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# Vertical alignment - II Notes | EduRev

## : Vertical alignment - II Notes | EduRev

``` Page 1

CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
Chapter 18
Vertical alignment -II
18.1 Overview
As discussed earlier, changes in topography necessitate the introduction of vertical curves. The second curve of
this type is the valley curve. This section deals with the types of valley curve and their geometrical design.
18.2 Valley curve
Valley curve or sag curves are vertical curves with convexity downwards. They are formed when two gradients
meet as illustrated in gure 18:1 in any of the following four ways:
(c)
(b)
(a)
(d)
N = n
1
+ n
2
N =n
1
N =(n
1
+ n
2
) N = (n
2
n
1
)
Figure 18:1: Types of valley curve
1. when a descending gradient meets another descending gradient [gure 18:1a].
2. when a descending gradient meets a
3. when a descending gradient meets an ascending gradient [gure 18:1c].
4. when an ascending gradient meets another ascending gradient [gure 18:1d].
Introduction to Transportation Engineering 18.1 Tom V. Mathew and K V Krishna Rao
Page 2

CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
Chapter 18
Vertical alignment -II
18.1 Overview
As discussed earlier, changes in topography necessitate the introduction of vertical curves. The second curve of
this type is the valley curve. This section deals with the types of valley curve and their geometrical design.
18.2 Valley curve
Valley curve or sag curves are vertical curves with convexity downwards. They are formed when two gradients
meet as illustrated in gure 18:1 in any of the following four ways:
(c)
(b)
(a)
(d)
N = n
1
+ n
2
N =n
1
N =(n
1
+ n
2
) N = (n
2
n
1
)
Figure 18:1: Types of valley curve
1. when a descending gradient meets another descending gradient [gure 18:1a].
2. when a descending gradient meets a
3. when a descending gradient meets an ascending gradient [gure 18:1c].
4. when an ascending gradient meets another ascending gradient [gure 18:1d].
Introduction to Transportation Engineering 18.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
B
L/2 L/2
C A
N
Figure 18:2: Valley curve details
18.2.1 Design considerations
There is no restriction to sight distance at valley curves during day time. But visibility is reduced during night.
In the absence or inadequacy of street light, the only source for visibility is with the help of headlights. Hence
valley curves are designed taking into account of headlight distance. In valley curves, the centrifugal force will
be acting downwards along with the weight of the vehicle, and hence impact to the vehicle will be more. This
will result in jerking of the vehicle and cause discomfort to the passengers. Thus the most important design
factors considered in valley curves are: (1) impact-free movement of vehicles at design speed and (2) availability
of stopping sight distance under headlight of vehicles for night driving.
For gradually introducing and increasing the centrifugal force acting downwards, the best shape that could
be given for a valley curve is a transition curve. Cubic parabola is generally preferred in vertical valley curves.
See gure 18:2.
During night, under headlight driving condition, sight distance reduces and availability of stopping sight
distance under head light is very important. The head light sight distance should be at least equal to the
stopping sight distance. There is no problem of overtaking sight distance at night since the other vehicles with
headlights could be seen from a considerable distance.
18.2.2 Length of the valley curve
The valley curve is made fully transitional by providing two similar transition curves of equal length The
transitional curve is set out by a cubic parabola y = bx
3
where b =
2N
3L
2
The length of the valley transition curve
is designed based on two criteria:
1. comfort criteria; that is allowable rate of change of centrifugal acceleration is limited to a comfortable
3
.
2. safety criteria; that is the driver should have adequate headlight sight distance at any part of the country.
Comfort criteria
The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort:
Let c is the rate of change of acceleration, R the minimum radius of the curve, v is the design speed and t is
Introduction to Transportation Engineering 18.2 Tom V. Mathew and K V Krishna Rao
Page 3

CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
Chapter 18
Vertical alignment -II
18.1 Overview
As discussed earlier, changes in topography necessitate the introduction of vertical curves. The second curve of
this type is the valley curve. This section deals with the types of valley curve and their geometrical design.
