleagues were laying such track before it became popular here.
Armstrong shows in his book that easement curves allow us to have tighter curves on our layouts that can handle long locomotives and rolling stock that would derail if even a larger radius curve met a perfect tangent with no easement curve in between. That meeting place between tangent and curve is a pinchpoint for our trains, more so than a perfect circle of track would present.
Fortunately for those like me who run kicking and screaming from formulas, there are purely mechanical means of replicating an easement curve that are entirely practical for model railroad purposes. One is the "bent stick" method, based on the notion that a small stick fasted at one end, and then bent, creates the sort of easement curve - sometimes called a French curve - that we seek. Draw a line along the bent stick and you have your own template. Another is the offset roadbed method which imagines that we do have a perfect tangent meeting a fixed radius curve -- and now move the tangent out a bit, an inch or two or whatever, and connect up the track creates its own easement curve
Splined roadbed, using three or four sheets of vertical homasote or whatever, creates its own natural easement curve. For that matter, using flex track that spans the area between tangent and fixed radius curve will also create its own natural easement curve -- you just can't bend a piece of flex track to do what a section straight piece mated with a sectional curved piece of track look like.
Still another method, which I have used, is to saw-in "kerf cuts" into the edges of plywood subroadbed which has been cut to create tangent as well as fixed radius curve, making it somewhat bendable at the ends. Woodland Scenics styrofoam "risers" work on this same property. So does the homabed product.
Regardless of method used, almost any layout can benefit from using easement curves, even if it is a mere approximation of an easement curve, such as introducing a piece of 22" radius section track between a tangent and 18" radius section track curves. The explanation and demonstrations in John Armstrong's book are recommended reading.
Dave Nelson
I used this system to lay out my easements:
https://www.jglrr.com/engineering/software/spiral/index.html
It gives you the offset for 10 equally spaced waypoints.
Ed
It's not so much that 'fixed-radius curves' aren't used as that a smooth transitioning from 'straight' track into the desired radius is made -- this usually following a section of a mathematical curve known as a 'spiral' [as noted above -- there is a colossal delay in making some postings, for which I offer apologies in lieu of Kalmbach "IT"...] I am a bit surprised that no one has mentioned the name 'Arthur Talbot' -- his book on the subject remains a suitable reference (and you can read the 1908 edition free on Hathitrust)
https://babel.hathitrust.org/cgi/pt?id=nnc1.cu55767079&view=1up&seq=8&skin=2021
Many people use and advocate this general approach in model trackwork, and we have had many threads here regarding the practice; in fact there is a current thread regarding wye curves that was adjacent to this one at the time I wrote this.
In practice the careful use of spiraling for lateral and vertical curvature and for superelevation is usually not as necessary as on the prototype, so some fairly crude approximations can usually be made to work. MR had an early article (1970s iirc) on the subject of curve easement in which they actually printed an HO 'transition spiral' suitable for tracing and gauge/template making... that fit conveniently on a single magazine page. Rather than chuckle, I'd point to this being all the 'transition' that many model layouts would need.
I don't think anyone actually lays out transition curves as the prototype does, which would involve very precise reading of scale rules where the geometry is sensitive to very small errors. Instead the approach is akin to using a French curve in drafting to transition smoothly between two curved segments or a curve and a 'tangent' line. In fact, this approach itself can be used if you have large enough ship curves: draw your track center lines directly on a paper template or the subroadbed, lay off the transition with the curve and trace it, and then if necessary translate the curve to an outside radius of roadbed or ties if you need a visible reference to follow.
And as simple as our model approximations may be, the advantages to both appearance and operation are dramatic.
Sheldon
ATLANTIC CENTRALAnd as simple as our model approximations may be, the advantages to both appearance and operation are dramatic.
Lastspikemike Easements can easily be created by eye in part because it is the eye you are catering to, not physics. As for so much of modelling what looks right, to you, is right because that's the objective.
Easements can easily be created by eye in part because it is the eye you are catering to, not physics. As for so much of modelling what looks right, to you, is right because that's the objective.
