Short Notes for Setback Distance and Curve Resistance | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Chapter 1 
¦
 Transportation Engineering | 3.925
Setback Distance and Curve Resistance
Introduction
Absolute minimum sight distance that is SSD should be 
available at every stretch of the highway and even on the 
horizontal curve sufficient clearance on the inner side of 
the curve should be provided which is called as setback 
distance.
Setback Distance (m) or Clearance
It is the distance required from the centre line of horizontal 
curve to an obstruction on the inner side of the curve to pro-
vide adequate sight distance on horizontal curve.
Factors Affecting Setback Distance
 1. Required sight distance (SSD, OSD, ISD)
 2. Radius of Horizontal curve (R)
 3. Length of the curve (L
c
)
Approximate formula: (From the concept of chords of 
circle)
Case 1: When L
c
 > S
m
S
R
=
2
8
Case 2: When L
c
 < S
m
LS L
R
=
- () 2
8
SD
F
m
D
R
O
R
C
A
a
2
Obstruction
Rational formulae: (As per IRC)
Case 1: L
c
 > S
From above figure CF = OF - OC
mR R =- cos
a
2
S
R
a = radians and 
a
2
180
2
=
?
S
R
 degrees.
 • For multi-lane roads, sight distance is measured along the 
middle of the inner side lane.
If ‘d’ is the distance between centre of multilane road and 
centre of the inner lane,
MR Rd
S
Rd
=- -
=
?-
()cos
()
a
a
2
2
180
2
degrees
R
d
S
C
m
F
Centre line
of inner
lane
Centre line
of road
Line of Sight
a
2
Case 2: If L
c
 < S
A
B
S
D
a
2
a
2
L
C
G
F
R R
O
mOCOGFG
mR R
SL
L
R
=- +
=- +
-
=
?
cos
()
sin
aa
a
22 2
2
180
2
degrees
 • For multi-lane roads
Part III_Unit 11_Chapter 01.indd   925 5/20/2017   7:22:20 PM
Page 2


Chapter 1 
¦
 Transportation Engineering | 3.925
Setback Distance and Curve Resistance
Introduction
Absolute minimum sight distance that is SSD should be 
available at every stretch of the highway and even on the 
horizontal curve sufficient clearance on the inner side of 
the curve should be provided which is called as setback 
distance.
Setback Distance (m) or Clearance
It is the distance required from the centre line of horizontal 
curve to an obstruction on the inner side of the curve to pro-
vide adequate sight distance on horizontal curve.
Factors Affecting Setback Distance
 1. Required sight distance (SSD, OSD, ISD)
 2. Radius of Horizontal curve (R)
 3. Length of the curve (L
c
)
Approximate formula: (From the concept of chords of 
circle)
Case 1: When L
c
 > S
m
S
R
=
2
8
Case 2: When L
c
 < S
m
LS L
R
=
- () 2
8
SD
F
m
D
R
O
R
C
A
a
2
Obstruction
Rational formulae: (As per IRC)
Case 1: L
c
 > S
From above figure CF = OF - OC
mR R =- cos
a
2
S
R
a = radians and 
a
2
180
2
=
?
S
R
 degrees.
 • For multi-lane roads, sight distance is measured along the 
middle of the inner side lane.
If ‘d’ is the distance between centre of multilane road and 
centre of the inner lane,
MR Rd
S
Rd
=- -
=
?-
()cos
()
a
a
2
2
180
2
degrees
R
d
S
C
m
F
Centre line
of inner
lane
Centre line
of road
Line of Sight
a
2
Case 2: If L
c
 < S
A
B
S
D
a
2
a
2
L
C
G
F
R R
O
mOCOGFG
mR R
SL
L
R
=- +
=- +
-
=
?
cos
()
sin
aa
a
22 2
2
180
2
degrees
 • For multi-lane roads
Part III_Unit 11_Chapter 01.indd   925 5/20/2017   7:22:20 PM
3.926 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
mR RD
SL
L
d
=- -+
-
=
?-
()cos
()
sin
(R )
aa
a
22 2
2
180
2
degrees.
Example 7
The following data are related to a horizontal curved por-
tion of a two lane highway: length of curve = 200 m, radius 
of curve = 300 m and width of pavement = 7.5 m. In order 
to provide a stopping sight distance (SSD) of 80 m, the set-
back distance (in m) required from the centre line of the 
inner lane of the pavement is  [GATE, 2012]
(A) 2.67 m (B) 4.55 m
(C) 7.10 m (D) 7.96 m
Solution
d = =
75
4
1875
.
.m (centre of road to centre of inner lane 
B
4
)
L
c
 = 200 m > S = 80 m
?m = R - (R - d) cos 
2
a
S
Rd
180
()
180
(300 1.875)
15.38 a =
?-
=
?-
=°
m = 300 - (300 - 1.875)cos
. 15 38
2
?
?
?
?
?
?
 = 4.56 m
*But setback distance asked from centre line of inner lane 
of pavement = m - d = 4.56 - 1.875 = 2.68 m
Hence, the correct answer is option (A).
