Horizontal alignment I Notes | EduRev

: Horizontal alignment I Notes | EduRev

 Page 1


CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Chapter 14
Horizontal alignment I
14.1 Overview
Horizontal alignment is one of the most important features in
uencing the eciency and safety of a highway. A
poor design will result in lower speeds and resultant reduction in highway performance in terms of safety and
comfort. In addition, it may increase the cost of vehicle operations and lower the highway capacity. Horizontal
alignment design involves the understanding on the design aspects such as design speed and the eect of
horizontal curve on the vehicles. The horizontal curve design elements include design of super elevation, extra
widening at horizontal curves, design of transition curve, and set back distance. These will be discussed in this
chapter and the following two chapters.
14.2 Design Speed
The design speed, as noted earlier, is the single most important factor in the design of horizontal alignment.
The design speed also depends on the type of the road. For e.g, the design speed expected from a National
highway will be much higher than a village road, and hence the curve geometry will vary signicantly.
The design speed also depends on the type of terrain. A plain terrain can aord to have any geometry, but
for the same standard in a hilly terrain requires substantial cutting and lling implying exorbitant costs as well
as safety concern due to unstable slopes. Therefore, the design speed is normally reduced for terrains with steep
slopes.
For instance, Indian Road Congress (IRC) has classied the terrains into four categories, namely plain,
rolling, mountainous, and steep based on the cross slope as given in table 14:1. Based on the type of road and
type of terrain the design speed varies. The IRC has suggested desirable or ruling speed as well as minimum
suggested design speed and is tabulated in table 14:2. The recommended design speed is given in Table 14:2.
Table 14:1: Terrain classication
Terrain classication Cross slope (%)
Plain 0-10
Rolling 10-25
Mountainous 25-60
Steep > 60
Introduction to Transportation Engineering 14.1 Tom V. Mathew and K V Krishna Rao
Page 2


CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Chapter 14
Horizontal alignment I
14.1 Overview
Horizontal alignment is one of the most important features in
uencing the eciency and safety of a highway. A
poor design will result in lower speeds and resultant reduction in highway performance in terms of safety and
comfort. In addition, it may increase the cost of vehicle operations and lower the highway capacity. Horizontal
alignment design involves the understanding on the design aspects such as design speed and the eect of
horizontal curve on the vehicles. The horizontal curve design elements include design of super elevation, extra
widening at horizontal curves, design of transition curve, and set back distance. These will be discussed in this
chapter and the following two chapters.
14.2 Design Speed
The design speed, as noted earlier, is the single most important factor in the design of horizontal alignment.
The design speed also depends on the type of the road. For e.g, the design speed expected from a National
highway will be much higher than a village road, and hence the curve geometry will vary signicantly.
The design speed also depends on the type of terrain. A plain terrain can aord to have any geometry, but
for the same standard in a hilly terrain requires substantial cutting and lling implying exorbitant costs as well
as safety concern due to unstable slopes. Therefore, the design speed is normally reduced for terrains with steep
slopes.
For instance, Indian Road Congress (IRC) has classied the terrains into four categories, namely plain,
rolling, mountainous, and steep based on the cross slope as given in table 14:1. Based on the type of road and
type of terrain the design speed varies. The IRC has suggested desirable or ruling speed as well as minimum
suggested design speed and is tabulated in table 14:2. The recommended design speed is given in Table 14:2.
Table 14:1: Terrain classication
Terrain classication Cross slope (%)
Plain 0-10
Rolling 10-25
Mountainous 25-60
Steep > 60
Introduction to Transportation Engineering 14.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Table 14:2: Design speed in km=hr as per IRC (ruling and minimum)
Type Plain Rolling Hilly Steep
NS&SH 100-80 80-65 50-40 40-30
MDR 80-65 65-50 40-30 30-20
ODR 65-50 50-40 30-25 25-20
VR 50-40 40-35 25-20 25-20
14.3 Horizontal curve
The presence of horizontal curve imparts centrifugal force which is a reactive force acting outward on a vehicle
negotiating it. Centrifugal force depends on speed and radius of the horizontal curve and is counteracted to a
certain extent by transverse friction between the tyre and pavement surface. On a curved road, this force tends
to cause the vehicle to overrun or to slide outward from the centre of road curvature. For proper design of the
curve, an understanding of the forces acting on a vehicle taking a horizontal curve is necessary. Various forces
acting on the vehicle are illustrated in the gure 14:1.
