Track curve sharpness is measured in units of degrees, of course, but exactly what does the measurement describe?

I understand it represents a rate of change [direction over unit distance?], but the specifics escape me. Help?

hi_plain@yahoo.com

This discussion should help...

Just how sharp is that curve?
eledbetter@rypn.org

The degree of curvature on RR track is the central angle of a 100 foot chord on the curved section of track, assumed to be circular.

PKurilecz
> Track curve sharpness is measured in units
> of degrees, of course, but exactly what does
> the measurement describe?

> I understand it represents a rate of change
> [direction over unit distance?], but the
> specifics escape me. Help?


pkurilecz@yahoo.com

> The degree of curvature on RR track is the
> central angle of a 100 foot chord on the
> curved section of track, assumed to be
> circular.

> PKurilecz

The formula relating the degree of curvature to radius is:

D=5730/R

where:

D - degree of curvature

R - radius in feet

Doug vV


douglasvanveelen@mindspring.com

> The formula relating the degree of curvature
> to radius is:

> D=5730/R

> where:

> D - degree of curvature

> R - radius in feet

This will give you a close approximation for small angles. The actual relationship is a trig function:

R*sin(D/2) = 100/2
R = 50/sin{D/2)
sin(D/2) = 50/R

A good source for all kinds of formula for laying track work is "Railroad Curves and Earthwork" by C. Frank Allen. This was first published in 1889. I'm looking at a 7th edition from 1931.

Brian Norden

bnorden49@earthlink.net

Brian,

You are absolutely correct about the chord definition.

Just to throw out the confusing munutea of a second definition, here are the derivations of both. I never could understand the arguments over which definition to use when the difference is so small.

Following the proof in "Surveying Sixth Edition" by Moffitt and Bouchard (1975):

Chord definition (AKA railroad):
Sin D/2 = 50/R
R = 50/Sin(D/2)

Arc definition (AKA highway):
There are 2 * PI radians in a circle and there are 360 degrees in a circle.

The relationship can be set up:

100 is to D as 2*3.14159 * R is to 360 degrees or:
100/D = 2*3.14159*R/360

This directly yields

R = 5729.58/D

The figure from which they got the initial relationship is on page 356.

I worked as a Civil Engineer while in Collage (7 years for BSEE and BSCE) and the State of Georgia at that time defined degree of curvature as D = 5730/R. If you did not use it, the state would not accept work. It may have changed since 1979. When I last worked CE in 1984/5, counties only used radius in feet for either RR or highway. Last State or Federal RR I did was in 1975.

Also according to Moffitt and Bouchard, for a 20 degree curve, the chord definition radius is 287.94 feet and the arc definition has a radius of 286.48 feet. About 1.5 feet in 288 feet or 0.5% error.

This is also backed up by Route Surveying and Design 4th edition, Carl F. Meyer, 1969, page 289.

I recall the argument at one museum when they were moving to a new site where a planned curve of 9 degrees caused a 6 hour argument over which definition to use and one claimed the RR definition was a shorter radius that the highway definition even though mathematically it was wrong.

Maybe someone will use this bit of trivia interesting.

8)

Doug

> This will give you a close approximation for
> small angles. The actual relationship is a
> trig function:

> R*sin(D/2) = 100/2
> R = 50/sin{D/2)
> sin(D/2) = 50/R

> A good source for all kinds of formula for
> laying track work is "Railroad Curves
> and Earthwork" by C. Frank Allen. This
> was first published in 1889. I'm looking at
> a 7th edition from 1931.

> Brian Norden


douglasvanveelen@mindspring.com

Doug,

I was surpised to read you reference to using degrees of curvature for highway construction.

I work with a water utility here in Southern California and make use of various street and road plans. What I have seen going back 50+ years has always defined road curves by radius, delta of central angle, arc length, and survey tangent. Stationing will be along the centerline arc. This is similar to the various subdivision maps that we have here.

Some recent plans for BNSF third tracking a line through our service area used feet of radius to define the curves.

CalTrans (California DOT) is now doing all their plans in metric and is requiring any plan submitted to them for an excavation permit or subdivision improvement to have metric dimensions. Just turn on the second units in the dimensioning function of the CAD program.

