Mostly true but if you read railroad engineering planning studies they often refer to the total degrees of curvature on a route before and after a proposed improvement or line change. The railroads did care about reducing the total degrees of curvature but as you stated it had to be balanced against other factors (costs, both construction and operating).
Thank you Tomikawa and others. Very interesting to me.
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Because from time to time I have read about sharp curves on prototype railroads stated in degrees and wondered how, in fact, they would compare to curves on a model layout.
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Because I recently read an article which said the Uintah Railway ran Malleys on 66 degree curves and wondered what the HO equivalent would be. (Apparently it’s about 14-15 inches, small enough for most layouts.) I guess the standard follow up is, “Yeah, but what would they look like?” and in this case, the answer is, “Just like the prototype.”
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Because knowledge is never wasted, and
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Because numerous people were kind enough to answer my question.
Sorry:
No insult was intended. I truly wondered what the purpose of the discussion is and why it is not in the prototype section. I did not mean to infer that those engaging in it were doing anything wrong.
Math is the same whether its 1:1 or 1:87 or 1:160. 
Dave H.
If you are using graph paper to construct a turnout template, it is much easier to plot a slope than it is to use a protractor and try to duplicate a quarter of a degree. The tangent function of an angle is the slope: rise over run.
The angle of a frog gives you angle of divergence, and if you are then wanting to reverse the angle with a reciprocal curve to bring the track parallel to the straight leg of the turnout (such as the last leg of a yard ladder) a curve of the same degree will do that.
Of course, I can freehand that reciprocal curve, but I want to insure that the radius is at or above my minimum, and balance that with my intended track spacing.
Likewise, if I have a dogbone-type reverse loop planned, I want to be able to calculate the smallest amount of real estate necessary, and that bringing that loop back into the divergent leg of the turnout where the loop begins and ends, requires an “S” turn which is dictated by my minimums. Mathematically treating the divergent leg of a turnout as a curve allows me to determine where that turnout is placed, and where the “s” breaks to reverse on itself.
The Atlas Custom turnouts for N=scale are interesting to do this math upon:
No problem.
Anyone notice there’s an error in the “Railway Track and Maintenance” book? It says to start at the Point of Tangency (Point B), then line back 100’ to Point A, then put the chain at A and move the other end in to point D. Actually the chain will need to be at B, and the other end swung in to point D. (This all makes sense if you look at the diagram!)
Okay, here is what I have tried.
I have a 5" section of 19" radius track, 24 of which makes a loop (2pi19=120” circumference, and 120”/5”= 24), so I know the angle is 15 degrees. (360 degrees/24 sections = 15 degrees). Is that right?
According to my divergence formula (previous post) the divergence should be .6696972202. I cannot measure to that mathematical precision, but that seems to fit.
I lay that out on graph paper, mark end points, mark center of curve, draw chord, measure distance between chord and midpoint of curve—It is about .5/32” = .156”
Tan A = .156/(1/2 * 5”) = .0624
That is not working for me.
Moving on…
Take the 15 degree angle
R = ½ * 5”/sin(15 degrees))
R= 2.5/.258819045
R=9.659 (nope)
BUT!
If I use C/sin(angle)
I get 5/.25881904= 19.31” A 19’ Radius. Yes.
So, what have I done wrong?
Crews
Oops. I made an error in the formula for the Radius (last formula). Correct one is:
Mark the center line of track at two places on the arc.
Measure the distance between those places. That’s the chord of the arc.
Bisect the chord and mark the center of the track to find the center of the arc.
Measure the offset from the center of the chord to the center of the arc.
Angle = A
Chord length = C
Offset = F
Radius = R
Tan A= F/(1/2 C) (to find the angle)
R= (1/2 C)/sin 2A
You have to double the angle.
In your case:
For a 15 degree section of 19" radius track, the chord is 4.96" (half chord is 2.48") and the offset is .163", so you were close.
Tan A = .163/2.48 = .0657
If you do this using the “scientific” view of the MS Windows calculator, check the “Inv” box and click the “tan” button.
That gives you an angle A of 3.76 degrees. Double A = 7.5 degrees.
Radius = 2.48/ sin 7.5 degrees
Radius = 19.00002 inches.
Dave H.
Guys, this is not a triangle. It’s a curve. The 10 degree angle isn’t achieved until the end of the segment of track, so the calculations based on a triangle are not accurate.
The math is actually not significantly much more complicated, we just need to use the right equations. The equation for a circle is (x-h)^2 + (y-k)^2 = r^2, where:
h and k are the x- and y- coordinates of the circle center and r is the radius.
You can algebraically solve for the arc length using the known coordinates, and can then calculate the angle, or work backwards using just the known arc length because you know the angle and the arc length- just solve the perimeter equation for radius.
Given 10 degree (= 1/36 of a circle) and the length of the arc is 100’, then the perimeter of the circle is 3600 feet. The diameter of a circle is (pi) * diameter. Divide 3600 feet by pi and you get about 1145 feet, so the radius is half of that or 572’.
In HO scale that comes out to 6.586 feet (79.03 inches).
There is also a convenient shortcut trackmen use in the field that can be useful for model railroad measurement. Stretch a 62-foot string as a chord between two points on the curve. Measure the offset at the midpoint of the string back over to the rail. Each inch of offset equates to 1 degree of curvature. So if the offset is 6 inches that is a 6 degree curve.