Highway Geometric Design—Transition 
Curves
Introduction
If a curve of radius R takes off from straight road, centrifu-
gal force suddenly acts on the vehicle just after the tangent 
point and a sudden jerk is felt on the vehicle. To avoid this 
a curve having a varying radius which decreases from infin-
ity at the tangent point to a designed radius of the circular 
curve.
Objectives of Providing Transition Curves
 1. To introduce gradually the centrifugal force between 
the tangent point and the beginning of circular curve 
for avoiding sudden jerk on the vehicle.
 2. To enable the driver to turn the steering gradually for 
his comfort and safety.
 3. To help in gradual introduction of super-elevation 
and extra width of pavement on circular curve.
 4. To improve aesthetic appearance of the road.
De?ection angle, (D)
Circular
curve
Transition curve (L
s
)
Straight
Straight
Shift
Shift
Transition
curve
Tangent
distance
O
D C
R
B
T.P
Transition curve in horizontal alignment
 • The rate at which the centrifugal force is introduced can 
be controlled by adopting suitable shape and designing 
its length.
Types of Transition Curves
 1. Spiral
 2. Lemniscate
 3. Cubic Parabola
MAJOR AXIS
Spiral
Lemniscate
Cubic parabola
45°
Ideal transition curve: The rate of change of centrifugal 
acceleration is uniform thought the curve.
Upto deflection angle of 4°, all three curves follow same 
path and practically all curves are same upto 9°.
Spiral (Clothoid/Glovers Spiral)
 • Ideal transition curve
 • Length of transition curve inversely proportional to radius 
R of circular curve. i.e., L
R
LR
ss
?? =
1
constant
 • IRC recommends spiral because
(a) It satisfies the properties of ideal transition curve.
(b) Geometric property is such that the calculations and 
setting out of curve is simple and easy.
Part III_Unit 11_Chapter 01.indd   926 5/20/2017   7:22:21 PM
Page 3


Chapter 1 
¦
 Transportation Engineering | 3.925
Setback Distance and Curve Resistance
Introduction
Absolute minimum sight distance that is SSD should be 
available at every stretch of the highway and even on the 
horizontal curve sufficient clearance on the inner side of 
the curve should be provided which is called as setback 
distance.
Setback Distance (m) or Clearance
It is the distance required from the centre line of horizontal 
curve to an obstruction on the inner side of the curve to pro-
vide adequate sight distance on horizontal curve.
Factors Affecting Setback Distance
 1. Required sight distance (SSD, OSD, ISD)
 2. Radius of Horizontal curve (R)
 3. Length of the curve (L
c
)
Approximate formula: (From the concept of chords of 
circle)
Case 1: When L
c
 > S
m
S
R
=
2
8
Case 2: When L
c
 < S
m
LS L
R
=
- () 2
8
SD
F
m
D
R
O
R
C
A
a
2
Obstruction
Rational formulae: (As per IRC)
Case 1: L
c
 > S
From above figure CF = OF - OC
mR R =- cos
a
2
S
R
a = radians and 
a
2
180
2
=
?
S
R
 degrees.
 • For multi-lane roads, sight distance is measured along the 
middle of the inner side lane.
If ‘d’ is the distance between centre of multilane road and 
centre of the inner lane,
MR Rd
S
Rd
=- -
=
?-
()cos
()
a
a
2
2
180
2
degrees
R
d
S
C
m
F
Centre line
of inner
lane
Centre line
of road
Line of Sight
a
2
Case 2: If L
c
 < S
A
B
S
D
a
2
a
2
L
C
G
F
R R
O
mOCOGFG
mR R
SL
L
R
=- +
=- +
-
=
?
cos
()
sin
aa
a
22 2
2
180
2
degrees
 • For multi-lane roads
Part III_Unit 11_Chapter 01.indd   925 5/20/2017   7:22:20 PM
3.926 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
mR RD
SL
L
d
=- -+
-
=
?-
()cos
()
sin
(R )
aa
a
22 2
2
180
2
degrees.
Example 7
The following data are related to a horizontal curved por-
tion of a two lane highway: length of curve = 200 m, radius 
of curve = 300 m and width of pavement = 7.5 m. In order 
to provide a stopping sight distance (SSD) of 80 m, the set-
back distance (in m) required from the centre line of the 
inner lane of the pavement is  [GATE, 2012]
(A) 2.67 m (B) 4.55 m
(C) 7.10 m (D) 7.96 m
Solution
d = =
75
4
1875
.
.m (centre of road to centre of inner lane 
B
4
)
L
c
 = 200 m > S = 80 m
?m = R - (R - d) cos 
2
a
S
Rd
180
()
180
(300 1.875)
15.38 a =
?-
=
?-
=°
m = 300 - (300 - 1.875)cos
. 15 38
2
?
?
?
?
?
?
 = 4.56 m
*But setback distance asked from centre line of inner lane 
of pavement = m - d = 4.56 - 1.875 = 2.68 m
Hence, the correct answer is option (A).
Highway Geometric Design—Transition 
Curves
Introduction
If a curve of radius R takes off from straight road, centrifu-
gal force suddenly acts on the vehicle just after the tangent 
point and a sudden jerk is felt on the vehicle. To avoid this 
a curve having a varying radius which decreases from infin-
ity at the tangent point to a designed radius of the circular 
curve.
