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Topic: Degrees and Percent (Read 173 times) 

ehbowen
Railfan
Posts: 242

Just returned home from a vacation in which I rode the Black Hills Central behind their unique 2662T compound Mallet and also the Cumbres & Toltec Scenic, both for the first time. The BHC touted their 4 to 6 percent grade as being the steepest ever operated by a standard gauge Class 1 railroad (Burlington), while the C&TS talked about narrow gauge's ability to operate on curves up to 30 degrees...but that there were no turns tighter than 20 degrees on their line. Grades in percent I understand. If there is a six foot rise in a 100 foot run of track, you have a six percent grade. (But is that 100 feet the base or the hypotenuse? Makes a (small) difference.) However, I'm missing a piece of information on understanding curves in degrees. I understand that degrees are a unit of circular measure, but without a corresponding tangent or arc measurement they are meaningless to me. Is that twenty degrees per thousand feet of track? Per hundred feet of track? What would the corresponding figures be in metric? Or am I lost completely here?

« Last Edit: Jun 26^{th}, 2011, 9:40pm by ehbowen » 
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Eric H. Bowen
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Norm_Anderson
Historian
Posts: 1726

Hi, Eric, Here are a couple of links: Gradient: http://en.wikipedia.org/wiki/Grade_(slope) This link will direct you to the "Grade" main page look under "Civil Engineering" and you will see the link to the "Grade (slope)" page. Curvature: http://mysite.du.edu/~jcalvert/railway/degcurv.htm Incidentally, at the bottom of Jim Calvert's Curvature page, there is a link to his main index, where you may find an hour or two of fascinating reading. Regards, Norm

« Last Edit: Jun 27^{th}, 2011, 1:47am by Norm_Anderson » 
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George_Harris
Moderator Historian
Posts: 3819

I have lived and breathed this stuff for so long, I tend to forget it is not obvious. In percent grade, the 100 feet is measured on the level. It is also true that the difference is insignificant in the normal range of railroad grades. So far as I know, the steepest grade on any railroad worked by adhesion, that is friction between wheels and rails instead of by rack or cable, is Southern's Saluda Grade which is called out as being 4.7%, but from what some have said, varies between about 4.5% and near 5.0%. (I would love to get out there someday with a surveying crew. It almost certainly has been done, and chances are the records are in the hands of some exSouthern Railway office or person somewhere.) As to the degree of curve: That is the change of direction turned in 100 feet. There are two ways of doing this: One is called the "Chord Definition" or "Railroad Definition" In it the 100 feet is measured by 100 feet chords, that is straight lines between points on the curve. The other is called the "Arc Definition" or "Highway Definition" In it the 100 feet is measured along the arc. In surveying texts, the older precomputed days ones at least, there are tables for the radii of the differing degrees, plus correction factors so you know what chord it takes to have 100 feet even on the arc, and for the chord definition, if you cannot do 100 feet chords, what lengths you have to use for half length, that is nominally 50 feet segments. For low degrees of curves, the difference between the two types is very small. The formula for the chord definition, knowing degree, to find radius is: Radius = 50 / sine(Degree/2) The formula for arc definition, knowing degree, to find the radius is: Radius = (18000 / pi) / Degree Churn these around using your algebra if you have radius and want degrees. I intend to follow up later with some examples, but at the moment I am supposed to be working.


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CHESSIEMIKE
Historian
Posts: 4302

on Jun 27^{th}, 2011, 6:28pm, George_Harris wrote: (Click here for original message)I have lived and breathed this stuff for so long, I tend to forget it is not obvious. In percent grade, the 100 feet is measured on the level. It is also true that the difference is insignificant in the normal range of railroad grades. So far as I know, the steepest grade on any railroad worked by adhesion, that is friction between wheels and rails instead of by rack or cable, is Southern's Saluda Grade which is called out as being 4.7%, but from what some have said, varies between about 4.5% and near 5.0%. (I would love to get out there someday with a surveying crew. It almost certainly has been done, and chances are the records are in the hands of some exSouthern Railway office or person somewhere.) As to the degree of curve: That is the change of direction turned in 100 feet. There are two ways of doing this: One is called the "Chord Definition" or "Railroad Definition" In it the 100 feet is measured by 100 feet chords, that is straight lines between points on the curve. The other is called the "Arc Definition" or "Highway Definition" In it the 100 feet is measured along the arc. In surveying texts, the older precomputed days ones at least, there are tables for the radii of the differing degrees, plus correction factors so you know what chord it takes to have 100 feet even on the arc, and for the chord definition, if you cannot do 100 feet chords, what lengths you have to use for half length, that is nominally 50 feet segments. For low degrees of curves, the difference between the two types is very small. The formula for the chord definition, knowing degree, to find radius is: Radius = 50 / sine(Degree/2) The formula for arc definition, knowing degree, to find the radius is: Radius = (18000 / pi) / Degree Churn these around using your algebra if you have radius and want degrees. I intend to follow up later with some examples, but at the moment I am supposed to be working. 
 What he said. I was going to answer, but I knew you would come through. CHESSIEMIKE


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George_Harris
Moderator Historian
Posts: 3819

Here is a table of degree and radii Degree
.Radius
....Radius of Curve
.Chord Def
..
Arc Def. .0d15m
.22,918.33
.22,918.31 .0d30m
.11,459.19
.11,459.16 .0d45m
7,639.49
.7,639.44 .1d00m
5,729.60
.5,729.58 .1d30m
3,819.83
.3,819.72 .2d00m
2,864.93
.2,864.79 .2d00m
2,292.01
.2,291.83 .3d00m
1,910.08
.1,909.86 .3d30m
1,637.28
.1,637.02 .4d00m
1,432.69
.1,432.39 .5d00m
1,146.28
.1,145.92 .6d00m
955.37
.954.93 .7d00m
819.02
.818.51 .8d00m
..716.78
..716.20 .9d00m
..637.27
..636.62 10d00m
573.69
..572.96 11d00m
..521.67
..520.87 12d00m
..478.34
..477.46

« Last Edit: Jan 31^{st}, 2012, 7:15pm by George_Harris » 
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