1. Tacheometric Surveying Mr. Vedprakash Maralapalle, Asst. Professor
Department: B.E. Civil Engineering
Subject: Surveying – II
Semester: IV
Teaching Aids Service by KRRC Information Section

2. Introduction Uses of Tacheometry
The primary object of tacheometry is the preparation of contoured maps or plans requiring both the horizontal as well as Vertical control. Also, on surveys of higher accuracy, it provides a check on distances measured with the tape.
Uses of Tacheometry
Preparation of topographic maps which require both elevations and horizontal distances.
Survey work in difficult terrain where direct methods are inconvenient.
Reconnaissance surveys for highways, railways, etc.
Checking of already measured distances
Hydrographic surveys.

3. INSTRUMENTS
Tacheometer An ordinary transit theodolite fitted with a stadia diaphragm is generally used for tacheometric survey.
- The stadia diaphragm essentially consists of one stadia hair above and the other an equal distance below the horizontal cross-hair, the stadia hairs being mounted in the ring.
Different forms of stadia diaphragm commonly used

4. The multiplying constant should have 100.
Characteristics of Tacheometer
The multiplying constant should have 100.
The telescope should be truly anallatic lens to make the additive constant zero
The telescope should be powerful having a magnification of 20 to 30 diameters.
The aperture of the objective should be 35 to 45 mm in diameter to have a sufficiently bright image.
The eye- piece should be greater magnifying power than usual, so that it is possible to obtain a clear staff reading from a long distance.

5. Levelling staff : For small distances (say up to 100 meters), ordinary levelling staff may be used. It is normally 4 m. long . Minimum reading is of 0.005m can be taken.
Stadia rod : For greater distances a stadia rod may be used. A stadia rod is usually of one piece, having 3 – 5 meters length. A stadia rod graduated in 5 mm (i.e m) for smaller distances and while for longer distances, the rod may be graduated in 1 cm (i.e m).

6. 10' 7"

7. Method of tacheometry The stadia method Fixed hair method
Moveable hair method
Tangential method

8. Fixed hair method
The distance between the stadia hairs is fixed in this method, which is the commonly used.
When the staff is sighted through the telescope a certain portion of the staff is intercepted by the upper and lower stadia.
The value of the staff intercept is varies with the distance.
The distance between the station and the staff can be obtained by multiplying the staff intercept by stadia constant.
This is the most common method used in tacheometry Surveying.

9. Moveable hair method
In this method the staff intercept is kept constant, but the distance between the stadia hairs is variable.
The diaphragm consists of a central wire fixed with the axis of the telescope.
The upper and lower stadia wires can be moved by micrometer screws in a vertical plane.
The distance by which the stadia wires are moved is measured according to the number of turns of the micrometer screws.

10. special type diaphragm of a moving hair theodolite

11. Tangential Method
The diaphragm of the tacheometer is not provided with stadia hair.
The reading are taken by the single horizontal hair.
To measure the staff intercept , two pointing are required. The angle of elevation or depression are measured and their tangents are used for finding the horizontal distances and elevations.
This method not use generally.

12. PRINCIPLE OF STADIA METHOD
The stadia method is based on the principle that the ratio of the perpendicular to the base is constant in similar isosceles triangles. A O A2 A1 B2 B1 B C2 C1 C ) β
In figure, let two rays OA and OB be equally inclined to central ray OC.
Let A2B2, A1B1 and AB be the staff intercepts. Evidently,
OC2
A2B2
OC1
A1B1 OC AB = β 2
= constant k = ½ cot
This constant k entirely depends upon the magnitude of the angle β.

13. In actual practice, observations may be made with either horizontal line of sight or with inclined line of sight.
In the later case the staff may be kept either vertically or normal to the line of sight.
First the distance-elevation formulae for the horizontal sights should be derived.
Horizontal Sights: i . f2 f1 s O d M b C B A c a D
Consider the figure, in which O is the optical centre of the objective of an external focusing telescope.
Let A, C, and B = the points cut by the three lines of sight corresponding to
three wires.
b, c, and a = top, axial and bottom hairs of the diaphragm.
ab = i = interval b/w the stadia hairs (stadia interval)
AB = s = staff intercept;
f = focal length of the objective

14. f i D = s + (f + d) = k . s + C 1 1 1 f f2 f1 f1 f2 s i f1 = f + f
f1 = horizontal distance of the staff from the optical centre of the objective
f2 = horizontal distance of the cross-wires from O.
d = distance of the vertical axis of the instrument from O.
D = horizontal distance of the staff from the vertical axis of the instruments.
M = centre of the instrument, corresponding to the vertical axis.
Since the rays BOb and AOa pass through the optical centre, they are straight so that AOB and aOb are similar. Hence,
f s
f i =
Again, since f1 and f2 are conjugate focal distances, we have from lens formula,
f f f1 + = f1 f2
Multiplying throughout by ff1, we get f1 = f + f
f s
f i =
Substituting the values of in the above, we get s i
f1 = f + f
Horizontal distance between the axis and the staff is D = f1 + d
D = s + (f + d) = k . s + C f i

15. Distance and Elevation formulae for Staff Vertical : Inclined Sight
Let P = Instrument station; Q = Staff station
M = position of instruments axis; O = Optical centre of the objective
A, C, B = Points corresponding to the readings of the three hairs
s = AB = Staff intercept; i = Stadia interval
Ө = Inclination of the line of sight from the horizontal
L = Length MC measured along the line of sight
D = MQ’ = Horizontal distance between the instrument and the staff
V = Vertical intercept at Q, between the line of sight and the horizontal line
h = height of the instrument; r = central hair reading
β = angle between the two extreme rays corresponding to stadia hairs.