Objectives of Providing Transition Curves
 1. To introduce gradually the centrifugal force between 
the tangent point and the beginning of circular curve 
for avoiding sudden jerk on the vehicle.
 2. To enable the driver to turn the steering gradually for 
his comfort and safety.
 3. To help in gradual introduction of super-elevation 
and extra width of pavement on circular curve.
 4. To improve aesthetic appearance of the road.
De?ection angle, (D)
Circular
curve
Transition curve (L
s
)
Straight
Straight
Shift
Shift
Transition
curve
Tangent
distance
O
D C
R
B
T.P
Transition curve in horizontal alignment
 • The rate at which the centrifugal force is introduced can 
be controlled by adopting suitable shape and designing 
its length.
Types of Transition Curves
 1. Spiral
 2. Lemniscate
 3. Cubic Parabola
MAJOR AXIS
Spiral
Lemniscate
Cubic parabola
45°
Ideal transition curve: The rate of change of centrifugal 
acceleration is uniform thought the curve.
Upto deflection angle of 4°, all three curves follow same 
path and practically all curves are same upto 9°.
Spiral (Clothoid/Glovers Spiral)
 • Ideal transition curve
 • Length of transition curve inversely proportional to radius 
R of circular curve. i.e., L
R
LR
ss
?? =
1
constant
 • IRC recommends spiral because
(a) It satisfies the properties of ideal transition curve.
(b) Geometric property is such that the calculations and 
setting out of curve is simple and easy.
Part III_Unit 11_Chapter 01.indd   926 5/20/2017   7:22:21 PM
Chapter 1 
¦
 Transportation Engineering | 3.927
 • Equation of spiral:
LR = L
s
 R
c
 = constant
Lm ? =
m = constant = 2RL
s
Where, ? = tangent deflection angle.
Bernoullie’s Lemniscate
 • Radius of curve decreases more rapidly with increase in 
length.
 • Mostly used in roads where deflection angle is large.
 • It is an autogenous curve (as it follows a path which is 
actually traced by a vehicle when turning freely).
 • The curve is set by polar coordinates.
Cubic Parabola (Froude’s Transition/Easement Curve)
 • It is set by simple cartesian coordinates.
 • It is used for valley curves (as radius reduces very fast 
with length)
 • Equation: y
l
Rl
S
=
3
6
Length of Transition Curve
 1. By rate of change of centrifugal acceleration:
 • Vehicle travels length L
s
 of transition curve with 
uniform speed ‘v’ m/s in time ‘t’
?= t
L
v
s
 • Maximum centrifugal acceleration is attained in 
time ‘t’.
Rate of change of centrifugal acceleration,
C
v
Rt
v
LR
s
= =
23
(m ) /s
3
IRC recommended value of C:
C
v
=
+
80
75
 m/s
3 
(V in km/h) and
0.5 < C < 0.8 m/s
3
 2. By rate of introduction of super-elevation:
 • e is rate of super-elevation.
 • W is width of pavement.
 • W
e
 is extra widening provided.
 • B is width of pavement.
 • E is total rise (with respect to inner/centre of road)
E = e 
.
 B = e (W + W
e
)
 • Allowing a rate of change of super-elevation of 1 
in N
L
s
 = EN = eN(W + W
e
)(E with inner edge)
L
EN eN
s
= =
22
 (W + W
e
)(E with respect to centre 
of road)
As per IRC:
N > / 
150–plain androllingterrains
100– built-up areas
60–hillroads
?
?
?
?
?
 3. By empirical formula:
 • For plain and rolling terrains:
L
V
R
V
s
=
27
2
.
in km/h
 • For mountainous and steep terrain:
L
V
R
V
s
= in km/h
(a)  The highest length of the curve obtained by all 
three methods is taken as length of transition 
curve.
(b)  For expressways, with minimum radius of hor-
izontal curve no transition curves are required.
For Design speeds 
V
R
R
120 km/h = 4000 m
100 km/h = 3000 m
min
min
=
?
?
?
?
?
?
?
Shift
The distance between the transition curve at middle and the 
original circular curve is called shift.
Shift,S
L
R
s
=
2
24
Example 8
At a horizontal curve portion of a 4 lane undivided carriage-
way, a transition curve is to be introduced to attain required 
super-elevation. The design speed is 60 km/h and radius of 
the curve is 245 m. Assume length of the wheel base of 
a longest vehicle as 6 m. Super-elevation rate as 5% and 
rate of introduction of this super-elevation as 1 in 150. The 
length of the transition curve (m) required, if the pavement 
is rotated about inner edge is  [GATE, 2006]
(A) 81.4 (B) 85.0
(C) 91.5 (D) 110.2
Solution
Width of road W = 3.5 × 4 = 14 m
Extra widening W
nl
R
V
R
e
=+
2
2
95 .
=
×
×
+
46
2 245
60
95 245
2
.
