unacademy appย ยท โขfor highway 2 ๐ =1 4 ... this equation is called as euler spiral, clothoid...
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Chapter 9: Curves
โข Curve are defined as Arc with some finite radius, provided between intersecting straight lines to gradually negotiate change in direction
โข This change in direction of straight line may be in a horizontal plane (or) Vertical plane, resulting in the provision of a horizontal (๐๐) vertical curve respectively.
5Civil Engineering by Sandeep Jyani
Horizontal Curves
โข A simple circular curve consist of an Arc of a circle which is tangential to the straight line at both the ends.
6
1. Simple Circular Curve 2. Compound Curve
โข A compound curve consist of two circular arcs of different radius with their centre of curvature on the same side.
R1
R2
O1
O2
Civil Engineering by Sandeep Jyani
3. Reverse Curve / S- Curve / Serpentine Curve
โข When two simple circular curves of equal (or) different radius having opposite direction of curvature join together, the resultant curve is called as โReverse curveโ
โข Reverse curves are provided between two parallel Lines (or) when angle between them is very small.
โข They are commonly used in railway yard but unsuitable for Highways.
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3. Reverse Curve / S- Curve / Serpentine Curve
R1
R2
Civil Engineering by Sandeep Jyani
4. Transition curve / Easement curve
โข Transition curve is usually introduced between a simple circular curve and a straight line, vice versa
โข Radius of Transition curve gradually varies from finite to infinite value and vice-versa.
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Note:
โ We have to provide a transition curve between two
branches of compound Curve and reverse curve
R1
R2
O1
O2
๐ซ = โ๐ซ = ๐น
Civil Engineering by Sandeep Jyani
SIMPLE CIRCULAR CURVE
โข ๐ฉ๐ป = ๐ฉ๐๐๐ ๐ป๐๐๐๐๐๐
โข ๐ญ๐ป = ๐ญ๐๐๐๐๐๐ ๐ป๐๐๐๐๐๐
โข ๐ท๐ช ๐ท๐๐๐๐ ๐๐ ๐๐๐๐๐(๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐)
โข โ= ๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐
โข ๐ฐ = ๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐
โข โ ๐ป๐ ๐ถ๐ป๐ = ๐๐๐๐๐๐๐ ๐๐๐๐๐ = โ
โข ๐ป = ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐
โข ๐ณ = ๐ป๐๐ป๐ = ๐๐๐๐ ๐๐๐๐๐
โข ๐ช๐ซ = ๐๐๐ ๐๐๐ ๐๐๐๐๐ = ๐ด
โข ๐ฌ = ๐๐๐๐ ๐ ๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐๐๐๐
โข ๐ = ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐
9
๐ฝ
โ
๐ฌ
๐ด๐ณ
๐
๐ณ
๐
๐น ๐น
๐
๐ป๐ ๐ป๐๐ซ
๐ฐ
โ
๐
โ
๐๐ฉ๐ป ๐ญ๐ป
๐ถ
๐ช
๐ท๐ช๐ท๐ป
Civil Engineering by Sandeep Jyani
Elements of Simple Circular Curve1. Length of curve (๐):
โข ๐ =๐๐ ฮ
180ยฐ(in radians)
2. Tangent Length (T)
โข ๐ป = ๐ป๐๐ฐ = ๐ป๐๐ฐ = ๐น tanโ
๐
3. Length of long chord (L) :โข ๐ฟ = ๐1๐2 = 2 ๐ . ๐ ๐๐
ฮ
2
4. Mid ordinate ๐ :โข ๐ = ๐ 1 โ cos
ฮ
2
5. External distance (E):
โข ๐ธ = ๐ ๐ ๐๐ฮ
2โ 1
โข ๐๐๐ ฮ
2=
๐
๐ธ+๐
6. Chainages of T1 and T2
โข ๐ถโ ๐๐ ๐1 = ๐โ ๐๐ก ๐ผ โ ๐๐๐๐๐กโ ๐
โข ๐ถโ ๐๐ ๐2 = ๐โ ๐๐ก ๐1+ ๐๐๐๐๐กโ ๐
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๐ฝ
โ
๐ฌ
๐ด๐ณ
๐
๐ณ
๐
๐น ๐น
๐
๐ป๐ ๐ป๐๐ซ
๐ฐ
โ
๐
โ
๐๐ฉ๐ป ๐ญ๐ป
๐ถ
๐ช
Civil Engineering by Sandeep Jyani
Note:
11Civil Engineering by Sandeep Jyani
Designation of Curveโข A curve can be designated by radius R (๐๐) Degree of curve (D).
