11 Geometric Design of Railway Track [Vertical Alignment] (Railway Engineering Lectures هندسة السكك الحديدية & Dr. Walied A. Elsaigh)

1. Dr. Walied A. Elsaigh
welsaigh@ksu.edu.sa
Asst. Prof. of Civil Engineering
CE 435
Railway Engineering
Geometric Design
of Railway Track
Vertical
Alignment

2. Text Book
Railway Engineering,
Satish and Agarwal,
2nd edition, 2013
Chapter 12 & Chapter 13
Page 197-258

3. Textbook
HSR Technical Specification for interoperability (INF-TSI)

4. Gradients

5. Types of Gradients
in Railway Tracks

6. Ruling Gradient

7. Momentum
Gradient

8. Pusher Gradient
(Helper)

9. Gradients in
Station Yards

10. Grade
Compensation

11. Example

12. Vertical Alignment
Gradient Design limits
Rail – rarely exceeds 1%
(2-2.5% for industry lines)
Highway – 4%
common
6% on ramps
Up to 8% on
county roads
LRT – maximum 4 to 6%
Up to 10% for short sections

13. • Ideal maximum for railway grade:
• Trains can roll safely down 0.3% grade without
wasting energy on brakes
• <0.1% for tracks for extensive storage
Vertical Alignment
Gradient Design limits

14. • Railway vertical curves – old formula:
L = D / R
D = algebraic difference of grade (ft. per 100-ft. station)
R = rate of change per 100-ft. station
• 0.05 ft. per station for crest on main track
• 0.10 ft. per station for sag on main track
• Secondary line may be twice those for main line
Vertical Alignment
Curve Length

15. Vertical Alignment
Curve Length
• Old railway formula developed in 1880’s for “hook and pin”
couplers in those days
• Present day couplers can accommodate shorter vertical
curves
• New formula developed in recent years:
L = 2.15 V2 D / A
V = train speed in mph
D = algebraic difference of grade in decimal
A = vertical acceleration in ft./sec2
0.1 ft./ sec2 for freight, 0.6 ft./ sec2 for passenger or transit
Task: Search AREMA and find the most recent vertical curve length formula

16. Vertical Curves
• The parabolic curve is used almost exclusively in
connecting profile grade tangents
• The primary reasons for the use of parabolic curve
in vertical curves is the convenient manner in which
the vertical offsets can be computed and the
smooth transitions created from tangent to curve
and then back to tangent
• Crest curve connects a positive grade with a
negative grade
• Sag curve connects a negative grade with a positive
grade

17. Types of Crest and Sag Vertical Curves

18. Properties of Typical Vertical Curve

20. Vertical Curves (continued)
• It is usually necessary to calculate elevations at
every even 50-ft (20-m) station, while some special
paving operations may require elevations at 25-ft
(10-m) intervals or less
• It is often necessary to compute other critical points
on the vertical curve in order to ensure proper
drainage, clearance, or connections to side streets
• The high point or low point of a parabolic curve is
seldom located at a point vertically above or below
the vertex of the intersecting tangent grades

21. Vertical Curves (continued)
• The discussion and formulas presented in this
section apply only for a symmetrical curve, that is,
one in which the tangents are of equal length in
the horizontal plane
• The unequal tangent or unsymmetrical vertical
curve is a compound vertical parabolic curve,
which is warranted only where a symmetrical
curve cannot meet imposed alignment conditions,
such as vertical clearance requirements

22. EXAMPLE
• A plus 3.0 percent grade intersects a minus 2.0
percent grade at station 3 + 20 and at an
elevation of 320.40 ft. Given that a 180-ft length
of curve is utilized,
(1) Determine the station and elevation of the PVC
and PVT.
(2) Calculate elevations at every even 25-ft station
(3) Compute the station and elevation of the high
point of the curve

23. EXAMPLE

24. EXAMPLE (continued)
• EPVC = EPVI - (G1/100)(L/2) = 320.40 - 0.03(90) = 317.70 ft
• EPVT = EPVI - (G2/100)(L/2) = 320.40 - 0.02(90) = 318.60 ft
• Location of high point:
• High point Sta = PVC Sta + Xm = 230 + 108 = 338 → Sta 3+
38
• Elevation of high point:

25. Calculations for point elevations at even
25-ft stations along the vertical curve
Station
x
(feet)
Elevation on
Initial Tangent
y
Final Elevation
on Curve
(Elev on tan - y)
2 + 50 20 318.30 -0.06 318.24
2 + 75 45 319.05 -0.28 318.77
3 + 00 70 319.80 -0.68 319.12
3 + 25 95 320.55 -1.25 319.30
3 + 50 120 321.30 -2.00 319.30
3 + 75 145 322.05 -2.92 319.13
4 + 00 170 322.80 -4.01 318.79
4 + 10 180 323.10 -4.50 318.60

26. AREMA -EXAMPLE

27. See how deep they cut!

28. Thank You
See how high they fill!