Covers superelevation, horizontal and vertical curves in Highway Engineering

2. Introduction
• The geometric design of highways deals with the
dimensions and layout of visible features of the
highway.

3. • Cross section elements
–Width of pavement
–Right of way
–Surface characteristics
–Cross slope of pavement

4. • Sight distance considerations
–Stopping Sight Distance
–Overtaking Sight Distance
–Sight distance at intersections
• Horizontal and vertical alignment details
–Horizontal curves
–Vertical curves
–Super-elevation

5. PRESENTATION COVERS
• Effect of centrifugal force
• Design of super elevation
• Achieving super elevation in the field
• Radius of Horizontal Circular curves
• Extra Widening of roads
• Transition Curves
• Vertical Curves

6. EFFECT OF CENTRIFUGAL
FORCE

7. LOCATION
• When a vehicle traverse a horizontal
curve, the centrifugal force act horizontally
outwards through the centre of gravity of
the vehicle.
The centrifugal force depends upon
1) radius of horizontal curve
2) speed of the vehicle negotiating the
curve

8. Centrifugal force P is given by the equation:
P = Wv2/gR
Where,
P = Centrifugal force, kg
W = wt. of vehicle, kg
R = radius of the circular curve, m
v = speed of vehicle, m/sec
g = acc. due to gravity = 9.8 m/sec2

9. CENTRIFUGAL RATIO or IMPACT RATIO
It is the ratio of the centrifugal force to the weight
of the vehicle, P/W
Centrifugal ratio = v2/gR

10. EFFECTS OF CENTRIFUGAL FORCE
• Tendency to overturn the vehicle outwards
about the outer wheels, and
• Tendency to skid the vehicle laterally
outwards.

11. C.G.
INNER SIDE
OF CURVE
OUTER SIDE
OF CURVE
A Bb/2 b/2
h
W
P
OVERTURNING EFFECT

12. Forces acting on the vehicle on curve
• Overturning moment due to centrifugal force ‘P’
is P x h
• Resisting moment due to the weight of the
vehicle ‘W’ is W x b/2
• At equilibrium
P x h = W x b/2
or P/W = b/2h
This means that there is a danger of overturning
when the centrifugal ratio P/W attains a value of
b/2h.

13. Skidding effect
INNER SIDE
OF CURVE
C.G.
A B
W
P
RA RB
FA=
f.RA
FB=
f.RB

14. Skidding effect
• The centrifugal force developed has tendency to
push the vehicle outwards in the transverse
direction.
• If the centrifugal force ‘P’ exceeds the maximum
possible transverse skid resistance due to
friction, the vehicle will start skidding in the
transverse direction.
• Equilibrium condition for the skid resistance is
given by:
• P = FA + FB = f(RA + RB) = fW
or P/W = f

15. To avoid overturning and lateral
skidding on a curve, the centrifugal ratio
should be less than ‘b/2h’ and also ‘f’.

16. SUPERELEVATION
In order to counteract the effect of
centrifugal force and to reduce the
tendency of the vehicle to overturn or skid,
the outer edge of the pavement is raised
w.r.t. the inner edge, providing a traverse
slope throughout the length of the
horizontal curve.

17. Superelevation
Wα
M
N
e
e = NL/ML = tan α
L

18. ANALYSIS OF SUPERELEVATION
• The forces acting on the vehicle while moving on
a circular curve of radius ‘R’ m, at speed of v
m/sec are
• The centrifugal force ‘P’
• The weight ‘W’ of the vehicle
• The frictional forces (FA and FB) developed
between the wheels and the pavement

19. Superelevation
W
α
W
Inner
side of
curve
For equilibrium condition,
P Cos α = W sin α + FA + FB

20. • FA = f RA
• FB = f RB
P cos α = W sin α + f (RA + RB)
= W sin α + f (W cos α + P sin α )
P (cos α – f sin α ) = W sin α + f W cos α
divide by Wcos α,
P/W(1 – f tan α ) = tan α + f
P/W = (tan α + f)/(1- f tan α)

21. This is an exact expression for superelevation.
But normally, f = 0.15 and α< 4o, 1 – f.tan α ≈ 1
and for small α , tan α ≈ sin α = E/B =
e, then equation becomes:
Therefore, P/W = tan α + f = e + f
v2/gR = e + f

22. RADIUS OF HORIZONTAL CURVE
Rruling = v2/(e+f)g
Rmin = v’2/ (e+f)g
Where,
v = design speed
v’ = min. design speed
f = coeff. of friction
e = rate of superelevation,
Maximum value of ‘e’ is taken as 0.07 at all the
regions except at hill roads where it is taken as
0.1

