**Introduction Of Curves**

During the survey of the alignrnent of a project involving roads or railways, the line direction may change thanks to some unavoidable circumstances.The angle of the change in direction is understood because the deflection angle.For it to be possible for a vehicle to run easily along the road or railway track, the 2 straight lines (the original line and also the deflected line) are connected by an arc (Fig. 10.1) which is understood because the curves of the road or track.

When the curve is provided within the horizontal plane, it's referred to as a horizontal curves.

Again, the character of the bottom might not be uniform along the alignment of any project and should contains different gradients (for instance, rising gradient could also be followed by falling gradient and vice versa).

In such a case, a parabolic curved path is provided within the vertical plane so as to attach the gradients for straightforward movement of the vehicles.

This curve is called as a *vertical curve*.

**Degree of Curves**

The angle a unit chord of 30 m length subtends at the centre of the circle formed by the curves referred to as the degree of the curve. it's designated as D (Fig. 10.2).

A curve is also designated consistent with either the radius or the degree of the curve.

When the unit chord subtends an angle of 1°, it's called a one-degree curve, when the angle is 2°, a two-degree curve, and so on.

It may be calculated that the radius of a one-degree curve is **1,719** m.

**Relation between Radius and Degree of Curve**

Let AB be the unit chord of 30 m, O the centre, R the radius and D the degree of the curve (Fig. 10.3).

Here, OA=R

AB=30 m

AC=15 m

⟨AOC=D/2

From triangle AOC,

sinD/2=AC/OA=15/R

R=15/sinD/2

When D is extremely small, sin D/ 2 could also be taken as D/2 radians.

R=15/(D/2)×(π/180)=15×360/πD=1718.9/D

=1719/D (approx.).

**Superelevations**

When a particle moves in an exceedingly circular path, then a force (known as centrifugal force) acts upon it, and tends to push it faraway from the centre.

Similarly, when a vehicle suddenly moves from a straight to a curved path, the force (centrifugal) tends to push the vehicle faraway from the road or track.

This is often because there's no component force to counterbalance this force (centrifugal).

To counterbalance the force (centrifugal), the fringes of the road or rail is raised to some height (with reference to the inner edge), in order that the sine component Of the load of the vehicle (W sinθ) may counterbalance the overturning force.

The height through which the fringes of the road or rail is raised is understood as superelevation or cant.

In Fig. 10.4, P is that the force , ** W sinθ** is that the component of the load of the vehicle, and h is that the superelevation given to the road or rail.

For equilibrium,

Wsinθ=WV²/ɡR

or Wxh/b=WV²/ɡR

(when θ isvery small, sinθ= tanθ h/b)

or h=bV²/ɡR for roads

or h=GV²/gR for railways

where b=width of the road in metres

G=distance between centres of rails gauge in metres.

R=radius of the curve in metres.

g=acceleration because of gravity 9.8 m/s².

V=speed of the vehicle in metres per second.

h=superelevation in metres.

**Centrifugal Ratio**

The ratio between the force (centrifugal) and therefore the weight of the vehicle is understood as centrifugal ratio.

Centrifugal ratio (CR)=P/W=WV²/gR×W=V²/gR

Allowable value for CR in roads=1/4

Allowable value for CR in railways=1/8.

**TYPES OF HORIZONTAL CURVES**

There are different types of horizontal curves:

**Simple Circular Curve**

When a curve consists of a with a continuing radius connecting the 2 tangents, it's said to be a circular curve (Fig. 10.5).

**Compound Curve**

When a curve consists of two or more arcs with different radii, it's called a compound curve.

Such a curve lies on an equivalent side of a standard tangent and centres of the various arcs lie on an equivalent side of their respective tangents (Fig. 10.6).

**Reverse Curve**

A reverse curve consists of two arcs bending in opposite directions. Their centres lie on converse sides of the curve.

Their radii could also be either equal or different, and that they have one common tangent (Fig. 10.7).

**Transition Curve**

A curve of variable radius is knoun as a transition curve. it's also called a spiral curve or easement curve.

In railways, such a curve is provided on each side of a circular curve to minimise superelevation.

Excessive superelevation may cause wear and tear of the rail section and discomfort to passengers (Fig. 10.8).

**Lemniscate Curve**

A lemniscate curve the same as a transition curve, generally adopted in city roads where the deflection angle is large.

In Fig. 10.9, OPD shows the form of such a curve. The curve is intended by taking a significant axis OD, axis PP'with origin O, and axes OA and OB. OP(p) is understood as polar ray, and ɑ because the polar angle.