W/2
A B
b/2
P
W/2
CG
W
R b R a
Figure 14:1: Eect of horizontal curve
They are the centrifugal force (P) acting outward, weight of the vehicle (W) acting downward, and the
reaction of the ground on the wheels (R
A
and R
B
). The centrifugal force and the weight is assumed to be from
the centre of gravity which is at h units above the ground. Let the wheel base be assumed as b units. The
centrifugal force P in kg=m
2
is given by
P =
Wv
2
gR
(14.1)
where W is the weight of the vehicle in kg, v is the speed of the vehicle in m=sec, g is the acceleration due to
gravity in m=sec
2
and R is the radius of the curve in m.
The centrifugal ratio or the impact factor
P
W
is given by:
P
W
=
v
2
gR
(14.2)
The centrifugal force has two eects: A tendency to overturn the vehicle about the outer wheels and a tendency
for transverse skidding. Taking moments of the forces with respect to the outer wheel when the vehicle is just
Introduction to Transportation Engineering 14.2 Tom V. Mathew and K V Krishna Rao
Page 3


CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Chapter 14
Horizontal alignment I
14.1 Overview
Horizontal alignment is one of the most important features in
uencing the eciency and safety of a highway. A
poor design will result in lower speeds and resultant reduction in highway performance in terms of safety and
comfort. In addition, it may increase the cost of vehicle operations and lower the highway capacity. Horizontal
alignment design involves the understanding on the design aspects such as design speed and the eect of
horizontal curve on the vehicles. The horizontal curve design elements include design of super elevation, extra
widening at horizontal curves, design of transition curve, and set back distance. These will be discussed in this
chapter and the following two chapters.
14.2 Design Speed
The design speed, as noted earlier, is the single most important factor in the design of horizontal alignment.
The design speed also depends on the type of the road. For e.g, the design speed expected from a National
highway will be much higher than a village road, and hence the curve geometry will vary signicantly.
The design speed also depends on the type of terrain. A plain terrain can aord to have any geometry, but
for the same standard in a hilly terrain requires substantial cutting and lling implying exorbitant costs as well
as safety concern due to unstable slopes. Therefore, the design speed is normally reduced for terrains with steep
slopes.
For instance, Indian Road Congress (IRC) has classied the terrains into four categories, namely plain,
rolling, mountainous, and steep based on the cross slope as given in table 14:1. Based on the type of road and
type of terrain the design speed varies. The IRC has suggested desirable or ruling speed as well as minimum
suggested design speed and is tabulated in table 14:2. The recommended design speed is given in Table 14:2.
Table 14:1: Terrain classication
Terrain classication Cross slope (%)
Plain 0-10
Rolling 10-25
Mountainous 25-60
Steep > 60
Introduction to Transportation Engineering 14.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Table 14:2: Design speed in km=hr as per IRC (ruling and minimum)
Type Plain Rolling Hilly Steep
NS&SH 100-80 80-65 50-40 40-30
MDR 80-65 65-50 40-30 30-20
ODR 65-50 50-40 30-25 25-20
VR 50-40 40-35 25-20 25-20
14.3 Horizontal curve
The presence of horizontal curve imparts centrifugal force which is a reactive force acting outward on a vehicle
negotiating it. Centrifugal force depends on speed and radius of the horizontal curve and is counteracted to a
certain extent by transverse friction between the tyre and pavement surface. On a curved road, this force tends
to cause the vehicle to overrun or to slide outward from the centre of road curvature. For proper design of the
curve, an understanding of the forces acting on a vehicle taking a horizontal curve is necessary. Various forces
acting on the vehicle are illustrated in the gure 14:1.
W/2
A B
b/2
P
W/2
CG
W
R b R a
Figure 14:1: Eect of horizontal curve
They are the centrifugal force (P) acting outward, weight of the vehicle (W) acting downward, and the
reaction of the ground on the wheels (R
A
and R
B
). The centrifugal force and the weight is assumed to be from
the centre of gravity which is at h units above the ground. Let the wheel base be assumed as b units. The
centrifugal force P in kg=m
2
is given by
P =
Wv
2
gR
(14.1)
where W is the weight of the vehicle in kg, v is the speed of the vehicle in m=sec, g is the acceleration due to
gravity in m=sec
2
and R is the radius of the curve in m.
