unit -2
curves
civil department
Definition
A transition curve differs from a circular curve in that its radius is alw
As one would expect, such curves involve more complex formulae
th with a constant radius and their design is more complex.
Because circular curves are easier to design compared to transition
they are easily set out on site, the question that naturally arises is
to transition necessary and why is it not possible to use circular
curves intersecting straights?
The need for Transition Curves
Circular curves are limited in road designs due to the forces which a
as they travel around a bend. Transition curves are used to introduc
gradually and uniformly thus ensuring the safety of passenger.
Transition curves have much more complex formulae and are more
out on site than circular curves as a result of the varying radius.
Radial Force and Design Speed
Radial forces act on a vehicle as it travels around a curve and this
is curves are necessary
A vehicle of mass m, travelling at a constant speed v, along a
curve subjected to a radial force P such that:
P
mv 2
r
This force acting on the vehicle is trying to push the vehicle back
on course. On a straight road where r = ∞, P = 0.
Roads are designed according to a ‘design speed’ which is constant
stretch of roadway. Thus a vehicle must be able to comfortable and
the length of a given stretch of road at the design speed regardless o
The mass of a vehicle is also assumed constant and consequently:
1
P∝
r
Thus the smaller the radius of a curve the greater the radial force
ac vehicle.
Any vehicle leaving a straight section of road and entering a circular
radius r will immediately experience the full radial for P. If the radius
and the thus P too large, the vehicle will skid off the roadway or over
Transition curves are curves in which the radius gradually changes f
a particular value R. The effect of this is to gradually increase the
rad zero to its maximum value, thereby reducing its effect.
To introduce P uniformly along the length of the transition curve, P
m proportional to the length of the transition curve l.
Therefore:
P∝
1
and
r
P∝ l
Combining the two requirements:
Thus
1
l∝
r
rl  K
Where K is a constant.
Therefore for each transition in a transition curve the radius R and
le be designed to equal to K over the whole length of the curve.
The use of Transition Curves
Transition curves can be used to join to straights in one of two ways:
- Composite curves
- Wholly transitional curves
Composite Curves
Here transition curves of equal length are used on either side of a
ce arc of radius R.
Wholly Transitional Curves
A wholly transitional curve consists of two transitional curves of
equa no central arc. The radius of this curve is constantly changing
and th force is constantly changing.
There is only one point Tc (the common tangent point) at which P is
This means wholly transitional curves are safer than composite curv
they cannot always be fitted between to straight due to minimum
rad requirements.
Superelevation
Although transition curves can reduce the effect of radial force on a
can also be further reduced or even eliminated by raising one side
o relative to the other. The difference in height between the two
sides known as the superelevation (SE).
In theory, by applying enough superelevation the resultant force
can act perpendicular to the surface of the road pushing the vehicle
dow throwing it off.
2
2
tan α  mv / R  v
mggR
and
SE  B tan α
Therefore:
Maximum theoretical superelevation, SEmax 
Bv 2
, where v is in gR
The maximum SE occurs where the radius r = R, along the central
a composite curve or at Tc on a wholly transitional curve.
In practice for roadways with high design speeds, wide
carriageways SEmax could be very large and would be alarming to
drivers approa dangerous with reduced speeds. Therefore the
following best practic generally applied:
Superelevation shall normally balance out only 45% of the radial
forc 0.45(Bv2/gR)).
In rural areas superelevation shall not exceed 7% (1 in 14.5 and
whe possible, radii should be chosen such that superelevation is
kept wit value of 5% (1 in 20)
In urban areas, superelevation shall not exceed 5%
The minimum allowable SE is 2.5% (1 in 40) to allow for drainage
Expressing v in kph and R in metres and substituting gravity for 9.81
2
Bv
Maximum allowable SE 
282.5R
Transition Curve Design Standards
The British standard for designing the radii of transitional and
circula depending on speed and superelevation is:
Types of Transition Curve
There are two types of curved used to form the transitional section
o or wholly transitional curve. These are:
-The clothoid
-The cubic parabola
For a transition curve the equation rl = K must apply i.e. the radius
m proportion to the length. This is the property of a spiral and one
curv this property is the clothoid.
