Introduction, classification of curves, Elements of a simple circular, designation of curve, methods of setting out a simple circular curve, elements of a compound and reverse curves, transition curve, types of transition curves, combined curve, types of vertical curves

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PREPARED BY : ASST. PROF. VATSAL D. PATEL
MAHATMA GANDHI INSTITUTE OF
TECHNICAL EDUCATION &
RESEARCH CENTRE, NAVSARI.

2.  Curves are generally used on highways and railways where it
is necessary to change the alignment.
 A curve is always tangential to the two straight directions.
 The two straight lines connected by a curve are called
tangents.
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3. Curve
Horizontal curve
Simple curve
Compound curve
Reverse curve
Transition curve
Combined curve
Vertical curve
Summit curve
Sag curve
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4.  There are three types of circular curves.
1. Simple curve
2. Compound curve
3. Reverse curve
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5.  A simple circular curve consists of a single arc of the circle.
 It is tangential to both the straight lines.
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6.  A compound curve consists of two or more simple arcs.
 The simple arcs turn in the same direction with their centres of
curvature on the same side of the common tangent.
 In figure, an arc of radius R1 has center O1 and the arc of
radius R2 has center O2.
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7.  A reverse curve consists of two circular arcs which have their
centres of curvature on the opposite side of the common
tangent.
 The two arcs turn in the opposite direction.
 Reverse curves are provided for low speed roads and railways.
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8.  Fig. shows a simple circular curve of radius R with centre at O
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9.  Back tangent:
 The tangent (AT1) previous to the curve is called the back
tangent or first tangent.
 Forward tangent:
 The tangent (T2B) following the curve is called the forward
tangent or second tangent.
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10.  Point of Intersection (P.I.):
 If the tangents AT1 and AT2 are produced they will meet in a
point, called the point of intersection. It is also called vertex
(V).
 Point of curve (P.C.):
 It is the beginning point T1 of a curve. At this point the
alignment changes from a tangent to a curve.
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11.  Point of tangency (P.T.):
 The end point of a curve (T2) is called the point of tangency.
 Intersection angle (ф):
 The angle AVB between tangent AV and tangent VB is called
intersection angle.
 Deflection angle (Δ):
 The angle at P.I. between the tangent AV produced and VB is
called the deflection angle.
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12.  Tangent distance:
 It is the distance between P.C. to P.I. It is also the distance
between P.I. to P.T.
 External distance (E):
 It is the distance from the mid. point of the curve to P.I.
 It is also called the apex distance.
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13.  Length of curve (l):
 It is total length of curve from P.C. to P.T.
 Long chord:
 It is chord joining P.C. to P.T. T1 T2 is a long chord.
 Normal chord:
 A chord between two successive regular stations on a curve is
called normal chord.
 Normally, the length of normal chord is 1 chain (20 m).
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14.  Sub chord:
 The chord shorter than normal chord (shorter than 20 m ) is
called sub chord.
 Versed sine:
 The distance between mid. point of long chord (D) and the
apex point C is called versed sine.
 It is also called mid-ordinate (M).
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15.  Right hand curve:
 If the curve deflects to the right of the direction of the progress
of survey, it is called right-hand curve.
 Left hand curve:
 If the curve deflects to the left of the direction of the progress
of survey, it is called left hand curve.
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16.  The sharpness of curvature of a curve may be expressed in
any of the following ways :
1. Radius of the curve
2. Degree of the curve
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17.  Radius of the curve (R) :
 In this method the curve is known by the length of its radius.
 For example, 200 m curve means the curve having radius
200 m. 6 chain curve means the curve having radius equal to
6 chain.
 This method is used in England.
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18.  Degree of the curve (D) :
 In this method the curve is designated by degree.
 The degree of curvature can be defined by two ways:
1. Chord definition
2. Arc definition
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19.  Chord definition :
 The angle subtended at the centre of curve by a chord of
20 m is called degree of curvature.
 For example, If an angle subtended at the centre of curve by
a chord of 20 m is 5˚, the curve is called 5˚ curve.
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20.  Arc definition:
 The angle subtended at the centre of curve by an arc of 20 m
length, is called degree of curve.
 This system is used in America, Canada, India, etc.
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21.  By chord definition :
 The angle subtended at the centre of curve by a chord of 20
m is called degree of curve.
 R = radius of curve
 D = degree of curvature
 PQ = 20 m = length of chord
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22.  By chord definition :
 From triangle PCO.
 When D is small, may be taken equal to
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23.  By chord definition :
(Where, D is in degree)
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24.  By arc definition :
 The angle subtended at the centre of curve by a chord of 20
m is called degree of curve.
(Where, D is in degree)
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25.  By arc definition :
 For 30 m arc
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26.  Based on the instruments used in setting out the curves on
the ground there are two methods:
1. Linear method
2. Angular method
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27.  Linear methods :
 In these methods only tape or chain is used for setting out the
curve. Angle measuring instrument are not used.
 These methods are used where a high degree of accuracy is
not required and the curve is not short.
 Main linear methods are: (1) By offsets from the long chord,