The Allen book has as its very first appendex a table that lists degree of curvature, the corresponding radius (to the second decimal place), and the logrithum of the radius. Boy, I'm glad for calculators, etc so that we don't have to use logs and slip sticks!

Brian


bnorden49@earthlink.net

Brian,

Not many know/recall the Degree of Curvature can be defined two ways and that it can be used in highway design.

> I was surprised to read you reference to
> using degrees of curvature for highway
> construction.

I've never seen Degree of Curvature (DoC) used in highway plans. My Dad worked as a CE from about 1948 to 1982 and he had seen it in the early expressway design days (if I recall correctly). As with most new technologies, they base as much on older ones as they can. Highways taking DoC from railroads and traffic control taking three-color signals from railroads.

The first commercial computer chip (as I have been told from profs in both CE and EE) was the 555 timer that was created for traffic signals.

> I work with a water utility here in Southern
> California and make use of various street
> and road plans. What I have seen going back
> 50+ years has always defined road curves by
> radius, delta of central angle, arc length,
> and survey tangent. Stationing will be along
> the centerline arc. This is similar to the
> various subdivision maps that we have here.

Absolutely. Very old plans sometimes show up with DoC. (1920's and 30's)

> Some recent plans for BNSF third tracking a
> line through our service area used feet of
> radius to define the curves.

That's the way we did when I helped layout several industrial parks here in Atlanta and that's how we did the new design of MARTA. The Marta section caused a relocation to the Georgia RR main and since it was part curve and part spiral easement and the GaRR plans had the curves in degrees, we used them also (mainly by changing to radius and then back when we were done).

> CalTrans (California DOT) is now doing all
> their plans in metric and is requiring any
> plan submitted to them for an excavation
> permit or subdivision improvement to have
> metric dimensions. Just turn on the second
> units in the dimensioning function of the
> CAD program.

My last working year during collage was 1979. it was about that time, several government agencies (who had insisted on the arc definition for all calculations) started allowing the correct curvature calculation for railroads.

> The Allen book has as its very first
> appendex a table that lists degree of
> curvature, the corresponding radius (to the
> second decimal place), and the logrithum of
> the radius. Boy, I'm glad for calculators,
> etc so that we don't have to use logs and
> slip sticks!

The use of calculators did cause the loss of trig and log and natural log tables (not much of a loss). Most do not know PI (3.14159) except as a button on a calculator (my sister teaches high school math) and do not recognize 2.7182818285. I've not found a person in years who knows how to take a square root with only a pencil and paper without going to trial and error.

I still have my slip stick. Haven't used it in +20 years.

If you are interested on my take on CE for Live Steam railroads (3.5” gauge to 7.5” gauge), I did a 4 part article for Live steam in late 2002 and 2003.

By the way, in 1970 or so, Dad and us took what was to be the only trip out west and we headed for a stop in San Francisco. Dad meet with someone with BART and we rode BART. The owner/pres or the company Dad worked for told him to visit and then write his portion of the gas and lodging off on the company :) They don't have as many good bosses like that any more.

Back to one last DoC item, the DoC for a 100 foot chord was to make it somple for the survey crew to stake the curve. the two chianmen would start at the PI and could swign a 100 foot radius. The insturment man knew he would backsite on the tangent and then turn 180 degrees and then turn 1/2 the DoC to the direction of the curve. The rodman would then put the level rod somewhere "on line" and the lead chainman would swing the 100 foot mark until he was also on line. After checking with the insturment man, a stake would be placed.

Now the insturment is moved to the first station in the curve. The backsite is to the last 100' stake (or the PI) and then the glass is turned 180 degrees. Now the insturment is turned the full DoC in the direction of the curve. The rest is like that above.

Field crews were 4-5 man teams. The fifth man would be the crew chief who would record measurments or direct the layout. I've been on a few pre-computer field crews :)

By the time I started learning CE, the field crews were not told the DoC, they just had angle and distance numbers that were worked out in the office before hand. With computers with printers, we could print out those notes fast and the partial stations at the TC and CT were very easy to generate.

I hope I do not sound like a know-it-all, I just enjoy sharing knowledge and especially interesting trivia.

Doug


douglasvanveelen@mindspring.com