Another common curve derived from the clothoid is the cubic
parabo a spiral i.e. rl is not always constant. However it can be used
over a and is less complex than the clothoid.
The clothoid
the equation of the clothoid can be derived from the above diagram,
two points close together (M and N) on a transition curve of lengt
Φ is the deviation angle between the tangent at M and the straight T
δis the tangential angle to M from T with reference to TI
x is the offset to M from the straight TI at a distance y from T
l is the length from point T to any point M on the curve (not shown)
δl is the length along the curve from M to N
Φl is the angle subtended by the arc δl of radius r
Derivation:
δl  rδφ (chord length), rl = K is required and substituting 1/r = (l/K)
δφ  l δl
K
Integration gives + constant. But when l = 0, , so constant = 0
Therefore:
2
l
φ 
(in radians)
2K
At the end of a transition r= R and l = LT, giving K = rl = RLT. This gi
φ
l2
(in radians)
2RLT
This is the basic equation of the clothoid. If its conditions are
satisfie constant radial force will be introduced uniformly.
The Cubic Parabola
The cubic parabola is not a true spiral and cannot always be used.
It very close to a spiral, however, and can be used within a
certain deviations angle.
In practice it is much easier set out a cubic parabola than a
clothoid more commonly used where appropriate.
The offset x is given by:
x
l
yl
3
and
6K
Substituting y = l in the expression for x gives
y3x
6K
Since
K  rl  RLT
It follows that:
y
3
x
6RLT
The length of a Transition Curve Required to Minimise Passeng
Transition curve lengths must be designed so that they minimise
pas discomfort and maximise safety. Consider the curve below:
At any point the radial force
P  mv
r
2
but since force = mass x acceleration the radial acceleration at any p
cv
r
2
i.e. the faster the change in radius the faster change in c an
the faster radial force is introduced resulting in passenger d
safety risks.
Design standards recommend a maximum value of c of 0.3m2/s,
ab passenger discomfort takes place.
The transition curve length LT can be determined from c as follows:
LT 
v3
cR
(where v is in m/s)
The Shift of a Cubic Parabola
In order that the tangent lengths can be calculated a parameter
know must also be calculated.
S
L 2
T
 VG
24R
VF 
S
 FG
2
Tangent Lengths and Total curve Lengths
In order for a composite cure to move vehicles through the deflection
transition curve must them through a further deflection φmax.
The tangent lengths are obtained as follows:
IT = IV + VT = IW + WU = IU
And
IT  ( R  S ) tan
θ
2

L
T
2
The Total curve length Ltotal is given by:
L
total
 2L  L
t
LCA  0
CA
LCA  R(θ − 2φmax )
Setting out composite and wholly transitional curves
The centre line provides an important reference on site from which
o can be established and it can be set out either by traditional or
coord methods.
Setting out using the tangential angles method
Setting out for tangential curves is similar to the method used for circ
This is the most accurate of the traditional methods and it can be us
transition curve. It is undertaken using a theodolite and a tape and,
circular curves method it is necessary to first set out the intersection
The method by which this is done is identical to that used for circular
the intersection point has been fixed, tangent points T and U
The entry and exit transition curves
I1 is chosen as a chord length such that it is ≤R/20, where R is the m of
curvature
δ1 is calculated from l1 using δ1 = (l21 /6 RLT )(180/ )
A theodolite is set at T, aligned to I with a reading of 00°00’00” and δ
A chord length of l1 is measured from T and lined in at point A using
A peg is driven into the ground at this point and a nail in this is used
The entry and exit transition curves
l2 is the distance around the curve from T to B.
2
δ 2 is calculated from l 2 using δ
2 = (l 2 / 6 RLT )(180 / π ) degrees
δ2 is set on the horizontal circle of the theodolite
A chord length (l2 − l1 ) is measured from A and lined in at point B
usin theodolite. A peg with a nail is used to locate B
This procedure is repeated for all subsequent setting out points up
to tangent point T at the end of the entry transition curve.
Setting out the pegs on the exit of a transition curve
The exit transition curve is set out from U to T2 with the theodolite se
aligned to I such that the horizontal circle is reading 00°00’00”. The
angles are the subtracted from 360°to give the required directions.
As for the entry transition curve, sub-chords are usually required at t
and end of the exit transition curve to ensure that pegs are placed
at multiples of through chainage.
