CE416:Lecture05

Geometric Design of Highways

Horizontal alignment 1

Horizontal alignment is a broad term that encompasses several aspects

of transportation engineering. In geometric design of highways,
horizontal alignment is determined by a series of tangents connected by
simple curves. The tangents set the general direction of the roadway.
The horizontal curves allow smooth transitions from one tangent to the
next.

Surveying and Stationing

Staking: route surveyors define the geometry of a highway by “staking”

out the horizontal and vertical position of the route and by marking of the
cross-section at intervals of 100 m.

Station: Start from an origin by stationing 0, regular stations are

established every 100 m., and numbered 0+000, 12 + 000 (=1200 m), 20
+ 45 (2000 m + 45) etc.

24
+0
23+00

Plan view and profile 0

22+00
21+00
20+00
19+00
18+00
17+ 0
16

0
+0

plan
0
15
+0
0

700
700
600
profile
500
400
300
200

15+00 16+00 17+00 18+00 19+00 20+00 21+00 22+00 23+00 24+00

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Horizontal Curves

Curves used in the horizontal alignment of highways are circular curves.

Easement or transition curves are used some times to introduce circular
curves and to allow for gradual change from tangent to circular curve.
The curve characteristics are determined by safety requirements
controlled by stopping sight distance and side friction characteristics.

Basics of the horizontal curve include radius, tangent distance,

stationing, deflection angle, length of curve and middle ordinate distance.

Type of Circular curves

R1

R2 R1

R2

(1) Simple horizontal curve (2) Broken back curve (3) Compound curve
R2
R1

R2
R1

(a) With tangent

(b) Without tangent

(4) Reverse Curve

Properties of Circular Curves

Traditionally, the steepness of the curvature is defined by either the

‘radius’ (R) or the ‘degree of curvature’ (D)
The degree of curvature is defined as the angle subtended by an arc of
length 20 m, (this is referred to as the arc definition of D - there is also a

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chord definition. The arc definition is the one used for highway design
applications. The chord definition is used in railroad design)
Based on the arc definition the relationship between R and D is as
follows:
R = 1145.916/D

Circular Horizontal Curve Definitions

PI

T
E
L

∆ M ∆
2 C 2
PC PT

∆ ∆
90 90
2 2

R R

∆ ∆
2 2

R = Radius, usually measured to the centerline of the road, in m.

∆ = Central angle of the curve in degrees
PC = point of curve (the beginning point of the horizontal curve)
PI = point of tangent intersection
PT = Point of tangent (the ending point of the horizontal curve)
T = tangent length in m.
M = middle ordinate from middle point of cord to middle point of curve in m.
E = External distance in m.
L = length of curve
D = Degree of curvature (the angle subtended by a 20-m arc* along the curve)
C = chord length from PC to PT

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 Key measures of the curve

M = R[1 cos(∆ / 2)]

1
E = R[( ) 1]
20(180 / π ) 3600 1145.916 cos(∆ / 2)
D= = =
R πR R

∆
T = R tan
2 ∆
C = 2 R sin
2
π
L= R∆
180

* Note 180 / π converts from radians to degrees

Angle measurement

90
60

30

180 0

(a) degree 1o = π / 180radians = 0.0174532 radians

o o
1radian = (180 / π ) = 57.2957

(b) Radian

Example: Circular Curve Data

In the horizontal alignment of a highway, a change in the direction of the

centerline of 20° has been decided at station 01+85.765. Using a curve
radius of 285 m, calculate the circular curve data?
Solution PI (01+85.765)

Given: ∆ = 20° PC ∆ = 20°

PT
R = 285 m

R= 285m

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 2πR∆ 2π (285)(20)
L= = = 99.484 m (00+99.484)
360 360
T = R tan(∆ / 2) = 285 tan(20/2) = 50.253 m (00+50.253)
E = R(sec(∆ 2) 1) = 285(sec(10)-1) = 4.397 m
M = R(1 cos(∆ / 2)) = 285(1- cos10) = 4.330 m
Station PI = (01+85.765)
Station PC = (01+85.765) - (00+50.253) = (01+35.512)
Station PI = (01+35.512) + (00+99.484) = (02+34.996)

Design of Horizontal Curve

The key steps in the design of horizontal curves are listed below.

1.) Determine a reasonable maximum superelevation rate.