โข Degree of curve is the angle subtended by an Arc (๐๐) a chord of specified length at the centre.
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1. Arc Definition:โข Case 1: Let arc length is 30m and radius of curve is R, the n degree of
curve is D
๐น ๐น๐ซ
๐๐๐๐ ๐น๐ซ
๐๐๐ยฐ= ๐๐๐
=> ๐ซ =๐๐ร๐๐๐
๐ ๐น
=> ๐ซ =๐๐๐๐.๐๐
๐น
โด ๐ซ =๐๐๐๐
๐นRemember
Civil Engineering by Sandeep Jyani
Designation of Curve
โข For 30m ๐ซ =๐๐๐๐
๐น
โข For 20m ๐ซ =๐๐๐๐
๐น
13
1. Arc Definition:โข Case 2: Let arc length is 20 m and radius of curve is R,
the n degree of curve is D
๐น ๐น๐ซ
๐๐๐๐ ๐น๐ซ
๐๐๐ยฐ= ๐๐๐
=> ๐ซ =๐๐ร๐๐๐
๐ ๐น
=> ๐ซ =๐๐๐๐.๐๐
๐น
โด ๐ซ =๐๐๐๐
๐นRemember
Civil Engineering by Sandeep Jyani
Designation of Curve
โข For 30m ๐ซ =๐๐๐๐
๐น
โข For 20m ๐ซ =๐๐๐๐
๐น
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2. Chord Definition:โข Case I: for 30m chord
๐น ๐น๐ซ
๐
๐๐๐
๐๐๐๐ซ
๐=๐๐
๐น
Since, ๐ซ
๐will be a small angle, therefore ๐๐๐๐ฝโ ๐ฝ
=>๐ซ
๐ร
๐
๐๐๐ยฐ=
๐๐
๐น
=> ๐ซ =๐๐ร๐ร๐๐๐ยฐ
๐ ๐น=
๐๐๐๐
๐น
๐๐๐ ๐๐๐
๐ซ
๐
โข Case II: for 20m chord
๐๐๐๐ซ
๐=๐๐
๐น
Since, ๐ซ
๐will be a small angle, therefore ๐๐๐๐ฝโ ๐ฝ
=>๐ซ
๐ร
๐
๐๐๐ยฐ=
๐๐
๐น
=> ๐ซ =๐๐ร๐ร๐๐๐ยฐ
๐ ๐น=
๐๐๐๐
๐น
๐น ๐น๐ซ
๐
๐๐๐
๐๐๐ ๐๐๐
๐ซ
๐
Civil Engineering by Sandeep Jyani
Note:
โข Since Degree of curve is inversely proportional to Radius, for sharp circles Degree of curve will be large, whereas for flat curve, Degree of curve will be small.
15Civil Engineering by Sandeep Jyani
Que : if Radius of curve is 1000 m, ฮ = 60ยฐ, chainage of P.I = 2000m
Determine
i) length of curve
ii) Tangent Length
iii) Long chord
iv) mid ordinate (M)
v) Apex distance
vi) Chainages of ๐1, ๐2vii) Degree of curve for 30 m Arc
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ii) ๐ป = ๐น๐๐๐๐
๐= ๐๐๐๐ ๐๐๐ ๐๐ยฐ = ๐๐๐. ๐๐๐
iii) ๐ณ = ๐ ๐น๐๐๐๐
๐= ๐ ร ๐๐๐๐๐๐๐๐๐ยฐ = ๐๐๐๐๐
iv) ๐ด = ๐น ๐ โ ๐๐๐๐
๐= ๐๐๐๐ ๐ โ ๐๐๐๐๐ยฐ = ๐๐๐. ๐๐ ๐
v) ๐ฌ = ๐น ๐บ๐๐๐
๐โ ๐ = ๐๐๐๐ ๐๐๐๐๐ยฐ โ ๐ = ๐๐๐ ๐๐๐
vi) ch of ๐ป๐ = ๐๐๐๐ โ ๐๐๐. ๐๐ = ๐๐๐๐. ๐๐ ๐
ch of ๐ป๐ = ๐๐๐๐. ๐๐ + ๐๐๐๐. ๐๐ = ๐๐๐๐. ๐๐ ๐
vii) ๐ซ =๐๐๐๐
๐๐๐๐= ๐. ๐๐๐
i) ๐ =๐ ๐น ๐
๐๐๐ยฐ=
๐ (๐๐๐๐)ร๐๐
๐๐๐= ๐๐๐๐. ๐๐ ๐
Civil Engineering by Sandeep Jyani
Setting out of Simple Circular Curveโข Setting out of a curve is a process of locating various points along the
length of the curve at equal and convenient distances.