If a wholly transitional curve is being set out, the common tangent
po the two transition curves is set out again, having already been
fixed the entry transition curve. The difference between the two
gives a m accuracy of the setting out.
Setting out the central circular arc
This only applies to composite curves since wholly transitional curve
central circular arc. The central circular arc is normally set out from T
is first necessary to establish the line of the common tangent at T1.
The next fig shows the entry transition curve and part of the central c
which the final tangential angle for T toT1 will be δ max φ max / 3
Move the theodolite to T1, align back to T with the horizontal circle re
180 − (2φmax / 3) The common tangent along T1N now corresponds
on the theodolite.
Rotate the telescope in azimuth until a reading of 00°00’00” is obtain
pegs on the circular arc from T1 to T2 using the tangential angles cir
Again, initial and final sub-chords are normally required to ensure
th located on the centre line at exact multiples of through chainage.
Finally, point T2, the second common tangent point is established. S
fixed when setting out the exit transition curve from U, the difference
two positions gives a measure of the accuracy of setting out.
In practice, the tangential angle and chord data are tabulated ready f
Setting out using offsets from the tangent lengths
This method is similar to that described for circular curves and
again the tangent points have been set out. Two tapes are required
and th best used for setting out short transition curves, since
accurate tapin more difficult as the curve gets longer.
In the case of a wholly transitional curve, the entry transition curve
is the tangent point on the entry straight and the exit transition curve
is the tangent point on the exit straight.
The next figure shows part of a cubic parabola transition curve. To s point Z on
the curve, the method involves choosing y and calculating x  y 3
For a complete curve, x and y values should be t
T
/ 6 RL
.
use on site.
In the case of a composite curve, the entry and exit transitions are
s same way as those for a wholly transitional curve and the central
cir then set out by offsets from the long chord.
Setting out using coordinate methods
The two traditional methods of establishing the centre lines of
compo wholly transitional curves have been described. Although
these meth used, they have been virtually superseded for all major
curves by co methods that use control networks.
In such methods, which are equally applicable to transition curves an
curves, the coordinates of points at regular intervals along the centre
calculated with reference to a site control network. The points are th
on site either using a total station set at points in the ground control
surrounding the scheme as shown in the next Figure.
These can also be set out by using a GPS receiver. In both cases
th coordinates of points to be fixed on the centre line and the
coordinat control network being used must be based on the same
site coordin system.
Nowadays, the coordinate calculations involved are usually done wit
computer software highway design packages and results of such
computations are normally presented in the form of computer printou
for immediate setting out use on site. The following table details a co
printout of the information required on the previous curve.
.
The curve is to be set out by bearing and distance from control
points with a total station, each centre line point being established
from one point and checked from another. The calculations required
to produce are as follows.
The coordinates of the control points are found from the control
surve The horizontal alignment is designed and the coordinates of
the inter tangent points are calculated.
Assuming that the centre line is to be pegged at exact multiples of
thr chainage, chord lengths and tangential angles are calculated for
the exit transition curves and the central circular arc.
The coordinates of the points to be established on the centre line
are using the chord lengths, tangential angles and the coordinates of
the and tangent points.
Control points which are visible from and which will give a good
inters the proposed centre line are found and the bearings/distances
are ca from the control points to the centre line points.
Coordinate methods compared with traditional methods
When compared with the traditional methods of setting out from the t
points, coordinate methods have a number of important advantages.
they are not always appropriate and some of the relative merits of
the categories of technique are listed below.
-Coordinate methods can be carried out by anyone who is capable
of station or a GPS receiver. Since the data is in the form of either
beari distances or coordinates, no knowledge of curve design is
necessary the case with traditional methods.
-The increased use of highway design computer software packages i
setting out data is presented ready for use in coordinate form has
pro corresponding increase in the adoption of such methods.
-The widespread use of computers has also greatly speeded up the
c procedures associated with coordinate methods, which were
always to be more difficult to perform by hand when compared with
those as with the traditional methods.
-Coordinate methods enable key sections of the centre line to be set
isolation, such as a bridge centre line, in order that work can
progres than one area of the site.
