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 δ_n 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:

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.