Considering the properties of polar coordinates , the polar equation of the curve is given by

r=*p/3sin*2ɑ

Where *p=*polar Ray of any point

r=radius of curvature at any point

ɑ=polar deflection angle

At the origin, the radius of curvature is infinity. It then gradually decreases and becomes minimum at the apex D.

Length of curve=1.31 15 K

K = 3r√sin2ɑ.

**Notations Used With Circular Curves**

1. AB and BC are known as the tangents to the curve (Fig. 10.10).

2. B is understood because the point of intersection or vertex.

3. The angle ø is known as the angle of deflection.

4. The angle I is called the angle of intersection.

5. Points T1 and T2 are known as tangent points.

6. Distances BT1 and BT2 are known as tangent lengths.

7. When the curve deflects to the right, it is called a right-hand curve, when it deflects to the left, it is said to be a left-hand curve.

8. AB is called the rear tangent and BC, the forward tangent.

9. The straight line T1DT2 is known as the long chord.

10. The curved line T1ET2 is said to be the length of the curve.

11. The mid-point E of the curve T1ET2 is known as the apex or summit of the curve.

12. The distance BE is known as the apex distclnce or external distance.

13. The distance DE is called the versed sine of the curve.

14. R is the radius of the curve.

15. ⟨T1OT2 is equal to the deflection angle** ø**.

16. The point T1 is known as the beginning of the curve or the point Of curve.

17. The end of the curve T2 is known as the point of tangency.

**Properties Of Simple Circular Curves**

Consider Fig. 10.10.

1. If the angle of intersection is given then

∅=180°-I (I = angle of intersection)

2. If radius is not given then

R=1,719 (D = degree of curve)

3. Tangent length BT1 or BT2=R tan∅/2

4. Length of curve=Length of arc T1ET2

=R×∅ radians

=πR∅°/180°

Again, length of curve=30∅°/D (if degree of curve D is given).

5. Length of long chord=2T1D=2OT1 sin∅/2=2Rsin∅/2 m.

6. Apex distance = BE=OB=OE

=Rsec∅/2-R=R(sec∅/2 — l) m

7. Versed sine of curve=DE=OF=OD

=R-Rcos∅/2=R(1 - cos ∅/2) m

8. **Full Chord (Peg Interval):**Pegs are fixed at regular intervals along the curve. Each interval is said to equal the length of a full chord or unit chord.

The curve is represented by a series of chords, instead of arcs. Thus, the length of the chord is practically equal to the length of the arc.

In usual practice, the length of the unit chord should not be more than 1/20th Of tfie radius of the curve.

In railway curves, the unit chords (peg intervals) are generally taken between 20 and 30 m. In road curves, the unit chord should be 10 m or less.

It should be remembered that the curve will be more accurate if short unit chords are taken.

9. ** Initial Sub-chord: **Sometimes the chainage of the first tangent point works out to be a very odd number.

To make it a round number, a short chord is introduced at the beginning. This short chord is known as the initial sub-chord.

10. ** Final Sub-chord**: Sometimes it is found that after introducing a number of full chords, some distance still remains to be covered in order to reach the second tangent point.

The short chord introduced for covering this distance is known as the final sub-chord.

l l . Chainagc of first tangent point

=Chainage of intersection point- Tangent length

12. Chainage of second tangent point

=Chainage of first tangent point + Curve length.

**Field Procedure Of Setting-out Curve**

1. In Fig. 10.19, AB and BC are two tangents intersecting at B. The tangents length and curve lengths are calculated, and the points T1 and T2 are fixed.

2. The length of the initial and final sub- chord, and the number of full chords are ascertained.

3. The deflection angles for the chords are calculated and verified by arithmetical check.

4. A setting-out table is prepared, depending on the least count of the theodolite. For setting the curve, only the angles from the angles to be set column should be taken.

5. The theodolite is centered over T1 and properly levelled. Then vernier A is set to 0° of the main scale. The upper clamp is fixed.

6. The lower clamp is released and the ranging rod at the intersection point B is perfectly bisected with the help of the lower tangent screw. The lower clamp is now tightened.

7. The upper clamp is released and the first deflection angle (δ1)is set on vernier A. The telescope is directed along the line T1E.

8. Now, the zero end of the tape is held at T1 and the distance T1P1 is measured equal to the length of the initial sub-chord in such as a way that the ranging rod at P1 is also bisected by the telescope.

Then the telescope is lowered to mark the base of ranging rod perfectly. So, P1 is a point on the curve which is marked by a nail or arrow.

