1. 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
2. INTRODUCTION
βTrigonometric levelling is the process of determining the differences of elevations of
stations from observed vertical angles and known distances.β
β’ The vertical angles are measured by means of theodolite.
β’ The horizontal distances by instrument
β’ Relative heights are calculated using trigonometric functions.
β’If the distance between instrument station and object is small, correction for earth's
curvature and refraction is not required.
β’ 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.
β’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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3. Method of Observation
β’ There are two methods of observation:
1. Direct Method
2. Reciprocal method
1. Direct Method:
β’ 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.
β’ 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.
β’ Sometimes, the instrument is set on any suitable station and the height of instrument
axis is determined by taking back sight on B.M.
β’ Then the observation of the top of the tower is taken.
β’ 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.
β’ 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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4. Method of Observation
2. Reciprocal method:
β’ In this method, the instrument is set on each of the two stations, alternatively and
observations are taken.
β’ Let, Difference in elevation between two stations A & B is to be determined.
β’ First, set the instrument on A and take observation of B. Then set the instrument at
B and take observation of A.
β’ Set the theodolite on A and measure the angle of elevation of B as L BAC = Ξ±
β’ Set the theodolite on B and measure the angle of depression of A as L ABCβ = Ξ²
β’ Measure the horizontal distance between A & B as D using tape.
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6. Method of Observation
β’ If Distance between
A & B is Small,
AB' = AC = D
L ACB = 900
Similarly,
BA' = BC' = D
L AC'B = 900
BC = D tan Ξ±
AC' = D tan Ξ²
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7. Method of Observation
β’ If Distance between A & B is large,
β’ The correction for earth's curvature and refraction is required.
β’ Combined correction = 0.0673 D2, Where D = distance in KM.
β’ The difference in elevation between A & B,
H=BB'
=BC + CB'
=D tan Ξ± + 0.0673 D2. ....(1)
β’ If the angle of Depression B to A is measured,
AC'=D tan Ξ² [ BC'= D ]
β’ True difference in elevation between A & B,
H=AA'
= ACβ β AβCβ
=D tan Ξ² - 0.0673 D2 ....(2)
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8. Method of Observation
β’ Adding equation (1) & (2),
2 H = D tan Ξ± + D tan Ξ²
H = D/2 [tan Ξ± + D tan Ξ²] ..... (3)
β’ From equation (3), it can be seen that by reciprocal method of observation, the
correction for earthβs 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 Ξ± + D tan Ξ²]
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9. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
β’ 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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10. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
Case 1: Base of the object accessible:
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11. METHODS OF DETERMINING THE ELEVATION OF A POINT
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.
Ξ± = Angle of elevation = L BAC
From fig.,
h = D tan Ξ± . ....(1)
R.L. of B = R.L. of B.M. + Bs + h
= R.L. of B.M. + Bs + D. tan Ξ± . ....(2)
β’ Now, when the distance between instrument station and object is large, then
correction for earth's curvature and refraction is required.
β’ combined correction = 0.0673 D2 Where D = distance in KM.
R.L. of B = R.L. of B.M. + Bs + D. tan Ξ± + 0.0673 D2
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12. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
Case 2: Base of the object inaccessible, Instrument stations in the vertical plane as the
elevated object:
β’ 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.
β’ In this method, the instrument is set on the two different stations and the observations
of the object are taken.
β’ 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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13. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
1. Instrument axes at the same level:
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14. METHODS OF DETERMINING THE ELEVATION OF A POINT
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
Ξ±1 = Angle of elevation from A to P
Ξ±2 = Angle of elevation from B to P
From, Ξ PAβPβ , h1 = D tan Ξ±1
. ....(1)
Ξ PBβPβ, h2 = (b + D) tan Ξ±2 ....(2)
β’ Equating (1) & (2),
D tan Ξ±1 = (b + D) tan Ξ±2
D tan Ξ±1 = b tan Ξ±2 + D tan Ξ±2
D (tan Ξ±1 - tan Ξ±2 ) = = b tan Ξ±2
D =
π πππ πΆ π
πππ πΆ πβ πππ πΆ π
....(3)
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15. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
Substitute value of D in equation (1),
D =
π πππ§ π π πππ§ π π
πππ§ π πβ πππ§ π π
....(4)
R.L. OF P = R.L. Of B.M. + Bs + h
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16. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
2. Instrument axes at different levels:
β’ In the field, it is very difficult to keep the same height of instrument axis, at
different stations.
