This document discusses various aspects of vertical alignment in transportation engineering. It describes how vertical alignment specifies the elevation of points along a roadway based on safety, comfort, drainage needs. Vertical curves are used to transition between different roadway grades and can be crest or sag curves. The coordination of vertical and horizontal alignment is also discussed to ensure driver safety and aesthetics. Maximum and minimum grades, as well as critical lengths of grades, are addressed based on truck performance.

1. Vertical Alignment
Transportation Engineering

2. Vertical Alignment
Vertical alignment specifies the elevation of points along roadway.
The elevation of these roadway points are usually determined by the
need to provide an acceptable level of driver safety, driver comfort,
cut to fill and proper drainage (form rainfall runoff).
A primary concern in vertical alignment is establishing the transition
of roadway elevations between two grades. This transition is
achieved by means of a vertical curve.
Each roadway is uniquely defined by stationing (which is measured
along a horizontal plane) and elevation.

3. Vertical Alignment
Vertical curve can be
broadly classified into
• Crest Vertical Curves
• Sag Vertical Curves
The proper relationships
between vertical and
horizontal curvature result
in an aesthetic and easy-
to-drive facility.
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4. Algebraic Difference of grades

5. Vertical Alignment
• Horizontal and vertical alignment should not be
designed independently. They complement each
other and poor design combinations can spoil the
good points and aggravate the deficiencies of
each. Properly coordinated horizontal and vertical
alignment can enhance community values,
increase utility and safety, encourage uniform
speed, and improve appearance.
• Sharp horizontal curvature should not be
introduced near bottom of steep grade near the
low point of a pronounced sag vertical curve
– Horizontal curves appear distorted
– Vehicle speeds (esp. trucks) are highest at the
bottom of a sag vertical curve
– Can result in erratic motion
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6. Coordination of Vertical and Horizontal
Alignment
• Curvature and grade should
be in proper balance
– Avoid
• Excessive curvature to achieve
flat grades
• Excessive grades to achieve flat
curvature
• Vertical curvature should be
coordinated with horizontal
• Sharp horizontal curvature should
not be introduced at or near the
top of a pronounced crest vertical
curve
– Drivers may not perceive
change in horizontal alignment
esp. at night
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7. Coordination of Vertical and Horizontal
Alignment
• On two-lane roads when passing
is allowed, need to consider
provision of passing lanes
– Difficult to accommodate with
certain arrangements of horizontal
and vertical curvature
– need long tangent sections to assure
sufficient passing sight distance
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8. Coordination of Vertical and Horizontal
Alignment
• At intersections where sight distance
needs to be accommodated, both
horizontal and vertical curves should be
as flat as practical
• In residential areas, alignment should
minimize nuisance to neighborhood
– Depressed highways are less visible
– Depressed highways produce less noise
– Horizontal alignments can increase the buffer
zone between roadway and cluster of homes
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9. Coordination of Vertical and Horizontal
Alignment
• When possible alignment should
enhance scenic views of the natural
and manmade environment
– Highway should lead into not away from
outstanding views
– Fall towards features of interest at low
elevation
– Rise towards features best seen from
below or in line against the sky
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10. Coordination of Horizontal
and Vertical Alignment
• Coordination of horizontal and vertical
alignment should begin with
preliminary design
• Easier to make adjustments at this
stage
• Designer should study long,
continuous stretches of highway in
both plan and profile and visualize the
whole in three dimensions

11. Coordination of Horizontal and Vertical
Alignment
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12. • Should be consistent with the
topography
• Preserve developed properties
along the road
• Incorporate community values
• Follow natural contours of the
land
Coordination of Horizontal and Vertical
AlignmentTruck

