1. CURVES
Lecture – 05

2. Curves
Curves are arcs provided between intersecting straights to negotiate a change in a direction.

3. Function of Curves
To avoid sudden change in the line of communication e.g., in roads, railways, canals etc it becomes necessary to provide curves. As shown in the figure, it is desired to go from direction AB to direction BC, it will be quite difficult for a vehicle to go up to point B and then take a turn in the direction of BC. The vehicle will have to slow down. This becomes even more difficult in case of long trains. But if these two points are joined by a curve, the change can be comfortably adopted by the vehicle. This curve provides a gradual change and makes the change safe, comfortable & easy.
Lines connected by the curves are tangential to it and are called tangents or straights. The curves are generally arcs, but parabolic arcs are often used in some countries for this purpose. B C A

4. Types of Circular Curves
Simple Curves.
Compound Curves.
Reverse Curves.

5. 1. Simple Curve
A simple curve consists of single arc connecting to straights. The radius of the curve throughout the curve remains the same.

6. 2. Compound Curve
A compound curve consists of two arcs of different radii bending in same direction. Their centers being on same side of the common tangent.

7. 3. Reverse Curve
A reverse curve is composed of two arcs of equal or different radii bending in opposite direction with a common tangent at their junction. This centers being on opposite side of the common tangent.

8. Degree of Curve:
Curves are designated by the angle (in degrees) at the center subtended by a chord of 100 ft length. The angle being called the degree (D) of the curve.

9. Relationship between degree (D) and Radius (R) of the Curve
Let
R = Radius of the curve in ft.
D = Degree of the curve.
MN = Chord of 100 ft length.
P = Mid point of chord
So; MP = 50 ft. 
In triangle OMP:
OM = R, MP = 50 ft Angle MOP = D/2
sin (D/2) = MP/OM = 50/R
R = 50 / sin (D/2)
When ‘D’ is small then sin (D/2) = D/2 in radians.
R = 50 / [(D/2) x (π/180)] = (50 x 360) / (π x D) = / D
R = 5730 / D M P N D O

10. Nomenclature of Circular Curve
The straight line AB and BC, which are connected by the curve are called the tangents of the curve.
The point B at which the two angles of the tangents AB & BC intersect is known as the point of intersection (P.I) or the vertex (V).
If curve deflects to the right of direction of the progress of survey AB; it is called Right Hand Curve and if to the left it is called Left Hand Curve.
Tangent line AB is called the First Tangent or the Rear Tangent (also called back tangent).
Tangent BC is the Second Tangent or Forward Tangent.
The point (T1 & T2) at whish curve touches the straights are called Tangent Points (T.P).
The angle ABC between the tangent lines AB and BC is called angle of intersection and the angle B'BC (i.e., the angle by which the forward tangent deflects from the rear tangent) is known as the deflection angle (Ф) of the curve.

11. Nomenclature of Circular Curve
The distance from the point of intersection to the tangent point is called tangent distance; tangent lengths (T1B & BT2).
The line T1T2 joining the two tangent points is known as “Long Chord (L)”.
The arc T1FT2 is called the length of curve (l).
Mid point (F) of the arc T1FT2 is known as Apex or summit of the curve and lies on the bisector of angle of intersection.
Distance BF from angle of intersection to the apex is known as “Apex distance” or External distance.
The angle T1OT2 subtended at the center of the curve by the arc T1FT2 is known as central angle and is equal to the deflection angle (Ф).
The intercept EF on the line OB between the apex F and point E of the long chord is called ‘Vosedsine of the curve’.

12. Elements of Simple Curve
T1BT2 + B'BT2 = 180o
Or I + Ф = 180o (1)
The angle T1OT2 = 180o – I
= 180o - (180o - Ф)
= Ф (2) 
Length of long chord = L = T1T2 = 2T1E
= 2(OT1 x sin Ф/2) (from ΔT1EO)
= 2R sin Ф/ (3)

13. Elements of Simple Curve
Tangent length = LT = BT1 = OT1 tan Ф/2
= Rtan Ф / (4) 
Length of the curve = (T1FT2) = l = Length of arc T1FT2
= R x Ф (in radians)
l = (R Ф x π) / 180o (5)

14. Elements of Simple Curve
If the curve is designated by degree (D) of the curves:
Then; l = (100 x Ф) / D (5-a) 
Apex distance = BF = BO – OF
= OT1 sec (Ф/2) – OF
= Rsec (Ф/2) –R
= R [sec (Ф/2) – 1] (6) 
Versed sine of the curve = EF = OF - OE
= R – Rcos (Ф/2)
= R [1 – cos (Ф/2)]
= R x versed sine (Ф/2) (7)
**NOTE:
(1 – cosθ) = versed sinθ
For l =RФ, 100=RD
So R=100/D
Therefore l =100Ф/D
From ΔBOT1
sec(Ф/2)=OT1/BO
or BO= OT1sec(Ф/2)