-Obstacles on the proposed centre line, which may be the subject of
can easily be by-passed using coordinate methods to allow work to
p while arbitration takes place. Once the obstacle is removed, it is an
e so establish the missing section of the centre line. This is not
usually with traditional methods.
-Coordinate methods have the disadvantage that there is very little
ch final setting out. Large errors will be noticed when the centre line
doe the designed shape, but small errors could pass unnoticed. In
the tan angles method, checks are provided by locating common
tangent poi two different positions.
-Although the widespread use of total stations and the increasing
use receivers on sites encourages the use of coordinate techniques,
such may not always be available and it may be simpler to use
traditional that work along the centre line. This will particularly be the
case wher cur1e are being set out, such as those used for roads on
housing est at roof intersections, short curves and boundaries.
Plotting the centre lines of composite and wholly transitional cu
Despite the widespread use of computer plotting facilities, there are s
occasions during the initial horizontal alignment design when it is to
undertake a hand drawing of the proposed centre line. For com
wholly transitional curves the following procedure is recommende
assumes that there is an existing plan of the area available.
1. Draw the intersecting straights in their correct relative positions
on tracing paper.
2 Calculate the length of each tangent using IT  IU  ( R  S )
tan(θ / 2
3 Plot the tangent points by measuring this distance along each straig
either side of the intersection point at the same scale as the existi
4 To plot the entry and exit transition curves, use the offsets from the
x y / 6RL
lengths. Use  3
T to prepare a table of offset values x for
sui values and ensure that the y values chosen will provide a
good de the centre line.
5 At the scale of the existing plan, plot the x and values on the tracing p
the tangent lengths to establish points on the entry and exit transition
cu shown in the next Figure.
6 To plot the central circular arc (where appropriate), carefully join the
p of the entry and exit transition curves. This is the long chord of the
centr arc.
7 Measure the offsets from the long chord method, prepare table of
offs for appropriate Y values. Again, ensure that the Y values chosen
will pr definition of the centre line.
8 At the scale of the existing plan, plot the X and Y values from the
long establish points on the centre line of the centre circular arc.
9 Carefully join all the points plotted to define the complete centre line.
French curves is useful for this purpose, although with care a
flexicurve used.
10 Superimpose the tracing paper on the existing plan and decide whet
the design is acceptable. If it is not, change the design and repeat the
p procedure.
Examples
On a proposed road having a design speed of 100 kph and a carriag
7.30 m, a composite curve consisting of two transition curves and a c
arc of radius 750 m is to join two intersecting straights having a defle
09°34”28’. The rate of change of radial acceleration for the road is to
The superelevation should be introduced at a rate of no more than 1
-Calculate the amount of superelevation that must be built into the
ce arc.
-Check that the transition curves are long enough for the
superelevati introduced.
Calculate the amount of superelevation that should be constructed
al transition curve at 20 m intervals from the entry tangent point.
1. The amount of superelevation that must be built into the central ci
2
Bv
maximum allowable SE =
282.8R
2
7.30
*100

282.2 * 750
 0.344m
then to express this as a %
s%=
v2

1002
2.828R 2.828 * 750
 4.71%
The radius of 750m is greater than the desirable min value of 720m
f SE of 4.71% is less than the value of 5%. Hence the 0.344m
SE into the central circular arc.
Checking that the transition curves are long enough
The length of each transition curve required for comfort and safety
is equation
LT

v
3
3.6 3 cR

100
3
 95.26m
3.6 3 * 0.3* 750
The superelevation value of 0.344 m must be introduced and
remove 95.26 m, which represents a gradient of
0.344
 0.36%
95.26
Since this is less that the maximum allowable rate of introduction of
1 transitions are long enough.
The amount of SE that should be constructed along the entry
transitio 20m intervals from the entry tangent point.
rl  K  RLT
K  95.26 * 750  71, 445
At 20m along the curve from entry tangent point
r  K  71, 445  3572.25m
20
20
(7.30 *1002 )
 0.07m
SE at 20m along the curve =
(282.8 * 3572.25)
1002
 0.99%
s% at 20m along the curve =
2.828 * 3572.25
Because this is less than the min allowable value of 2.5% for
drainag 2.5% must be used therefore
SE built at 20m along the curve = 2.5% of B = 0.025 * 7.30 = 0.18m