(2) By successive bisection of arcs or chords, (3) By offsets
from the tangents
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28.  By offsets from the long chord
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29.  R = Radius of the curve
 O0 = Mid ordinate
 Ox = Ordinate at distance x from the mid point of the chord
 T1 and T2 = Tangent points
 L = Length of long chord
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30.  To obtain equation O0 :
 From triangle OT1D :
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31.  In order to calculate Ox to any point E, draw the line EE1,
parallel to the long chord T1 T2. Join EO to cut the long
chord in G.
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32.  By successive bisection of arcs or chords
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33.  Join the tangent points T1 T2 and bisect the long chord at D.
 Erect perpendicular DC at D equal to the mid ordinate (M).
 Mid ordinate,
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34.  Join T1C and T2C and bisect them at D1 and D2 respectively.
 At D1 and D2 set out perpendicular offsets
and obtain points C1 and C2 on the curve.
 By the successive bisection of these chords more points may
be obtained.
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35.  By offsets from the tangents
 The offsets from the tangent can be of two types :
1. Radial offsets
2. Perpendicular offsets
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36.  Radial offsets :
 Let Ox = Radial offset DE at any distance x from T1 along
the tangent.
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37.  Perpendicular offsets :
 Ox = offset perpendicular to the tangent, so DE = Ox
 T1 D = x, measured along the tangent,
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38.  Angular method :
 In this method instruments like theodolite are used for
setting out the curves. Sometimes chain or tape is also used
with the theodolite.
 These methods are used when the length of curve is large.
 These methods are more accurate than Linear methods.
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39.  The Angular methods are :
1. Rankine method of tangential angles
2. Two theodolite method
3. Tacheometric method
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40.  Rankine method of tangential angles :
 This method is most frequently used for setting out the
circular curves of large radius and considerable length.
 This method is useful for setting out the curves for Railway,
Highway and Expressway with more accuracy.
 In this method, only one theodolite is used, hence it is called
One Theodolite Method.
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41.  Rankine’s Principle:
 “A deflection angle to any point on the curve is the angle at
P.C. between the back tangent and the chord from P.C. to that
point.”
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42.  Rankine’s method (Deflection angle method):
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43.  Procedure:
1. Set out 𝑇1 and 𝑇2.
2. Set the theodolite at P.C. 𝑇1.
3. With both the plates clamped to zero, direct the theodolite to
bisect the point of intersection (V).
4. Release the upper clamp screw and set angle ∆1 on the
vernier. The line of sight is thus directed along the chord
T1A.
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44.  Procedure:
5. With zero end of the tape pointed at T1 and an narrow held at
a distance T1A=C1, swing the tape around T1 till the arrow is
bisected by the cross hairs. Thus, the first point A is fixed.
6. Now Release the upper plate and set the second deflection
angle ∆2 on the vernier so that the line of sight is directed
along T1B.
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45.  Procedure:
7. With the zero end of the tape pinned at Aand an arrow held
at a distance AB = C2 swing the tape around A till the
narrow is bisected by the cross hairs. Thus, the second point
B is fixed.
8. Repeat the steps 6,7 till the last point T2 is reached.
9. Join the points T1,A,B,C….T2 to obtain the required curve.
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46.  Two theodolite method :
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47.  Two theodolites are used in this, one at P.C. and at P.T.
 In this method, tape/chain is not required.
 This method is used when the ground is unsuitable for
chaining.
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48.  Procedure:
 Setup one theodolite at P.C. (𝑇1) and the other at P.T. (𝑇2)
 Clamp both plates of each transit to zero reading.
 With zero reading, direct the line of sight of the transit at 𝑇1
towards V. Similarly direct the line of sight of the other
transit at 𝑇2 towards 𝑇1. Vernier A of both the theodolites will
show zero reading.
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49.  Procedure:
 Set the reading of each of the transits to the deflection angle
for the first point A equal to ∆1. The line of sight of theodolite
at 𝑇1 will be along 𝑇1A and the line of sight of theodolite at 𝑇2
will be along 𝑇2A.
 Move a ranging rod or an arrow in such a way that it is
bisected simultaneously by cross hairs of both the instruments.
Thus, point A is fixed.
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50.  Procedure:
 Now, to fix the second point B, set reading ∆2 on both the
instruments and bisect the ranging rod.
 Repeat the above steps to obtain other points.
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51.  Tacheometric method :
 In this method, the angular and linear measurements are made
by using a tacheometer.
 This method is less accurate than Rankine’s method but the
advantage is that, chaining is completely eliminated.
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52.  Tacheometric method :
 In this method, a point on the curve is fixed by the deflection
angle from, the rear tangent and by using tacheometrically, the
distance of that point from P.C. (𝑇1) and not from the
preceding point as in Rankine’s method.
 Thus, each point is fixed independently and the error in setting
out is not carried forward.
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53.  Tacheometric method :
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54.  Procedure:
 Set the tacheometer at 𝑇1 and sight the point of intersection
(V) when the reading is zero.
 Set the deflection angle ∆1 on the vernier, thus directing the
line of sight along 𝑇1A.
 Direct the staff man to move in the direction 𝑇1A till the
calculated staff intercept 𝑆1 is obtained. The staff is generally
held vertical. First point A is fixed.
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55.  Procedure:
 Set the deflection angle ∆2 directing the line of sight along
𝑇1B. Move the staff backward or forward until the staff
intercept 𝑆2 is obtained thus fixing the point B.
 Similarly, other points are fixed.
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56.  A compound curve consists of two or more circular arcs of
different radii with their centres of curvature on the same side
of the common tangent.
 Compound curves are required when space restrictions
preclude a single circular curve and when there are property
boundaries.
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57.  Elements of a compound curve :
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58.  Fig. shows a two centred compound curve T1T3T2 having two
circular arcs T1T3 and T3T2 meeting at common point T3
known as the point of compound curvature (P.C.C).
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59.  T1 is the point of curve(P.C)
 T2 is the point of tangency(P.T)
 RS, RL = the radius of thecurve
 ΔS, ΔL = the deflection angle
 lS, lL = length of curve
 tS, tL = the tangent length
 TS, TL = the tangent length
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60.  Tangent length of circular curve :
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61.  Total deflection angle :
Δ = Δs +ΔL
 Tangent length of compound curve :
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62.  Tangent length of compound curve :
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63.  Length of compound curve :
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64.  Chainages :
Chainage of T1 = Chainage of B –TS
Chainage of T2 = Chainage of T1 +lS
Chainage of T3 = Chainage of T3 +lL
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65.  Reversed curve, though pleasing to the eye, would bring
discomfort to motorist running at design speed.
 Despite this fact, reversed curves are being used with great
success on park roads, formal paths, waterway channels.
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66.  Elements of reverse curve :
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67.  Radii R1 and R2 of the two circular arcs.
 Angle of total deflection Δ between straights.
 Central angle or angle of deflection (Δ1 and Δ2) of the
common tangent.
 Angle ( δ1, δ2 ) between the straights and the joining the points
of commencement and tangency.
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68.  When a vehicle moves on a curve, there are two forces acting
1. Weight of the vehicle (W)
2. Centrifugal force (P)
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69.  The centrifugal force is given by,
 Where, P = Centrifugal force in kg or N
 W = Weight of the vehicle in kg or N
 V = Speed of the vehicle, m/sec2
 g = Acceleration due to gravity, m/sec2
 R = Radius of curve
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70.  The centrifugal force (P) is inversely proportional to the
radius of the curve (R).
 As the radius decrease, centrifugal force increases. Straight
road has infinite radius of curvature. Hence, centrifugal force
on vehicles moving on straight road is zero.
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71.  When a vehicle enters from straight road to the curve, its
radius changes from infinite to R, resulting in sudden
centrifugal force P on the vehicle. It causes the vehicle to
sway outwards. If this exceeds a certain value the vehicle may
overturn.
 To avoid these effects, a curve of changing radius must be
introduced between the straight and the circular curve. Such a
curve, is known as transition curve.
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72.  It should be tangential to the straight.
 It should meet the circular curve tangentially.
 Its curvature should be zero at the origin on straight.
 Its curvature at the junction with the circular curve should be
the same as that of the circular curve.
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73.  The rate of increase of curvature along the transition should
be the same as that of increases of super elevation.
 The length should be such that full super-elevation (Cant) is
attained at the junction with the circular curve.
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74.  To accomplish gradually the transition from the straight to the
circular so that the curvature is increased gradually from zero
to a specified value.
 To provide a medium for the gradual introduction of super
elevation.
 To provide extra widening on the circular curve gradually.
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75.  Reduce the discomfort.
 Reduce the chances of overturning of the vehicles.
 Allows higher speed at curve.
 Wear on running gears is reduce.
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76.  Super elevation is defined as the rising of the outer edge of a
road respect to its inner edge.
 It is provided to counteract the centrifugal force acting on the
vehicle at circular curves.
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77.  The centrifugal force is given by,
 Where, P = Centrifugal force in kg or N
 W = Weight of the vehicle in kg or N
 V = Speed of the vehicle, m/sec2
 g = Acceleration due to gravity, m/sec2
 R = Radius of curve
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79.  Length of the transition curve introduce between straight and
circular curve is calculated following consideration.
1. By rate of super elevation
2. By time rate
3. By rate of change of radial acceleration
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80.  BY RATE OF SUPER ELEVATION :
 The length of the transition curve is given by
 L = ne
 The value of n may vary from 300 to 1200.
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81.  BY TIME RATE :
 The time taken by a vehicle over the transition curve of length
L with speed v is
 The super elevation is attained in this time is attained in this
time is
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82.  BY TIME RATE :
 Substituting the value of e in the equation,
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83.  BY RATE OF CHANGE OF RADIAL ACCELERATION
 In this method, the length of transition curve is decided based
on the basis of the comfort of the passenger.
 If α is the rate of change of radial acceleration, the radial
acceleration attain during the time vehicle passes over the
transition curve is given by
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84.  Radial acceleration is given by
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85.  There are mainly three types of transition curves :
1. Cubic spiral
2. Cubic parabola
3. The lemniscates curve
85

