Lec 06 Vertical Alignment (Transportation Engineering Dr.Lina Shbeeb)

1. Dr. Lina Shbeeb
Vertical alignment
Transportation Engineering

2. Dr. Lina Shbeeb
Vertical alignment
Vertical Curves are used to provide a gradual change from
one tangent grade to another, so that vehicles may smoothly
navigate changes in grade as they travel the highway.

3. Dr. Lina Shbeeb
Vertical alignment
• Vertical alignment consists of grade tangents
connected with parabolic curves.
• The desirable maximum grades and gradient
changes depend on the facility type and vehicle
characteristics.
• The desirable grade as function of facility type:
– 2% for freeways
– 6% for local street
– Higher grades are unavoidable at location with difficult
topography.
• The length of vertical curve is measured along the
horizontal alignment. A point on the curve is
specified by its station location on the horizontal
alignment and its elevation from a datum.

4. Dr. Lina Shbeeb
Maximum grades by highway function

5. Dr. Lina Shbeeb
Vertical alignment
• Grades – vertical slope from reference station
– upgrade – positive
– downgrade - negative
• Crest, sag curves

6. Dr. Lina Shbeeb
Vertical alignment
• Vertical point of intersection (VPI): The point where the grade
line intersect
• Vertical point of tangency (VPT): The point where the grade
vertical curve ends
• Vertical point of curvature (VPC) The point where the grade
vertical curve beigns

7. Dr. Lina Shbeeb
Vertical design element
• It important to find the vertical elevation at each station
particularly VPC,VPI and VPT.
• External distance E needs to be calculated to estimate
their elevation.

8. Dr. Lina Shbeeb
Vertical alignment
• Total change in grade
• Vertical curvature
• External distance
• Vertical offset
• High(low) point of curve
• Elevation of any P
12 GGA 
|| A
L
K 
ft
AL
E
800

2
4 






L
x
Ey
0X 


21
1
GG
LG
X
yx
G













100
1
VPCofelevation
X is the distance along
the horizontal alignment
from PVC to the point of
interest

9. Dr. Lina Shbeeb
General considerations
• Reasonable upgrades without significant
loss in speed is 4-5 %
• @ 70 mph design speed max grade is
5%
• @ 30 mph design speed max grade is 7-
12%

10. Dr. Lina Shbeeb
Vertical alignment example
• A 600-ft curve connects a +4% grade to a –2% grade
at station 25+60.55 and elevation 648.64 ft. Calculate
the location and elevation of the VPC, the middle of the
curve, the VPT, and the curve elevation at stations
24+00 and 27+00
?
? ? ?
?

11. Vertical alignment example
• Total change in grade
• Vertical curvature
• External distance
• Vertical offset
• High(low) point of curve
• Elevation of any P
%612  GGA
100
6
600

|| A
L
K
ft
AL
E 54
800
6006
800
.


2
4 






L
x
Ey
400
21
1





6
(4)600
GG
LG
X
2
1
4
100



















L
x
Ex
G
VPCofelevation

12. Dr. Lina Shbeeb
Vertical alignment example

13. Dr. Lina Shbeeb
Calculation of vertical curve length
• In most cases, sight distance will govern for
highways. The equations used to calculate
minimum lengths of vertical curves based on
sight distance depend on whether the sight
distance is greater than or less than the
vertical curve length.

14. Dr. Lina Shbeeb
Crest Vertical Curves, S > L

15. Dr. Lina Shbeeb
Crest Vertical Curves, S < L

16. Dr. Lina Shbeeb
Crest Vertical Curves
• For first condition: S < L
• For second condition: S > L
– where L = minimum length of vertical curve
– S = sight distance
– A = algebraic difference in grades (gradient)
– h1 = height of eye above roadway surface
– h2 = height of object above roadway surface
A
HH
SL
2
21 )(200
2


2
21
2
)(200 HH
AS
L



17. Dr. Lina Shbeeb
Crest Vertical Curves
• If the height of eye, h1, is 3.5 ft and height of object,
h2, is 0.5 ft respectively, as used for SSD, then
• For first condition: S < L
• For second condition: S > L
A
SL
1329
2 
1329
2
SA
L 

18. Dr. Lina Shbeeb
Sag vertical curve: headlight sight distance
• Headlight beam of automobile at 2 ft ( or 600 mm),
with 1o upward divergence from longitudinal axis of
roadway.
• “S” is actually the distance between the vehicle and
point where the 1o angle of light beam intersects the
surface of the roadway.
• For first condition: S < L
• For second condition: S > L
A
Sh
SL
)tan(200
2


)tan(200
2
Sh
SA
L



19. Dr. Lina Shbeeb
Headlight Sight Distance
• For safety the length of a sag vertical curve
should be long enough so that the light beam
distance is nearly the same as the SSD.
• Therefore SSD is used for "S" in the above
equations.

20. Dr. Lina Shbeeb
Vertical Curves
• The procedures for designing a crest vertical curve
are the following:
• Step 1: Determine the minimum length of curve to
satisfy sight distance requirements
• Step 2: Determine from the layout plans the station
and elevation of the PVI, that is the point where the
grades intersect.
• Step 3: Compute the elevations of the PVC (or BVC)
and end of vertical curve (EVC or PVT).
• Step 4: Compute the offsets “y” from the tangent to
the curve at equal distances, usually 100 feet or 20 m
apart.
• Step 5: Compute elevations on the curve.