18.2 Valley curve
Valley curve or sag curves are vertical curves with convexity downwards. They are formed when two gradients
meet as illustrated in gure 18:1 in any of the following four ways:
(c)
(b)
(a)
(d)
N = n
1
+ n
2
N =n
1
N =(n
1
+ n
2
) N = (n
2
n
1
)
Figure 18:1: Types of valley curve
1. when a descending gradient meets another descending gradient [gure 18:1a].
2. when a descending gradient meets a
3. when a descending gradient meets an ascending gradient [gure 18:1c].
4. when an ascending gradient meets another ascending gradient [gure 18:1d].
Introduction to Transportation Engineering 18.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
B
L/2 L/2
C A
N
Figure 18:2: Valley curve details
18.2.1 Design considerations
There is no restriction to sight distance at valley curves during day time. But visibility is reduced during night.
In the absence or inadequacy of street light, the only source for visibility is with the help of headlights. Hence
valley curves are designed taking into account of headlight distance. In valley curves, the centrifugal force will
be acting downwards along with the weight of the vehicle, and hence impact to the vehicle will be more. This
will result in jerking of the vehicle and cause discomfort to the passengers. Thus the most important design
factors considered in valley curves are: (1) impact-free movement of vehicles at design speed and (2) availability
of stopping sight distance under headlight of vehicles for night driving.
For gradually introducing and increasing the centrifugal force acting downwards, the best shape that could
be given for a valley curve is a transition curve. Cubic parabola is generally preferred in vertical valley curves.
See gure 18:2.
During night, under headlight driving condition, sight distance reduces and availability of stopping sight
distance under head light is very important. The head light sight distance should be at least equal to the
stopping sight distance. There is no problem of overtaking sight distance at night since the other vehicles with
headlights could be seen from a considerable distance.
18.2.2 Length of the valley curve
The valley curve is made fully transitional by providing two similar transition curves of equal length The
transitional curve is set out by a cubic parabola y = bx
3
where b =
2N
3L
2
The length of the valley transition curve
is designed based on two criteria:
1. comfort criteria; that is allowable rate of change of centrifugal acceleration is limited to a comfortable
3
.
2. safety criteria; that is the driver should have adequate headlight sight distance at any part of the country.
Comfort criteria
The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort:
Let c is the rate of change of acceleration, R the minimum radius of the curve, v is the design speed and t is
Introduction to Transportation Engineering 18.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
the time, then c is given as:
c =
v
2
R
0
t
=
v
2
R
0
L
v
=
v
3
LR
L =
v
3
cR
(18.1)
For a cubic parabola, the value of R for length L
s
is given by:
R =
L
N
(18.2)
Therefore,
L
s
=
v
3
cLs
N
L
s
=
2
r
Nv
3
c
L = 2
2
r
Nv
3
c
(18.3)
where L is the total length of valley curve, N is the deviation angle in radians or tangent of the deviation angle
or the algebraic dierence in grades, and c is the allowable rate of change of centrifugal acceleration which may
be taken as 0:6m=sec
3
.
Safety criteria
Length of the valley curve for headlight distance may be determined for two conditions: (1) length of the
valley curve greater than stopping sight distance and (2) length of the valley curve less than the stopping sight
distance.
Case 1 Length of valley curve greater than stopping sight distance (L > S)
The total length of valley curve L is greater than the stopping sight distance SSD. The sight distance available
will be minimum when the vehicle is in the lowest point in the valley. This is because the beginning of the
curve will have innite radius and the bottom of the curve will have minimum radius which is a property of the
transition curve. The case is shown in gure 18:3. From the geometry of the gure, we have:
h
1
+ S tan  = aS
2
=
NS
2
2L
L =
NS
2
2h
1
+ 2S tan
(18.4)
where N is the deviation angle in radians, h
1
in degrees and S is the sight distance. The inclination  is 1 degree.