The centrifugal ratio or the impact factor
P
W
is given by:
P
W
=
v
2
gR
(14.2)
The centrifugal force has two eects: A tendency to overturn the vehicle about the outer wheels and a tendency
for transverse skidding. Taking moments of the forces with respect to the outer wheel when the vehicle is just
Introduction to Transportation Engineering 14.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
about to override,
Ph = W
b
2
or
P
W
=
b
2h
At the equilibrium over turning is possible when
v
2
gR
=
b
2h
and for safety the following condition must satisfy:
b
2h
>
v
2
gR
(14.3)
The second tendency of the vehicle is for transverse skidding. i.e. When the the centrifugal force P is greater
than the maximum possible transverse skid resistance due to friction between the pavement surface and tyre.
The transverse skid resistance (F) is given by:
F = F
A
+ F
B
= f(R
A
+ R
B
)
= fW
where F
A
and F
B
is the fractional force at tyre A and B, R
A
and R
B
is the reaction at tyre A and B, f is the
lateral coecient of friction and W is the weight of the vehicle. This is counteracted by the centrifugal force
(P), and equating:
P = fW or
P
W
= f
At equilibrium, when skidding takes place (from equation14.2)
P
W
= f =
v
2
gR
and for safety the following condition must satisfy:
f >
v
2
gR
(14.4)
Equation 14.3 and 14.4 give the stable condition for design. If equation 14.3 is violated, the vehicle will overturn
at the horizontal curve and if equation 14.4 is violated, the vehicle will skid at the horizontal curve
14.4 Analysis of super-elevation
Super-elevation or cant or banking is the transverse slope provided at horizontal curve to counteract the cen-
trifugal force, by raising the outer edge of the pavement with respect to the inner edge, throughout the length of
the horizontal curve. When the outer edge is raised, a component of the curve weight will be complimented in
counteracting the eect of centrifugal force. In order to nd out how much this raising should be, the following
analysis may be done. The forces acting on a vehicle while taking a horizontal curve with superelevation is
shown in gure 14:2.
Forces acting on a vehicle on horizontal curve of radius R m at a speed of v m=sec
2
are:
Introduction to Transportation Engineering 14.3 Tom V. Mathew and K V Krishna Rao
Page 4


CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Chapter 14
Horizontal alignment I
14.1 Overview
Horizontal alignment is one of the most important features in
uencing the eciency and safety of a highway. A
poor design will result in lower speeds and resultant reduction in highway performance in terms of safety and
comfort. In addition, it may increase the cost of vehicle operations and lower the highway capacity. Horizontal
alignment design involves the understanding on the design aspects such as design speed and the eect of
horizontal curve on the vehicles. The horizontal curve design elements include design of super elevation, extra
widening at horizontal curves, design of transition curve, and set back distance. These will be discussed in this
chapter and the following two chapters.
14.2 Design Speed
The design speed, as noted earlier, is the single most important factor in the design of horizontal alignment.
The design speed also depends on the type of the road. For e.g, the design speed expected from a National
highway will be much higher than a village road, and hence the curve geometry will vary signicantly.
The design speed also depends on the type of terrain. A plain terrain can aord to have any geometry, but
for the same standard in a hilly terrain requires substantial cutting and lling implying exorbitant costs as well
as safety concern due to unstable slopes. Therefore, the design speed is normally reduced for terrains with steep
slopes.
For instance, Indian Road Congress (IRC) has classied the terrains into four categories, namely plain,
rolling, mountainous, and steep based on the cross slope as given in table 14:1. Based on the type of road and
type of terrain the design speed varies. The IRC has suggested desirable or ruling speed as well as minimum
suggested design speed and is tabulated in table 14:2. The recommended design speed is given in Table 14:2.
Table 14:1: Terrain classication
Terrain classication Cross slope (%)
Plain 0-10
Rolling 10-25
Mountainous 25-60
Steep > 60
Introduction to Transportation Engineering 14.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Table 14:2: Design speed in km=hr as per IRC (ruling and minimum)
Type Plain Rolling Hilly Steep
NS&SH 100-80 80-65 50-40 40-30
MDR 80-65 65-50 40-30 30-20
ODR 65-50 50-40 30-25 25-20
VR 50-40 40-35 25-20 25-20
14.3 Horizontal curve
The presence of horizontal curve imparts centrifugal force which is a reactive force acting outward on a vehicle
negotiating it. Centrifugal force depends on speed and radius of the horizontal curve and is counteracted to a
certain extent by transverse friction between the tyre and pavement surface. On a curved road, this force tends
to cause the vehicle to overrun or to slide outward from the centre of road curvature. For proper design of the
curve, an understanding of the forces acting on a vehicle taking a horizontal curve is necessary. Various forces
acting on the vehicle are illustrated in the gure 14:1.