W
e
 = 0.697 m
Part III_Unit 11_Chapter 01.indd   927 5/20/2017   7:22:22 PM
Page 4


Chapter 1 
¦
 Transportation Engineering | 3.925
Setback Distance and Curve Resistance
Introduction
Absolute minimum sight distance that is SSD should be 
available at every stretch of the highway and even on the 
horizontal curve sufficient clearance on the inner side of 
the curve should be provided which is called as setback 
distance.
Setback Distance (m) or Clearance
It is the distance required from the centre line of horizontal 
curve to an obstruction on the inner side of the curve to pro-
vide adequate sight distance on horizontal curve.
Factors Affecting Setback Distance
 1. Required sight distance (SSD, OSD, ISD)
 2. Radius of Horizontal curve (R)
 3. Length of the curve (L
c
)
Approximate formula: (From the concept of chords of 
circle)
Case 1: When L
c
 > S
m
S
R
=
2
8
Case 2: When L
c
 < S
m
LS L
R
=
- () 2
8
SD
F
m
D
R
O
R
C
A
a
2
Obstruction
Rational formulae: (As per IRC)
Case 1: L
c
 > S
From above figure CF = OF - OC
mR R =- cos
a
2
S
R
a = radians and 
a
2
180
2
=
?
S
R
 degrees.
 • For multi-lane roads, sight distance is measured along the 
middle of the inner side lane.
If ‘d’ is the distance between centre of multilane road and 
centre of the inner lane,
MR Rd
S
Rd
=- -
=
?-
()cos
()
a
a
2
2
180
2
degrees
R
d
S
C
m
F
Centre line
of inner
lane
Centre line
of road
Line of Sight
a
2
Case 2: If L
c
 < S
A
B
S
D
a
2
a
2
L
C
G
F
R R
O
mOCOGFG
mR R
SL
L
R
=- +
=- +
-
=
?
cos
()
sin
aa
a
22 2
2
180
2
degrees
 • For multi-lane roads
Part III_Unit 11_Chapter 01.indd   925 5/20/2017   7:22:20 PM
3.926 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
mR RD
SL
L
d
=- -+
-
=
?-
()cos
()
sin
(R )
aa
a
22 2
2
180
2
degrees.
Example 7
The following data are related to a horizontal curved por-
tion of a two lane highway: length of curve = 200 m, radius 
of curve = 300 m and width of pavement = 7.5 m. In order 
to provide a stopping sight distance (SSD) of 80 m, the set-
back distance (in m) required from the centre line of the 
inner lane of the pavement is  [GATE, 2012]
(A) 2.67 m (B) 4.55 m
(C) 7.10 m (D) 7.96 m
Solution
d = =
75
4
1875
.
.m (centre of road to centre of inner lane 
B
4
)
L
c
 = 200 m > S = 80 m
?m = R - (R - d) cos 
2
a
S
Rd
180
()
180
(300 1.875)
15.38 a =
?-
=
?-
=°
m = 300 - (300 - 1.875)cos
. 15 38
2
?
?
?
?
?
?
 = 4.56 m
*But setback distance asked from centre line of inner lane 
of pavement = m - d = 4.56 - 1.875 = 2.68 m
Hence, the correct answer is option (A).
Highway Geometric Design—Transition 
Curves
Introduction
If a curve of radius R takes off from straight road, centrifu-
gal force suddenly acts on the vehicle just after the tangent 
point and a sudden jerk is felt on the vehicle. To avoid this 
a curve having a varying radius which decreases from infin-
ity at the tangent point to a designed radius of the circular 
curve.
Objectives of Providing Transition Curves
 1. To introduce gradually the centrifugal force between 
the tangent point and the beginning of circular curve 
for avoiding sudden jerk on the vehicle.
 2. To enable the driver to turn the steering gradually for 
his comfort and safety.
 3. To help in gradual introduction of super-elevation 
and extra width of pavement on circular curve.
 4. To improve aesthetic appearance of the road.
De?ection angle, (D)
Circular
curve
Transition curve (L
s
)
Straight
Straight
Shift
Shift
Transition
curve
Tangent
distance
O
D C
R
B
T.P
Transition curve in horizontal alignment
 • The rate at which the centrifugal force is introduced can 
be controlled by adopting suitable shape and designing 
its length.
Types of Transition Curves
 1. Spiral
 2. Lemniscate
 3. Cubic Parabola
MAJOR AXIS
Spiral
Lemniscate
Cubic parabola
45°
Ideal transition curve: The rate of change of centrifugal 
acceleration is uniform thought the curve.
Upto deflection angle of 4°, all three curves follow same 
path and practically all curves are same upto 9°.
Spiral (Clothoid/Glovers Spiral)
 • Ideal transition curve
 • Length of transition curve inversely proportional to radius 
R of circular curve. i.e., L
R
LR
ss
?? =
1
constant
 • IRC recommends spiral because
(a) It satisfies the properties of ideal transition curve.
(b) Geometric property is such that the calculations and 
setting out of curve is simple and easy.
Part III_Unit 11_Chapter 01.indd   926 5/20/2017   7:22:21 PM
Chapter 1 
¦
 Transportation Engineering | 3.927
 • Equation of spiral:
LR = L
s
 R
c
 = constant
Lm ? =
m = constant = 2RL
s
Where, ? = tangent deflection angle.
