4.1 Introduction to Horizontal and Vertical Curve

Unit - 4

Curves

4.1 Introduction to Horizontal and Vertical Curve:

Horizontal curve:

Definition: The curves which are provided at turning points so as to get gradual change in the direction of alignment of a road or a track are termed as horizontal curves.

Note that these curves are in the horizontal plan. In railway track, the minimum radius of a horizontal curve should be 175 m for broad gauge.

Horizontal as well as vertical, curves in case of road must ensure safety, comfort and convenience of the traffic.

Vertical Curve:

Definition: The curves which are provided in the alignment of road or track at change of gradient is termed as vertical curves.

Necessity of vertical curves: It gives adequate visibility and safety to the traffic.

It gives gradual change in grade or slope.

It gives adequate comfort to the passengers.

Key Takeaways:

The curves which are provided at turning points so as to get gradual change in the direction of alignment of a road or a track are termed as horizontal curves.

The curves which are provided in the alignment of road or track at change of gradient is termed as vertical curves.

4.2 Different types of Curves and their applications:

There are four types of horizontal curves:

Simple curve

Compound curve

Reverse curve

Transition curve

Simple curve:

The curve which consists of a single arc of a circle of which two straight tangents further connects and brings about a deflection of the road through an angle '0' is called as simple curve. See Fig. for better understanding.

Simple curve is normally represented by the length of its radius or by the degree of curve.

This type of curve is provided at every change in alignment of the road or railway à track in a plain and in hilly areas.

$H:\unit 4 survey\IMG_20210528_183612.jpg$

Fig.4.1: Simple curve

Compound curves:

A curve of having the series of two and more simple curves of different radii curving in the same direction is called as compound curves. Refer to the Fig.

for better understanding.

$H:\unit 4 survey\IMG_20210528_183634.jpg$

Fig.4.2: Compound curve

In compound curves, the two adjacent curves will have a common tangent "BC" as shown in Fig.

The centers of two adjacent curve lie on the same side of the curve as shown in Fig.

To avoid the cutting through hard rocks, heavy cutting or filling in the alignment of road or track, compound curves are provided.

Reverse curve:

The curve which consists of two simple curves having equal or different radii turning in opposite direction is called as reverse curve. Refer to Fig. 6.4.3 for better understanding.

The two centres of curves are on opposite sides of a common tangent 'BD'.

Reverse curves are necessary on hill roads where frequently changes in the direction of travel is required.

$H:\unit 4 survey\IMG_20210528_183702.jpg$

Fig.4.3: Reverse curve

Reverse curves are also necessary for cross-overs in station yards and in the alignment of the railway tracks in hilly areas.

Advantages of reverse curve:

This type of curve is oftenly used in the alignment of a hill road.

It is also used in the layout of railway spur tracks and cross-over.

Disadvantages of reverse curve:

The use of reverse curve is not suitable on highways where the speeds are high.

This type of curve is also not suitable in the main railway lines due to high speed.

There is sudden change of direction in reverse curve and this makes passengers more uncomfortable.

Steering is dangerous on highways and driver has to be very cautious.

Transition curve:

The curve in which radius varies gradually from infinity to a finite value equal to that of the circular curve to be connected and vice versa is termed as transition curve.

These curves are commonly used in railway tracks between the circular curve and a tangent so as to provide easy and gradual change in direction.

There are Two types of vertical curve:

Summit Curve

Valley Curve

Summit curves:

Definition: The curves which are having convex surface on upward side is termed as summit curves.

Vertical curves are used manly when rising gradient intersects a falling grade or when rising gradient meets another rising gradient or when rising gradient meets a horizontal road or when falling gradient meets at a steeper falling gradient. Refer Fig. for better understanding.

Fig. shows the different situations where summit curves are provided.