โข Distance between two successive points is called as โpeg intervalโ Generally peg interval is 20 m (๐๐) 30 m, but for sharp curves it may be further reduced.
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Linear Methods (Only chain (๐๐) Tape) Angular Methods (Theodolite with (๐๐)
without chain (๐๐) Tape)
1. Perpendicular offset from long chord 1. Deflection angle method
2. Perpendicular offset from Tangent
3. Radial offset from Tangent
2. Two theodolite method
4. Successive bisection of Arc offset from
chord produced
3. Tacheometric distance method.
Civil Engineering by Sandeep Jyani
Transition Curves
โข Transition curve is a curve of varying radius introduced between a straight line and a circular curve.
โข Transition curve provides a gradual change from straight line to the circular curve and from circular curve to the straight line also
18Civil Engineering by Sandeep Jyani
โข Basic criteria for design of Transition Curve:
1. It should be tangential to the straight line and also meet the circular curve tangentially at the junction
2. Its Curvature should be zero (๐ = โ) at one
end and its curvature should be equal to ๐
๐นwhere it meets the circular curveโข ๐น โ Radius of circular curve
3. Rate of increase of curvature along the Transition curve should be equal to Rate of increase of Super Elevation.
19
๐ซ = โ๐ซ = ๐น
๐ = ๐ ๐
Civil Engineering by Sandeep Jyani
Transition Curves
Super Elevation
โข Super Elevation:โข Super elevation is the vertical distance by
which outer end of the road is raised above the inner one
โข For equilibrium condition:
โ ๐ ๐๐๐ ๐ผ = ๐ ๐ ๐๐๐ผ
โ tan๐ผ =๐
๐=๐๐2
๐๐๐=๐2
๐๐
[๐ =๐๐ฃ2
๐, ๐ = ๐๐]
โข Since, ๐ผ will be very small angle therefore, tan ฮฑ tends to sinฮฑ
(๐. ๐, tan ๐ผ โ sin ๐ผ ๐๐๐ ๐ ๐๐๐ผ =๐2
๐๐)
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๐ถ๐พ๐๐๐๐ถ
๐ท๐๐๐๐ถ
๐ท ๐ถ
๐พ
๐๐ถ
๐พ๐๐๐๐ถ
๐. ๐, . ๐๐๐๐ถ =๐ยฒ
๐๐๐๐๐๐ถ =
๐
๐ฎ
๐
๐ฎ=
๐ฝยฒ
๐๐(๐ < ๐% ๐๐๐๐๐ ๐๐ ๐ฐ๐น๐ช)
๐ =๐ฎ๐๐
๐๐
The value of super elevation (e) cannot be as high
as possible because high super elevation can cause
Toppling of vehicle in presence of cross winds. In
such case, either large radius is provided or velocity
is reducedCivil Engineering by Sandeep Jyani
โข Maximum Centrifugal ratio: โ๐
๐=
๐2
๐๐
โข To avoid inconvenience to the passengers, the maximum value of centrifugal ratio is generally specified as
โข For highway๐2
๐๐ =
1
4โ ๐ =
๐๐
4
โข For Railways ๐2
๐๐ =
1
8โ ๐ =
๐๐
8
21Civil Engineering by Sandeep Jyani
Ideal Transition Curve Equation
โข A curve of variable radius of required length is inserted between straight road and a circular curve such that centrifugal force increases uniformly and gradually along the length of Transition Curve, so that lateral shock and discomfort is minimized
๐ โ ๐ i. e.