Bernoullie’s Lemniscate
 • Radius of curve decreases more rapidly with increase in 
length.
 • Mostly used in roads where deflection angle is large.
 • It is an autogenous curve (as it follows a path which is 
actually traced by a vehicle when turning freely).
 • The curve is set by polar coordinates.
Cubic Parabola (Froude’s Transition/Easement Curve)
 • It is set by simple cartesian coordinates.
 • It is used for valley curves (as radius reduces very fast 
with length)
 • Equation: y
l
Rl
S
=
3
6
Length of Transition Curve
 1. By rate of change of centrifugal acceleration:
 • Vehicle travels length L
s
 of transition curve with 
uniform speed ‘v’ m/s in time ‘t’
?= t
L
v
s
 • Maximum centrifugal acceleration is attained in 
time ‘t’.
Rate of change of centrifugal acceleration,
C
v
Rt
v
LR
s
= =
23
(m ) /s
3
IRC recommended value of C:
C
v
=
+
80
75
 m/s
3 
(V in km/h) and
0.5 < C < 0.8 m/s
3
 2. By rate of introduction of super-elevation:
 • e is rate of super-elevation.
 • W is width of pavement.
 • W
e
 is extra widening provided.
 • B is width of pavement.
 • E is total rise (with respect to inner/centre of road)
E = e 
.
 B = e (W + W
e
)
 • Allowing a rate of change of super-elevation of 1 
in N
L
s
 = EN = eN(W + W
e
)(E with inner edge)
L
EN eN
s
= =
22
 (W + W
e
)(E with respect to centre 
of road)
As per IRC:
N > / 
150–plain androllingterrains
100– built-up areas
60–hillroads
?
?
?
?
?
 3. By empirical formula:
 • For plain and rolling terrains:
L
V
R
V
s
=
27
2
.
in km/h
 • For mountainous and steep terrain:
L
V
R
V
s
= in km/h
(a)  The highest length of the curve obtained by all 
three methods is taken as length of transition 
curve.
(b)  For expressways, with minimum radius of hor-
izontal curve no transition curves are required.
For Design speeds 
V
R
R
120 km/h = 4000 m
100 km/h = 3000 m
min
min
=
?
?
?
?
?
?
?
Shift
The distance between the transition curve at middle and the 
original circular curve is called shift.
Shift,S
L
R
s
=
2
24
Example 8
At a horizontal curve portion of a 4 lane undivided carriage-
way, a transition curve is to be introduced to attain required 
super-elevation. The design speed is 60 km/h and radius of 
the curve is 245 m. Assume length of the wheel base of 
a longest vehicle as 6 m. Super-elevation rate as 5% and 
rate of introduction of this super-elevation as 1 in 150. The 
length of the transition curve (m) required, if the pavement 
is rotated about inner edge is  [GATE, 2006]
(A) 81.4 (B) 85.0
(C) 91.5 (D) 110.2
Solution
Width of road W = 3.5 × 4 = 14 m
Extra widening W
nl
R
V
R
e
=+
2
2
95 .
=
×
×
+
46
2 245
60
95 245
2
.
W
e
 = 0.697 m
Part III_Unit 11_Chapter 01.indd   927 5/20/2017   7:22:22 PM
3.928 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
If pavement is rotated about inner edge, Length of transition 
curve 
m = e N (W + W
e
)
=
?
?
?
?
?
?
+
5
100
150 14 0 697 ()(. )
m = 110.22 m
Hence, the correct answer is option (D).
Highway Geometric Design—Summit 
Curves
Introduction
There will be variation in grade in vertical alignment of 
highway, which cause discomfort to passengers if roads are 
laid according to that grade. To smoothen out the vertical 
profile and thus reduce the variation in grades for comfort 
of passengers. Vertical curves are designed for sight dis-
tance and comfort of passengers.
Vertical Curves (Valley Curve)
A valley curve is a curve along the longitudinal profile of 
the road provided to smoothen out the vertical profile.
 1. Summit curves/crest curves with convexity upwards
 2. Valley/sag curves with concavity upwards.
Summit Curves
 • When a vehicle moves on summit curve, centrifugal force 
acts upwards against gravity and hence reduces pressure 
on tyres. Therefore no problem of discomfort.
 • Design of summit curves are governed by absolute mini-
mum sight distance as on all highways and no transition 
curves (for comfort condition) are required.
 • Simple parabola is used as summit curve as it gives good 
riding comfort, simple calculation and uniform rate of 
change of grade throughout the parabola. 
Length of vertical curve:
 • Equation of y
N
L
x =
2
2
N = difference of grades = n
1
 - n
2
Length of curve L =  Totalchange of grade
Rate of change of grade
 1. Length of summit curve (for SSD): Criteria for 
design is sight distance.
  Case 1: L = SSD (Length of curve = L)
L
NS
Hh
=
+
2
2
22 ()
  Where
   L = Length of summit curve, m
   S = Stopping sight distance, SSD (m).
   N = Deviation angle, (n
1
 - n
2
)
   H = Height of drivers eye level
   h = Height of object above road surface
  As per IRC: Usually H = 1.2 m and h = 0.15 m
L
NS
=
2
44 .