β’ 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.
β’ 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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17. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
A. Height of instrument axis never to the object is lower:
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18. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
From, Ξ PAβPβ , h1 = D tan Ξ±1
. ....(1)
Ξ PBβPββ, h2 = (b + D) tan Ξ±2 ....(2)
Deducting equation (2) from equation (1)
h1 β h2 = D tan Ξ±1 - (b + D) tan Ξ±2
= D tan Ξ±1 - b tan Ξ±2 β D tan Ξ±2
hd = D (tan Ξ±1 - tan Ξ±2 ) βb tan Ξ±2 (hd = h1 - h2 )
hd + b tan Ξ±2 = D (tan Ξ±1 - tan Ξ±2 )
D =
π‘ π+ π πππ§ π π
πππ§ π πβ πππ§ π π
D =
(π + π‘ π ππ¨π π π) πππ§ π π
πππ§ π πβ πππ§ π π
...(3)
Substitute value of D in equation (1),
h1 = D tan Ξ±1
h1 =
(πβ π π πππ πΆ π) πππ πΆ π πππ πΆ π
πππ πΆ πβ πππ πΆ π
...(4)
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19. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
B. Height of instrument axis to the object is higher:
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20. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
From, Ξ PAβPβ , h1 = D tan Ξ±1
. ....(1)
Ξ PBβPββ, h2 = (b + D) tan Ξ±2 ....(2)
Deducting equation (1) from equation (2)
h2 β h1 = (b + D) tan Ξ±2 - D tan Ξ±1
= b tan Ξ±2 + D tan Ξ±2 β D tan Ξ±1
hd = b tan Ξ±2 + D (tan Ξ±2 - tan Ξ±1) (hd = h1 - h2 )
hd - b tan Ξ±2 = D (tan Ξ±2 - tan Ξ±1)
- hd + b tan Ξ±2 = D (tan Ξ±1 - tan Ξ±2)
D =
π πππ πΆ πβπ π
πππ πΆ πβ πππ πΆ π
D =
(πβ π π πππ πΆ π) πππ πΆ π
πππ πΆ πβ πππ πΆ π
...(3)
Substitute value of D in equation (1),
h1 = D tan Ξ±1
h1 =
(πβ π π πππ πΆ π) πππ πΆ π πππ πΆ π
πππ πΆ πβ πππ πΆ π
...(4)
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21. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
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In above two cases, the equations of D and h1 are,
D =
(πΒ± π π πππ πΆ π) πππ πΆ π
πππ πΆ πβ πππ πΆ π
h1 =
(πΒ± π π πππ πΆ π) πππ πΆ π πππ πΆ π
πππ πΆ πβ πππ πΆ π
Use (+) sign if A is lower & Use (-) sign if A is higher.
22. METHODS OF DETERMINING THE ELEVATION OF A POINT
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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23. METHODS OF DETERMINING THE ELEVATION OF A POINT
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:
β’ Select two survey stations A and B on the level ground and measure b as the horizontal
distance between them.
β’ 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.
β’ Measure the angle of elevation Ξ±1 to P.
β’ Measure the horizontal angle at A, L BAC = ΞΈ
β’ Shift the instrument to B and Measure the angle of elevation Ξ±2 to P.
β’ Measure the horizontal angle at B as Ξ±.
24. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
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Ξ±1 = angle of elevation from A to P
Ξ±2 = angle of elevation from B to P
Ξ = Horizontal angle L BAC at station A (clockwise)
Ξ± = 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 Ξ ABC, L BAC = Ξ
L ABC = Ξ±
So, L ACB = 180 β (Ξ + Ξ±)
AB = b
25. METHODS OF DETERMINING THE ELEVATION OF A POINT
BY THEODOLITE
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We know, three angles and one side of Ξ ABC. Therefore using Sine Rule, we can calculate distance AC & BC
as below.