13. Good Coordination of Horizontal and
Vertical Alignment
• Does not affect
aesthetic, scenic,
historic, and
cultural resources
along the way
• Enhances attractive
scenic views
– Rivers
– Rock formations
– Parks
– Historic sites
– Outstanding
buildings
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20. Vertical Curves
• Connect roadway grades
(tangents)
• Grade (rise over run)
– 10% grade increases 10' vertically
for every 100 ' horizontal
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21. Vertical Curves
Ascending grade:
– Frequency of
collisions increases
significantly when
vehicles traveling
more than 10 mph
below the average
traffic speed are
present in the
traffic stream
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22. Example
• If a highway with
traffic normally
running at 65 mph
has an inclined
section with a 3%
grade, what is the
maximum length
of grade that can
be used before the
speed of the larger
vehicles is reduced
to 55 mph?
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23. Example
• a 3%
grade
causes a
reductio
n in
speed of
10 mph
after
1400
feet
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24. Continuity of Vertical
Curve
• Continuity of
forms
For vertical curves continuity
of form is not a problem.
There is no sudden break in
grade at the PC or PT since
the parabolic curve provides a
gradual transition in grade
• Continuity of
scale
Continuity of scale relates to
the relative lengths of the
tangent and curves
From an esthetic point of view, one of the goal in alignment design
is to achieve a continuous alignment. Continuity is desirable so
that the road fit the terrain, does not have a jarring aspect and
present a path which is easy to follow

26. Climbing Lanes
• When flatter grades cannot be
accommodated, consider climbing lane
when all 3 of the following criteria are met
(AASHTO):
– Upgrade traffic flow rate in excess of 200
vehicles per hour.
– Upgrade truck flow rate in excess of 20
vehicles per hour.
– One of the following conditions exists:
• A 15 km/h or greater speed reduction is expected
for a typical heavy truck.
• Level-of-service E or F exists on the grade.
• A reduction of two or more levels of service is
experienced when moving from the approach
segment to the grade.
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27. Critical Lengths of Grade
for Design
• Maximum length of a designated upgrade upon
which a loaded truck can operate without an
unreasonable reduction in speed,
• For a given grade, lengths less than critical results in
acceptable operation in the desired range of speeds.
where the desired freedom of operation is to be
maintained on grades longer than critical, design
adjustments such as change in location to reduce grades
or addition of extra lanes should be made.
• A separate climbing lane exclusively for slow moving
vehicles is preferred to the addition of an extra lane
carrying mixed traffic. The large speed differential
between trucks and passenger cars traveling up grade in
the same lanes greatly increases the accident potential.

28. Critical Lengths of Grade
for Design
Design values for critical lengths of grade were established
based on the size and power of a representative truck or
truck combination, the gradability data for this vehicle, the
speed at entrance to critical length of grade, and the
minimum speed on the grade, below which interference to
following vehicles is considered unreasonable.
A loaded truck with a weight-power ratio of approximately
400 was used as a nationally representative truck. The
speed of vehicles beginning at uphill climb approximates
the average running speed of the highway.
On most of the arterial streets where the running speed will
be 40 mph or less, truck speeds of 20 to 35 mph are not
unreasonably annoying to following drivers. The developed
control basis for determining critical length of grades and
the need for climbing lanes is normally a reduction of 15
miles per hour in speed of trucks below the average running
speed.

29. Descending Grades
• Problem is increased speeds and loss of control for
heavy trucks
• Runaway vehicle ramps are often designed and
included at critical locations along the grade
• Ramps placed before each turn that cannot be
negotiated at runaway speeds
• Ramps should also be placed along straight
stretches of roadway, wherever unreasonable speeds
might be obtained
• Ramps located on the right side of the road when
possible
• Have detrimental effect on capacity and safety on
urban facilities with high traffic volumes and
numerous heavy trucks. Although criteria are not
established for these conditions., in some cases
consideration should be given to providing a truck
lane for downhill traffic.
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30. Maximum Grades
Criteria for maximum grades are based
mainly on studies of the operating
characteristics of typical heavy trucks.
• Passenger vehicles can easily negotiate 4
to 5% grade without appreciable loss in
power
• Upgrades: trucks average 7% decrease
in speed
• Downgrades: trucks average speed
increase 5%
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31. Minimum Grades
• Minimum grades are primarily related to the need for
adequate drainage.
• For uncurbed pavements that are adequately crowned to
drain laterally, relatively flat or even level profile grades
may be used.
• With curbed pavement, the minimum longitudinal grade
in usual cases should be 0.5 percent.
• With a high-type pavement accurately crowned on a firm
subgrade, a longitudinal grade of about 0.3 percent may
be used.
• Even on uncurbed pavements, it is desirable to provide a
minimum of about 0.3 percent longitudinal grade
because the lateral crown slope originally constructed
may subsequently be reduced as a result of irregular
swell, consolidation, maintenance operations or
resurfacing.
• Equalizing situations such as lakebeds or slough areas a
0.0 % grade may be used.
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32. Minimum Ditch Grades
• Special attention should be directed to
minimum ditch grades in areas of expansive
soils.
• These areas will be identified in the soils design.
Any pounding of water has a very detrimental
effect on the subgrade.
• To ensure continuing flow, ditch grades should
be sloped at least 0.3 percent -- preferably 0.5
percent or steeper.
• This may require some special warping of ditch
grades when the roadway profile cannot be
adjusted accordingly.