W/2
A B
b/2
P
W/2
CG
W
R b R a
Figure 14:1: Eect of horizontal curve
They are the centrifugal force (P) acting outward, weight of the vehicle (W) acting downward, and the
reaction of the ground on the wheels (R
A
and R
B
). The centrifugal force and the weight is assumed to be from
the centre of gravity which is at h units above the ground. Let the wheel base be assumed as b units. The
centrifugal force P in kg=m
2
is given by
P =
Wv
2
gR
(14.1)
where W is the weight of the vehicle in kg, v is the speed of the vehicle in m=sec, g is the acceleration due to
gravity in m=sec
2
and R is the radius of the curve in m.
The centrifugal ratio or the impact factor
P
W
is given by:
P
W
=
v
2
gR
(14.2)
The centrifugal force has two eects: A tendency to overturn the vehicle about the outer wheels and a tendency
for transverse skidding. Taking moments of the forces with respect to the outer wheel when the vehicle is just
Introduction to Transportation Engineering 14.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
about to override,
Ph = W
b
2
or
P
W
=
b
2h
At the equilibrium over turning is possible when
v
2
gR
=
b
2h
and for safety the following condition must satisfy:
b
2h
>
v
2
gR
(14.3)
The second tendency of the vehicle is for transverse skidding. i.e. When the the centrifugal force P is greater
than the maximum possible transverse skid resistance due to friction between the pavement surface and tyre.
The transverse skid resistance (F) is given by:
F = F
A
+ F
B
= f(R
A
+ R
B
)
= fW
where F
A
and F
B
is the fractional force at tyre A and B, R
A
and R
B
is the reaction at tyre A and B, f is the
lateral coecient of friction and W is the weight of the vehicle. This is counteracted by the centrifugal force
(P), and equating:
P = fW or
P
W
= f
At equilibrium, when skidding takes place (from equation14.2)
P
W
= f =
v
2
gR
and for safety the following condition must satisfy:
f >
v
2
gR
(14.4)
Equation 14.3 and 14.4 give the stable condition for design. If equation 14.3 is violated, the vehicle will overturn
at the horizontal curve and if equation 14.4 is violated, the vehicle will skid at the horizontal curve
14.4 Analysis of super-elevation
Super-elevation or cant or banking is the transverse slope provided at horizontal curve to counteract the cen-
trifugal force, by raising the outer edge of the pavement with respect to the inner edge, throughout the length of
the horizontal curve. When the outer edge is raised, a component of the curve weight will be complimented in
counteracting the eect of centrifugal force. In order to nd out how much this raising should be, the following
analysis may be done. The forces acting on a vehicle while taking a horizontal curve with superelevation is
shown in gure 14:2.
Forces acting on a vehicle on horizontal curve of radius R m at a speed of v m=sec
2
are:
Introduction to Transportation Engineering 14.3 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
 W sin
F
b
W cos 
P sin 
P cos
P
W
F
A
Figure 14:2: Analysis of super-elevation
 P the centrifugal force acting horizontally out-wards through the center of gravity,
 W the weight of the vehicle acting down-wards through the center of gravity, and
 F the friction force between the wheels and the pavement, along the surface inward.
