Curve Surveying - Circular Curves

Most of the alignment in highways and railways is straight or tangent. Whenever there is a requirement to change the direction, a gradual change is brought in the alignment, which results in curves. 

Curves are used in surveying alignments to change the direction of motion to lessen the forces involved when a vehicle changes direction is necessary. 

Take a Quick Video: Components of Circular Curves

Types of Curves in Surveying

The three main types of curves used in surveying are:

1. Circular or Simple Curves
2. Spiral or Transition Curves
3. Vertical or Parabolic Curves

Circular or Simple Curves

A circular or simple curve is the simplest curve and it forms a segment of a circle. Circular curves are used for horizontal alignments as they can be laid out on the ground using basic surveying tools and techniques.

A circular curve is laid using a chain or EDM to measure the distance along the arc of the curve. A theodolite or transit is used to measure the horizontal angles from a reference line to the station to be set. 

The parameter required for laying a circular curve are:

1. The radius of the curve
2. The beginning of the station
3. The distance along the arch between the instrument and the points to be set

Geometry of  a Curve

The main components of a curve is given the figure below:

From the figure above, important curve parameters can be determined. The given details of the curve is the radius (R).

1. Subtangent T

tan (Δ/2) = T/R

T = Rtan (Δ/2)

2. Long Chord (C)

Sin (Δ/2) = (1/2C)/R

C= 2R sin(Δ/2)

3. Mid Ordinate (M)

cos (Δ/2) = OB/R

OB = Rcos (Δ/2)

But, OB = R - M

Rcos (Δ/2) = R - M

M = R { 1- cos ( Δ/2) }

4. External Distance (E)

Consider triangle O-PI-BC,

cos (Δ/2) = R/ (R+E)

E = R { sec (Δ/2)) -1 }

5. Length of Curve ( L)

L/2πR = Δ/360

L = 2πR (Δ/360)

6. Fractional Portion  of Curve

From the figure, we can derive the following relation:

360/ Δ = πR²/As= 2πR/L

Δ/360 = As/ πR²= L/2πR

Where, As is the area of the sector and L is the length of chord of sectior.

Hence the fractional part of the curve is:

Fractional Part =  Δ/360 = L/2πR = As/ πR²

7. Tangent Deflection Angle Δ/ Central Angle of the Curve

Δ = (L x 360 ) 2πR

8. Degree of Curve (D)

As per Highway, D is the central angle subtended by a 100' arc and as per Railroad, it is the central angle subtended by a 100; chord.

From the figure above:

D/360 = 100/ 2πR

D = 5729.58/R and L/100 = Δ/D

Also Read: Trilateration Surveying - Working, Applications & Advantages