  Case 2: When < SSD
LS
Hh
N
=-
+
2
22
2
()
   As per IRC,
   H = 1.2 m and h = 0.15 m
LS
N
=- 2
44 .
 2. Length of summit Curve for OSD /ISD:
  Case 1: When L > OSD (or) ISD
 • In OSD, height of object and height of driver are 
equal H = h
L
NS
H
=
2
8
L
NS
=
2
96 .
(as H = 1.2 m as per IRC)
  Case 2: When L < OSD or ISD
LS
H
N
=- 2
8
LS
N
=- 2
96 .
(as H = 1.2 m)
 • Minimum radius of parabolic summit curve
R
L
N
=
 • On humps, where sight distance is not a problem, 
simple transition curve is appropriate for comfort 
riding. 
 3. Highest point on the summit curve: The highest 
point is at distance of  
Ln
N
1
 from the tangent point of 
first grade n
1
.
Example 9
Given the sight distance as 120 m. The height of driver’ s eye 
as 1.5 m and height of object is 0.15 m. Grade difference of 
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Page 5


Chapter 1 
¦
 Transportation Engineering | 3.925
Setback Distance and Curve Resistance
Introduction
Absolute minimum sight distance that is SSD should be 
available at every stretch of the highway and even on the 
horizontal curve sufficient clearance on the inner side of 
the curve should be provided which is called as setback 
distance.
Setback Distance (m) or Clearance
It is the distance required from the centre line of horizontal 
curve to an obstruction on the inner side of the curve to pro-
vide adequate sight distance on horizontal curve.
Factors Affecting Setback Distance
 1. Required sight distance (SSD, OSD, ISD)
 2. Radius of Horizontal curve (R)
 3. Length of the curve (L
c
)
Approximate formula: (From the concept of chords of 
circle)
Case 1: When L
c
 > S
m
S
R
=
2
8
Case 2: When L
c
 < S
m
LS L
R
=
- () 2
8
SD
F
m
D
R
O
R
C
A
a
2
Obstruction
Rational formulae: (As per IRC)
Case 1: L
c
 > S
From above figure CF = OF - OC
mR R =- cos
a
2
S
R
a = radians and 
a
2
180
2
=
?
S
R
 degrees.
 • For multi-lane roads, sight distance is measured along the 
middle of the inner side lane.
If ‘d’ is the distance between centre of multilane road and 
centre of the inner lane,
MR Rd
S
Rd
=- -
=
?-
()cos
()
a
a
2
2
180
2
degrees
R
d
S
C
m
F
Centre line
of inner
lane
Centre line
of road
Line of Sight
a
2
Case 2: If L
c
 < S
A
B
S
D
a
2
a
2
L
C
G
F
R R
O
mOCOGFG
mR R
SL
L
R
=- +
=- +
-
=
?
cos
()
sin
aa
a
22 2
2
180
2
degrees
 • For multi-lane roads
Part III_Unit 11_Chapter 01.indd   925 5/20/2017   7:22:20 PM
3.926 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
mR RD
SL
L
d
=- -+
-
=
?-
()cos
()
sin
(R )
aa
a
22 2
2
180
2
degrees.
Example 7
The following data are related to a horizontal curved por-
tion of a two lane highway: length of curve = 200 m, radius 
of curve = 300 m and width of pavement = 7.5 m. In order 
to provide a stopping sight distance (SSD) of 80 m, the set-
back distance (in m) required from the centre line of the 
inner lane of the pavement is  [GATE, 2012]
(A) 2.67 m (B) 4.55 m
(C) 7.10 m (D) 7.96 m
Solution
d = =
75
4
1875
.
.m (centre of road to centre of inner lane 
B
4
)
L
c
 = 200 m > S = 80 m
?m = R - (R - d) cos 
2
a
S
Rd
180
()
180
(300 1.875)
15.38 a =
?-
=
?-
=°
m = 300 - (300 - 1.875)cos
. 15 38
2
?
?
?
?
?
?
 = 4.56 m
*But setback distance asked from centre line of inner lane 
of pavement = m - d = 4.56 - 1.875 = 2.68 m
Hence, the correct answer is option (A).
Highway Geometric Design—Transition 
Curves
Introduction
If a curve of radius R takes off from straight road, centrifu-
gal force suddenly acts on the vehicle just after the tangent 
point and a sudden jerk is felt on the vehicle. To avoid this 
a curve having a varying radius which decreases from infin-
ity at the tangent point to a designed radius of the circular 
curve.
Objectives of Providing Transition Curves
 1. To introduce gradually the centrifugal force between 
the tangent point and the beginning of circular curve 
for avoiding sudden jerk on the vehicle.
 2. To enable the driver to turn the steering gradually for 
his comfort and safety.
 3. To help in gradual introduction of super-elevation 
and extra width of pavement on circular curve.
 4. To improve aesthetic appearance of the road.
De?ection angle, (D)
Circular
curve
Transition curve (L
s
)
Straight
Straight
Shift
Shift
Transition
curve
Tangent
distance
O
D C
R
B
T.P
Transition curve in horizontal alignment
 • The rate at which the centrifugal force is introduced can 
be controlled by adopting suitable shape and designing 
its length.