16. Draw a line A’CB’ normal to the line of sight OC.
Angle AA`C = β/2, being the exterior angle of the ∆COA`.
Similarly, from ∆COB`, angle OB`C = angle BB`C = 900 – β/2. h B A O D P A` C B` Q Q` M r V Ө β L

17. D = k s cos2Ө + C cosӨ . . . . . . (1) sin2Ө V = k s + C sinӨ 2
Since β/2 is very small (its value being equal to 17’ 11” for k = 100), angle AA’C and angle BB’C may be approximately taken equal to 900.
∟AA’C = ∟BB’C = 900
From ∆ ACA’, A’C = AC cos Ө or A’B’ = AB cos Ө = s cos Ө ……….(a)
Since the line A’B’ is perpendicular to the line of sight OC, equation D = k s + C is directly applicable. Hence, we have
MC = L = k . A’B’ + C = k s cosӨ + C (b)
The horizontal distance
D = L cosӨ = (k s cosӨ + C) cosӨ
D = k s cos2Ө + C cosӨ (1)
Similarly, V = L sin Ө = (k s cosӨ + C) sinӨ = k s cosӨ . sinӨ + C sinӨ
(2)
V = k s C sinӨ
sin2Ө 2
Thus equations (1) and (2) are the distance and elevation formulae for inclined line of sight.

18. Elevation of Q = Elevation of P + h – V - r
(a) Elevation of the staff station for angle of elevation
If the line of sight has an angle of elevation Ө, as shown in the figure, we have
Elevation of staff station = Elevation of instrument station + h + V – r.
(b) Elevation of the staff station for the angle of depression:
Elevation of Q = Elevation of P + h – V - r

19. Distance and Elevation formulae for Staff Normal : Inclined SightӨ
rcosӨ B V β Q O Ө M C` Q` h P
L cosӨ
rsinӨ D
Figure shows the case when the staff is held normal to the line of sight.

20. D = MC’ + C’Q’ = L cosӨ + r sinӨ = (k s + C) cosӨ + r sinӨ
Case (a): Line of Sight at an angle of elevation Ө
Let AB = s = staff intercept;
CQ = r = axial hair reading
With the same notations as in the last case, we have
MC = L = K s + C
The horizontal distance between P and Q is given by
D = MC’ + C’Q’ = L cosӨ + r sinӨ
= (k s + C) cosӨ + r sinӨ
(3)
Similarly, V = L sinӨ = (k s + C) sinӨ
(4)

21. Case (a): Line of Sight at an angle of depression Ө
rsinӨ M P h Ө A C C’ Q Q’ B C1 D
L cosӨ
rcosӨ V O
Figure shows the line of sight depressed downwards,
MC = L = k s + C
D = MQ’ = MC’ – Q’C’
= L cosӨ - r sinӨ
D = (k s + C) cosӨ - r sinӨ
(5)
(6)
V = L sinӨ = (k s + C) sinӨ
Elevation of Q = Elevation of P + h – V – r cosӨ

22. Subtense Bar
The subtense bar is an instrument used for measuring the horizontal distance between the instrument station and a station where the subtense bar is to be set up.
Substense method is an indirect method of distance determination.
The subtense bar is a metal bar of length varying from 3 to 4 m.

24. There are two discs of diameter about 20 cm at both ends of the bar.
The alidade is made perpendicular to the axis of the bar.
A spirit level is included for levelling. The bar is mounted on a tripod stand which contains a ball and socket arrangement for levelling.

25. Procedure of observation
Tacheometer is set up at T. the subtense bar is set up and level at C. the point B and D are the discs 3m apart . H is the required horizontal distance.
With the help of alidade the line of sight of the telescope is made perpendicular to the axis of the bar.
The horizontal distance <BTD is measured by the repetition method. Let this angle be θ.
Now TC is perpendicular to BD and C is the middle point of BD .
Tan θ./2 = s/2/H
H tan θ./2 = s/2 Or
H = ½ (S cot θ./2 )

26. References
Surveying and Levelling: Vol-I and II: Kanetkar and Kulkarni, Pune Vidyarthi Griha, Pune.
Surveying and Levelling: N N Basak, Tata McGraw Hill, New Delhi.
Surveying: R. Agor, Khanna Publishers.
Surveying: Vol-I: Dr K.R. Arora, Standard Book House.
Kanpur and IIT Madras.
Google images.