At equilibrium, by resolving the forces parallel to the surface of the pavement we get,
P cos = W sin  + F
A
+ F
B
= W sin  + f(R
A
+ R
B
)
= W sin  + f(W cos + P sin )
where W is the weight of the vehicle, P is the centrifugal force, f is the coecient of friction,  is the transverse
slope due to superelevation. Dividing by W cos, we get:
P cos
W cos
=
W sin
W cos
+
fW cos
W cos
+
fPsin
W cos
P
W
= tan  + f + f
P
W
tan 
P
W
(1 f tan ) = tan  + f
P
W
=
tan  + f
1 f tan
(14.5)
We have already derived an expression for P/W.By substituting this in equation 14.5, we get:
v
2
gR
=
tan  + f
1 f tan 
(14.6)
Introduction to Transportation Engineering 14.4 Tom V. Mathew and K V Krishna Rao
Page 5


CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Chapter 14
Horizontal alignment I
14.1 Overview
Horizontal alignment is one of the most important features in
uencing the eciency and safety of a highway. A
poor design will result in lower speeds and resultant reduction in highway performance in terms of safety and
comfort. In addition, it may increase the cost of vehicle operations and lower the highway capacity. Horizontal
alignment design involves the understanding on the design aspects such as design speed and the eect of
horizontal curve on the vehicles. The horizontal curve design elements include design of super elevation, extra
widening at horizontal curves, design of transition curve, and set back distance. These will be discussed in this
chapter and the following two chapters.
14.2 Design Speed
The design speed, as noted earlier, is the single most important factor in the design of horizontal alignment.
The design speed also depends on the type of the road. For e.g, the design speed expected from a National
highway will be much higher than a village road, and hence the curve geometry will vary signicantly.
The design speed also depends on the type of terrain. A plain terrain can aord to have any geometry, but
for the same standard in a hilly terrain requires substantial cutting and lling implying exorbitant costs as well
as safety concern due to unstable slopes. Therefore, the design speed is normally reduced for terrains with steep
slopes.
For instance, Indian Road Congress (IRC) has classied the terrains into four categories, namely plain,
rolling, mountainous, and steep based on the cross slope as given in table 14:1. Based on the type of road and
type of terrain the design speed varies. The IRC has suggested desirable or ruling speed as well as minimum
suggested design speed and is tabulated in table 14:2. The recommended design speed is given in Table 14:2.
Table 14:1: Terrain classication
Terrain classication Cross slope (%)
Plain 0-10
Rolling 10-25
Mountainous 25-60
Steep > 60
Introduction to Transportation Engineering 14.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
Table 14:2: Design speed in km=hr as per IRC (ruling and minimum)
Type Plain Rolling Hilly Steep
NS&SH 100-80 80-65 50-40 40-30
MDR 80-65 65-50 40-30 30-20
ODR 65-50 50-40 30-25 25-20
VR 50-40 40-35 25-20 25-20
14.3 Horizontal curve
The presence of horizontal curve imparts centrifugal force which is a reactive force acting outward on a vehicle
negotiating it. Centrifugal force depends on speed and radius of the horizontal curve and is counteracted to a
certain extent by transverse friction between the tyre and pavement surface. On a curved road, this force tends
to cause the vehicle to overrun or to slide outward from the centre of road curvature. For proper design of the
curve, an understanding of the forces acting on a vehicle taking a horizontal curve is necessary. Various forces
acting on the vehicle are illustrated in the gure 14:1.
W/2
A B
b/2
P
W/2
CG
W
R b R a
Figure 14:1: Eect of horizontal curve
They are the centrifugal force (P) acting outward, weight of the vehicle (W) acting downward, and the
reaction of the ground on the wheels (R
A
and R
B
). The centrifugal force and the weight is assumed to be from
the centre of gravity which is at h units above the ground. Let the wheel base be assumed as b units. The
centrifugal force P in kg=m
2
is given by
P =
Wv
2
gR
(14.1)
where W is the weight of the vehicle in kg, v is the speed of the vehicle in m=sec, g is the acceleration due to
gravity in m=sec
2
and R is the radius of the curve in m.
The centrifugal ratio or the impact factor
P
W
is given by:
P
W
=
v
2
gR
(14.2)
The centrifugal force has two eects: A tendency to overturn the vehicle about the outer wheels and a tendency
for transverse skidding. Taking moments of the forces with respect to the outer wheel when the vehicle is just
Introduction to Transportation Engineering 14.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
about to override,
Ph = W
b
2
or
P
W
=
b
2h
At the equilibrium over turning is possible when
v
2
gR
=
b
2h
and for safety the following condition must satisfy:
b
2h
>
v
2
gR
(14.3)
The second tendency of the vehicle is for transverse skidding. i.e. When the the centrifugal force P is greater
than the maximum possible transverse skid resistance due to friction between the pavement surface and tyre.
