Trigonometric levelling

TRIGONOMETRIC LEVELING
Mahatma Gandhi Institute Of
Technical Education
& Research Centre, Navsari (396450)
SURVEYING
4TH SEMESTER
CIVIL ENGINEERING
PREPARED BY:
Asst. Prof. GAURANG PRAJAPATI
CIVIL DEPARTMENT

INTRODUCTION
“Trigonometric levelling is the process of determining the differences of elevations of
stations from observed vertical angles and known distances.”
• The vertical angles are measured by means of theodolite.
• The horizontal distances by instrument
• Relative heights are calculated using trigonometric functions.
•If the distance between instrument station and object is small, correction for earth's
curvature and refraction is not required.
• If the distance between instrument station and object is large, the combined correction =
0.0673 D2 for earth's curvature and refraction is required. Where D = distance in KM.
•If the vertical angle is (+ve), the correction is taken as (+ve) & If the vertical angle is (-ve),
the correction is taken as (-ve)
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Method of Observation
• There are two methods of observation:
1. Direct Method
2. Reciprocal method
1. Direct Method:
• This method is useful Where is not possible to set the instrument over the station
whose elevation is to be determined. i.e. To determine height of a tower.
• In this method, the instrument is set on the station on the ground whose elevation is
known. And the observation of the top of the tower is taken.
• Sometimes, the instrument is set on any suitable station and the height of instrument
axis is determined by taking back sight on B.M.
• Then the observation of the top of the tower is taken.
• In this method, the instrument can not be set on the top of the tower and observation
can not be taken of the point on the ground.
• If the distance between instrument station and object is large, the combined
correction = 0.0673 D2 for earth's curvature and refraction is required. Where D =
distance in KM.
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2. Reciprocal method:
• In this method, the instrument is set on each of the two stations, alternatively and
observations are taken.
• Let, Difference in elevation between two stations A & B is to be determined.
• First, set the instrument on A and take observation of B. Then set the instrument at
B and take observation of A.
• Set the theodolite on A and measure the angle of elevation of B as L BAC = α
• Set the theodolite on B and measure the angle of depression of A as L ABC’ = β
• Measure the horizontal distance between A & B as D using tape.
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Correction for Curvature and Refraction
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• If Distance between
A & B is Small,
AB' = AC = D
L ACB = 900
Similarly,
BA' = BC' = D
L AC'B = 900
BC = D tan α
AC' = D tan β
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• If Distance between A & B is large,
• The correction for earth's curvature and refraction is required.
• Combined correction = 0.0673 D2, Where D = distance in KM.
• The difference in elevation between A & B,
H=BB'
=BC + CB'
=D tan α + 0.0673 D2. ....(1)
• If the angle of Depression B to A is measured,
AC'=D tan β [ BC'= D ]
• True difference in elevation between A & B,
H=AA'
= AC’ – A’C’
=D tan β - 0.0673 D2 ....(2)
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• Adding equation (1) & (2),
2 H = D tan α + D tan β
H = D/2 [tan α + D tan β] ..... (3)
• From equation (3), it can be seen that by reciprocal method of observation, the
correction for earth’s curvature and refraction can be eliminated.
R.L of station B = R.L. of station A + H
= R.L. of station A + D/2 [tan α + D tan β]
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METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
• In order to calculate the R.L. Of object, we may consider the following cases.
Case 1: Base of the object accessible
Case 2: Base of the object inaccessible, Instrument stations in the vertical plane as the
elevated object.
Case 3: Base of the object inaccessible, Instrument stations not in the same vertical
plane as the elevated object.