9. The next deflection angle (δ₂) is set on vernier A and the point P2 is so marked that P1P2 is equal to the length of a full chord, and the ranging rod at P2 is perfectly bisected by the telescope. So, P2 is the next point on the curve.

10. This process is continued until all the deflection angles are set out and all the points on the curve are marked. Finally, the last point should coincide with T2.

If it does not, the amount of error is found out. If this error is small, it is distributed among the last few pegs.

If the error is large, the entire operation should be repeated. Ultimately, the every points P1, P2, P3 ... are marked by stout pegs.

**Procedure for Setting Deflection Angles**

1. The theodolite is centered and levelled at the first tangent point and the lower clamp is fixed. The upper clamp is loosened and vernier A is set approximately to the zero of the main scale.

After that, the upper clamp is tightened and by turning the upper tangent screw the arrow of vernier A is brought into exact coincidence with the zero of the main scale.

2. Now, the lower clamp is loosened and the ranging rod at the intersection point is perfectly bisected with the help of the lower tangent screw. Then both the clamps are tightened.

3. Suppose the deflection angle 0°48'20" is to be set. By turning the upper tangent screw very slowly, the arrow of vernier A is made to cross two small divisions (i.e. 40') of the main scale.

Then, looking through the divisions of the vernier scale carefully, the first small division after eight big divisions (i.e. *8'2"*) to the vernier scale is made to coincide with any division of the main scale.

Thus, Deflection angle = 0°48'0"+ 0°8'20"

=0°48'20"

4. Similarly, by turning the upper tangent screw very slowly, subsequent deflection angles are set out one by one according to the entries in the "angles to be set" column of the setting out table.

**Field Procedure of Two-theodolite Method**

This method is generally employed in railway curve setting, as it gives the correct location of points.

In this method, no chain or tape is required to fix the points on the curve. It is mostly suitable when the ground surface is not favourable for chaining along the curve due to undulations.

The two-theodolite method involves the following procedure:

1. All the necessary data for setting out the curve are calculated in the usual manner. The setting out table is also prepared.

2. Tangent points T1 and T2 are marked on the ground (Fig. 10.20).

3. A theodolite is centred over T2 and levelled properly. Vernier A is set to 0° and the upper clamp is tightened.

The lower clamp is released and by turning the telescope the ranging rod at T1 is perfectly bisected. The lower clamp is now fixed.

4. The ranging rod at T1 is taken off and another theodolite is centered over this point and levelled.

Vernier A is set to 0°. The upper clamp is tightened.The lower clamp is released and the ranging rod at B is perfectly bisected The lower clamp is now fixed.

5. The upper clamps of both theodolites are released. The first deflection angle (δ1) is set on vernier A of both theodolites.

6. Then a point P1 is so located that the lines of sight of both instruments intersect at it. So, P1 is a point on the curve.

7. The next deflection angle (δ₂) is set on the vernier A of both instruments. Again a point is so found that the lines of sight of both instruments intersect at it. So, P2 is the next point on the curve.

8. This process is continued until all the deflection angles are set out, and all the points are marked.

9. Finally, when the total deflection angle (δn) is set out in both instruments, the line of sight of the theodolite at T1 should bisect T2 and that of the theodolite at T2 should bisect B.

**Compound Curve—Calculation Of Data And**** Setting-Out**

When it is not possible to connect the two tangents by one circular curve, it becomes necessary to take a suitable common tangent.

And set out two curves of different radii to connect the rear and forward tangents. This curve is known as a compound curve [Fig. 10.21].

**
Notation**

AB=rear tangent

BC=forward tangent

DE=common tangent

∅=deflection angle between rear and forward tangent

∅1=deflection angle between rear and common tangent

∅2=deflection angle between common and forward tangent

O1=centre of short curve

O2=centre of long curve

Rs=radius of short curve

RL=radius of long curve

T1 and T2= tangent points for short curve

T2 and T3 = tangent points for long curve

Ts=total tangent length of shortest side (BT1)

TL=total tangent length of long side (BT3)

ts=tangent length of shot curve

tL=tangent length of long curve.

