AN EDUSAT LECTURE ON
1
CURVES
Curves are regular bends provided in the lines
of communication like roads, railways and
canals etc. to bring about gradual change of
direction.
2
CURVES
They
enable
the
vehicle to pass from
one path on to another T1
when the two paths A
meet at an angle. They
are also used in the
vertical plane at all
changes of grade to
avoid
the
abrupt
change of grade at the
apex.
B
T2
C
O
Fig 1. A CURVE
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HORIZONTAL CURVES
Curves provided in the horizontal plane to have
the gradual change in direction are known as
horizontal curves.
VERTICAL CURVES
Curves provided in the vertical plane to obtain
the gradual change in grade are called as
vertical curves.
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5
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NEED OF PROVIDING CURVES
Curves are needed on Highways, railways
and canals for bringing about gradual change
of direction of motion. They are provided for
following reasons:i) To bring about gradual change in
direction of motion.
ii) To bring about gradual change in grade
and for good visibility.
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NEED OF PROVIDING CURVES
iii) To alert the driver so that he may not fall
asleep.
iv) To layout Canal alignment.
v) To control erosion of canal banks by the
thrust of flowing water in a canal.
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CLASSIFICATION OF CIRCULAR CURVES
Circular curves are classified as :
(i)
Simple Curves.
(ii) Compound Curves.
(iii) Reverse Curves.
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B
i) Simple Curve:
A simple curve
Consists of a
single arc of
circle connecting
two straights. It
has radius of the
same magnitude
throughout.
T2
T1
A
C
R
R
O
Fig. 2. A SIMPLE CURVE
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ii) COMPOUND CURVE
M
P
R1
O1
N
T1
R2
A
O2
Fig.3 Compound Curve
C
A compound Curve consists of two or
more simple curves having different radii
bending in the same direction and lying on
the same side of the common tangent. Their
centres lie on the same side of the curve.
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iii) REVERSE CURVE
A reverse curve is made up of two
arcs having equal or different
radii bending in opposite direction
with a common tangent at their
junction .
B
O2
R2
T2
R2
M
p
Their centres lie on
opposite sides of the curve. T1
R
Reverse curves are used
R
when the straights are A
parallel or intersect at a
O1
Fig. 4. A Reverse Curve.
very small angle.
N
1
1
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REVERSE CURVE
They are commonly used
in railway sidings and
sometimes on railway
tracks and roads meant
for low speeds. They
should be avoided as far
as possible on main lines
and highways where
speeds are necessarily
high.
B
O2
T2
P
M
N
T1
A
O1
Fig.5 A Reverse Curve.
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B’
B
φ
I
F
T1
E
A
T2
C
R
φ/2
φ
O
SIMPLE CIRCULAR CURVE
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NAMES OF VARIOUS PARTS OF CURVE
(i) The two straight lines AB and BC which
are connected by the curve are called the
tangents or straights to the curve.
(ii) The point of intersection of the two
straights (B) is called the intersection point
or the vertex.
(iii) When the curve deflects to the right side of
the progress of survey ,it is termed as right
handed curve and when to the left , it is
termed as left handed curve.
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NAMES OF VARIOUS PARTS OF CURVE
(iv) The lines AB and BC are tangents to the
curve. AB is called the first tangent or the
rear tangent . BC is called the second
tangent or the forward tangent.
(v) The points ( T1 and T2 ) at which the
curve touches the tangents are called
the tangent points. The beginning of
the curve ( T1) is called the tangent
curve point and the end of the curve
(T2) is called the curve tangent point.
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NAMES OF VARIOUS PARTS OF CURVE
(vi) The angle between the lines AB and BC
( ABC) is called the angle of intersection
└
(I).
(vii) The angle by which the forward tangent
deflects from the rear tangent (└B’BC) is
called the deflection angle (φ) of the curve.
(viii) The distance from the point of intersection
to the tangent point is called tangent length
( BT1 and BT2).
(ix)The line joining the two tangent points (T1
and T2) is known as the long chord.
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(x) The arc T1FT2 is called the length of curve.
(xi) The mid point(F) of the arc (T1FT2) is called
the summit or apex of the curve.
(xii) The distance from the point of intersection
to the apex of the curve BF is called the
apex distance.
(xiii) The distance between the apex of the curve
and the mid point of the long chord (EF) is
called versed sine of the curve.
(xiv) The angle subtended at the centre of the
curve by the arc T1FT2 is known as
central angle and is equal to the deflection
angle (φ) .
