# Curvature vs. Minimum Radius



## rhyman (Apr 19, 2009)

The term _“minimum radius”_ can be very confusing to model railroaders, particularly those of us who want to make our outdoor layouts look more prototypical and less toy-like in appearance. As a necessity, manufacturers talk about minimum radius when they specify how their products will operate on our layouts. Sectional curved track are specified in radius. Most of us use radii when we are laying out our curves for our layouts. 

Unfortunately, the real railroads do not measure curves in terms of radius. That would be impractical for them. Rather, railroad surveyors use basic trigonometry to layout prototype curves. They reference curves using a term called "Degree of Curvature". 


A 1° curve has a radius of 5729.65 feet. Curves of 1° or 2° are found on high-speed lines. A 6° curve, about the sharpest that would be generally found on a main line, has a radius of 955.37 feet. On early American railroads, some curves were as sharp as 400 ft radius, or 14.4°. Street railways have even sharper curves. The sharpest curve that can be negotiated by normal diesel locomotives is not less than 250 feet radius, or 23°. Even narrow gauge lines like the Rio Grande Southern had maximum curvatures of 24°, and there were only two curves that sharp … one at the Ophir Loop and the other at Trout Lake. 


To convert degree of curvature into an actual curve radius, simply take the sine of ½ the degree of curvature and divide it into 50 feet. The result will be the prototype radius in feet. To find the model radius, just divide the prototype radius by the model scale. 


Sound confusing? It is … unless you are a math major. A much simpler approximation is to divide the degree of curvature into 5729.65. The resulting number is the prototype radius in feet. Then divide that number by your scale and you have the model radius. 


The following table does the math for you. I have converted standard track radii in one foot increments from 4 to 20 feet into degree of curvature for the five most common large scales. The curvatures are rounded to two decimal places. 


Degree of Curvature Table
As an example, if you are using 10 foot as your radius and you are modeling in 1:29 scale, then your curvature is slightly less than 20°. If you model in 1:20.32 scale and you want to layout a 20° curve, then your radius should be approximately 14 feet. 

Hope this helps take some of the mystery out of curvature for both newbies and seasoned veterans alike.


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## jimtyp (Jan 2, 2008)

Thanks for the info Bob, and the link to the table, I was looking for that a while back.


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## East Broad Top (Dec 29, 2007)

Narrow gauge curves could be quite extreme. The Uintah Railway quite famously had a 60-degree curve (that's 5' radius in 1:20), and other curves nearly as tight. No, not all their locos could make it around those curves, but their famous articulated 2-6-6-2 locos were designed specifically to do so. (It was their 8-coupled locos that couldn't make the curves). While the Uintah was decidedly the tight-curve champion in the narrow gauge world, it was not uncommon to find 30- to 35-degree curves on some roads. Those railroads were usually limited in terms of the locomotives they could run, as the long wheelbase 2-8-0s and 2-8-2s wouldn't make it around them. 

Incidentally, the reverse loop at Antonito that they use to turn their Ks is 225' radius (11' in 1:20.3). That's a 25-degree curve. 

Later, 

K


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## ralphbrades (Jan 3, 2008)

I am going to be polite and say : "None of the Above is applicable"... 

There is no method of transposing the gauge, nor even the shape of the profile of the flange, nor the diameter of the driver into the above calculations, nor is there any method of analysing the axle spacing to radius reqirements.

Most of the world outside of North America use *CHAINS* to specify radius and thus the super elevation of the curves with respect to them...

I would not class myself as a "Math Major" -just a person who in their normal daily life does *thousands* of calculations. This has enabled me to design locomotive chassis of 2-C0-2 and 2-D0-2 that can take 2.2m radius curves. The former uses a combination of "colonial profile" drivers with coned bogie wheels the latter is pure coned wheels and tapered flanges. I model in 16mm scale on 32mm track and 13.5mm scale on 63.5mm track. The curves for the former are 1.25m (1.18 chains) and 2.2m (2.46 chains) respectively and are typical of the type of track work that they would have run over in real life and I do not find them them to be "toy train like" in the least.

ralph


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## Greg Elmassian (Jan 3, 2008)

I took Bob's post to be as it was advertised, a conversion of degrees of curvature to radius. Very helpful. He gave examples for standard gauge and some US narrow gauge. Puts it in great perspective of how far our model curves can be from the prototype. A simple way to convert from degrees to radius is great, I even quoted him on my site in my Beginner FAQs.

Saying "None of the Above is applicable" is only true if you believe the entire world is in chains. Why say this, when you could have just _added _in the information about chains? 

You said "NONE" like not at all... that is not correct, nor fair. 

You have added a dimension in as part of your justifying "NONE is applicable", but the points you bring up do not "invalidate" the original post. 

Interesting to consider all the things you bring up, but the original post was clearly more of the "mainstream" standard and some limited amount of narrow gauge. 

I'm just reacting to "I'm going to be polite"...... 

Regards, Greg


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## 3lphill (Feb 22, 2008)

Good Morning,
Respectfully setting aside the brewing conflict above. I am interested in modeling Urban and interurban light-rail, dose anyone have any info as to what radii they were using in the 20s?
FYI if you are interested in light rail and street cars Portland is a good place to visit. 
Phillip


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## ralphbrades (Jan 3, 2008)

Ok then(!)

Let me turn the data into something that is "APPLICABLE" in my viewpoint...

In order to do this I have to reverse all his calculations to return to a figure in real world sizes, and then process all the figures to arrive at the figure in chains to begin all the calculation.