**Calculation Of Data**

1. ∅=∅1+∅2

2. Ts=BD+DT1=BD+ts

=DE×sin∅2/sin∅2 +Rs tan∅1/2

3. TL=BE+ET3=BE+tL

=DE×sin∅1/sin∅2+RL ∅2/2

4. Common tangent, DE=ts+tL=Rs tan∅1/2+RL tan∅2/2

Where ts =Rs tan∅1/2. tL=RL tan∅2/2

From trianɡle BDE,

BD/sin∅2=BE/sin∅1. or BE=DE×sin∅/sin∅2

5. Curve length (short curve)=πRs∅1/180°

6. Curve length(long curve)=πRL∅2/180°

7. Deflection angle (short curve)

=δs=1718.9×Cs/Rs mins

Where, Cs=chord of the short curve

Deflection angle of long curve

δL=1718.9×CL/RLmins

Where, CL=chord of the long curve

8. Chainage of T1=Chainage of B-Ts

9. Chainage of T2=Chainage of T1+short curve length.

10. Chainage of T3=chainage of T2+long curve length.

**Reverse Curve—Calculation Of**

**Datat And Setting-Out**

A reverse curve consists of two circular arcs of equal or different radii returning opposite directions with a standard tangent at the junction of the arcs.

The junction point is claimed to possess reverse curvature, The reverse curve is additionally called a serpentine curve.

Reverse curves are generally wont to connect two parallel roads or railway lines, or when two lines intersect at a really small angle.

These curves are compatible for railway sidings, city roads, etc.

But they ought to be avoided as far as possible for important tracks or highways for the subsequent reasons:

1. Superelevation can't be provided at the purpose of reverse curvature.

2. A snappy changing of direction would be dangerous for a vehicle.

3. A snappy changing of cant causes discomfort to passengers.

4. Carvlessness of the driving force may cause the vehicle to overturn over a reverse curve.

5. Reverse curves are generally short. and hence they're started out by the chain and tape method.

**Notation**

1. In Fig. 10.23. AB and also the straight lines, BE is that the common tangent and C is that the point of reverse curvature.

2. and T1 are the tangent points.

3. ∅ is the angle of intersection between the straight lines.

4. ∅1 and ∅2 are the deflection angles of the common tangent.

5. R1 and R2 are the radii of the arcs.

Reverse curves may involve various cases. Here, we shall illustrate two of them.

**Case I—When the Straights are Non-parallel**

Suppose AB, BC and CD are lines of an open traverse along the alignment of a road (Fig. 10.24).

AB and CD when produced meet at a point E, where is the angle of intersection. It is necessary to join the lines AB and CD by a reverse curve with BC as the common tangent.

Let

∅1=angle of deflection for first arc

∅2=angle of deflection for second arc

∅=angle of intersection between AB and DC

T1 and T2 = tangent points

F=point of reverse curve

R=common radius for the arc

**The following data have to be calculated for setting out the curve**

1. Tangent length of first arc, T1B=BF=Rtan∅1/2

2. Tangent length of second arc, T2C=CF=Rtan∅2/2

3. Length of the common tangent, BE=BF+CF

=Rtan∅1/2+Rtan∅2/2

4. Length of first curve, T1F=πR∅1/180°

5. Length of second curve, T2F=πR∅2/180°

6. Chainage of T1=chainage of B-T1B.

7. Chainage of F=chainage of T1+1st curve length.

8. Chainage of T2=chainage of F+2nd curve length.

The length of the reverse curve is normally small. So, the may be set out by taking offset from .

(i) the long chord.

(ii) the chord produced.

If the length of curve becomes large and chaining along it difficult, the curve may be set out by the deflection angle method (Rankine's method).

**CASE II—When the straight lines are Parallel**

In Fig. 10.25, PQ and RS are two parallel lines a distance y apart.

It is necessary to join the lines PQ and RS by a reverse curve of equal radii. Line AB is drawn parallel to PQ or RS through point C.

R=common radius.

C=point of reverse curve.

T1 and T2=tangent points.

∅=angle subtended at the centre by the curve.

T1T2=l=length of line joining T1 and T2.

x=perpendicular distance between T1 and T2.

y=perpendicular distance between lines PQ and RS.

The following data have to be calculated for the setting out the curve.

1. Long chord for first curve, T1C=2Rsin∅/2.

2. Long chord for second curve, T2C=2Rsin∅/2.

3. Length T1T2=2×(2Rsin∅/2)=√4Ry.

4. y=T1A+T2B=2R(1-cos∅).

5. x=AB=CA+CB=Rsin∅+Rain∅=2Rsin∅

**VERTICAL CURVES**

**Definition**

**Definition**

When two different gradients meet at some extent along a surface of the road, they form a pointy point at the apex.

Unless this apex point is rounded off to create a smooth curve, no vehicle can move along that portion of the road.

So, for the graceful and safe running of vehicles, the meeting point of the gradients is rounded off to fom a smooth curve during a vertical plane.