The transverse skid resistance (F) is given by:
F = F
A
+ F
B
= f(R
A
+ R
B
)
= fW
where F
A
and F
B
is the fractional force at tyre A and B, R
A
and R
B
is the reaction at tyre A and B, f is the
lateral coecient of friction and W is the weight of the vehicle. This is counteracted by the centrifugal force
(P), and equating:
P = fW or
P
W
= f
At equilibrium, when skidding takes place (from equation14.2)
P
W
= f =
v
2
gR
and for safety the following condition must satisfy:
f >
v
2
gR
(14.4)
Equation 14.3 and 14.4 give the stable condition for design. If equation 14.3 is violated, the vehicle will overturn
at the horizontal curve and if equation 14.4 is violated, the vehicle will skid at the horizontal curve
14.4 Analysis of super-elevation
Super-elevation or cant or banking is the transverse slope provided at horizontal curve to counteract the cen-
trifugal force, by raising the outer edge of the pavement with respect to the inner edge, throughout the length of
the horizontal curve. When the outer edge is raised, a component of the curve weight will be complimented in
counteracting the eect of centrifugal force. In order to nd out how much this raising should be, the following
analysis may be done. The forces acting on a vehicle while taking a horizontal curve with superelevation is
shown in gure 14:2.
Forces acting on a vehicle on horizontal curve of radius R m at a speed of v m=sec
2
are:
Introduction to Transportation Engineering 14.3 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
 W sin
F
b
W cos 
P sin 
P cos
P
W
F
A
Figure 14:2: Analysis of super-elevation
 P the centrifugal force acting horizontally out-wards through the center of gravity,
 W the weight of the vehicle acting down-wards through the center of gravity, and
 F the friction force between the wheels and the pavement, along the surface inward.
At equilibrium, by resolving the forces parallel to the surface of the pavement we get,
P cos = W sin  + F
A
+ F
B
= W sin  + f(R
A
+ R
B
)
= W sin  + f(W cos + P sin )
where W is the weight of the vehicle, P is the centrifugal force, f is the coecient of friction,  is the transverse
slope due to superelevation. Dividing by W cos, we get:
P cos
W cos
=
W sin
W cos
+
fW cos
W cos
+
fPsin
W cos
P
W
= tan  + f + f
P
W
tan 
P
W
(1 f tan ) = tan  + f
P
W
=
tan  + f
1 f tan
(14.5)
We have already derived an expression for P/W.By substituting this in equation 14.5, we get:
v
2
gR
=
tan  + f
1 f tan 
(14.6)
Introduction to Transportation Engineering 14.4 Tom V. Mathew and K V Krishna Rao
CHAPTER 14. HORIZONTAL ALIGNMENT I NPTEL May 7, 2007
This is an exact expression for superelevation. But normally, f = 0:15 and  < 4
o
, 1 f tan  1 and for small
, tan  sin = E=B = e, then equation 14.6 becomes:
e + f =
v
2
gR
(14.7)
where, e is the rate of super elevation, f the coecient of lateral friction 0:15, v the speed of the vehicle in
m=sec
2
, R the radius of the curve in m and g = 9:8m=sec
2
.
Three specic cases that can arise from equation 14.7 are as follows:
1 If there is no friction due to some practical reasons, then f = 0 and equation 14.7 becomes e =
v
2
gR
. This
results in the situation where the pressure on the outer and inner wheels are same; requiring very high
super-elevation e.
2 If there is no super-elevation provided due to some practical reasons, then e = 0 and equation 14.7 becomes
f =
v
2
gR
. This results in a very high coecient of friction.
3 If e = 0 and f = 0:15 then for safe traveling speed from equation 14.7 is given by v
b
=
p
fgR where v
b
is
the restricted speed.
14.5 Summary
Design speed plays a major role in designing the elements of horizontal alignment. The most important element is
superelevation which is in
uenced by speed, radius of curve and frictional resistance of pavement. Superelevation
is necessary to balance centrifugal force. The design part is dealt in the next chapter.
14.6 Problems
1. The design speed recommended by IRC for National highways passign through rolling terrain is in the
range of
(a) 100-80
(b) 80-65
(c) 120-100
(d) 50-40
2. For safety against skidding, the condition to be satised is
(a) f>
v
2
gR
(b) f<
v
2
gR
(c) f>
v
gR
(d) f=
v
2
gR
Introduction to Transportation Engineering 14.5 Tom V. Mathew and K V Krishna Rao
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