33. • The “roller coaster” or “ hidden –dip” type profile
generally occur on relatively straight horizontal
alignment where the roadway profile closely follows a
rolling natural ground line.
• A sky line horizon at crest disappear into the sky and can
be disconcerting to the driver. It would be appealing to
have the view above the crest backed by a back drop of
landscape.
• A profile on a pronounced crest along a horizontal curve,
particularly in flat terrain, tends to produce in a
foreshortened view a disturbingly awkward appearance.
Use of longer and flatter approach gradients, coupled with
special gradient and planting on inside of the curve may
reduce the undesirable effect.

34. Vertical Curves
• Parabolic shape
• VPI, VPC, VPT, +/- grade, L
• Types of crest and sag curves
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35. Vertical Curves
• Crest – stopping, or passing sight
distance controls
• Sag – headlight/SSD distance,
comfort, drainage and appearance
control
• Green Book vertical curves defined by
K = L/A = length of vertical
curve/difference in grades (in percent)
= length to change one percent in
grade
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36. Parabola
y = ax2
+ bx + c
Where:
y = roadway elevation at distance x
x = distance from beginning of
vertical curve
a = G2 – G1
L
b = G1
c = elevation of PVC
Vertical Curve Equations
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37. Vertical Curve AASHTO Controls
(Crest)
• Minimum length must provide stopping
sight distance S
• Two situations (both assume h1=3.5’
and h2=2.0’)
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38. Assistant with Target Rod (2ft object height)
Observer with
Sighting Rod (3.5
ft)

39. Vertical Curve AASHTO
Controls (Crest)
Note: for passing sight distance,
use 2800 instead of 2158
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40. Example: Try SSD > L,
Design speed is 60 mph
G1 = 3% and G2 = -1%,
what is L?
(Assume grade = 0% for SSD)
SSD = 570feet ( see: Table 3.4 of text)
Lmin = 2 (570’) – 2158’ = 600.5’
|(-1-3)|
S < L, so it doesn’t match condition
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41. Example: Assume SSD < L,
Design speed is 60 mph
G1 = 3% and G2 = -1%,
what is L?
Assuming average grade = 0%
SSD = 570 feet - ( Table 3.4 of text)
Lmin = |(-3 - 1)| (570 ft)2
= 602 ft
2158
SSD < L, equation matches condition
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42. Evaluation of example:
• The AASHTO SSD distance equations
provided the same design length from
either equation in this special case.
(600 compared to 602 - this is not
typical)
• Garber and Hoel recommend using the
most critical grade of - 1% for SSD
computation.
– Resulting SSD would be: d = 573 ft
– Resulting minimum curve: L = 608 ft
• Difference between 602 and 608 is too
small to worry about
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43. Text example : g1 = + 3% g2 = -3%
Design speed of 60 mph
If SSD = 570’ (AASHTO – no grade consideration)
Resulting minimum curve: L = 903 ft (S < L)
Consider grade per Garber and Hoel
SSD, using - 3% grade, = 595’
Resulting minimum curve L = 984 ft
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44. Assessment of Grade
Adjustment
If sight distance is less than curve length, the driver
will be on an upgrade a greater portion of the
distance than on a down grade
(for eye ht = 3.5’ and object ht = 2.0 ft, 68% of the
distance between eye and object will be on + grade.)
For crest vertical curve, selecting a curve length
based on down grade SSD may produce an overly
conservative design length.
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45. AASHTO design tables
• Vertical curve length can also be
found in design tables
L = K *A
Where
K = length of curve per percent
algebraic difference in intersecting
grade
Charts from Green Book
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46. From Green book