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Fig.4.4: A rising gradient intersecting a falling gradient

$H:\unit 4 survey\IMG_20210528_183734.jpg$

Fig.4.5: A rising gradient meeting another rising gradient

$H:\unit 4 survey\IMG_20210528_183752.jpg$

Fig.4.6: A rising gradient meeting at horizontal road

Valley curves:

Definition: The curves which are having the convex surface on downward side Fig. shows the different situations where valley curves are termed as valley curve.

$H:\unit 4 survey\IMG_20210528_183808.jpg$

Fig.4.7: A falling gradient intersecting a rising gradient

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Fig.4.8: A falling gradient meeting another falling gradient

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Fig.4.9: A falling gradient meeting a horizontal road

Key Takeaways:

There are four types of horizontal curves:

Simple curve

Compound curve

Reverse curve

Transition curve

There are Two types of vertical curve:

Summit Curve

Valley Curve

4.3 Simple Circular Curve:

Refer the Fig. for understanding the various terms of notation for a simple circular curve.

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Fig.4.10: Simple circular curve (Right hand curve)

The common notation which are normally preferred for simple circular curve are as follows:

BC and AB are the two tangents drawn to the curve at T1, and T2, respectively. In short, BC and AB are two straight called as tangents.

The two tangents AB and BC are intersected at the B which is called as point of intersection. In short, B is the point of intersection of tangents AB and BC.

In Fig., the curve deflects to right side. Therefore, it is called as right-hand curve.

$H:\unit 4 survey\IMG_20210528_183940.jpg$

Fig.4.11: Simple circular curve (left hand curve)

4. In Fig., the curve deflects to left side. Therefore, it is called as left-hand curve.

5. In Fig., BC is first tangent and also called as back tangent. AB is second tangent and also called as forward tangent. In Fig. AB is first or back tangent at T₁ and BC is second or forward tangent at T₁.

6. T1, and T2, are the tangent points. T1, shows the starting of the curve and T2, shows the end of the curve.

7. In fig., I is an angle of intersection and 0 is a deflection angle.

8. BT1, and B T2, are the lengths of tangent.

9. T1, T2, is a long chord of length 'L'.

10. The arc 'T1ET2' is a length of curve.

11. The midpoint 'E' is apex of summit of the curve T1ET2

12. BE is the distance of apex.

13. The angle 'T1OT2' = The deflection angle ''.

14. DE is a versed sine of the curves.

Key Takeaways:

Elements of the Simple of the Curve:

Refer the Fig. to obtain the various elements of the simple circular curve.

'=180° ∴

'= 180°-I

Tangent length = B T₁ = B T₂ = OT, tan = R tan

Length of long chord (L) = 2R sin

Length of the curve (I) = R0 in radians =

rad

Apex distance = BE = R

4.4 Compound circular Curve:

A curve of having the series of two and more simple curves of different radii curving in the same direction is called as compound curves. Refer to the Fig. for better understanding.

In compound curves, the two adjacent curves will have a common tangent 'BC' as shown in Fig.

The centers of two adjacent curve lie on the same side of the curve as shown in Fig.

To avoid the cutting through hard rocks, heavy cutting or filling in the alignment of road or track, compound curves are provided.

$H:\unit 4 survey\IMG_20210528_184002.jpg$

Fig.4.12: Compound curve

Fig. shows T1D T2 is a two centred compound curve having two circular arcs T1D and DT2 meeting at a common point D known as the point of compound curvature.

$H:\unit 4 survey\IMG_20210528_184022.jpg$

Fig.4.13: Two centered compound curve

T1 is the point of curve and T, is the point of tangency.

O1 and O2 are the centres of the two areas:

Let,

Rs =the smaller radius (T1O1)

R2= the longer radius (T2O2)

BC= common tangent

deflection angle between the rear and the common tangent.

=deflection angle between the common and the forward tangent.

total deflection angle.

ts=the length of the tangent to the arc (T, D) having a smaller radius.

Ts = tangent distance T1A corresponding to the shorter radius.

TL = tangent distance T2A corresponding to the longer radius.