Rich
Alton Junction
Lastspikemike...they are really only worthwhile at the most visible parts of the layout where your trains run fast...
Lastspikemike...it is the eye you are catering to, not physics...
If all of your curves are wide enough to not have any derailment concerns without easements, then I suppose they would be considered just "visual"; however, many people use them to have smooth trouble-free running on curves tighter than they could otherwise.
Lastspikemike In practice quite short transitions provide most of the desired effect. Any easement necessarily tightens the remainder of the curve. For most home layouts where we are trying for maximum radius curves in the space available you are well advised to make your easements as short as possible and they are really only worthwhile at the most visible parts of the layout where your trains run fast. Within tunnels you can make easements very short indeed with no effect on running. Model trains have massive flanges running on tall rail sections and the forces developed are tiny compared to prototype. The materials used are massively stronger than prototype and the trains are much lighter. Actual speeds are trivial. Easements can easily be created by eye in part because it is the eye you are catering to, not physics. As for so much of modelling what looks right, to you, is right because that's the objective.
In practice quite short transitions provide most of the desired effect.
Any easement necessarily tightens the remainder of the curve.
For most home layouts where we are trying for maximum radius curves in the space available you are well advised to make your easements as short as possible and they are really only worthwhile at the most visible parts of the layout where your trains run fast.
Within tunnels you can make easements very short indeed with no effect on running.
Model trains have massive flanges running on tall rail sections and the forces developed are tiny compared to prototype. The materials used are massively stronger than prototype and the trains are much lighter. Actual speeds are trivial.
Or, deticate more space, or build a less complex layout in the space you have.
That's what I am doing. My new layout could easily be built in 1/3 less space with 30" radius and a little less real-estate for scenery.
But large curves are a "must have" for me.. the extra inch of width for the easement is no problem because the layout is not crowded in to begin with.
Lastspikemike Whatever the space you have an easement always makes the curve tighter than if you used all the width for the constant radius. That's just geometry. The key aspect to using an easement is that the resulting tighter curve works better than using all the space you have to maximize the radius. Quite short easements will get you that effect. The rest is how it looks. If you've got the space by all means lengthen the easements at the ends of the curves. If you are tight for space shorten up the easements. A few inches of easement works fine and looks better both with and without a train running through it. But the theoretical ideal easement just isn't possible to use in an average home layout and trying to use them will probably foul up the rest of the plan.
Whatever the space you have an easement always makes the curve tighter than if you used all the width for the constant radius. That's just geometry.
The key aspect to using an easement is that the resulting tighter curve works better than using all the space you have to maximize the radius. Quite short easements will get you that effect. The rest is how it looks. If you've got the space by all means lengthen the easements at the ends of the curves. If you are tight for space shorten up the easements. A few inches of easement works fine and looks better both with and without a train running through it. But the theoretical ideal easement just isn't possible to use in an average home layout and trying to use them will probably foul up the rest of the plan.
Lastspikemike...The key aspect to using an easement is that the resulting tighter curve works better than using all the space you have to maximize the radius...
Exactly! Since you can get smoother, more reliable running in the same space using a slightly tighter curve and easments, the easment is not just visual.
Lastspikemike In practice quite short transitions provide most of the desired effect. ....you are well advised to make your easements as short as possible.
....you are well advised to make your easements as short as possible.
I disagree for the following reasons:
I used Kato's large radius track (31.33" Radius I believe) that actually comes in superelevated transition sections (you can buy full superelevation track and the transition pieces). On a purely flat table top, that track would probably be just fine. However, to secure the track to the pink insulation foam sheets that represent my table top, I used a combination of Liquid Nails and/or joint compound. In one area, it later became obvious AFTER the Liquid Nails/Joint Compount was all set up that I had completed one superelevation transition TOO FAST.
Many of today's diesels, from Rapido RS-11's up to big GEVO's only have so much horizontal and vertical motion built into the truck swiveling mechanism. If your "easement" track section combined with any superelevation transition (in real life they are usually the same length, ie the superelevation transition to full super takes place within the easement length) is too short you will have derailments specifically because the rate of change of cross slope exceeds all the combined horizontal and vertical motion designed into the typical diesel power truck. The track can still look good, but it isn't good.
