Setting out simple circular curve- two theodolite method

Objective:

The main objective of this survey job is to set out a horizontal simple circular curve on the ground utilizing two theodolite methods in the field.

Apparatus:

  • Two theodolites
  • Two ranging rods

Least count of the instrument:

Least count of theodolite = 00°00’05’’

Related theory:

Principle of two theodolite method

This method is based on the principle that the deflection angle between a tangent and a chord is equal to the angle which the chord subtends in the alternate segment.

Mathematical explanation

Suppose, two tangent points are marked on the ground such that T1 is the point of commencement and T2 be the point of tangency. Let, I be their point of intersection and O is the center of the simple circular curve as shown in the figure:

Let, T1 a be the first sub-chord of length C1 and δ1 be its deflection angle. As we know from the properties of the circle:

∠ I T1 a = ½ ∠ T1O a

But;

∠ T1T2 a = ½ ∠ T1O a

∠ I T1 a = ∠ T1T2 a = δ1

Similarly, it can be shown that:

∠ I T1 b = ∠ T1T2 b = δ2

Practical application of two theodolite method

  • The two theodolite method is widely used in setting out railway curves because it gives the correct location of the points.
  • This method does not require any tape or chain to fix points on the curve.
  • Each point is set irrespective of other points so this method provides more accuracy in fieldwork.
  • If any error occurs at one point of the curve, it doesn’t affect the accuracy of other points.
  • When it is unfavorable to perform chaining because the ground surface is uneven from undulations, two theodolite method can be very conveniently employed.

Procedure:

  • The following steps are performed to set out a simple circular curve using two theodolite method:
  • Mark T1 and T2 tangent points on the ground.
  • Make a table for deflection angles for the first sub-chord, normal chords, and the last sub-chord.
  • Center and level the theodolite over T2 properly. Set vernier A to 0° and tighten the upper clamp.
  • Release the lower clamp and slowly turn the telescope towards the ranging rod such that the ranging rod at is perfectly T1 bisected. Now fix the lower clamp at this position.
  • Take off the ranging rod from point T1. Center and level the second theodolite over this point. Set the vernier A to 0°. First, tighten the upper clamp then release the lower clamp and bisect the ranging rod at B. Again, the lower clamp is fixed is now fixed at this point.
  • Release the upper clamps of both theodolites and set the first deflection angle δ1 on vernier A of both theodolites.
  • Locate a point P1 such that the lines of sight of both theodolites intersect at this point. Thus, P1 is a point on the curve.
  • Now, set the next deflection angle δ2 on vernier A of both the instruments. Again, locate a point P2, such that the lines of sight of both instruments intersect at it, hence, P2 is the next point on the curve.
  • Continue this procedure till all the deflection angles are set out and all the points are marked on the ground.
  • Set out the total deflection angle δn on both the theodolites such that the line of sight at T1 should bisect T2 and the line of sight of theodolite at T2 should bisect the point B.

Observations & Calculations:

Points Chainage (m) Chord length (m) Deflection angle δ
T1
P1
P2
T2

Results & Discussions:

A simple circular curve has been set out using two theodolite method.

Precautions:

  • Theodolite should be adjusted and leveled properly and tightly clamped for precise readings.
  • Cover theodolite with an umbrella to protect it from direct sunlight and rain.
  • Helmet, closed-toe shoes, and safety vests must be worn by all field workers.
  • Pegs should be laid vertically by mallet into the ground.

Precautions:

  • In centering and levelling, make sure that the tribrach is made as horizontally as possible.
  • Make sure the legs of the stand stay firmly fixed into the ground.
  • Theodolite should be adjusted and leveled properly and tightly clamped for precise readings.
  • Cover theodolite with an umbrella to protect it from direct sunlight and rain.
  • Helmet, closed-toe shoes, and safety vests must be worn by all field workers.
  • Follow the proper step-by-step procedure to avoid any errors.
  • Establish theodolite station points on firm ground.
  • Keep ranging rods exactly vertical by holding it with both hands.
  • Place the ranging rods on clean and leveled ground.
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