From Fig. we have,

ts= T1B= BD= Rs

tl=T2C=CD+RL

From triangle ABC we have,

Ts=T1B+BA

=ts+(ts+tl)

Key Takeaways:

A curve of having the series of two and more simple curves of different radii curving in the same direction is called as compound curves.

4.5 Elements and setting out by linear methods such as radial and perpendicular offsets:

Radial offsets:

Let Ox=Radial offset PN at any distance x along the tangent

T1P = x

From Δ T1 PO,

PO2 = T1O2+T1 P²

Ox= –R …... (1)

Fig.4.14: setting out by radial offsets

In order to get an approximate expression for Ox, expand

Thus,

Ox=R-R

Neglecting the other terms except the first two, we get

Or Ox=x2/ 2R

When the radius is large

T1P2=PN(2R+PN)

X2=Ox(2R+Ox)

Neglecting Ox in comparison to 2R we get

Ox=x2/2R

Perpendicular offsets:

$H:\unit 4 survey\IMG_20210528_184105.jpg$

Fig.4.15: setting out by perpendicular offsets

Let DN=Ox= offset perpendicular to the tangent

T1D=x= measured along the tangent

Draw NN1 parallel to the tangent

From leEE1O, we have

E1O2=EO2-E1E2

(R-Ox)2=R2-x2

From which

Ox=R-

The corresponding approximate expression for Ox may be obtained by expanding the term. Thus,

Ox=R-R

Neglecting the other terms except the first two of the expressions

Ox=R-R+

4.6 Offsets from long chord:

Offset from long chord is calculated as:

L=2R

Where R =Radius of curve

4.7 Successive bisection of chords:

Join the tangent point T₁, T, and bisect the long chord at D.

$H:\unit 4 survey\IMG_20210528_184124.jpg$

Fig.4.16: Successive bisection of arcs

Erect the perpendicular DC and make it equal to the versed since of the curve. Thus

NO= R (1-)

=R-^2

Join T1N and T2N and bisect them at O1 and O2 respectively. At O1 and O2, set out perpendicular offsets N1O1=N2O2=R (1-

)

By the successive bisection of these chords, more points may be obtained.

4.8 Offsets from chord produced:

$H:\unit 4 survey\IMG_20210528_184141.jpg$

Fig.4.17: setting out the curve by deflection distance

When the curve is long, this method is useful (generally on highway curves) when a theodolite is not available.

Let T1L1 = T1L = initial sub-cord= C1

L, M, W are the points on curve

LM = C2, MN = C3 etc

T1x = rear tangent

∠ L1 T1 L = = deflection angle of the first chord

L1 L =O1 = first offset

M2 M = O2 = second offset

N2 N = O3 = Third offset etc.

Now Arc L1 L = O1 = T1 L ……(1)

Since T1x is the tangent to the circle at T₁

∠T O L = 2∠L T L

= 2

∴ T L = R 2

Substituting the value of in (1) we get

Arc L1 L = O1 = T1L

= T1 L2

Taking arc T1L=chord T1L (very nearly) we get

O1=C1^2/2R

Thus, to obtain the value of the second offset O2, for getting the point M on the curve, draw a tangent LM1, to the curve at L to cut the rear tangent in L. Join T1L and prolong it to point M2 such that LM2=LM=C2, length of the second chord.

The O2= M2M

As from equation No. 3, the offsets M1M from the tangent LM1 is given by

L1L =C22 /2R

Angle M2LM1=angle L’LT1 being opposite angles

Again,

Since T1L' and L'L are both tangents, they are equal in length.

Arc M2M1=LM2=C2

Generally, the first chord is a sub-chord say of length c, and the intermediate chords are normal chords, say of length C. In that case the above formulae reduce to

O12=C2/2R

O₂ = C/2R(C+C)

03= 04=C/2R(2C)

=C2/R

4.9 Angular method:

Angular methods which are most commonly used for setting out a circular curve are:

Rankine's method of tangential (or deflection) angle.