2.) Decide upon a maximum side-friction factor.
3.) Calculate the minimum radius for your horizontal curve.
4.) Iterate and test several different radii until you are satisfied with your
design.
5.) Make sure that the stopping sight distance is provided throughout the
length of your curve. Adjust your design if necessary.
6.) Design the transition segments.

Determining the Minimum Radius

One design issue is to determine appropriate values for the following:

o external angle
o curve length or curve radius

The starting point is to determine the minimum radius for design

Once Rmin is determined, the designer can use any value of R > Rmin

The Minimum Radius is derived by considering the forces acting on the

occupants of a vehicle negotiating a curve and the resulting comfort level
of the occupants.

Minimum Radius Horizontal Curve :

1. No Super Elevation R mV 2
R
For a vehicle traveling along a curved path
f .m.g
the forces acting on the vehicle are as shown

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in the figure; the centripetal force (mV2/R) pushing

the vehicle outside the curve resisted by the

friction force (f.m.g) generated by the side friction of the tires of the
vehicle with the pavement surface. Equating theses force for
equilibrium, radius R could be determined in terms of V & f.

mV 2
= f .m.g
R

V2
R=
g f

Typical values of f are given below

Speed, Mph f Speed, Km/hr f

30 0.16 30 0.17
40 0.15 40 0.17
45 0.145 50 0.16
50 0.14 60 0.15
60 0.12 70 0.14
65 0.11 80 0.14
70 0.1 90 0.13
80 0.08 100 0.12
120 0.09

2. With Super Elevation

For high speeds and/or low coefficient of side friction, vehicles can't
attain equilibrium along sharp curves, and the solution is tilting the road
surface at curved path to utilize part of the acting forces in acheaving
equilibrium and preventing vehicles from sliding putward away from the
center of the curve. This tiliting of the road surface is called
"Superelevation".

In this case the force acting on the vehicle are analysed as follows:
mV 2 mV 2
cos θ = mg sin θ + f ( mg sin θ + sin θ )
R R
mV 2
cos θ
R mV 2
sin θ
R
And, therefore,
mV 2
V2 fV 2
= tan θ + f + ( ) tan θ mg mg sin θ R
gR gR θ

mg sin θ

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 fV 2
the quantity tan θ is so small that it can be neglected. If
gR
tan θ is expressed in terms of the crossfall slope (or superelevation), e,
then

V 2 (e + f )
=
R g

where V2/R is the centerfugal acceleration. If V in m/s is

replaced by V Km/h and g=9.81 m/s2, then

V2
= e+ f
127 R
This is known as the minimum radius equation, and could be
re-written as:

V2
Rmin =
e%
127( f + )
100

The radius calculated in the equations above is the minimum. It is only

use in cases where there are severe restrictions on the alignment. The
usual case is to use a radius that is well above the calculated value.

Minimum Radius Calculations

Calculating the minimum radius for a horizontal curve is based on three

factors: the design speed, the superelevation, and the side-friction factor.
The minimum radius serves not only as a constraint on the geometric
design of the roadway, but also as a starting point from which a more
refined curve design can be produced.

For a given speed, the curve with the smallest radius is also the curve
that requires the most centripetal force. The maximum achievable
centripetal force is obtained when the superelevation rate of the road is
at its maximum practical value, and the side-friction factor is at its
maximum value as well. Any increase in the radius of the curve beyond
this minimum radius will allow you to reduce the side-friction factor, the
superelevation rate, or both.

Using the equations for circular motion, friction, and inclined plane
relationships, the following equation has been derived.

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 V2
Rmin =
emax %
127( f max + )
100

Where:
Rmin = Minimum radius of the curve (m)
V = Design velocity of the vehicles (km/h)
emax = Maximum superelevation rate as a percent
fmax = Maximum side-friction factor

This equation allows the engineer to calculate the minimum radius for a
horizontal curve based on the design speed, the superelevation rate, and
the side friction factor.

Superelevation and Side-Friction

Most highways will change directions several times over the course of
their lengths. These changes may be in a horizontal plane, in a vertical
plane, or in both. The engineer is often charged with designing curves
that accommodate these transitions, and consequently must have a
good understanding of the physics involved.

The superelevation of the highway cross-section and the side-friction

factor are two of the most crucial components in the design of horizontal
curves. The superelevation is normally discussed in terms of the
superelevation rate, which is the rise in the roadway surface elevation as
you move from the inside to the outside edge of the road. For example, a
superelevation rate of 10% implies that the roadway surface elevation
increases by 1 m for every 10 m of roadway width. The side-friction
factor is simply the coefficient of friction between the design vehicle's
tires and the roadway.