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BY THEODOLITE
Case 1: Base of the object accessible:
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BY THEODOLITE
A = Instrument station
B = Point to be observed
h = Elevation of B from the instrument axis
D = Horizontal distance between A and the base of object
h1 = Height of instrument (H. I.)
Bs = Reading of staff kept on B.M.
α = Angle of elevation = L BAC
From fig.,
h = D tan α . ....(1)
R.L. of B = R.L. of B.M. + Bs + h
= R.L. of B.M. + Bs + D. tan α . ....(2)
• Now, when the distance between instrument station and object is large, then
correction for earth's curvature and refraction is required.
• combined correction = 0.0673 D2 Where D = distance in KM.
R.L. of B = R.L. of B.M. + Bs + D. tan α + 0.0673 D2
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BY THEODOLITE
Case 2: Base of the object inaccessible, Instrument stations in the vertical plane as the
elevated object:
• This method is used When it is not possible to measure the horizontal; distance (D)
between the instrument station and the base of the object.
• In this method, the instrument is set on the two different stations and the observations
of the object are taken.
• There may be two cases:
1. Instrument axes at the same level
2. Instrument axes at different levels
A. Height of instrument axis never to the object is lower
B. Height of instrument axis to the object is higher
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BY THEODOLITE
1. Instrument axes at the same level:
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BY THEODOLITE
A, B = Instrument station
h = Elevation of top of the object (P) from the instrument axis
b = Horizontal distance between A & B
D = Horizontal distance between A and the base of the object
α1 = Angle of elevation from A to P
α2 = Angle of elevation from B to P
From, Δ PA’P’ , h1 = D tan α1
. ....(1)
Δ PB’P’, h2 = (b + D) tan α2 ....(2)
• Equating (1) & (2),
D tan α1 = (b + D) tan α2
D tan α1 = b tan α2 + D tan α2
D (tan α1 - tan α2 ) = = b tan α2
D =
𝒃 𝒕𝒂𝒏 𝜶 𝟐
𝒕𝒂𝒏 𝜶 𝟏− 𝒕𝒂𝒏 𝜶 𝟐
....(3)
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BY THEODOLITE
Substitute value of D in equation (1),
D =
𝐛 𝐭𝐚𝐧 𝛂 𝟏 𝐭𝐚𝐧 𝛂 𝟐
𝐭𝐚𝐧 𝛂 𝟏− 𝐭𝐚𝐧 𝛂 𝟐
....(4)
R.L. OF P = R.L. Of B.M. + Bs + h
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BY THEODOLITE
2. Instrument axes at different levels:
• In the field, it is very difficult to keep the same height of instrument axis, at
different stations.
• Therefore, the instrument is set at two different stations and the height of
instrument axis in both the cases is determined by taking back sight on B.M.
• There are two cases:
A. Height of instrument axis never to the object is lower
B. Height of instrument axis to the object is higher
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BY THEODOLITE
A. Height of instrument axis never to the object is lower:
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BY THEODOLITE
. ....(1)
Δ PB’P’’, h2 = (b + D) tan α2 ....(2)
Deducting equation (2) from equation (1)
h1 – h2 = D tan α1 - (b + D) tan α2
= D tan α1 - b tan α2 – D tan α2
hd = D (tan α1 - tan α2 ) –b tan α2 (hd = h1 - h2 )
hd + b tan α2 = D (tan α1 - tan α2 )
D =
𝐡 𝐝+ 𝐛 𝐭𝐚𝐧 𝛂 𝟐
D =
(𝐛 + 𝐡 𝐝 𝐜𝐨𝐭 𝛂 𝟐) 𝐭𝐚𝐧 𝛂 𝟐
...(3)
h1 = D tan α1
h1 =
(𝒃− 𝒉 𝒅 𝒄𝒐𝒕 𝜶 𝟐) 𝒕𝒂𝒏 𝜶 𝟏 𝒕𝒂𝒏 𝜶 𝟐
...(4)
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BY THEODOLITE
B. Height of instrument axis to the object is higher:
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BY THEODOLITE
. ....(1)
Δ PB’P’’, h2 = (b + D) tan α2 ....(2)
Deducting equation (1) from equation (2)
h2 – h1 = (b + D) tan α2 - D tan α1
= b tan α2 + D tan α2 – D tan α1
hd = b tan α2 + D (tan α2 - tan α1) (hd = h1 - h2 )
hd - b tan α2 = D (tan α2 - tan α1)
- hd + b tan α2 = D (tan α1 - tan α2)
D =
𝒃 𝒕𝒂𝒏 𝜶 𝟐−𝒉 𝒅
D =
(𝒃− 𝒉 𝒅 𝒄𝒐𝒕 𝜶 𝟐) 𝒕𝒂𝒏 𝜶 𝟐
...(3)
h1 = D tan α1
h1 =
(𝒃− 𝒉 𝒅 𝒄𝒐𝒕 𝜶 𝟐) 𝒕𝒂𝒏 𝜶 𝟏 𝒕𝒂𝒏 𝜶 𝟐
...(4)
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BY THEODOLITE
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In above two cases, the equations of D and h1 are,
D =
(𝒃± 𝒉 𝒅 𝒄𝒐𝒕 𝜶 𝟐) 𝒕𝒂𝒏 𝜶 𝟐
h1 =
(𝒃± 𝒉 𝒅 𝒄𝒐𝒕 𝜶 𝟐) 𝒕𝒂𝒏 𝜶 𝟏 𝒕𝒂𝒏 𝜶 𝟐
Use (+) sign if A is lower & Use (-) sign if A is higher.

