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 C.G
Where, P
h
P = centrifugal force, kg A B
W = weight of the vehicle, kg b/2
R= Radius of circular curve, m
b/2

v = speed of vehicle, m/sec W

g = acceleration due to gravity, m/sec2
Wv 2( m / s )
P/W = Centrifugal Ratio
P=
or Impact Factor
gR
P v 2( m / s )
=
W gR
Dr. Rizwan Memon 2
The overturning moment due to
centrifugal force is; P * h
C.G
P
The restoring moment due to weight
h
is equal to; W * b/2
A B
The stability condition for overturning b/2 b/2

would occur when;

W
P*h = W*b/2 or P/W = (b*h)/2

when the centrifugal ratio (P/W) attains a value

b/2h, there is a danger of overturning

Dr. Rizwan Memon 3

The latral skidding will occur if;

P ≥ FA + FB
C.G
P
or P ≥ f ( RA + RB) =fW
FA FB
or P/W ≥ f

when the centrifugal ratio attains the value RA RB

equal to the coefficient of lateral friction W
(f), there is danger of lateral skidding.

Thus to avoid overturning and lateral skidding in a

horizontal curve, the centrifugal ratio should always
be less than “b/2h” and also “f”
Dr. Rizwan Memon 4
Horizontal Design Iterations

◼ Design baseline
◼ Curve radius above the minimum
◼ Superelevation and side-friction factor not exceeding
the maximum values

◼ Design is revised to consider:

cost, environmental impacts, sight distances,
aesthetic consequences, etc.

Dr. Rizwan Memon 5

Superelevation is the banking of the roadway along a
horizontal curve so that the drivers can negotiate the
curve at safe and comfortable speed.

Normal crown

Dr. Rizwan Memon Fully superelevated 6

Superelevation “e” and side friction coefficient “f” on horizontal curves

Dr. Rizwan Memon 7

 From the laws of mechanics, the basic formula that governs vehicle
Operation on a curve is:

0.01e + f v2
=
1 − 0.01ef gR

In practice: 1 − 0.01ef  1
v2
0.01e + f =
gR
v : vehicle speed, (m/s or ft/s)
R: radius of curve, (m or ft)
e: rate of superelevation, percent
f: side friction factor (lateral ratio, cornering ratio, unbalanced
Dr. Rizwan Memon centrifugal ratio) 8
 Minimum Curve Radius R

◼ Curve requiring the most centripetal force for

the given design speed
◼ Given emax, fmax, Vdesign

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

three factors:
the design speed, the superelevation, and the side-friction factor.

◼ The minimum radius serves not only as a constraint on the

geometric design of the roadway, but also as a starting point from
which a more refined curve design can be produced.
◼ Any increase in the radius of the curve beyond this minimum radius
will allow you to reduce the side-friction factor, the superelevation
rate, or both.Memon
Dr. Rizwan 9
 Minimum Radius (Rmin)

It is minimum value of curvature for a given design speed and is

determined from emax and fmax

V2
Rmin =
127(0.01emax + f max )
v : vehicle speed, km/h
R: radius of curve, m V2
Rmin =
e: rate of superelevation, percent 15(0.01emax + f max )
f: side friction factor (lateral ratio)
v : vehicle speed, mph
R: radius of curve, ft
e: rate of superelevation, percent
Dr. Rizwan Memon f: side friction factor (lateral ratio)
10
 Get familiar with GB tables

Minimum radius with design speed

Dr. Rizwan Memon 11

Dr. Rizwan Memon 12
For a given design speed v, there are multiple combinations of e and f
for sustaining centripetal acceleration on curves. Green book
recommends 5 methods.

Method 1: Superelevation and side friction are directly proportional to

the inverse of the radius

Method 2: Start with using side friction f up to fmax, then f remains

fmax, then e is used until emax

Method 3: start with using e, until emax, then e remains emax, and f is
used until fmax

Method 4: same as method 3, but using average running speed instead

of design speed

Method 5: e and f are in a curvilinear relation with 1/R

Dr. Rizwan Memon 13
Dr. Rizwan Memon 14
Superelevation
Road Section View Road Plan View

