NICE LECTURE FOR ROAD ENGINEERS.

1. ROAD GEOMETRIC DESIGN
TECHNICAL SECTION
NRAP/MoPW KABUL-AFGHANISTAN
PART : 03
BY: MALYAR TALASH
ROAD DESIGN ENGINEER
malyar@nrap.gov.af +93786575033

2. VERTICAL ALIGNMENT
• The topography of the land traversed
has an influence on the alignment of
roads and streets. Topography affects
horizontal alignment, but has an even
more pronounced effect on vertical
alignment. To characterize variations in
topography, engineers generally
separate it into three classifications
according to terrain—level, rolling, and
mountainous.

3. VERTICAL ALIGNMENT
• The two major aspects of vertical
alignment are vertical curvature, which is
governed by sight distance and comfort
criteria, and gradient which is related to
vehicle performance and level of service.
• Vertical curves are applied to effect the
transition between straight gradients.
There are two types of vertical curves,
namely crest vertical curves and sag
vertical curves.

4. VERTICAL ALIGNMENT
Crest and Sag Curves

5. PLAIN TERRAIN
The plain terrain is defined as area which
maintains between 0 and 10 No of Five
meter contour lines per km plain or gently
rolling tertian, this offers few obstacles to the
construction of a road or in other words, the
natural ground, cross slopes (ie.
perpendicular to natural ground contours) in
a flat terrain are generally below 3 %

6. PLAIN TERRAIN

7. ROLLING TERRAIN
Between11 to 25 No Five meter contour
lines per km. rolling, hilly or Foothill County
where the slope is generally rise and fall
moderately and where occasional steep
slopes are encountered, resulting in some
restrictions in alignment, in other words, the
natural ground cross slopes in rolling terrain
is generally between 3 – 25 %.

8. ROLLING TERRAIN

9. MOUNTAINOUS TERRAIN
Greater than 25 No .five meter contour lines
per km rugged, hilly and mountainous
county and river gorges this class of terrain
imposes definite restrictions on the
alignment and often involves long steep
grades and limited sight distance. In other
words, the naturals ground cross slopes in
mountainous terrain are generally above 25
%

10. MOUNTAINOUS TERRAIN

11. VERTICAL ALIGNMENT
• In level terrain, highway sight distances,
as governed by both horizontal and
vertical restrictions, are generally long or
can be made to be so without construction
difficulty or major expense.
• In rolling terrain, natural slopes
consistently rise above and fall below the
road or street grade, and occasional steep
slopes offer some restriction to normal
horizontal and vertical roadway alignment.

12. VERTICAL ALIGNMENT
In mountainous terrain, longitudinal and
transverse changes in the elevation of the
ground with respect to the road or street are
abrupt, and benching and side hill
excavation are frequently needed to obtain
acceptable horizontal and vertical alignment.

13. VERTICAL GRADES
Grades for local residential streets should
be as level as practical, consistent with the
surrounding terrain. The gradient for local
streets should be less than 15 %. Where
grades of 4 % or steeper are necessary, the
drainage design may become critical. On
such grades special care should be taken to
prevent erosion on slopes and open
drainage facilities.

14. VERTICAL GRADES
For streets in commercial and industrial
areas, gradient design desirably should be
less than 8 present, grades should be
desirable less than 5 present, and a flatter
grades should be encouraged.
To provide for proper drainage, the
desirable minimum grade for streets with
outer curb should be 0.30 percent, but
minimum grade of 0.20 percent maybe
used.

15. VERTICAL GRADES
The second issue to be considered for
vertical alignment is the maximum gradient.
Recommendations for maximum gradients
using the design speed approach are given
in below

16. VERTICAL GRADES

17. 17
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

18. 18
Maximum Grades
• Passenger cars – 4% to 5% no problem
• Upgrades: trucks average 7%
decrease in speed
• Downgrades: trucks average speed
increase 5%

19. 19
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

20. 20
Vertical Curve AASHTO Controls (Crest)
• Minimum length must provide stopping sight
distance (SSD) or “S” in equations
• Two situations (both assume h1=3.5’ and
h2=2.0’)
Source: Transportation Engineering On-line Lab
Manual, http://www.its.uidaho.edu/niatt_labmanual/

21. 21
Assistant with Target Rod (2ft object height)
Observer with
Sighting Rod (3.5
ft)

23. Notation
• Curve point naming is similar to horizontal
curves, with addition of V for vertical
– PVC: Point of Vertical Curvature
– PVI: Point of Vertical Intersection
(of initial and final tangents)
– PVT: Point of Vertical Tangency
• Curve positioning and length usually
referenced in stations
– Stations represent 1000 m or 100 ft
– e.g., 1258.5 ft → 12 + 58.5
(i.e., 12 stations & 58.5 ft)