BY THEODOLITE
Case 3: Base of the object inaccessible, Instrument stations not in the same vertical
plane as the elevated object:
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BY THEODOLITE
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Let A and B be the two instrument stations not in the same vertical plane as that of P.
Procedure:
• Select two survey stations A and B on the level ground and measure b as the horizontal
distance between them.
• Set the instrument at A and level it accurately. Set the vertical Vernier to 0 degree. Bring
the altitude level bubble at the center and take a back sight hs on the staff kept at B.M.
• Measure the angle of elevation α1 to P.
• Measure the horizontal angle at A, L BAC = θ
• Shift the instrument to B and Measure the angle of elevation α2 to P.
• Measure the horizontal angle at B as α.

BY THEODOLITE
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α1 = angle of elevation from A to P
α2 = angle of elevation from B to P
Θ = Horizontal angle L BAC at station A (clockwise)
α = Horizontal angle L CBA at station B (clockwise)
B = Horizontal distance between A & B
h1= PP1= Height of object P from instrument axis of A
h2= PP2= Height of object P from instrument axis of B
In Δ ABC, L BAC = Θ
L ABC = α
So, L ACB = 180 – (Θ + α)
AB = b

BY THEODOLITE
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We know, three angles and one side of Δ ABC. Therefore using Sine Rule, we can calculate distance AC & BC
as below.
BC =
𝒃 𝒔𝒊𝒏 𝜽
𝐬𝐢𝐧 [𝟏𝟖𝟎− 𝜽+𝛂 ]
……(1)
Ac =
𝒃 𝒔𝒊𝒏 𝜶
𝐬𝐢𝐧 [𝟏𝟖𝟎− 𝜽+𝛂 ]
……(2)
Now, h1= AC tan α1 & h2= BC tan α2
Values of AC & BC are obtained from equation (1) & (2) as above.
R.L. of P = Height of instrument axis at A + h1 OR R.L. of P = Height of instrument axis at B + h2
Height of instrument axis at A,
= R.L. of B.M. + B.S.
= R.L. of B.M. + hs
Height of instrument axis at B = R.L. of B.M. + B.S.

INDIRECT LEVELLING ON A ROUGH TERRAIN
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On a rough terrain, indirect levelling can be used to determine the difference of
elevations of two point which are quite apart.
Let difference of elevation of two points P and Q is required.

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• Set up the theodolite at some convenient point O1 midway between P and Q.
• Measure the vertical angle α1 to the station P. Also measure the horizontal distance
D1 between O1 and P.
• Similarly, measure the vertical angle β1 to the station Q. also measure the
horizontal distance D2 between O1 and Q.
• Determine the difference in elevation H1 between P and Q as explained below.
• Let us assume that,
α1 = angle of depression
β1 = angle of elevation
H1 = PP” +QQ”
= (PP’ – P’P”) + (QQ’ + Q’Q”)
= (D1 tan α1 - C1) + (C2 + D2 tan β1) … … … (1)
• Where C1 and C2 are the corrections due to curvature of earth and refraction.

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• As the distances D1 and D2 are nearly equal, the corrections C1 and C2 are also approximately
equal.
∴ H1 = D1 tan α1 + D2 tan β1 … … … (2)
• Now shift the instrument to the station O2 midway between Q and R.
• Measure the vertical angles α2 and β2 to the stations Q and R and respective
horizontal distance D3 and D4.
∴ Difference of elevations between Q and R is,
H2 = D3 tan α2 + D4 tan β2 … … … (3)
• Repeat the above process at the station O3
∴ H3 = D5 tan α3 + D6 tan β3 … … … (4)
• Determine the difference in elevations of P and S as,
H = H1 + H2 + H3
• ∴R.L. of S = R.L. of P + H

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• Indirect levelling is not as accurate as direct levelling with a levelling instrument. The
method is used in rough country.
• If back sight and foresight distances are approximately equal, the effect of curvature and
refraction is eliminated.

INDIRECT LEVELLING ON A STEEP TERRAIN
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• If the ground is quite steep, the method of indirect levelling can be used with
advantage.

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• The following procedure can be used to determine the difference of elevations between P
and R.
• Set up the instrument at a convenient station O1 on the line PR.
• Make the line of collimation roughly parallel to the slope of the ground. Clamp the
telescope.
• Take a back sight PP’ on the staff held at P. Also measure the vertical angle α1 to P’.
Determine R.L. of P’ as.
R.L. of P’ = R.L. of P + PP’
• Take a foresight QQ’ on the staff held at the turning point Q. without changing the vertical
angle α1. Measure the slope distance PQ between P and Q.
R.L. of Q = R.L. of P’ + PQ sin α1 – QQ’
OR
R.L. of Q = R.L. of P + PP’ + PQ sin α1 – QQ’

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• Shift the instrument to the station O2 midway between Q and R. Make the line of
collimation roughly parallel to the slope of the grounds. Clamp the telescope.
• Take a back sight QQ” on the staff held at the turning point Q. Measure the vertical
angle α2.
R.L. of Q” = R.L. of Q + QQ”
• Take a foresight RR’ on the staff held at the point R without changing the vertical
angle α2. Measure the sloping distance QR.
∴ R.L. of R = R.L. of Q” + QR sin α2 – RR’
Thus,
R.L. of R = (R.L. of P + PP’ + PQ sin α1 – QQ’) + QQ” + (QR sin α2 – RR’)

Trigonometric levelling

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