This curve is understood as a vertical curve.

Generally, the parabolic curves are preferred because it is simple to figure out the minimum sight distance in their case, and therefore the minimum sight distance is a crucial factor to be considered whilst computing the length of the vertical curve.

**Gradient**

**Gradient**

The gradient is expressed in two ways:

1. As a percentage, e.g. 1%, 1.5%, etc.

2. As 1 in n, where n is that the horizontal distance and 1 represents vertical distance, e.g. 1 in 100, 1 in 200, etc.

Again, the gradient could also be 'rise' or 'fall'. An up gradient is understood as 'rise' and is denoted by a positive sign.

A down gradient is understood as 'fall' and is indicated by a negative sign.

**Rate of Change of Grade**

**Rate of Change of Grade**

The characteristic of a parabolic curve is that the gradient changes from point to point but the speed of change of grade remains constant.

Hence, for locating the length of the vertical curve, the speed of change of grade should be a very important consideration as this factor remains constant throughout the length of the vertical.

Generally, the recommended rate of change of grade is 0.1% per 30 m at summits and 0.05% per 30 m at sags.

**Length of Vertical Curve**

**Length of Vertical Curve**

The length of the vertical curve is calculated by considering the sight distance.

To produce minimum sight distance, a particular permissable rate of change of grade is decided and also the length of the vertical curve is calculated as follows:

Length of vertical curve

=change of grade/rate of change of grade

=Algebraic difference of grade/rate of change of grade

=g1-g2/r.

Where, g1 and g2=percentage of grade and r=rate of change of grade.

**Types of Vertical Curves **

**Types of Vertical Curves**

The are the various sorts of vertical curves which will occur.

**Summit Curve **

**Summit Curve**

Figure 10.36(a) a summit curve where an up gradient is followed by a down gradient.

Figure 10.36(b) shows a summit curve where a down gradient is followed by another down gradient.

**Sag Curve**

Figure 10.36(c) shows a sag curve where a down gradient is followed by an up gradient.

Figure 10.36(d) shows a sag curve where an up gradient is followed by another up gradient.

**Setting Out Vertical Curve**

The vertical curve could also be started out by the subsequent two methods:

(a) The tangent correction method.

(b) The chord gradient method.

The tangent correction method is preferred in practical situations, because it involves simple calculations and curve setting.

**Tangent Correction Method **

In Fig. 10.37 the tangent correction or tangent offset is the difference of elevation between points P and P1, P being a point on the curve, P1 a point on the gradient.

Then

y=RL=P1— RL of P tangent correction.

Let, x be the horizontal distance of point P from the origin. is the sloping distance along the gradient of the point P1.

Here, x is taken to be approximately equal to X1.

The equation of the curve is

y=Cx².

Where, C=constant=g1-g2/400×*l*

*l*= half-length of the vertical curve tangent correction at any point.

y=(g1-g2)×X₁²/400×*l X=X₁*

y₂=(g1-g2)×X₂²/400×*l*

*Where, X1*,X2 ...=distance taken along the slope measured from tangent point.

*l=*half-length of the curve.

g1 and g2 = percentage of grade.

**Problems Faced In Curve Setting**

The process of curve setting may involve various problems, which have to be suitably tackled depending on the site condition. We shall now discuss a few of them.

The following are the different problems that occur:

l . The point of intersection may be inaccessible.

2. Both tangent points may be inaccessible.

**3. It may not be possible to set out the full curve from one point**

The tangent points T1 and T2 are marked in the usual way. The theodolite is set up at T1 and the points on the curve are set out as usual upto point P. Let total deflection angle be ∆p see (Fig. 10.41).

The theodolite is shifted and set up at P. Vernier A is set at zero, and the ranging rod at T1 is bisected.

Then the angle **∆p** is set on vernier A, and a point T'1 is marked. The line T'1P is the tangent to the curve at P.

Vernier A is again set at zero and the ranging rod at T1 is bisected. The telescope is now transited.

The deflection angle δn for the next point N is set and marked on the ground. The process is repeated until all the points are located.

The calculation of deflection angle and mode of setting out are the same as in Rankine's method.

4. *There may an obstacle across the curve.*

Suppose a building comes across the curve (Fig. 10.42). From T1, points P1, ...,P4 are marked. Then the total deflection angle for Ps is set out. Let this angle be θ.

The length of the long chord T1P5 is calculated as follows:

T1P5=2Rsinθ

This calculated length is measured along the line T1P5 to locate the point P5 on the curve.

Then normal procedure is followed in order to locate the remaining point on the curves.

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