It would have been far more useful to produce a table which said :

10 feet 203' 225' 240' 290' 320'
10 feet 9.22ch 10.22ch 10.90ch 13.18ch 14.54ch

This provides APPLICABLE data that can be used in the design of locomotives and layouts. 
By knowing the gauge and the angle of attack on the flange and feeding that into the standard "Greenly" formulae:
1:The minimum spacing between axles for a given diameter of driver can be calculated.
2:The super elevation of the curve for the speed of cornering is calculated.
3:The maximum length between bogies for articulated rolling stock can be calculated. 

If you would like to examine the world rolling stock for multiplars and divisors of a chain, (22 feet), you might find something interesting.... 

ralph


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## East Broad Top (Dec 29, 2007)

There is no method of transposing the gauge, nor even the shape of the profile of the flange, nor the diameter of the driver into the above calculations, nor is there any method of analysing the axle spacing to radius reqirements. 
In terms of the mathematical relationship between the prototype curve (expressed however you choose to express it) and the model, what physically runs over the rails doesn't enter the equation. It's simply (a) a conversion of units to units, and (b) the application of a scale ratio. What actually can run on the curves has to do with the geometry of the equipment and how it interacts with a given curve. History has long demonstrated that the miniature world is far more forgiving in terms of allowable curvature than the prototype world ever could dream of being. It has to be, otherwise we'd never be able to fit model railroads into our basements, let alone back yards or even neighborhoods. In that regard, the design criteria as it applies to the prototype equipment and curves really gets thrown out the window in the model world. If we can manufacture a GG1 to successfully fit around a 2' radius curve, then pretty much anything goes. 

Later, 

K


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## Greg Elmassian (Jan 3, 2008)




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## ralphbrades (Jan 3, 2008)

Ok then...

http://www.cabbagepatchrailway.co.uk/mls/DSCS0020.JPG

That is a scale model at 16mm scale taking a scale correct curve in the prototypically correct manner. The model is 115cm long and the curve 2 feet 6 inches radius.

Since you have introduced the GG-1 and quite graphically demonstrated the poor design of the chassis -perhaps you might like to sit down and work out a working model chassis for
one. *After all that is what I did.... *

http://www.cabbagepatchrailway.co.uk/gg1pic2.png

http://www.cabbagepatchrailway.co.uk/gg1pic1.jpg 

ralph


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## Greg Elmassian (Jan 3, 2008)

Very cool Ralph, do you expect to build a GG-1? 

Do you have a spec for what kind of curve (degrees, radius, chains, furlongs per fortnite) it will take? 

The only one made that seems to be not a toy is the USAT one, but have never run one, just seen it on display. 

Regards, Greg


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## East Broad Top (Dec 29, 2007)

Greg, it was bad enough I had to visualize it in my head. Did you _really_ have to go and provide a photo? That's just low, man. Really low.  

Ralph, I'm not sure why you describe the chassis as a "poor design." Yeah, it looks like a monstrosity doing so, but it fits, and fits well. In my book, that makes it quite a successful design. 

Later, 

K


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## SteveC (Jan 2, 2008)

Posted By Greg Elmassian on 13 May 2010 03:50 PM 
Very cool Ralph, do you expect to build a GG-1? {snip...}[/i] Actually, if I remember correctly it was the PRR R1 built by Baldwin Locomotive that Ralph has or is going to build.


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## Greg Elmassian (Jan 3, 2008)

Kevin, it's from my site about not using tight curves on large locomotives... it is painful to look at, but LGB did achieve what they said they would do. 

Regards, Greg


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## ralphbrades (Jan 3, 2008)

Steve -you are quite correct the "*R-1*" is the selected loco for my "to build list" and yes -it will take a 2.2m radius curve... However that does not stop the world needing an articulated chassis for a GG-1 at various gauges and I do know that there are three of them under construction here in the UK using that chassis design. However we all face the one problem that has defied us all -getting the noses right(!) 

As to the minimum radius and curvature table which was the starting point of this thread -that is not applicable data that can be used in locomotive or railway calculations -*it is a map surveying method*.

The following data tables relate to Gauge '3' but it does show the use of radii rather than "degrees of curvature", and yes -they are all my own work...

http://www.cabbagepatchrailway.co.uk/mls/curves1.pdf
http://www.cabbagepatchrailway.co.uk/mls/curves2.pdf
http://www.cabbagepatchrailway.co.uk/mls/pointwork.pdf
http://www.cabbagepatchrailway.co.uk/mls/pointwork2.pdf 


Kevin, as to why it is a poor design -the stress on the front bogie does not even begin to steer the loco around the corner. I would class it as an "ornamental" rather than functional bogie...
Greg, all the Gauge '3' chassis are designed to take 2m (7 feet 6 inches) radius curves. 

ralph


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## East Broad Top (Dec 29, 2007)

Kevin, as to why it is a poor design -the stress on the front bogie does not even begin to steer the loco around the corner. 
Hardly unique in terms of model engineering. Very few models have front pilot trucks which actually do any "piloting" at all. At most, they have springs that "kind-of" keep them centered and/or apply a bit of downward pressure. None that I have seen (in commercial models) are actually linked prototypically to the frame to where they actually function as per the prototype. (If they were, they wouldn't remotely fit around the curves which we typically run.) 

"Poor" design in terms of prototype fidelity, perhaps, but in terms of meeting the physical requirements of the environment in which the models themselves will be run, the manufacturers are surprisingly ingenious and very successful on that front. Since I operate in the inherently compromised environment of the model railway, I would find anything that specifically replicates the prototype practice to the detriment of being able to operate in the garden to be of "poor" design. It's purely a matter of perspective. 

Later, 

K


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