Two theodolite method.

Tacheometric method.

4.10 Rankine’s method of deflection angle:

Transit theodolite and chain or tape are used in this method.

Since this method gives more precise results, it is mostly used for setting out the curves in railways and other important work.

Derivation of formula:

Let R be the radius of the curve and

be the angle of deflection.

Refer the Fig. for better understanding and further simplification of derivation.

Fig.4.18: Rankine’s method of tangential angle or deflection angle

AB is the first tangent to the curve.

T1 and T2 are the tangent points.

E, D, T1, etc. are the successive points on the curve.

3, etc. are the tangential angles or deflection angles made by chords T1E, ED, DT2 etc. w.r.t. tangents.

₁,

, etc. are total tangential angles or deflection angles between:

first tangent AB and T1E

first tangent AB and ED

first tangent AB and DT2, respectively.

C₁, C₂, C3, etc. are the lengths of chords T1E, ED, DT2, etc. Chord T1E = Arc T1 E assumed to be C

BT1E = 8, = ½ T1OE

T1OE = 2,

Now,=

T1OE =C1

2=C1

minutes

Similarly,

Total tangential or deflection angles are

, etc. Total deflection angle for first chord

∴

Total deflection angle

, for the second chord ED= BT1D. but BT1D= BT1E+ET1+DET1D is the angle subtended by the chord 'ED' in the opposite segment and hence equal to the tangential angle 'S,' between the tangent at E and chord ED.

Hence,

Check: Total deflection angle (BT1 T2) =

Stepwise procedure for setting the curve by deflection angles:

Set up a transit theodolite over the first tangent point "T1, Centering and leveling is done precisely and set the vernier A to 360°.

Direct the telescope to A so as to bisect the ranging rod held at B.

Set the vernier A to the first deflection angle A, then direct the telescope along T, E.

Take one end of the chain or tape at T, and hold the arrow on the chain at a distance equal to the length of first sub-chord. Swing the chain around 'T, and bisect the arrow so as to fix the position on first point E on the curve.

Loose the upper plate and set the vernier to the second deflection angle "A". Then direct the line of sight along 'T, D'.

Now take the one end of the chain or tape at 'D' and swing the other end. Bisect the arrow, which is already held at the other end. In this way the second point on the chain D is located.

In this way, establish points on the curve until the end of the curve i.e., T₂: The slight deflection can be adjusted on the last few pegs.

Note that if error goes beyond the permissible limit, whole procedure should be checked or repeated.

Note that in left hand curve, each of the calculated values of the deflection angles A, A, etc. should be minus from 360°. In this way, the angles obtained will be set on the vernier of theodolite for setting out the curve. Enter the reading in following field note.

Peg or point	Chainage in m	Chord length in m	Deflection angle	Total deflection angle	Actual theodolite reading	Offset in m	Remark

Key Takeaways:

Transit theodolite and chain or tape are used in this method. Since this method gives more precise results, it is mostly used for setting out the curves in railways and other important work.

4.11 Two theodolite method:

$H:\unit 4 survey\IMG_20210528_184224.jpg$

Fig.4.19: Two theodolite method

In this method, two theodolites are used. One is at T1 and other at T2.

When the ground is unsuitable for chaining, this method is used.

This method is based on the principle that "The angle between the tangent and the chord is equal to the angle which that chord subtends in the opposite segment.

Thus, in Fig.

<PT1L= A=Deflection angle for L.

But LT2T1, is the angle subtended by the chord T, L in the opposite segment.

∴ angle LT2T1= angle PT1M =

Similarly, angle PT1L= = angle T1T2M

Hence the angle between the long chord and the line joining any point to T2 is equal to the deflection angle to the point measured with respect to the rear tangent.

Method of setting out the Curve:

Set up one transit theodolite at T1, and the other at T2.

Clamp both plates of each transit theodolite to zero reading.