CL
2% 2%
Superelevation
Road Section View Road Plan View

CL
1.5% 2%
Superelevation
Road Section View Road Plan View

CL
1% 2%
Superelevation
Road Section View Road Plan View

CL
0.5% 2%
Superelevation
Road Section View Road Plan View

CL
-0.0% 2%
Superelevation
Road Section View Road Plan View

CL
-0.5% 2%
Superelevation
Road Section View Road Plan View

CL
-1% 2%
Superelevation
Road Section View Road Plan View

CL
-1.5% 2%
Superelevation
Road Section View Road Plan View

CL
-2% 2%
Superelevation
Road Section View Road Plan View

CL
-3% 3%
Superelevation
Road Section View Road Plan View

CL
-4% 4%
Superelevation
Road Section View Road Plan View

CL
-3% 3%
Superelevation
Road Section View Road Plan View

CL
-2% 2%
Superelevation
Road Section View Road Plan View

CL
-1.5% 2%
Superelevation
Road Section View Road Plan View

CL
-1% 2%
Superelevation
Road Section View Road Plan View

CL
-0.5% 2%
Superelevation
Road Section View Road Plan View

CL
-0.0% 2%
Superelevation
Road Section View Road Plan View

CL
0.5% 2%
Superelevation
Road Section View Road Plan View

CL
1% 2%
Superelevation
Road Section View Road Plan View

CL
1.5% 2%
Superelevation
Road Section View Road Plan View

CL
2% 2%
Tangent runout (crown runoff) section :

Length of roadway needed to accomplish a change in out-

side cross slope from normal cross slope rate to zero

Runoff section :

Length of roadway needed to accomplish a change in

out-side cross slope from zero to full superelevation

Dr. Rizwan Memon 36

 Tangent Run-out Profile

0%

Normal crown

Dr. Rizwan Memon 37

 Supperelevation runoff

0% 
Normal crown Relative gradient

Dr. Rizwan Memon 38

Dr. Rizwan Memon 39
Minimum Length of Tangent Runout

eNC  Lr
Lt =
ed
where
◼ Lt = minimum length of tangent

runout (ft or m)
◼ eNC = normal cross slope rate (%)

◼ ed = design superelevation rate

◼ Lr = minimum length of superelevation

runoff (ft or m)
Dr. Rizwan Memon 40
Minimum Length of Superelevation Runoff

( wn1 )ed 12 e
Lr = (bw ) Lr = 
where  G
◼ Lr = minimum length of superelevation

runoff (ft)
◼ Δ or G = maximum relative gradient(%)

◼ n1 = number of lanes rotated

◼ bw or α= adjustment factor for number

of lanes
◼ w = width of one traffic lane

◼ ed = design superelevation rate

Dr. Rizwan Memon 41
Keep water drainage in mind while considering all of the
available cross-section options

Dr. Rizwan Memon 42

Axis of rotation:

Undivided highways are usually superelevated with the axis of rotation

at the roadways centerline

Axis of rotation

Superelevated section
Normal cross section

Dr. Rizwan Memon 43

Muti-lane highways with depressed medians are usually superelevated with the axis of
Rotation at the median edges of the traveled way.

Axis of rotation

Median width

Dr. Rizwan Memon 44

 Transition Curves

◼ Gradually changing the curvature from tangents to circular

curves
Without Transition Curves

With Transition Curves

Dr. Rizwan Memon 45

Spiral Curve:

Spiral curves are curves with a continuously

changing radii, they are sometimes used on
high-speed roadways with sharp horizontal
curves and are sometimes used to gradually
introduce the superelevation of an upcoming
horizontal curve

Dr. Rizwan Memon 46

Spiral Transition:

1. Spiral transition tends to promote uniformity in speed as a

vehicles enters and leaves a circular curve

2. The transition curve length provides a suitable location for

the superelevation runoff.

3. Use of spiral transition provides flexibility in accomplishing

the widening of sharp curves

4. The appearance of the highway or street is enhanced by

the application of spiral transition curves
Dr. Rizwan Memon 47
Dr. Rizwan Memon 48
 length of spiral – should be larger of 1, 2

V 3 V3
Ls ,min = 0.0214 OR Ls ,min = 3.15 (1)
RC RC
(2)
Ls ,min = 24( pmin ) R
Maximum length of spiral: Ls ,max = 24( Pmax ) R (3)
Where,

Ls,min = minimum length of spiral, (m or ft)

Ls,max = maximum length of spiral, m
Pmin = minimum lateral shift that occurs as a result of the natural steering
behavior of most drivers (0.20m or 0.66ft)
Pmax = maximum lateral shift that occurs as a result of the natural steering
behavior of most drivers (1.0 m or 3.3 ft)
R = radius of circular curve, m
V = design speed, (kmph or mph)
C = maximum rate Memon
Dr. Rizwan of change in lateral acceleration, (1.2 m/s3 or 4 ft/s3) 49