BC =
π πππ π½
π¬π’π§ [πππβ π½+π ]
β¦β¦(1)
Ac =
π πππ πΆ
π¬π’π§ [πππβ π½+π ]
β¦β¦(2)
Now, h1= AC tan Ξ±1 & h2= BC tan Ξ±2
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.
26. 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.
27. INDIRECT LEVELLING ON A ROUGH TERRAIN
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β’ Set up the theodolite at some convenient point O1 midway between P and Q.
β’ Measure the vertical angle Ξ±1 to the station P. Also measure the horizontal distance
D1 between O1 and P.
β’ Similarly, measure the vertical angle Ξ²1 to the station Q. also measure the
horizontal distance D2 between O1 and Q.
β’ Determine the difference in elevation H1 between P and Q as explained below.
β’ Let us assume that,
Ξ±1 = angle of depression
Ξ²1 = angle of elevation
H1 = PPβ +QQβ
= (PPβ β PβPβ) + (QQβ + QβQβ)
= (D1 tan Ξ±1 - C1) + (C2 + D2 tan Ξ²1) β¦ β¦ β¦ (1)
β’ Where C1 and C2 are the corrections due to curvature of earth and refraction.
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β’ As the distances D1 and D2 are nearly equal, the corrections C1 and C2 are also approximately
equal.
β΄ H1 = D1 tan Ξ±1 + D2 tan Ξ²1 β¦ β¦ β¦ (2)
β’ Now shift the instrument to the station O2 midway between Q and R.
β’ Measure the vertical angles Ξ±2 and Ξ²2 to the stations Q and R and respective
horizontal distance D3 and D4.
β΄ Difference of elevations between Q and R is,
H2 = D3 tan Ξ±2 + D4 tan Ξ²2 β¦ β¦ β¦ (3)
β’ Repeat the above process at the station O3
β΄ H3 = D5 tan Ξ±3 + D6 tan Ξ²3 β¦ β¦ β¦ (4)
β’ Determine the difference in elevations of P and S as,
H = H1 + H2 + H3
β’ β΄R.L. of S = R.L. of P + H
29. INDIRECT LEVELLING ON A ROUGH TERRAIN
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β’ Indirect levelling is not as accurate as direct levelling with a levelling instrument. The
method is used in rough country.
β’ If back sight and foresight distances are approximately equal, the effect of curvature and
refraction is eliminated.
30. INDIRECT LEVELLING ON A STEEP TERRAIN
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β’ If the ground is quite steep, the method of indirect levelling can be used with
advantage.
31. INDIRECT LEVELLING ON A STEEP TERRAIN
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β’ The following procedure can be used to determine the difference of elevations between P
and R.
β’ Set up the instrument at a convenient station O1 on the line PR.
β’ Make the line of collimation roughly parallel to the slope of the ground. Clamp the
telescope.
β’ Take a back sight PPβ on the staff held at P. Also measure the vertical angle Ξ±1 to Pβ.
Determine R.L. of Pβ as.
R.L. of Pβ = R.L. of P + PPβ
β’ Take a foresight QQβ on the staff held at the turning point Q. without changing the vertical
angle Ξ±1. Measure the slope distance PQ between P and Q.
R.L. of Q = R.L. of Pβ + PQ sin Ξ±1 β QQβ
OR
R.L. of Q = R.L. of P + PPβ + PQ sin Ξ±1 β QQβ
32. INDIRECT LEVELLING ON A STEEP TERRAIN
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β’ 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.
β’ Take a back sight QQβ on the staff held at the turning point Q. Measure the vertical
angle Ξ±2.
R.L. of Qβ = R.L. of Q + QQβ
β’ Take a foresight RRβ on the staff held at the point R without changing the vertical
angle Ξ±2. Measure the sloping distance QR.
β΄ R.L. of R = R.L. of Qβ + QR sin Ξ±2 β RRβ
Thus,
R.L. of R = (R.L. of P + PPβ + PQ sin Ξ±1 β QQβ) + QQβ + (QR sin Ξ±2 β RRβ)