In order to orient two transit theodolites correctly, theodolite at T1, must sight towards P with zero reading on vernier. Similarly, direct the line of sight of the other transit theodolite at T2, towards T1, when the reading is zero.

Set the reading of each of theodolite to the deflection angle (

) for the first point L. The line of sight of both the theodolites are thus directed towards L along T1L and T2 L respectively.

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 L. is bisected.

To fix the second point M, set reading to A, on both the theodolite o bisect the ranging rod.

In order to locate all other points, repeat steps (4) and (5)

Advantage of two theodolite method:

This method is more accurate, since each point is fixed independently of the other

An error in setting out one point is not carried right through the curve as in the method of tangential angles.

Disadvantage of two theodolite method:

The only disadvantage of this method is it is costlier since two theodolites two surveyors are required.

Key Takeaways:

This method is based on the principle that "The angle between the tangent and the chord is equal to the angle which that chord subtends in the opposite segment.

4.12 Numerical on simple circular curve and compound curve:

Two straights intersect at a chainage of 2610 m, the deflection angle being 36 A circular curve of 400 m radius is to be set. Calculate the chainages of the tangent point of right-handed circular curve. Peg interval is 30 m.

Given: Deflection angle (0) = 36°.

BT₁ Tangent length = R tan=400 129.967 =129.97 m

= (4+9.97) chains

Length of curve == =251.327=251.33m

= (8+11.33) chains

Chainage of the second tangent point "T1':

=-(4+9.97) chains

= (87) -(4+9.97) chains

= (83+9.97) chains

Chainage of the second tangent point ‘T2':

= (Chainage of T,) + (length of curve)

= (83 +9.97) + (8 +11.33)

= (91 +21.3) chains

Two roads meet at an angle of 150. Calculate the data necessary for setting out curve of 10 chains radius to connect the two straight roads if it intended to set out curve by chain and offset from long chord, chain used is a 30 m chain only. Calculate two offsets from centre of long chord.

Given:

I =point of intersection

=deflection angle = 180° - 150° = 30°

R = radius of curve = 10 chains = 10x30=300 m (since 30 m chain used)

Length of long chord (L) =2 R sin2

L = 2x300 x sin 30°/2 = 155.29 m

Note: Length of long chord is taken in round figures as 160 m.

∴L = 160 m

Ordinate at the middle of the long chord (O)= versed sine

O0 = R-

=300-

O0=300-289.136

∴O0=10.86 m

The various ordinates can be calculated by the following formula as,

Ox= –(R-O0)

Consider the offset at 20 m from the centre.

O20=-(300-10.86)

=299.33-289.14=10.19 m

(It is the first offset from the centre of long chord)

O40=-(300-10.86)

=297.32-289.14=8.18m

(It is the second offset from the centre of long chord.)

Hence the two offsets from the center of long chord are 10.19 m and 8.18 m.

Extension of problem:

Remaining offsets till curve ends can be calculated as follows:

-(300-10.86)

=293.938-289.14

=4.8 m

O80=-(300-10.86)

=289.136-289.14

=0 m

4.13 Transition Curve: Necessity

The curve in which radius varies gradually from infinity to a finite value equal to that of the circular curve to be connected and vice versa is termed as transition curve.

These curves are commonly used in railway tracks between the circular curve and tangent so as to provide easy and gradual change in direction.

Necessity:

To provide gradual change in the radius of curvature.

To provide smooth entry of vehicle from a straight portion to curved portion.

To permit gradual application of the superelevation and extra widening at horizontal curves.

To provide safety to the vehicular traffic.

Key Takeaways:

The curve in which radius varies gradually from infinity to a finite value equal to that of the circular curve to be connected and vice versa is termed as transition curve.

References:

Surveying and leveling by r. Subramanian, Oxford Publication

GPS Satellite Surveying-Alfred Leick-Wiley

Surveying and leveling Vol.1 and 2 by T.P. Kanetkar and S.V. Kulkarni Pune vidyarthi Griha Prakashan

Surveying by B.C. Punmia

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