Types of Transition Curves
 1. Spiral
 2. Lemniscate
 3. Cubic Parabola
MAJOR AXIS
Spiral
Lemniscate
Cubic parabola
45°
Ideal transition curve: The rate of change of centrifugal 
acceleration is uniform thought the curve.
Upto deflection angle of 4°, all three curves follow same 
path and practically all curves are same upto 9°.
Spiral (Clothoid/Glovers Spiral)
 • Ideal transition curve
 • Length of transition curve inversely proportional to radius 
R of circular curve. i.e., L
R
LR
ss
?? =
1
constant
 • IRC recommends spiral because
(a) It satisfies the properties of ideal transition curve.
(b) Geometric property is such that the calculations and 
setting out of curve is simple and easy.
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Chapter 1 
¦
 Transportation Engineering | 3.927
 • Equation of spiral:
LR = L
s
 R
c
 = constant
Lm ? =
m = constant = 2RL
s
Where, ? = tangent deflection angle.
Bernoullie’s Lemniscate
 • Radius of curve decreases more rapidly with increase in 
length.
 • Mostly used in roads where deflection angle is large.
 • It is an autogenous curve (as it follows a path which is 
actually traced by a vehicle when turning freely).
 • The curve is set by polar coordinates.
Cubic Parabola (Froude’s Transition/Easement Curve)
 • It is set by simple cartesian coordinates.
 • It is used for valley curves (as radius reduces very fast 
with length)
 • Equation: y
l
Rl
S
=
3
6
Length of Transition Curve
 1. By rate of change of centrifugal acceleration:
 • Vehicle travels length L
s
 of transition curve with 
uniform speed ‘v’ m/s in time ‘t’
?= t
L
v
s
 • Maximum centrifugal acceleration is attained in 
time ‘t’.
Rate of change of centrifugal acceleration,
C
v
Rt
v
LR
s
= =
23
(m ) /s
3
IRC recommended value of C:
C
v
=
+
80
75
 m/s
3 
(V in km/h) and
0.5 < C < 0.8 m/s
3
 2. By rate of introduction of super-elevation:
 • e is rate of super-elevation.
 • W is width of pavement.
 • W
e
 is extra widening provided.
 • B is width of pavement.
 • E is total rise (with respect to inner/centre of road)
E = e 
.
 B = e (W + W
e
)
 • Allowing a rate of change of super-elevation of 1 
in N
L
s
 = EN = eN(W + W
e
)(E with inner edge)
L
EN eN
s
= =
22
 (W + W
e
)(E with respect to centre 
of road)
As per IRC:
N > / 
150–plain androllingterrains
100– built-up areas
60–hillroads
?
?
?
?
?
 3. By empirical formula:
 • For plain and rolling terrains:
L
V
R
V
s
=
27
2
.
in km/h
 • For mountainous and steep terrain:
L
V
R
V
s
= in km/h
(a)  The highest length of the curve obtained by all 
three methods is taken as length of transition 
curve.
(b)  For expressways, with minimum radius of hor-
izontal curve no transition curves are required.
For Design speeds 
V
R
R
120 km/h = 4000 m
100 km/h = 3000 m
min
min
=
?
?
?
?
?
?
?
Shift
The distance between the transition curve at middle and the 
original circular curve is called shift.
Shift,S
L
R
s
=
2
24
Example 8
At a horizontal curve portion of a 4 lane undivided carriage-
way, a transition curve is to be introduced to attain required 
super-elevation. The design speed is 60 km/h and radius of 
the curve is 245 m. Assume length of the wheel base of 
a longest vehicle as 6 m. Super-elevation rate as 5% and 
rate of introduction of this super-elevation as 1 in 150. The 
length of the transition curve (m) required, if the pavement 
is rotated about inner edge is  [GATE, 2006]
(A) 81.4 (B) 85.0
(C) 91.5 (D) 110.2
Solution
Width of road W = 3.5 × 4 = 14 m
Extra widening W
nl
R
V
R
e
=+
2
2
95 .
=
×
×
+
46
2 245
60
95 245
2
.
W
e
 = 0.697 m
Part III_Unit 11_Chapter 01.indd   927 5/20/2017   7:22:22 PM
3.928 | Part III 
¦
 Unit 11 
¦
 Transportation Engineering
If pavement is rotated about inner edge, Length of transition 
curve 
m = e N (W + W
e
)
=
?
?
?
?
?
?
+
5
100
150 14 0 697 ()(. )
m = 110.22 m
Hence, the correct answer is option (D).
Highway Geometric Design—Summit 
Curves
Introduction
There will be variation in grade in vertical alignment of 
highway, which cause discomfort to passengers if roads are 
laid according to that grade. To smoothen out the vertical 
profile and thus reduce the variation in grades for comfort 
of passengers. Vertical curves are designed for sight dis-
tance and comfort of passengers.
Vertical Curves (Valley Curve)
A valley curve is a curve along the longitudinal profile of 
the road provided to smoothen out the vertical profile.
