# 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.

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 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 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 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
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 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).