24. Notation
• G1 is initial roadway grade
– Also referred to as initial tangent grade
• G2 is final roadway (tangent) grade
• A is the absolute value of the difference in
grades (generally expressed in percent)
– A = |G2 – G1|
• L is the length of the vertical curve measured
in a horizontal plane (not along curve center
line, like horizontal curves)

25. Fundamentals
• Parabolic curves are generally used for
design
– Parabolic function → y = ax 2
+ bx + c
y = roadway elevation
x = distance from PVC
c = elevation of PVC
– Also usually design for equal-length tangents
• i.e., half of curve length is before PVI and half after

26. First Derivative
• First derivative gives slope
–
• At PVC, x = 0, so , by definition
• G1 is initial slope (in ft/ft or m/m) as previously
defined
bax
dx
dy
+= 2
1G
dx
dy
b ==

27. Second Derivative
• Second derivative gives rate of change
of slope
• However, the average rate of change of
slope, by observation, can also be
written as
• Giving,
a
dx
yd
22
2
=
L
GG
dx
yd 12
2
2
−
=
L
GG
a
2
12 −
=

28. Offsets are vertical distances from initial
tangent to the curve
Offsets

29. • For an equal tangent parabola,
– Y = offset (in m or ft) at any distance, x, from the
PVC
– A and L are as previously defined
• It follows from the figure that,
2
200
x
L
A
Y =
200
800
AL
Y
AL
Y
f
m
=
=
Offset Formulas

30. “K” Values
• The rate of change of grade at successive
points on the curve is a constant amount for
equal increments of horizontal distance, and
• Equals the algebraic difference between
intersecting tangent grades divided by the
length of curve, or A/L in percent per ft (m)
• The reciprocal L/A is the horizontal distance
required to effect a 1% change in gradient
and is, therefore, a measure of curvature
• The quantity L/A is termed ‘K’

31. “K” Values
• The K-value can be used directly to compute
the high/low points for crest/sag vertical
curves (provided the high/low point is not at a
curve end) by,
– xhl = K × |G1|
– Where x = distance from the PVC to the high/low
point
• Additionally, K-values have important
applications in the design of vertical curves,
which we will see shortly

32. Vertical Curves
• Controlling factor: sight distance
• Stopping sight distance should be provided
as a minimum
• Rate of change of grade should be kept
within tolerable limits
• Drainage of sag curves is important
consideration, grades not less than 0.5%
needed for drainage to outer edge of roadway

33. Vertical Alignment Relationships
1
2
200
800
200
GKx
A
L
K
AL
Y
AL
Y
x
L
A
Y
hl
f
m
×=
=
=
=
=
L
GG
a
a
dx
yd
G
dx
dy
b
xatPVC
bax
dx
dy
cbxaxy
2
2
:0,
2
12
2
2
1
2
−
=
=
==
=
+=
++=

34. Stopping Sight Distance &
Crest Curves
• Two different factors are important for
crest curves
– The driver’s eye height in vehicle, H1
– Height of a roadway obstruction object, H2

35. SSD & Curve Design
• It is necessary, when designing vertical
curves, to provide adequate stopping-
sight distance (SSD)
• Because curve construction is
expensive, we want to minimize curve
length, subject to adequate SSD

36. Minimum Curve Length
• By using the properties of a parabola
for an equal tangent curve, it can be
shown that the minimum length of
curve, Lm, for a required SSD is
LSfor3.14Eq
)(200
2
LSfor3.13Eq
)(200
2
21
2
21
2
>
+
−=
<
+
=
A
HH
SL
HH
AS
L
m
m

37. Minimum Curve Length
• For the sight distance required to provide
adequate SSD, current AASHTO design
standards use the following specifications:
– H1 (driver’s eye height) = 3.5 ft (1080 mm)
– H2 (object height) = 2.0 ft (600 mm)

38. Minimum Curve Length
• Substituting these values into previous
two equations yields:
LSSDfor
404
2
LSSDfor
404
2
>−×=
<
×
=
A
SSDL
SSDA
L
m
m
Since using these equations can be cumbersome, tables have been
developed, utilizing K=L/A (discussed earlier)

39. Example

40. Sag Vertical Curves
• Four criteria for establishing length of
sag curves
– Headlight sight distance
– Passenger comfort
– Drainage control
– General appearance

41. Headlight Sight Distance
• At night, the portion of highway that is visible
to the driver is dependent on the position of
the headlights and the direction of the light
beam
• Headlights are assumed to be 2 ft (600 mm)
and 1-degree upward divergence of the light
beam from the longitudinal axis of the vehicle
• Equations 3-19 through 3-23 describe the
required sight distance for sag curves

42. Sag Vertical Curve Length
• The most controlling factor is headlight
sight distance
• If for economic reasons such lengths
cannot be provided, fixed source
lighting should be provided to assist the
driver.