Introduction to Transportation Engineering 18.3 Tom V. Mathew and K V Krishna Rao
Page 4

CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
Chapter 18
Vertical alignment -II
18.1 Overview
As discussed earlier, changes in topography necessitate the introduction of vertical curves. The second curve of
this type is the valley curve. This section deals with the types of valley curve and their geometrical design.
18.2 Valley curve
Valley curve or sag curves are vertical curves with convexity downwards. They are formed when two gradients
meet as illustrated in gure 18:1 in any of the following four ways:
(c)
(b)
(a)
(d)
N = n
1
+ n
2
N =n
1
N =(n
1
+ n
2
) N = (n
2
n
1
)
Figure 18:1: Types of valley curve
1. when a descending gradient meets another descending gradient [gure 18:1a].
2. when a descending gradient meets a
3. when a descending gradient meets an ascending gradient [gure 18:1c].
4. when an ascending gradient meets another ascending gradient [gure 18:1d].
Introduction to Transportation Engineering 18.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
B
L/2 L/2
C A
N
Figure 18:2: Valley curve details
18.2.1 Design considerations
There is no restriction to sight distance at valley curves during day time. But visibility is reduced during night.
In the absence or inadequacy of street light, the only source for visibility is with the help of headlights. Hence
valley curves are designed taking into account of headlight distance. In valley curves, the centrifugal force will
be acting downwards along with the weight of the vehicle, and hence impact to the vehicle will be more. This
will result in jerking of the vehicle and cause discomfort to the passengers. Thus the most important design
factors considered in valley curves are: (1) impact-free movement of vehicles at design speed and (2) availability
of stopping sight distance under headlight of vehicles for night driving.
For gradually introducing and increasing the centrifugal force acting downwards, the best shape that could
be given for a valley curve is a transition curve. Cubic parabola is generally preferred in vertical valley curves.
See gure 18:2.
During night, under headlight driving condition, sight distance reduces and availability of stopping sight
distance under head light is very important. The head light sight distance should be at least equal to the
stopping sight distance. There is no problem of overtaking sight distance at night since the other vehicles with
headlights could be seen from a considerable distance.
18.2.2 Length of the valley curve
The valley curve is made fully transitional by providing two similar transition curves of equal length The
transitional curve is set out by a cubic parabola y = bx
3
where b =
2N
3L
2
The length of the valley transition curve
is designed based on two criteria:
1. comfort criteria; that is allowable rate of change of centrifugal acceleration is limited to a comfortable
3
.
2. safety criteria; that is the driver should have adequate headlight sight distance at any part of the country.
Comfort criteria
The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort:
Let c is the rate of change of acceleration, R the minimum radius of the curve, v is the design speed and t is
Introduction to Transportation Engineering 18.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
the time, then c is given as:
c =
v
2
R
0
t
=
v
2
R
0
L
v
=
v
3
LR
L =
v
3
cR
(18.1)
For a cubic parabola, the value of R for length L
s
is given by:
R =
L
N
(18.2)
Therefore,
L
s
=
v
3
cLs
N
L
s
=
2
r
Nv
3
c
L = 2
2
r
Nv
3
c
(18.3)
where L is the total length of valley curve, N is the deviation angle in radians or tangent of the deviation angle
or the algebraic dierence in grades, and c is the allowable rate of change of centrifugal acceleration which may
be taken as 0:6m=sec
3
.
Safety criteria
Length of the valley curve for headlight distance may be determined for two conditions: (1) length of the
valley curve greater than stopping sight distance and (2) length of the valley curve less than the stopping sight
distance.
Case 1 Length of valley curve greater than stopping sight distance (L > S)
The total length of valley curve L is greater than the stopping sight distance SSD. The sight distance available
will be minimum when the vehicle is in the lowest point in the valley. This is because the beginning of the
curve will have innite radius and the bottom of the curve will have minimum radius which is a property of the
transition curve. The case is shown in gure 18:3. From the geometry of the gure, we have:
h
1
+ S tan  = aS
2
=
NS
2
2L
L =
NS
2
2h
1
+ 2S tan
(18.4)
where N is the deviation angle in radians, h
1
in degrees and S is the sight distance. The inclination  is 1 degree.