Whenever a body changes directions, it does so because of the

application of an unbalanced force. In the special case of a body moving
in a circular path, the force required to keep that body traveling in a
circular path is called the centripetal force. When vehicles travel over a
horizontal curve, it is this centripetal force that keeps the vehicles from
sliding to the outside edge of the curve. In the simplest case, where the
road is not banked, the entire centripetal force is provided by the friction
between the vehicle's tires and the roadway. If we add some side-slope
or superelevation to the cross-section of the roadway, some of the
centripetal force can be provided by the weight of the car itself.

High rates of superelevation that make cornering more comfortable

during the summer by requiring less frictional force, can make winter
driving ponderous by causing slow-moving vehicles to slip downhill
toward the inside of the curve. Because of this, there are practical

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maximum limits for the rate of superelevation. In areas where ice and
snow are expected, a superelevation rate of 8% seems to be a
conservative maximum value. In areas that are not plagued by ice and
snow, a maximum superelevation rate of 10-12% seems to be a practical
limit. Both modern construction techniques and driver comfort limit the
maximum superelevation rate to 12%.

The side-friction factor has practical upper limits as well. As was

discussed in the braking distance, the coefficient of friction is a function
of several variables, including the pavement type and the vehicle speed.
In every case, the side-friction factor that is used in design should be well
below the side-friction factor of impending release. In addition to the
safety concerns, drivers don't feel comfortable if the roadway seems to
rely heavily on the frictional force. Several studies aimed at determining
the maximum side-friction factors that are comfortable for drivers have
been conducted.

The side-friction factors that are employed in the design of horizontal

curves should accommodate the safety and comfort of the intended
users.

Example: Horizontal Curve Radius Calculations

A new transportation engineer is charged with the design of a horizontal

curve for a new highway. His final design calls for a curve with a radius
of 520 meters. Would you sign your name to his plans?

Assume that the design speed for the highway is 110 km/h. You can also
assume that snow and ice will be present on the roadway from time to
time.

Solution

The first step in a review of his plans would be to make sure that the
curve radius as designed is greater than the minimum curve radius. For
a design speed of 110 km/h, the comfortable side-friction factor is
0.10. In addition, since the roadway will be covered with snow and ice
from time to time, the maximum superelevation rate is 8%. With this
information we can go ahead and calculate the minimum curve radius
using the equation below.

V2
Rmin = Where:
emax %
127( f max + )
100

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Rmin = Minimum radius (m)
V = Design speed,110 km/hr
emax = Maximum superelevation rate, 8%
fmax = Maximum side-friction factor, 0.10

Substituting and solving yields a minimum radius of 530 meters. The 520
meter radius that is called for in the plans would probably work, but it
might be uncomfortable for the vehicle occupants. A larger radius would
be more appropriate.

Design Iterations
In many ways, horizontal alignment is an art form. The goal is to produce
a horizontal curve that is comfortable and safe to use, and also cost
efficient and aesthetically pleasing. The first step is to calculate the
radius of the horizontal curve. We can calculate the radius for any
combination of superelevation and side-friction factors using the equation
given above.

As long as the radius of your curve is above the minimum radius as

described in the minimum radius module, and as long as you haven't
exceeded the practical values for the superelevation or for the side-
friction factor, you know that your design is acceptable .

You will probably need to test several different curve radii before you
select your final design. While iterating, you also need to consider other
factors: the cost, environmental impacts, sight distances, and, of course,
the aesthetic consequences of your curve. Most surveying books contain
a complete chapter on the layout of horizontal curves, and consequently,
we won't delve into the surveying issues. Please refer to your surveying
texts for this information.

Horizontal Curve Sight Distance

Objects such as cut slopes, walls, buildings, bridge piers, and
longitudinal barriers can create sight obstructions on the inside of curves
(or the inside of a median lane on a divided highway). If removal of the
object is not a possibility, then the alignment of the roadway may need to
be altered to provide adequate sight distance.
For the purpose of designing a horizontal curve, the sight line is a chord
of the curve. Sight distance is measured along the centerline of the

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inside lane around the curve. The distance M is the minimum distance an
object needs to be located from the center of the inside lane to provide
adequate sight distance.