So on my layout, if I back engines up through the one easement section that is superelevated (with the correct arrangement of Kato superelevated track pieces) they derail. It doesn't matter if it's a Rapido RS-11 or a big Gevo, most still have a tendency to derail.
Outside that curve (which is glued down hard) I have another track curve with NO EASEMENT whatsoever and NO SUPERELEVATION whatsoever, Kato 34.13" Radius sectional track. Guess what? Nothing ever derails on that curve, in either direction.
I say use easements and superelevation VERY carefully. If the cross slope of the rails in final glued or nailed down position changes too rapidly, the diesel loco trucks cannot handle excessive rocking motion and will derail even at speed step 1.
In my case I have not relaid that section of track yet because as long as I don't backup through that curve, I don't have problems. Also, relaying it will be quite destructive. So I live with it.
I think in the model world it is better to have the larger radius and forget about the superelevation and/or easements as for me on my layout they actually have caused more trouble than they are worth. It is hard to control radius change and cross slope change in the model world. It's easy to mess it up.
As always, your mileage may vary.
John
PRR8259...I think in the model world it is better to have the larger radius and forget about the superelevation and/or easements as for me on my layout they actually have caused more trouble than they are worth...
What caused your trouble is the transition to superelevation, not the easement. In my experience, a slightly tighter curve with easements has always been smoother and more reliable than the larger curve with no easmenent, but for comparisons sake they both have to be level or both have superelevation. If you're comparing one that's flat to one that has superelevation it's not a fair comparison.
CSX RobertWhat caused your trouble is the transition to superelevation
my understanding is that the change in super-elevation goes hand-in-hand with the curve easement -- just as there is a gradual change in curvature, there is a gradual change in super-elevation.
greg - Philadelphia & Reading / Reading
gregc CSX Robert What caused your trouble is the transition to superelevation my understanding is that the change in super-elevation goes hand-in-hand with the curve easement -- just as there is a gradual change in curvature, there is a gradual change in super-elevation.
CSX Robert What caused your trouble is the transition to superelevation
Yes, when you have both. When you only have an easement, you only have an easement. Not all trackage, even mainline trackage is/was superelevated.
The decision to superelevate, and how much, is speed dependent.
The decision to have easements is not strictly based on speed. As I said recently, the two are not joined at the hip in all cases.
And as it relates ot our models superelevation is a visual luxury, easement are a true prefromance improvement.
Our train don't have enough mass relative to wheel friction for superelevation to work correctly.
Physics relating to weight/mass typically to not scale. Physics related to angles and truck/coupler swing scale perfectly.
IAAHMnobody will notice any effect from superelevation of model railroad track
Superelevated model railroad trackage on curves looks amazing and is worth the effort.
-Kevin
Living the dream.
PRR8259So on my layout, if I back engines up through the one easement section that is superelevated (with the correct arrangement of Kato superelevated track pieces) they derail. It doesn't matter if it's a Rapido RS-11 or a big Gevo, most still have a tendency to derail.
The other thing I forgot to mention - the Kato "easement" tracks are not spiral easements, they are constant radius pieces that transition from level to superelevated.
Splined roadbed, eased curves (because of the splines), and superelevated.
Speaking of curves, our hosts this minute show a user online (in the margin at right) who is rather over-well endowed...if you knowhaddamean. What's up with her?
LastspikemikeThere is no operational advantage to including superelevation in an HO railroad....
Probably not on my layout, as the track is mostly curves (with easements) and grades, most of it meant for low or moderate speeds.
LastspikemikeFrom any kind of visual distance you don't notice superelevation in typical prototype freight track, the speeds just aren't that high.
Back in the days of the original photobucket, I posted a video which had a camera on a flatcar, immediately behind the tender of the steamer pulling the train, and it was plainly visible when the train entered the vertical easement into the superelevation, and through the transition into the greatest portion of superelevation and then out of it. The current photobucket has the photos representing the beginning of various videos, but the videos are no longer operational.