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ELEMENTS of a Simple Circular Curve
(i) Angle of intersection +Deflection angle = 1800.
or I + φ = 1800
(ii)
T1OT2 = 1800 - I = φ
└
i.e the central angle = deflection angle.
(iii)Tangent length = BT1 =BT2= OT1 tan φ/2
= R tan φ/2
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ELEMENTS of a Simple Circular Curve
(iv) Length of long chord =2T1E
=2R sin φ/2
(v) Length of curve = Length of arc T1FT2
= R X φ (in radians)
= πR φ/1800
(vi) Apex distance = BF = BO – OF
= R sec. φ/2 - R
= R (1 – cos φ/2 )=R versine φ/2
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DESIGNATION OF CURVE
A curve may be designated either by
the radius or by the angle subtended at the
centre by a chord of particular length.
a curve is designated by the angle (in
degrees) subtended at the centre by a chord of
30 metres (100 ft.) length. This angle is called
the degree of curve (D).
The degree of the curve indicates the
sharpness of the curve.
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DESIGNATION OF CURVES.
In English practice , a curve is defined
by the radius of the curve in terms of chains,
such as a six chain curve means a curve having
radius equal to six full chains, chain being 30
metres unless otherwise specified.
In America,Canada,India and some
other countries a curve is designated by the
degree of the curve
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RELATION between the Radius of curve and
Degree of Curve.
The relation between the radius
and the degree of the curve may
be determined as follows:-
M
Let R = the radius of the curve in metres.
D = the degree of the curve.
MN = the chord, 30m long.
P = the mid-point of the chord.
P
N
D
R
R
D/2
O
Degree of Curve
In
OMP,OM=R,
MP= ½ MN =15m
MOP=D/2
Then, sin D/2=MP/OM= 15/R
PTO
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RELATION between the Radius of curve and
Degree of Curve.
M
Then,sin D/2=MP/OM= 15/R
P
N
Or
R = 15
(Exact)
sin D/2
D
But when D is small, sin D/2 may be
R
R
assumed approximately equal to
D/2
D/2 in radians.
Therefore:
O
R = 15 X 360
Degree of Curve
πD
= 1718.87
D
0
Or say ,
R = 1719  This relation holds good up to 5
curves.For higher degree curves the
D
(Approximate) exact relation should be used.
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METHODS OF CURVE RANGING
A curve may be set out
(1) By linear Methods, where chain and tape
are used or
(2) By Angular or instrumental methods,
where a theodolite with or without a chain is
used.
Before starting setting out a curve by any
method, the exact positions of the tangents
points between which the curve lies ,must be
determined. Following procedure is adopted:25
METHODS OF SETTING OUT A CURVE
Procedure :i) After fixing the directions of the straights,
produce them to meet in point (B)
ii) Set up the Theodolite at the intersection
point (B) and measure the angle of
intersection (I) .Then find the deflection
angle ( φ ) by subtracting (I) from 1800 i.e
φ=1800 – I.
iii) Calculate the tangent length from the
following equation
Tangent length = R tanφ/2
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METHODS OF SETTING OUT A CURVE
Procedure :iv)
Measure the tangent length (BT1)
backward along the rear tangent BA from
the intersection point B, thus locating the
position of T1.
vi)
Similarly, locate the position of T2 by
measuring the same distance forward
along the forward tangent BC from B.
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METHODS OF SETTING OUT A CURVE
Procedure (contd…) :After locating the positions of the tangent
points T1 and T2 ,their chainages may be
determined. The chainage of T1 is obtained by
subtracting the tangent length from the known
chainage of the intersection point B. And the
chainage of T2 is found by adding the length
of curve to the chainage of T1.
Then the pegs are fixed at equal intervals
on the curve.The interval between pegs is
usually 30m or one chain length.
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METHODS OF SETTING OUT A CURVE
Procedure (contd…) :This distance should actually be measured along
the arc ,but in practice it is measured along
the chord ,as the difference between the chord
and the corresponding arc is small and hence
negligible. In order that this difference is
always small and negligible ,the length of the
chord should not be more than 1/20th of the
radius of the curve. The curve is then obtained
by joining all these pegs.
