What is formula for calculating the compensated gradient of a curve?

 
  tom9876543 Chief Train Controller

The Curve and Gradient diagram for Wynyard to the Sydney Harbour Bridge has 1/30 compensated.
Would anyone have the exact formula John Bradfield used to calculate the compensated grade?
What is the uncompensated grade on the curves?
Also there is a 7.5 chain curve between Town Hall and Central, with an uncompensated grade of 1/40. What would the compensated grade be? This is probably the tightest curve currently in use on NSW railways.

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  tom9876543 Chief Train Controller

Thank you 3l diesel, it looks like you have found the answer.

We will have to assume John Bradfield used the imperial version of the formula:

3/R %, where 'R' is the curve radius in chains

If my calculations are correct, a 7.5 chain curve at 1/40 uncompensated is about 1/34.5 compensated.
  cityrail-rulez Chief Train Controller

This topic is quite interesting, tom9876543 I am glad that you have asked. Sorry I have to highjack your thread though lol Smile The thing I would like to understand about gradients is "E" or Easement grades, there are quite a few located throughout NSW and I have been wanting to know how are these gradients calculated into  1 in grades?

If someone could explain this to me about easement grades, how to calculate them into 1 in grades this would be very useful
these gradients has me puzzled for years especially when I am trying to build routes in MSTS I run into easement gradients and have no idea what to do next lol
  cityrail-rulez Chief Train Controller

Thanks NSWRcars, but I already know what easement gradients are used for
I mean how to calculate them into a 1 in grade value

"E" doesn't mean ease as you have stated, it clearly shows in this gradient profile of the City Circle "E" means Easement Smile


  cityrail-rulez Chief Train Controller

Yes, I believe the answer is in the "vertical curve" section 5.3.2 has something in there I wouldn't quite understand it lol

I am thinking on how to split the easement gradient into a 1 in value so it be easier to understand them
If my calculations are correct starting with the previous grade of 1 in 30 and then I think it would be

1 in 40, 1 in 50, 1 in 100, LEVEL, 1 in 100, 1 in 50, 1 in 40 and then the 1 in 43 they are just a fair guess as to how easement gradients are calculated into a 1 in value by splitting it into sections as shown below

What do you guys think? If I am not correct that's fine but it be a fair guess if it is Smile


  tom9876543 Chief Train Controller

Thank you for explaining what E means on curve a gradient diagrams.
E will always be less steep than the ruling gradient of the track.
  62440 Chief Commissioner

Actually I might have an answer "When designing grades within 1 in 5 of the ruling grade the grade shall be compensated for curvature by an amount: 60/R %" 'R' is the curve radius in metres. From http://www.asa.transport.nsw.gov.au/sites/default/files/asa/railcorp-legacy/disciplines/civil/esc-210.pdf .
"3l diesel"

I think you'll find the figure is 0.6/R%, similar to the NG handbook 1/1.65R%. Heavy haul lines were based on American practice, some are 0.035%/degree curvature, some are 0.04%/deg. A degree of curvature is the angle subtended on 100 feet of track, and is R/1746 in metric. On a 0.33% ruling grade, a curve of 1000m would be compensated to 0.26% for instance where .035/deg is used.
  c3526blue Deputy Commissioner

Location: in the cuckoos nest
Yes, I believe the answer is in the "vertical curve" section 5.3.2 has something in there I wouldn't quite understand it lol

I am thinking on how to split the easement gradient into a 1 in value so it be easier to understand them
If my calculations are correct starting with the previous grade of 1 in 30 and then I think it would be

1 in 40, 1 in 50, 1 in 100, LEVEL, 1 in 100, 1 in 50, 1 in 40 and then the 1 in 43 they are just a fair guess as to how easement gradients are calculated into a 1 in value by splitting it into sections as shown below

What do you guys think? If I am not correct that's fine but it be a fair guess if it is Smile


cityrail-rulez
Hi cityrail-rulez,

Going back to my pure and applied maths at uni it is a very simple exercise.  You have decided (why I don't know) to split the easement into 7 segments (n = 7).  You could choose any number of segments you wish between 1 and infinity.  An infinite number of segments will give you a constantly changing gradient (don't go there, it's called calculus). Going from one segment to the next each change will share an equal portion of the total change between 1:30 and -1:43 (note the - sign which signifies downhill instead of uphill), that is 8 changes in gradient (n+1).

For ease of calculation I have converted the gradients into percentages (US term for gradient), 1:30 is 3.33%, -1:43 is -2.33% (to two decimal places).  Thus with eight changes in gradient each changes is -0.707% (to 3 decimal places).  Converting back from percentages to 1:g we get: -
  1:30, 1:38, 1:52, 1:83, 1:198, -1:491, -1:110, -1:62, -1:43 (all to nearest whole number).

Happy arithmeticking,

John

PS; You were mostly on the right track (pun intended).

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