 1. Summit curves/crest curves with convexity upwards
 2. Valley/sag curves with concavity upwards.
Summit Curves
 • When a vehicle moves on summit curve, centrifugal force 
acts upwards against gravity and hence reduces pressure 
on tyres. Therefore no problem of discomfort.
 • Design of summit curves are governed by absolute mini-
mum sight distance as on all highways and no transition 
curves (for comfort condition) are required.
 • Simple parabola is used as summit curve as it gives good 
riding comfort, simple calculation and uniform rate of 
change of grade throughout the parabola. 
Length of vertical curve:
 • Equation of y
N
L
x =
2
2
N = difference of grades = n
1
 - n
2
Length of curve L =  Totalchange of grade
Rate of change of grade
 1. Length of summit curve (for SSD): Criteria for 
design is sight distance.
  Case 1: L = SSD (Length of curve = L)
L
NS
Hh
=
+
2
2
22 ()
  Where
   L = Length of summit curve, m
   S = Stopping sight distance, SSD (m).
   N = Deviation angle, (n
1
 - n
2
)
   H = Height of drivers eye level
   h = Height of object above road surface
  As per IRC: Usually H = 1.2 m and h = 0.15 m
L
NS
=
2
44 .
  Case 2: When < SSD
LS
Hh
N
=-
+
2
22
2
()
   As per IRC,
   H = 1.2 m and h = 0.15 m
LS
N
=- 2
44 .
 2. Length of summit Curve for OSD /ISD:
  Case 1: When L > OSD (or) ISD
 • In OSD, height of object and height of driver are 
equal H = h
L
NS
H
=
2
8
L
NS
=
2
96 .
(as H = 1.2 m as per IRC)
  Case 2: When L < OSD or ISD
LS
H
N
=- 2
8
LS
N
=- 2
96 .
(as H = 1.2 m)
 • Minimum radius of parabolic summit curve
R
L
N
=
 • On humps, where sight distance is not a problem, 
simple transition curve is appropriate for comfort 
riding. 
 3. Highest point on the summit curve: The highest 
point is at distance of  
Ln
N
1
 from the tangent point of 
first grade n
1
.
Example 9
Given the sight distance as 120 m. The height of driver’ s eye 
as 1.5 m and height of object is 0.15 m. Grade difference of 
Part III_Unit 11_Chapter 01.indd   928 5/20/2017   7:22:23 PM
Chapter 1 
¦
 Transportation Engineering | 3.929
international gradient is 0.09. The required length of sum-
mit parabolic curve is [GATE, 1991]
(A) 25 m (B) 125 m
(C) 250 m (D) 500 m
Solution
Sight distance S = 120 m
Deflection angle N = 0.09
Length of summit curve,  
L =
+
NS
Hh
2
2
22 ()
=
×
×+ ×
=
009 120
20 15 20 15
250
2
2
.
(. .)
m
Hence, the correct answer is option (C).
Highway Geometric Design—Valley 
Curves
Valley curves or sag curves are formed when 
 1. Descending gradient meets milder descending 
gradient.
 2. Descending gradient meets level gradient.
 3. Descending gradient meets ascending gradient
 4. Ascending gradient meets steeper ascending gradient
Centrifugal force developed acts downward in addition to 
self weight and increases pressure on the tyres and causes 
discomfort topassengers due to impact.
Factors Considered for Designing 
Valley Curves
 1. Comfort to passengers 
 2. Adequate sight distance for vehicles using head lights 
at night.
 3. Locating lowest point of valley curve for cross 
drainage.
Cubic parabola is generally preferred in valley curves (as 
per IRC) and transition curve is used.
There is no problem of overtaking sight distance at night, 
as opposite vehicles with head lights can be seen from con-
siderable distance.
From Comfort Condition
LL
Nv
C
s
==
?
?
?
?
?
?
22
3
12 /
Where
 v = Design speed m/s
 N = Deviation angle in radians 
 L = Total length of valley curve
 C = Allowable rate of change of centrifugal acceleration
 C = 0.6 m/s
3
 is preferred for comfort condition
 L
s
 = Length of transition curve
Minimum radius of valley curve for cubic parabola,
R
L
N
L
N
s
min
= =
2
From Headlight Sight Distance
Case 1: L = SSD 
Sight distance will be minimum when the vehicle is at 
lowest point on the sag curve
y
NS
L
=
2
2
hS
NS
L
1
2
2
+=
tan
a
L
NS
hS 22 tan
2
1
a
=
+
h
1
h
1
S
S tana
As per IRC:
h
1
= height of head light = 0.75 m
ma = Head light beam angle = 1°
L
NS
S
=
+
2
15 0 035 (. .)
Case 2: L < SSD
 • hS S
L
N
1
2
+-
?
?
?
?
?
?
tana =
LS
hS
N
2
(2 2tan )
1
a
=-
+
LS
S
N
=-
+
2
15 0 035 (. .)
The higher of the above two values is taken as length of 
valley curve.
Lowest point on valley curve:
 • x
NL
N
=
1
Part III_Unit 11_Chapter 01.indd   929 5/20/2017   7:22:24 PM