Introduction to Transportation Engineering 18.3 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
I
A
B
E G
N
D
S
C
F
h
1
h
1
Stan

Figure 18:3: Valley curve, case 1, L > S
L=2 S L=2
h
D
N B
L
h
S
C
A

S tan
Figure 18:4: Valley curve, case 2, S > L
Case 2 Length of valley curve less than stopping sight distance (L < S)
The length of the curve L is less than SSD. In this case the minimum sight distance is from the beginning of
the curve. The important points are the beginning of the curve and the bottom most part of the curve. If the
vehicle is at the bottom of the curve, then its headlight beam will reach far beyond the endpoint of the curve
whereas, if the vehicle is at the beginning of the curve, then the headlight beam will hit just outside the curve.
Therefore, the length of the curve is derived by assuming the vehicle at the beginning of the curve . The case
is shown in gure 18:4.
From the gure,
h
1
+ s tan  =

S
L
2

N
L = 2S
2h
1
+ 2S tan
N
(18.5)
Note that the above expression is approximate and is satisfactory because in practice, the gradients are
very small and is acceptable for all practical purposes. We will not be able to know prior to which case to be
adopted. Therefore both has to be calculated and the one which satises the condition is adopted.
Introduction to Transportation Engineering 18.4 Tom V. Mathew and K V Krishna Rao
Page 5

CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
Chapter 18
Vertical alignment -II
18.1 Overview
As discussed earlier, changes in topography necessitate the introduction of vertical curves. The second curve of
this type is the valley curve. This section deals with the types of valley curve and their geometrical design.
18.2 Valley curve
Valley curve or sag curves are vertical curves with convexity downwards. They are formed when two gradients
meet as illustrated in gure 18:1 in any of the following four ways:
(c)
(b)
(a)
(d)
N = n
1
+ n
2
N =n
1
N =(n
1
+ n
2
) N = (n
2
n
1
)
Figure 18:1: Types of valley curve
1. when a descending gradient meets another descending gradient [gure 18:1a].
2. when a descending gradient meets a
3. when a descending gradient meets an ascending gradient [gure 18:1c].
4. when an ascending gradient meets another ascending gradient [gure 18:1d].
Introduction to Transportation Engineering 18.1 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
B
L/2 L/2
C A
N
Figure 18:2: Valley curve details
18.2.1 Design considerations
There is no restriction to sight distance at valley curves during day time. But visibility is reduced during night.
In the absence or inadequacy of street light, the only source for visibility is with the help of headlights. Hence
valley curves are designed taking into account of headlight distance. In valley curves, the centrifugal force will
be acting downwards along with the weight of the vehicle, and hence impact to the vehicle will be more. This
will result in jerking of the vehicle and cause discomfort to the passengers. Thus the most important design
factors considered in valley curves are: (1) impact-free movement of vehicles at design speed and (2) availability
of stopping sight distance under headlight of vehicles for night driving.
For gradually introducing and increasing the centrifugal force acting downwards, the best shape that could
be given for a valley curve is a transition curve. Cubic parabola is generally preferred in vertical valley curves.
See gure 18:2.
During night, under headlight driving condition, sight distance reduces and availability of stopping sight
distance under head light is very important. The head light sight distance should be at least equal to the
stopping sight distance. There is no problem of overtaking sight distance at night since the other vehicles with
headlights could be seen from a considerable distance.
18.2.2 Length of the valley curve
The valley curve is made fully transitional by providing two similar transition curves of equal length The
transitional curve is set out by a cubic parabola y = bx
3
where b =
2N
3L
2
The length of the valley transition curve
is designed based on two criteria:
1. comfort criteria; that is allowable rate of change of centrifugal acceleration is limited to a comfortable
3
.