47. From Green book

48. Vertical Curve AASHTO
Controls (Crest)
Since you do not at first know L, try
one of these equations and compare
to requirement, or use L = KA (see
tables and graphs in Green Book for
a given A and design speed)
Note min. L(ft) = 3V(mph) – Why?
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49. Chart vs computed
From chart
V = 60 mph K = 151 ft / % change
For g1 = 3 g2 = - 1
A = | -1 – 3 | = 4
L = ( K * A) = 151 * 4 = 604
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50. Sag Vertical Curves
• Sight distance is governed by
nighttime conditions
– Distance of curve illuminated by
headlights need to be considered
• Driver comfort
• Drainage
• General appearance
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51. Vertical Curve AASHTO
Controls (Sag)
Headlight Illumination sight distance
S < L: L = AS2
400 + (3.5 * S)
S > L: L = 2S – (400 + 3.5S)
A
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52. Vertical Curve AASHTO Control
(Sag)
• For driver comfort use:
L > AV2
46.5
(limits g force to 1 fps/s)
• To consider general appearance
use:
L > 100 A
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53. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –3% to
a +3% grade. Design speed is 40 mph. What is L?
Skipping steps: SSD = 313.67 feet S > L
Determine whether S<L or S>L
L = 2(313.67 ft) – (400 + 2.5 x 313.67) = 377.70 ft
[3 – (-3)]
313.67 < 377.70, so condition does not apply
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54. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –
3% to a +3% grade. Design speed is 40 mph.
What is L?
Skipping steps: SSD = 313.67 feet
L = 6 x (313.67)2
= 394.12 ft
400 + 3.5 x 313.67
313.67 < 394.12, so condition applies
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55. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –3% to a +3%
grade. Design speed is 40 mph. What is L?
Skipping steps: SSD = 313.67 feet
Testing for comfort:
L = AV2
= (6 x [40 mph]2
) = 206.5 feet
46.5 46.5
Testing for appearance:
L = 100A = (100 x 6) = 600 feet
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56. Vertical Curve AASHTO
Controls (Sag)
• For curb drainage, want min. of
0.3 percent grade within 50’ of low
point = need Kmax
= 167 (US units)
• For appearance on high-type
roads, use min design speed of 50
mph (K = 100)
• As in crest, use min L = 3V
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57. Other important
issues:
• Use lighting if need to use shorter
L than headlight requirements
• Sight distance at under crossings
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59. Example: A crest vertical curve joins a +3% and –4% grade.
Design speed is 75 mph. Length = 2184.0 ft. Station at
PVI is 345+ 60.00, elevation at PVI = 250 feet. Find
elevations and station for PVC (BVC) and PVT (EVC).
L/2 = 1092.0 ft
Station at PVC = [345 + 60.00] - [10 + 92.00] = 334 + 68.00
Vertical Diff PVI to PVC: -0.03 x (2184/2) = -32.76 feet
ElevationPVC = 250 – 32.76 = 217.24 feet
Station at PVT = [345 + 60.00] + [10 + 92.00] = 357 + 52.00
Vertical Diff PVI to PVT = 0.04 x (2184/2) = 43.68 feet
Elevation PVT = 250 – 43.68 = 206.32 feet

60. Example: A crest vertical curve joins a +3% and –
4% grade. Design speed is 75 mph. Length =
2184.0 ft. Station at PVI is 345+ 60.00, elevation
at PVI = 250 feet. Station at BVC (PVC) is 334 +
60.00, Elevation at BVC: 217.24 feet.
Calculate points along the vertical curve.
X = distance from BVC
Y = Ax2
200 L
Elevationtangent = elevation at BVC + distance x
grade
Elevationcurve = Elevationtangent - Y

61. Example: A crest vertical curve joins a +3%
and –4% grade. Design speed is 75 mph.
Length = 2184.0 ft. Station at PVI is 345+
60.00, elevation at PVI = 250 feet. Find
elevation on the curve at a point 400 feet from
BVC.
Y = A x 2
= - 7 x (400 ft)2
= - 2.56 feet
200L 200 (2814)
Elevation at tangent = 206.32 + (400 x 0.03) =
218.32
Elevation on curve = 218.32 – 2.56 feet =
226.68’

62. Calculating x from BVC, calculating tangent
elevation along +3% tangent
Y

64. Calculating x from EVC, calculating tangent
elevation along +4% tangent
Y

65. Source: Iowa DOT
Design Manual

66. Source: Iowa DOT Design Manual

67. Note: L is measured from here to here
Not here