LastspikemikeAt the very low highway speeds we are allowed to drive at superelevation is not there for safety related to vehicle speed, there isn't enough to make a difference.
Perhaps not in Alberta, but superelevation on highways, especially in southern Ontario, is quite common and speeds on major highways (when not clogged by traffic jams) seems to be around 130-140km/hr (roughly 81-87mph), even though the speed limits on major highways is 100-110km/hr (62-68mph)
Many years ago, I somehow got onto a newly-built highway the night before the day that it was scheduled to open. Obviously, there was no other traffic, so it took only a few seconds to get up to 100mph, and as I approached a long rising and rather severely superelevated curve, was on the gas with a vengance, running through that curve at 145mph...just as smooth as a marble down a drainpipe. I then took the first option to leave that road, surprised that an exit was open, and that I'd seen neither a roadblock nor a cop car...no point in pressing my luck.We also have roads (not highways) with reverse superelevation on curves to accommodate drainage, and for inexperienced drivers (who think they're hotshots), it can be pretty scary if you happen to be beside them when they suddenly find themselves in an unexpected situation. I'm not sure, but those places were likely built that way as a cost-cutting measure...just plain dumb, in my opinion.
When I began adding superelevation into my layouts' curves (most of the track and especially the curves, are on riser-supported 3/4" sub-roadbed) I simply parked a train which covered the length of the curve, then selected the riser closest to the mid-point of the curve, pushing the bottom of it outward until the train looked like it was suitably superelevated. All of the risers on either side of that one were automatically offset to diminishing amounts as they moved towards the beginning and end of the curve, and were then screwed securely to the open grid that forms the base of the layout.
It's very easy to do, and I've done it through a fairly broad-radius S-curve, too...it's especially nice to watch a train thread its way through, the locos and cars gently tilting left and right as they make their way along the route.
Wayne
LastspikemikeThere is no operational advantage to including superelevation in an HO railroad... ... but nobody will notice any effect from superelevation of model railroad track
Lastspikemike At the very low highway speeds we are allowed to drive at superelevation is not there for safety related to vehicle speed, there isn't enough to make a difference.
"One difference between pessimists and optimists is that while pessimists are more often right, optimists have far more fun."
LastspikemikeThere is no operational advantage to including superelevation in an HO railroad.
True, in fact, in our modeling scales superelevation increases the chances of stringlining or derailing.
Lastspikemikenobody will notice any effect from superelevation of model railroad track
Many people do find it noticeable, if they didn't, why would they bother doing it?
LastspikemikeFrom any kind of visual distance you don't notice superelevation in typical prototype freight track
There's a curve on the freight only CSX mainline near my house that has quite noticeable superelevation.
CSX Robert ... Lastspikemike From any kind of visual distance you don't notice superelevation in typical prototype freight track There's a curve on the freight only CSX mainline near my house that has quite noticeable superelevation.
...
Lastspikemike From any kind of visual distance you don't notice superelevation in typical prototype freight track
Our friend would probably be amazed at the super on the tracks in and around the Blue Ridge where slow-moving Y6b's slogged up the grade heavily canted to one side. I know I was when I first saw the images.
Lastspikemike I'm a bit surprised nobody posted the link to the NMRA item on easement and superelevation: https://www.nmra.org/sites/default/files/d3b1.pdf Assertions that prototype superelevation is noticeable induced me to take a look at some of the photos in Morant's book about the CPR. If you really look you can just see the effect if you already know you're looking for it. Both railroads and highways use similar guidelines for calculating maximum recommended superelevation. Not that much. Modelling superelevation is virtually impossible. There's not going to be enough space. Superelevation needs transition sections also. Long ones. Camber on roads is different to superelevation. Camber is for drainage. Reverse camber will be on all roads on the outside of the bend so you'd better not let it unnerve you at speed. Of course, superelevation reduces negative camber on the outside of a bend. For one way sets of lanes reverse camber is sometimes completely eliminated by superelevation. One aspect of superelevation that causes problems is when transitioning from left to right curves or vice versa. One place that occurs in my City is in the middle of a bridge....not such a great idea.