23. Superelevation Transition

24. STEPS FOR SUPERELEVATION DESIGN
(i) The superelevation for 75 % of design
speed is calculated neglecting the friction
e = (0.75 v)2/gR
(ii) If the calculated value of ‘e’ is less than 0.07
or 7% the value so obtained is provided.
If the value of ‘e’ exceeds 0.07 then
provide the max. superelevation equal to
0.07 and proceed with steps (iii) or (iv)

25. (iii) Check the coeff. of friction developed for the
max. value of e = 0.07 at the full value of
design speed
f = v2/gR – 0.07
If the value of f calculated is less than 0.15, the
superelevation of 0.07 is safe for the design
speed. If not, calculate the restricted speed as
given in step (iv).

26. (iv) The allowable speed at the curve is
calculated by considering the design
coeff. of lateral friction and the max.
superelevation i.e.,
e + f = 0.07+ 0.15 = 0.22
Calculate the safe allowable speed,
va = √(0.22gR) m/sec

27. • If the allowable speed, as calculated in step (iv)
is higher than the design speed, then the design
is adequate and provide a superelevation equal
to 0.07. If the allowable speed is less than the
design speed, the speed is limited to the
allowable speed va calculated above.
• Appropriate warning sign and speed limit
regulation sign are installed to restrict and
regulate the speed at such curves when the safe
speed va is less than the design speed.

28. ATTAINMENT OF SUPERELEVATION IN THE
FIELD
• Elimination of crown of the cambered
section
• Rotation of pavement to attain
superelevation

29. Extra Widening
• Extra widening refers to the additional width of
carriageway that is required on a curved section
of a road. This widening is done due to two
reasons:
• the first and most important is the additional
width required for a vehicle negotiating a
horizontal curve (off tracking), and
• the second is due to the tendency of the drivers
to ply away from the inner edge of the
carriageway as they drive on a curve.
The first is referred as the mechanical
widening and the second is called the
psychological widening.

30. Extra Widening

31. Mechanical Widening
The reasons for the mechanical widening are:
• When a vehicle negotiates a horizontal curve, the
rear wheels follow a path of shorter radius than
the front wheels. This phenomenon is called off –
tracking.
• In addition speeds higher than the design speed
causes transverse skidding which requires
additional width for safety purpose.

32. Mechanical Widening
• If the road has ‘n’ lanes, ‘R’ is the radius of
the curve and ‘l’ is the length of wheel
base of longest vehicle then the
mechanical widening (Wm) of a road is
given by,
Wm = nl2/2R

33. Psychological Widening
IRC proposed an empirical relation for the
psychological widening at horizontal curves Wps:
Wps = V/9.5√R
Where, V = design speed, kmph
R = radius of the curve, m

34. The total widening needed at a horizontal curve We
is:
We = Wm +Wps
= nl2/2R + V/9.5√R
Where,
n = number of traffic lanes
l = length of wheel base of longest vehicle, m
V = design speed, kmph
R = radius of horizontal curve, m

35. Extra Widening

36. TRANSITION CURVES
A transition curve has a radius which decreases
from infinity at the tangent point to a designed
radius of the circular curve. It is introduced between
straight and a circular curve.
Functions of transition curve
 To introduce gradually the centrifugal force between
the tangent point and the beginning of the circular
curve to avoid a sudden jerk on the vehicle.
 To enable the driver turn the steering wheel
gradually for his own comfort and security
 To enable gradual introduction of the designed
superelevation and extra widening of pavement at
the start of the circular curve.

37. With Transition Curves
Transition Curves
• Gradually changing the curvature from
tangents to circular curves
Without Transition Curves

38. Types of transition curves
• SPIRAL
• LEMNISCATE
• CUBIC PARABOLA
IRC recommends the use of spiral curve as
transition curve in the horizontal alignment of
highways.
Reasons
1 The spiral curve satisfies the requirement
of an ideal transition
2 The geometry properties of spiral is such
that the calculations and setting out the
curve in the field is simple and easy.