86.  CUBIC SPIRAL :
 The Cubic spiral is best suited on railways.
 The equation of a cubic spiral is,
 Where, Y = perpendicular offset from the tangent
 l = distance measured along the curve
 R = radius of the circular curve
 L = length of the transition curve
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87.  CUBIC PARABOLA :
 This type of curve is used in railway line construction.
 The equation of a cubic parabola is,
 Where, Y = perpendicular offset from the tangent
 x = distance measured along the curve
 R = radius of the circular curve
 L = length of the transition curve
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88.  THE LEMNISCATES CURVE :
 Instead of providing intermediate circular curve, when entire
curve is provided in the form of transition curve, it is known
as lemniscates.
 This type of curve is used on highways.
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89.  THE LEMNISCATES CURVE :
 The equation of Bernoulli's lemniscates curve is,
 Where, P = polar distance of any point
 α = polar deflection angle for any point
 k = constant
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90.  When a vehicles moving on a straight takes a circular path,
super elevation is required to be introduce uniformly from
zero to its maximum design value.
 On the other hand speed of the vehicle keep constant speed.
 Requirements are fulfil by
1. Increase the centrifugal force at a constant rate.
2. Varying the distance travelled along transition curve with
time.
90

91.  Centrifugal force is directly proportional to the length of
transition curve :
91

92.  Super elevation is proportional to the length of transition
curve :
92

93.  It is provided when there is a sudden change in gradient of a
highway or a railway.
 It is provided when a highway or railway crosses a ridge or a
valley.
 It smoothens the change in gradient so that there is no
discomfort to the passengers travelling in vehicles.
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94.  ADVANTAGES :
 Due to vertical curve, change in gradient is gradual.
 It improves the appearance of roads.
 Road and railway journey becomes comfortable.
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95.  There are mainly two types of curves :
1. Summit Curve (Convex curve)
2. Valley Curve (Concave curve)
95

96.  It is provided in following situations:
 An upgrade (+g1) followed by a down grade (-g2)
 An upgrade (+g1) followed by another upgrade (+g2). (g1>g2)
 A down grade (-g1) followed by another down grade (-g2).
(g2>g1)
 A plane surface followed by down grade (-g1)
96

97.  It is provided in following situations:
 A down grade (-g1) followed by a upgrade (+g2).
 A down grade (-g1) followed by another down grade (-g2).
(g1>g2)
 An upgrade (+g1) followed by another upgrade (+g2). (g2>g1)
 A plane surface followed by upgrade (+g1)
97

98.  The length of vertical curve can be obtained by dividing the
algebraic difference of the two grades by the rate of change of
grade.
= g2 – g1
r
 Where, g1 , g2 = grades in %
 r = rate of change of grade (%)
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