๐ =๐๐ฃ2
๐and for constant mass and velocity,
๐ โ1
๐๐๐
๐๐ = ๐๐๐๐ ๐ก๐๐๐ก
This equation is called as Euler Spiral, Clothoid Curve, Glover Spiral curve and Ideal Transition Curveโข At the end of transition curve, ๐ = ๐ฟ ๐๐๐ ๐ = ๐ โข Therefore at the end of the transition curve ๐ฟ๐ = ๐๐๐๐ ๐ก๐๐๐ก
22
๐ซ
๐
Civil Engineering by Sandeep Jyani
Length of Transition Curve1. Arbitrary value from past experience
2. Such that super elevation is applied at 1
๐๐๐๐ ๐ is the total super elevation to be
provided the end of the curve
๐ฟ =๐
1๐
3. Such that rate of change of radial acceleration is within the desired limit
โ=
๐2
๐ ๐กโ โ=
๐2
๐ ๐ฟ๐
โ ๐ณ =๐ฝ๐
โ ๐น
23๐ซ = โ
๐ซ = ๐น
๐2
๐=0
๐2
๐
Civil Engineering by Sandeep Jyani
Que: A transition curve is required for a radius of 30m, gauge length is 1m and maximum super elevation is restricted to 100mm. Permissible value of rate of change of radial acceleration = 30cm/sec2. Determine length of required transition curve.
Solution:
๐ณ =๐ฝ๐
โ ๐น
โ ๐. ๐ =๐ ร ๐ฝ๐
๐. ๐๐ ร ๐๐๐โ ๐ฝ = ๐๐. ๐๐๐๐/๐๐๐
๐ณ =๐ฝ๐
โ ๐น
โ ๐ณ =(๐๐. ๐๐๐)๐
๐. ๐ ร ๐๐๐= ๐๐. ๐๐๐
24
And we know that ๐ =๐ฎ๐๐
๐๐
Civil Engineering by Sandeep Jyani
Cubic Spiral Curve
โข Ideal transition curve is a cubic Spiral Curve
25
๐ ๐ =๐ณ๐
๐๐น๐ณ
๐ฉ๐ป
๐ =๐๐
๐๐น๐ณ
๐ฉ๐ป
๐ =๐ณ๐
๐๐น๐ณ
Cubic Parabola
โข Also known as โFroude Transition Curveโ
โข Cubic parabola more resembles Ideal transition curve in comparison to cubic parabola
โข Setting out cubic parabola is easy than cubic spiral, so cubic parabola is commonly used
โข But after invention of electronic equipment like total station, nowadays any curve can be set out so cubic parabola is obsolete.
Civil Engineering by Sandeep Jyani
Vertical Curve
โข A vertical curve is used to connect two different gradients of Highway and Railway.
โข Vertical curve can be Parabolic (๐๐) circular
โข Parabolic curve is preferred over circular curve because.โข It is flatter at top and provides longer sight distance
โข It is simple to layout
โข Rate of change of gradient is constant.
26Civil Engineering by Sandeep Jyani
Vertical Curves
When down gradient is followed by Up gradient
27
SUMMIT CURVESAG CURVE/Valley Curve
Steep down gradient is followed by mild down gradient
Mild up gradient is followed by steep up gradient
When up gradient is followed by down gradient
Mild down gradient is followed by steep down gradient
Steep gradient followed by Mild up gradient
Civil Engineering by Sandeep Jyani
โข Total change of grade: is the algebraic difference of two gradients
โข Length of Vertical Curve:
โข ๐ฟ๐๐๐๐กโ ๐๐ ๐ฃ๐๐๐ก๐๐๐๐ ๐๐ข๐๐ฃ๐ =๐๐๐ก๐๐ ๐โ๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐ก
๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐๐๐ก๐ ๐๐ ๐โ๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐ก
Que: A parabolic curve is to be set out connecting two uniform gradients of +1.6% and +1.0%. The permissible rate of change of gradient is 0.1 % per 30m chain length. Length of vertical curve will be?
Solution: ๐ฟ๐๐๐๐กโ ๐๐ ๐ฃ๐๐๐ก๐๐๐๐ ๐๐ข๐๐ฃ๐ =1.6%โ1.0%
0.1%
30
= 180๐
28
Vertical Curves
+๐๐% โ๐๐%
+๐๐ โ (โ๐๐)
Civil Engineering by Sandeep Jyani
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Civil Engineering by Sandeep Jyani
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