MS

Rv Rv

The relationship between R and M is given as:

28.65S
M S = Rv (1‫ ـ‬cos )
Rv

Where: Rv is the radius of the curve based on design speed and S is

the stopping sight distance.
The quantity (28.65S/R) is expressed in degrees.

This equation applies when the length of the curve is greater than the
stopping sight distance.

Once M has been determined, other issues such as clear zone distances
must also be considered when determining the distance an object needs
to be from the roadway.
In cut areas, a cut slope on the inside of a curve may obstruct sight
distance. Considering the driver’s eye to be at a height of 1080 mm and
the object to be at 150 mm, a height of 600 mm may be used as the
midpoint of the sight line where the cut slope may present an obstacle
(assuming little or no vertical curvature).

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Since passing sight distance is much greater than stopping sight
distance, clear sight areas on the inside of curves will require much
greater widths than compared to widths required for stopping sight
distance.
Although the equation for finding M may be used for passing sight
distance as well, the results are practical only for longer curves (radii of
three or more times Rmin). These results however could be used to
demonstrate the need for very flat curves if passing sight distance is
required, which in turn may be valuable information when choosing
between design alternatives.

Example: Horizontal Curve Sight Distance (Sound Barrier)

A horizontal curve for a four-lane section of a freeway in an urban area is

to have a design speed of 100 Kmph. Site conditions necessitate a curve
radius of 381 m. A 6-m high sound barrier is required in this area. If the
freeway has a 19.5-m median, how far from the centerline of the freeway
will the barrier need to be located to satisfy minimum stopping sight
distance requirements?
The center of the inside lane is located 15.25 m from the centerline of the
freeway, as shown in Figure. This is determined by adding ½ the width of
the median (9.75 m) plus the width of the outside lane (3.6 m) plus ½ the
width of the inside lane (1.8 m): 9.75 m + 3.6 m + 1.8 m = 15.15 m. Thus
Rv = 381 m – 15.15 m = 365.85 m. Using a design speed of 100 Kmph,
stopping sight distance is 205 m.

9.75 m
15.25 m

3.6 m 1.8 m

From this,
28.65S 28.65 x 205
M S = Rv (1‫ ـ‬cos ) = 381(1‫ ـ‬cos 381
)
Rv
=20.325 m= 20.5 m

The distance the barrier needs to be from the freeway centerline is about
20.5 m (barrier to center of inside lane) + 15.25 m (center of inside lane

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to centerline of freeway) = 35.75 m. This value represents the minimum
distance needed to satisfy stopping sight distance requirements.
Using stopping sight distance requirements places the barrier 35.75 m
(barrier to center of inside lane) – 1.8 m (center of inside lane to edge of
traveled way) = 33.95 m from the edge of traveled way.

Example: Horizontal Curve Sight Distance (Median Barrier)

Suppose another section of the freeway in the previous example requires

a 4.8 m paved median with 1.10-m tall concrete median barrier rail.
Assuming a 1.8-m inside shoulder, what is the minimum radius required
to avoid sight distance problems?
The face of the barrier rail is located about 0.3 m from the centerline of
the roadway. The center of the inside lane is located 6 m from the
centerline, as shown in Figure. This is determined by adding ½ the width
of the median (2.4 m) plus the 1.8-m shoulder plus ½ the width of the
lane adjacent to the median (1.8 m): 2.4 m + 1.8 m + 1.8 m = 6 m. The
distance from the center of the lane to the face of the barrier rail is MS in
Figure. For this example, MS = 6 m (the distance to the centerline from
the center of the lane) – 0.3 m (the distance to the centerline from the
face of the barrier rail) = 5.7 m. Stopping sight distance is 205 m.

1.8 m
1.8 m
2.4 m

From this,
28.65 x 205
M = 5.7 = R(1‫ ـ‬cos )
R

This cannot be solved directly for R, so different values of R are tried

until a value of M equal to or slightly less than 5.7 m is achieved. Using
an M slightly less than 5.7 will produce an R slightly more than the
minimum required for sight distance. For R = 920 m, M = 5.705 m, which
is a bit too large. For R = 925 m, M = 5.675 m, so use 925 m. To find the
radius to the centerline, subtract out the distance from the center of the
inside lane to the centerline, which is 6 m as shown Figure. The radius to
the centerline is 925 m – 6 m = 919 m.

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