I'm a bit surprised nobody posted the link to the NMRA item on easement and superelevation:
https://www.nmra.org/sites/default/files/d3b1.pdf
Assertions that prototype superelevation is noticeable induced me to take a look at some of the photos in Morant's book about the CPR. If you really look you can just see the effect if you already know you're looking for it.
Both railroads and highways use similar guidelines for calculating maximum recommended superelevation. Not that much.
Modelling superelevation is virtually impossible. There's not going to be enough space. Superelevation needs transition sections also. Long ones.
Camber on roads is different to superelevation. Camber is for drainage. Reverse camber will be on all roads on the outside of the bend so you'd better not let it unnerve you at speed. Of course, superelevation reduces negative camber on the outside of a bend. For one way sets of lanes reverse camber is sometimes completely eliminated by superelevation. One aspect of superelevation that causes problems is when transitioning from left to right curves or vice versa. One place that occurs in my City is in the middle of a bridge....not such a great idea.
This may come as some sort of surprise to you Mike, but lots of people build model railroads with 36" radius and larger curves, 15" to 18" long easements, superelevation, #6 and #8 turnouts, and actually consider such standards as the "minimum acceptable" requirements for good modeling.
Sure, not everybody has the room, and not all choose to use their space that way, but I could take you to a couple dozen layouts, all in one day, within an hour of my home, with such features.
Not everybody is doing what you are doing. Starting with my second layout at age 13, I was using standards similar to those I listed above. 50 years later I still am.
I think the eye's ability to detect things may be surprising to many. They may not be able to pinpoint exactly what it is that they're seeing, but they know it looks better.
I can tell which color of paint has been applied over the other. Or how an SD40-2 more sharply diverges away from tangent on a PECO #6 turnout compared to an Atlas #6, which to my knowledge, is something nobody else has ever noticed.
- Douglas
LastspikemikeAssertions that prototype superelevation is noticeable induced me to take a look at some of the photos in Morant's book about the CPR. If you really look you can just see the effect if you already know you're looking for it.
Different people have an "eye" for different things, so for some it may be more noticeable than for others, but when I look at a picture like the folowing, it's plainly obvious that the locomotive is leaning without me knowing what I'm looking for:
LastspikemikeModelling superelevation is virtually impossible.
Again, if that were the case there would not be so many modelers successfully doing it .
LastspikemikeThere's not going to be enough space. Superelevation needs transition sections also. Long ones.
My transitions are in my easements and they are not any longer than the easements would be without the transition to superelevation.
Wish you would have told me this sooner. I laid my track in 1995 and superelevated all the visible mainline curves.
NYC_L3a_Pacemaker by Edmund, on Flickr
PRR_T-E7_sm by Edmund, on Flickr
Guess I really missed the boat on that one...
Why did I waste my time?
NJ Transit at elizabeth
Just a thought. Is there a timeframe when easements became the norm?
Here in the U.K. when the East & West Yorkshire Union Railway built their line in 1883, they would not know what you are talking about.
David
To the world you are someone. To someone you are the world
I cannot afford the luxury of a negative thought
NorthBrit Just a thought. Is there a timeframe when easements became the norm? Here in the U.K. when the East & West Yorkshire Union Railway built their line in 1883, they would not know what you are talking about. David
I have no working knowledge of railways in the UK, but here in North America by 1883 passenger cars were approaching 80' long, trains were longer and heavier, locomotives getting bigger, and air brakes were about to be required. Speeds were increasing, curves broadened, and I suspect the easement was already in play as a standard design feature on most main lines.
Surely by 1910, with air brakes mandated and all steel 85' cars about to become the norm, and much larger locos being developed by the day, modern trackwork was also following suit.
Again, my knowledge of UK or European railroading is extremely limited, but I have never understood the slow development of things like air brakes, automatic couplers, or larger equipment in other parts of the world?
While not considered safe enough for current use, by 1910 North American equipment had reached interchange standards we still use today.
And freight and passenger cars had reached sizes and weights that remained typical until the 50's, when they grew even more.
So here in North America, easements would have been just as desired and necessary in 1890, or 1900, or 1910 as they are today.