2. safety criteria; that is the driver should have adequate headlight sight distance at any part of the country.
Comfort criteria
The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort:
Let c is the rate of change of acceleration, R the minimum radius of the curve, v is the design speed and t is
Introduction to Transportation Engineering 18.2 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
the time, then c is given as:
c =
v
2
R
0
t
=
v
2
R
0
L
v
=
v
3
LR
L =
v
3
cR
(18.1)
For a cubic parabola, the value of R for length L
s
is given by:
R =
L
N
(18.2)
Therefore,
L
s
=
v
3
cLs
N
L
s
=
2
r
Nv
3
c
L = 2
2
r
Nv
3
c
(18.3)
where L is the total length of valley curve, N is the deviation angle in radians or tangent of the deviation angle
or the algebraic dierence in grades, and c is the allowable rate of change of centrifugal acceleration which may
be taken as 0:6m=sec
3
.
Safety criteria
Length of the valley curve for headlight distance may be determined for two conditions: (1) length of the
valley curve greater than stopping sight distance and (2) length of the valley curve less than the stopping sight
distance.
Case 1 Length of valley curve greater than stopping sight distance (L > S)
The total length of valley curve L is greater than the stopping sight distance SSD. The sight distance available
will be minimum when the vehicle is in the lowest point in the valley. This is because the beginning of the
curve will have innite radius and the bottom of the curve will have minimum radius which is a property of the
transition curve. The case is shown in gure 18:3. From the geometry of the gure, we have:
h
1
+ S tan  = aS
2
=
NS
2
2L
L =
NS
2
2h
1
+ 2S tan
(18.4)
where N is the deviation angle in radians, h
1
in degrees and S is the sight distance. The inclination  is 1 degree.
Introduction to Transportation Engineering 18.3 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
I
A
B
E G
N
D
S
C
F
h
1
h
1
Stan

Figure 18:3: Valley curve, case 1, L > S
L=2 S L=2
h
D
N B
L
h
S
C
A

S tan
Figure 18:4: Valley curve, case 2, S > L
Case 2 Length of valley curve less than stopping sight distance (L < S)
The length of the curve L is less than SSD. In this case the minimum sight distance is from the beginning of
the curve. The important points are the beginning of the curve and the bottom most part of the curve. If the
vehicle is at the bottom of the curve, then its headlight beam will reach far beyond the endpoint of the curve
whereas, if the vehicle is at the beginning of the curve, then the headlight beam will hit just outside the curve.
Therefore, the length of the curve is derived by assuming the vehicle at the beginning of the curve . The case
is shown in gure 18:4.
From the gure,
h
1
+ s tan  =

S
L
2

N
L = 2S
2h
1
+ 2S tan
N
(18.5)
Note that the above expression is approximate and is satisfactory because in practice, the gradients are
very small and is acceptable for all practical purposes. We will not be able to know prior to which case to be
adopted. Therefore both has to be calculated and the one which satises the condition is adopted.
Introduction to Transportation Engineering 18.4 Tom V. Mathew and K V Krishna Rao
CHAPTER 18. VERTICAL ALIGNMENT -II NPTEL May 3, 2007
18.3 Summary
The valley curve should be designed such that there is enough headlight sight distance. Improperly designed
valley curves results in extreme riding discomfort as well as accident risks especially at nights. The length
of valley curve for various cases were also explained in the section. The concept of valley curve is used in
underpasses.
18.4 Problems
1. A valley curve is formed by descending gradient n
1
= 1 in 25 and ascending gradient n
2
= 1 in 30. Design the
length of the valley curve for V =80kmph. (Hint: c=0.6 m/cm
3
, SSD=127.3m) [Ans: L=max(73.1,199.5)]
Introduction to Transportation Engineering 18.5 Tom V. Mathew and K V Krishna Rao
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