11. Tacheometric Surveying

 

TACHEOMETRIC SURVEYING

1. Definition

Tacheometric surveying is a rapid method of surveying in which horizontal distance and difference in elevation of points are determined indirectly by optical means using a theodolite (tacheometer) instead of direct chaining.

The word tacheometry comes from Greek:
“Tachos” = speed, “Metron” = measurement
 Meaning: rapid measurement.

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2. Purpose of Tacheometry

Tacheometric survey is used when:

• The ground is rough, hilly, or inaccessible

• Chaining is difficult or impossible

• A quick topographic survey is required

• Contouring work is needed rapidly

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3. Instruments Used

• Tacheometer (theodolite fitted with stadia hairs)

• Levelling staff

• Tripod

• Measuring tape (occasionally)

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4. Principle of Tacheometry

The principle is based on the fact that:

The distance between two stadia hairs in the telescope subtends a fixed angle at the instrument.
Hence, the distance to the staff is proportional to the staff intercept.

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5. Stadia System

The telescope has three horizontal hairs:

• Upper stadia hair

• Central hair

• Lower stadia hair

Staff intercept (s):

$$s = \text{Upper reading} - \text{Lower reading}$$

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6. Distance Formula in Tacheometry

General equation:

$$D = Ks + C$$

Where:

• $D$ = horizontal distance

• $s$ = staff intercept

• $K$ = multiplying constant (≈ 100)

• $C$ = additive constant (≈ 0)

For modern instruments:

$$D = 100s$$

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7. Types of Tacheometric Survey

1. Stadia Method (most common)

Distance is obtained from stadia hair readings.

2. Tangential Method

Used when stadia hairs are not present.
Distance is found using vertical angles.

3. Subtense Bar Method

A bar of known length is observed from distance.

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8. Working of Stadia Method

Case 1: Line of sight horizontal

$$D = Ks + C$$

If $K = 100$ and $C = 0$:

$$D = 100s$$

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Case 2: Line of sight inclined

$$D = Ks \cos^2 \theta + C \cos \theta$$

And vertical difference:

$$V = \frac{Ks \sin 2\theta}{2} + C \sin \theta$$

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9. Numerical Example

Example 1 – Horizontal Distance

Given:

• Upper stadia reading = 2.850 m

• Lower stadia reading = 2.350 m

• Instrument constants:
  $K = 100$, $C = 0$

• Line of sight is horizontal

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Step 1: Find staff intercept

$$s = 2.850 - 2.350 = 0.500 \text{ m}$$

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Step 2: Apply distance formula

$$D = 100 \times 0.5 = 50 \text{ m}$$

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Final Answer

Horizontal distance = 50 m

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10. Numerical Example (Height Difference)

Given:

• Distance $D = 50$ m

• Angle of elevation $\theta = 10^\circ$

Find difference in elevation.

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Formula:

$$h = D \tan \theta$$ $$h = 50 \times \tan 10^\circ$$ $$h = 50 \times 0.1763 = 8.82 \text{ m}$$

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Final Answer

Point is 8.82 m above instrument level.

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11. Advantages of Tacheometric Survey

• Very fast method

• No need for chaining

• Best for hilly and rough terrain

• Requires less manpower

• Suitable for contour surveying

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12. Disadvantages

• Less accurate than chaining

• Needs skilled observer

• Affected by instrument errors

• Not suitable for very precise work

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13. Applications in Civil Engineering

• Topographic surveys

• Contour mapping

• Route surveys (roads, railways, canals)

• Preliminary surveys

• Hydropower and irrigation projects

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14. Difference: Tacheometry vs Chain Survey

Feature	Tacheometry	Chain Survey
Speed	Very fast	Slow
Accuracy	Moderate	High
Terrain	Rough, hilly	Flat ground
Instruments	Theodolite + staff	Chain, tape
Cost	Moderate	Low