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METHODS OF SETTING OUT A CURVE
Procedure (contd…) :The distances along the centre line of the
curve are continuously measured from the
point of beginning of the line up to the end .i.e
the pegs along the centre line of the work
should be at equal interval from the beginning
of the line up to the end. There should be no
break in the regularity of their spacing in
passing from a tangent to a curve or from a
curve to the tangent. For this reason ,the first
peg on the curve is fixed
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METHODS OF SETTING OUT A CURVE
Procedure (contd…) :at such a distance from the first tangent point (T1)
that its chainage becomes the whole number of
chains i.e the whole number of peg interval. The
length of the first sub chord is thus less than the
peg interval and it is called a sub-chord. Similarly
there will be a sub-chord at the end of the curve.
Thus a curve usually consists of two sub-chords
and a no. of full chords.
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LINEAR METHODS of setting out Curves
The following are the methods of setting out
simple circular curves by the use of chain
and tape :(i)
By offsets from the tangents.
(ii) By successive bisection of arcs.
(iii) By offsets from chords produced.
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LINEAR METHODS of setting out Curves
1. By offsets from the tangents. When the
deflection angle and the radius of the
curve both are small, the curves are set out
by offsets from the tangents.
Offsets are set out either
(i) radially or
(ii) perpendicular to the tangents
according as the centre of the curve is
accessible or inaccessible
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LINEAR METHODS of setting out Curves
B’
B
P Ox
x
P1
T1
φ
T2
900
A
C
R
O
Fig. By Radial Offsets
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LINEAR METHODS of setting out Curves
B’
Offsets is given by :
Ox = R2 +x2 – R …….. (Exact relation.)
When the radius is large ,the offsets may be
calculated by the approximate formula
which is as under
Ox = x 2
……… (Approximate )
2R
By Radial Offsets
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LINEAR METHODS of setting out Curves
B’
B
P
x
Ox
P1
T2
T1
A
P2
B
O
(ii) By offsets perpendicular to the Tangents
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LINEAR METHODS of setting out Curves
1. (ii) By offsets perpendicular to the Tangents
Ox= R – R2 – x2
Ox = x 2
2R
…………… (Exact)
……… (Approximate )
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LINEAR METHODS of setting out Curves
By offsets from the tangents: Procedure
(i)
Locate the tangent points T1 and T2.
(ii) Measure equal distances , say 15 or 30 m
along the tangent fro T1.
(iii) Set out the offsets calculated by any of
the above methods at each distance ,thus
obtaining the required points on the
curve.
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LINEAR METHODS of setting out Curves
By offsets from the tangents: Procedure….
(iv) Continue the process until the apex of
the curve is reached.
(v) Set out the other half of the curve from
second tangent.
(vi) This method is suitable for setting out
sharp curves where the ground outside
the curve is favourable for chaining.
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Example. Calculate the offsets at 20m intervals along
the tangents to locate a curve having a radius of
400m ,the deflection angle being 600 .
Solution . Given:
Radius of the curve ,R = 400m
Deflection angle, φ = 600
Therefore tangent length = R. tan φ/2
= 400 x tan 600
= 230.96 m
Radial offsets. (Exact method)
Ox= R2 + x2 - R
…………… (Exact)
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Radial offsets. (Exact method)
Ox= R2 + x2 - R
…………… (Exact)
O20 = 4002+202 - 400 = 400.50 - 400 = 0.50 m
O40 = 4002+402 - 400 = 402.00 - 400 = 2.00 m
O60 = 4002+602 - 400 = 404.47 - 400 = 4.47 m
O80 = 4002+802 - 400 = 407.92 - 400 = 7.92 m
O100 = 4002+1002- 400 = 412.31 - 400 = 12.31 m
And so on….
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B) Perpendicular offsets
Ox = R
– R2 – x2
(Exact method)
…………… (Exact)
O20 = 400 -
4002 - 202 = 400 -399.50 = 0.50 m
O40 = 400 -
4002 - 402 = 400 -398.00 = 2.00 m
O60 = 400 -
4002 - 602 = 400 -395.47 = 4.53 m
O80 = 400 -
4002 - 802 = 400 -391.92 =8.08 m
O100 = 400 -
4002 -1002 = 400 -387.30 =12.70 m
And so on…..
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B) By the approximate Formula
(Both radial and perpendicular offsets)
Ox =
x2
2R
Therefore O20 = 202
= 0.50 m
2x400
O40 = 402
= 2.00 m
2x400
O60 = 602
= 4.50 m
2x400
O80 = 802
= 8.00 m
2x 400
O100 = 1002
= 12.50 m
2 x 400
and so on….
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