44. Min Sag Curve Length
• Like crest curves, we need expressions
for determining the minimum length of
crest curve required for adequate SSD
LSfor
tan(200
2
3.20Eq
LSfor
tan(200
3.19Eq.
2
>
)+
−=
<
)+
×
=
A
SH
SL
SH
SSDA
L
m
m
β
β

45. 45
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)

46. 46
Chart vs computed
From chart
V = 60 mph K = 151 ft / % change
For g1 = 3 g2 = - 1
A = |g2 – g1| = |-1 – 3| = 4
L = ( K * A) = 151 * 4 = 604

47. 47
Sag Vertical Curves
• Sight distance is governed by night-
time conditions
– Distance on curve illuminated by
headlights need to be considered
• Driver comfort
• Drainage
• General appearance

48. 48
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

49. 49
Vertical Curve AASHTO Controls (Sag)
• For driver comfort use:
L > AV2
46.5
(limits g force to 1 fps/s)
• To consider general appearance use:
L > 100 A

50. 50
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

51. 51
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

52. 52
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

53. Minimum Curve Length
For the sight distance required to provide
adequate SSD, current AASHTO design
standards use the following specifications:
H (headlight height) = 2.0 ft (600 mm)
β (headlight angle) = 1°

54. Minimum Sag Curve Length
US Customary Metric
For SSD < L
SSD53400
SSD2
×
×
=
.+
A
Lm
SSD53120
SSD2
×
×
=
.+
A
Lm (3.21)
For SSD > L
A
.+
Lm
SSD53400
SSD2
×
−×=
A
.+
Lm
SSD53120
SSD2
×
−×= (3.22)
Substituting the recommended values for beta and H
gives:
If not sure which equation to use, assume
SSD < L first (for either sag or crest
curves)

55. K Values for Adequate SSD
US Customary Metric
Rate of vertical
curvature, Ka
Rate of vertical
curvature, KaDesign
speed
(mi/h)
Stopping
sight
distance
(ft)
Calculated Design
Design
speed
(km/h)
Stopping
sight
distance
(m)
Calculated Design
15 80 9.4 10 20 20 2.1 3
20 115 16.5 17 30 35 5.1 6
25 155 25.5 26 40 50 8.5 9
30 200 36.4 37 50 65 12.2 13
35 250 49.0 49 60 85 17.3 18
40 305 63.4 64 70 105 22.6 23
45 360 78.1 79 80 130 29.4 30
50 425 95.7 96 90 160 37.6 38
55 495 114.9 115 100 185 44.6 45
60 570 135.7 136 110 220 54.4 55
65 645 156.5 157 120 250 62.8 63
70 730 180.3 181 130 285 72.7 73
75 820 205.6 206
80 910 231.0 231
a
Rate of vertical curvature, K, is the length of curve per percent algebraic difference in
intersecting grades (A). K = L/A
Design Controls for Sag Vertical Curves Based on SSD
Table 3.3

56. Passing Sight Distance & Crest
Vertical Curve Design
• Only a factor for vertical curves
• A consideration for two-lane highways
• Sag curves have unobstructed sight
distance
• Assume driver eye height and height of
object on roadway surface both 3.5’

57. Stopping Sight Distance &
Horizontal Curve Design
• Adequate sight distance must be provided in
the design of horizontal curves
• Cost of right of way or the cost of moving
earthen materials often restrict design options
• When such obstructions exist, stopping sight
distance is checked and measured along the
horizontal curve from the center of the
traveled lane

59. Sight Distance Relationships





 −
=
=−=
=∆
∆
∆=
−
)(cos
90
SSDforsolving),
90
cos1(
Mforsolvecan3.38)(eqcurvehorizontalsimpleof
ordinatemiddleforequationgeneralintongSubstituti
180
angle)central(not thedistancesightstoppingrequired
thetoequallengtharcanforangleThe?isWhat
*
180
1
s
s
v
svv
v
vs
v
s
sv
R
MRR
SSD
R
SSD
RM
R
SSD
RSSD
π
π
π
π

60. Sight Distance Example
Horizontal curve with 2000’ radius;
12’lanes; 60mph design speed.
Determine the distance that must be
cleared from the inside edge of the
inside lane to provide sufficient stopping
sight distance.

61. Sight Distance Example
Continued
ftM
RR
R
SSD
RM
s
v
v
vs
33.20)
)1994(1417.3
)570(90
cos1(1994
1994620002/12
)
90
cos1(
=−=
=−=−=
−=
π
*SSD is determined from Table 3.1 for 60mph design speed