39. CALCULATION OF LENGTH OF TRANSITION CURVE
The length of transition curve is designed to
fulfill three conditions:
1. Rate of change of centrifugal acceleration
(v2/R) to be developed gradually
2. Rate of introduction of superelevation to be at
a reasonable rate
3. Min. length by IRC empirical formula

40. Rate of change of centrifugal acceleration
• Length of transition curve is calculated using
formula
Ls = v3/CR
here C = 80/(75 + V) m/sec3
[0.5< C < 0.8]
Here, Ls = length of transition curve, m
v = design speed in m/sec
V = design speed in Kmph
C = allowable rate of change of
centrifugal acc., m/sec3
R = radius of the circular curve
N = gradient

41. Rate of introduction of superelevation
• When the pavement is raised about centre line
Ls = EN/2 = (eN/2)* (W + We)
• When the pavement is rotated about the inner edge
Ls = EN = eN* (W + We)
Here, Ls = length of transition curve, m
E = total raising of pavement, m
W = width of pavement, m
We = extra widening, m
e = rate of superelevation

42. By IRC empirical formula
• For plain and rolling terrain:
Ls = 2.7 V2/R
• For mountainous and steep terrains:
Ls = V2/R
The length of transition curve for the
design should be the highest of the three
values.

43. Vertical Alignment
The vertical alignment of a road consists of
gradients (straight lines in a vertical plane) and
vertical curves.
Just as a circular curve is used to connect
horizontal straight stretches of road, vertical curves
connect two gradients. When these two curves
meet, they form either convex or concave.
The former is called a summit curve, while the
latter is called a valley curve.

44. Summit Curve
Summit curves are vertical curves with gradient
upwards. They are formed when two gradients
meet in any of the following four ways:
• when a positive gradient meets another positive
gradient.
• when positive gradient meets a at gradient.
• when an ascending gradient meets a descending
gradient.
• when a descending gradient meets another
descending gradient.

45. TYPE OF SUMMIT CURVES

46. Length of Summit Curve for SSD
Two cases are to be considered in
deciding the length
• When the length of the curve is greater
than the sight distance (L > SSD)
• When the length of the curve is less than
the sight distance (L < SSD)

47. SUMMIT CURVES OR CREST VERTICAL CURVES
n1
n2
PVI
PVTPVC
h2
h1 L
SSD
For SSD < L For SSD > L
Line of Sight
( )
( )2
21
2
22 hh
SSDN
L
+
= ( )
( )
N
hh
SSDL
2
212
2
+
-=
N = deviation angle = (n1 – n2)

48. Crest Vertical Curves
• Assumptions for design
– h1 = driver’s eye height =
– h2 = height of obstruction =
Simplified Equations
For SSD < L For SSD > L
( )
4.4
2
SSDN
L = ( )
N
SSDL
4.4
2 -=
1.2 m
0.15 m

49. Length of Summit Curve for safe OSD or ISD
• Assumptions for design
– h1 = driver’s eye height = 1.2 m
– h2 = height of obstruction = 1.2 m
• Simplified Equations
For SSD < L For SSD > L
( )
9.6
2
SSDN
L = ( )
N
SSDL
9.6
2 -=

50. Valley Curve
Valley curve or sag curves are vertical curves with convexity
downwards. They are formed when two gradients meet in
any of the following four ways:
• when a descending gradient meets another
descending gradient.
• when a descending gradient meets a at gradient.
• when a descending gradient meets an ascending
gradient.
• when an ascending gradient meets another
ascending gradient.

52. Length of Valley Curve
The length of transition valley curve is
designed based on two criteria:
1. The allowable rate of change of
centrifugal acceleration of 0.6
m/sec3, and
2. The head light sight distance
Usually the second criterion of head light
sight distance is higher and governs the
design.

53. Rate of change of Centrifugal Acceleration
or
Comfort Criteria
L = 2[Nv3/C]1/2
Where, L = total length of valley curve,
N = deviation angle
v = design speed, m/sec
C = is the allowable rate of change
of centrifugal acceleration
which may be taken as
0.6m/sec3.

54. Head Light Sight Distance criteria
n1
n2
PVI
PVTPVC
h2=0h1
L
Light Beam Distance (SSD)
For SSD < L For SSD > L
headlight beam (diverging from LOS by β degrees)
( )
( )btan2 1
2
Sh
SSDN
L
+
= ( ) ( )( )
N
SSDh
SSDL
btan2
2 1
+
-=

55. Sag Vertical Curves
• Assumptions for design
– h1 = headlight height = 0.75 m.
– β = 1 degree
Simplified Equations
For SSD < L
( )
( )SSD
SSDN
L
0.0351.5
2
+
=
For SSD > L
( )
( )





 +
-=
N
SSD
SSDL
0.0351.5
2