On a model railroad, a solid straight edge 62 scale feet long would be more convenient. Then measure the offset of the midpoint in scale inches and you have the translation into degrees.
It’s not that you would build a model railroad that way, just that it can be interesting to some to know how it compares to your prototype.
My EMD SD40-2 Operators Manual specifies that a single unit with single shoe or clasp brakes could negotiate a 193’ radius or 30 degree curve. Two units coupled could negotiate a 22 degree curve but a unit coupled to a standard 50’ boxcar could only handle 16 degrees (or like a model railroad, the wider swing of the pilot would pull the boxcar off the track sideways).
My EMD F7 Enginemen’s Operating Manual simpy says that the minimum radius is 274 feet.
My recollection reading somewhere is that the GE C-truck was rated at 21 or 22 degrees, and a DRGW K-36 was rated at 24 degrees.
Jeff Otto
Missabe Northern Ry.
As an old exsurveyor, a chain was a unit of measure used in surveying, is still found on old plans, and is 66’ and yes old chains for did stretch. As for his curves, just use inches of radious, or are we just complicating things,
For measuring simple curves, one can measure the degree of the curve using the length of the chord (triangle method) or by the arc. Using the chord method, the curve is the angle of the center for a given 100-foot chord (a 100-foot straight line crossing the curve at two points). Using the arc method, the curve is the angle at the center for an arc of 100 feet. Practical methods of measuring the length of an arc is not practical before construction, so the chord method was what was used by railroads. (Source: Clement C. Williams’s [Professor of Civil Engineering, University of Illinois] book The Design of Railway Location, copyright 1917, 1924.) Therefore, a an X-degree curve using the chord method has a different radius than an X-degree curve using the arc method.
Mark
I beg to differ. Every prototype railroad for the last hundered years has used the methods I described to measure curves. the method that Jeff Otto outlined is essentially the same thing i described, its just that by using a 62 ft chord, the offset is proportional to the degree of curvature in inches. Same math.
And that is the benefit of the methods that “use triangles”. All you need are two measurements which are easily obtained with simple measuring tools, the chord length and the offset.
That’s why real railroaders have used the chord method for decades. You can find tables of offsets in most engineering manuals.
Dave H.
I wish I could figure out what you mean, here. I can not get .0657 to turn into 3.76 degrees to save my live-- not using inverse function, not using tangent, not by holding my eyes just right.
But (!)
I do get the formula
Radius = 1/2 chord length / SIN (1/2 angle in degrees)
4.19" section (of known radius of 24") is a 10 degree section (one of 36 to make a circle) and to test:
Radius = 2.095/SIN(5 degrees) = 24
That works! Cool! Thank you
Crews
When building a railroad through mountainous country, or any rough country for that matter, it would clearly be impossible for the surveyor to swing a trammel bar in order to produce a curve of a known sharpness. This is why the method of measuring curves using the 100 foot cord was used. Laying out a curve consisted of moving down the rough roadbed in 100 foot intervals and swinging the transit at each interval through a certain angle. Then the track layers would essentially “follow the stakes” when placing the ties and rail. Since rail isn’t exactly real flexible, simply placing the rail along the stake line created very smooth curves of constant radius, which was the object of the exercise.
Steam locomotives were not so forgiving of rough track and sharp curves as a four axle diesel with two swiveling trucks. Nevertheless, all Baldwin standard gauge locomotives had to be able to traverse a curve of sixteen degrees in order to be able to manage turnouts and other special trackwork. This would often be done at quite slow speeds to avoid derailments. Some locomotives, of course, could handle much sharper curves. For example, the Denver and Salt Lake Railway, which I model, had curves of twenty degrees on its “temporary” line over the main range west of Denver. This line crossed the continental divide at an elevation of 11,660 feet and was used from 1904 until the Moffat tunnel was built many years later. The line was sharp enough that consolidations, ten wheelers and unique 2-6-6-0 mallets were used here. Some low wheeled mikado locomotives were too stiff for this line even though they had only 55" drivers and quite short wheelbases.
The grades on this line were four percent so the speeds were of necessity very slow in order to be able to operate safely. A speed of twenty miles an hour was pretty fast for this line.
The 100 foot cord method was used for curves up to about twenty degrees or so and perhaps even sharper in some situations. When the radius began
The LION makes simple curves. 24" radius is my minimum curve at the edge of the table, so any curve I lay on the table will be more than that. (snicker snicker!) The LION then lays out the tangent tracks along the table edge, and curves it around the curve. He flexes it a few times until it is resting naturally, and then spikes it down.
The LION has nice smooth trackwork, but it is a good thing that this LION models subway trains with 50’ cars. Some of the curves are sharper than his design thought for.
BTW: the real IRT subway cars can make some pretty darn sharp turns underground. The South Ferry Station is a loop, just like in HO, and is actually, just as tight.
Good thing your life doesn’t depend on it. It doesn’t turn into it - it IS it.
0.0657 radians = 3.76 degrees.
I can’t think of how many times I’ve made that same umm ummm “oversight” [:-^].
I see! The INV with the TAN function on the MS Windows calculator is not a 1/x inverse, it is a programmed function to __re__verse the calculation.
My TI hand held calculator has the same function (TAN^-1). From that, I get the 3.758930464.
I do wish that I had a formula as that will allow me to understand the mathematical relationships whereas a button on a keypad looks simply like magic.
So, how was this done before electronic calculators? Slide rule? And before a slide rule?
Crews