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CIVIL ENGINEERING DEPARTMENT

SURVEYING (3140601)

Trigonometric levelling

Prepared by

Ajay B. Patel

Assistant Professor

Civil Engineering Department

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Introduction�

�This is an indirect method of levelling.

In this method the difference in elevation of the points is determined from the observed vertical angles and measured distances.

The vertical angles are measured with a transit theodolite and

The distances are measured directly (plane surveying) or computed trigonometrically (geodetic survey).

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Trigonometric levelling is commonly used in topographical work to find out the elevation of the top of buildings, chimneys, church spires, and so on.

Also, it can be used to its advantage in difficult terrains such as mountainous areas.

Depending upon the field conditions and the measurements that can be made with the instruments available, there can be innumerable cases.

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The principle of trigonometric heighting�

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HEIGHTS And DISTANCES

When the distance btw the stations is not large, the distance btw the stations measured on the surface of the earth or computed trigonometrically may be assumed as a plane distance.
The amount of correction due to curvature of the earth surface an refraction just be ignored.
Depending on field conditions, the following three cases are involved

Case 1)Base of the object is accessible

Case 2)Base of the object inaccessible and instruments station are in the same vertical plane

Case 3)Base of the object inaccessible and instruments are not in same vertical plane

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(Case 1)�Determination of elevation of object when the base is accessible�the object is Vertical��It is assumed that the horizontal distance between the instrument and the object can be measured accurately. In Fig. 1, let B = instrument station F = point to be observed = center of the instrument AF = vertical object D = CE = horizontal distance 1�= height of the instrument at Bh = height FES = reading on the levelling staff held vertical on the Bench Mark (B.M)��= angle of elevation of the top of the object�so, H=D tan z

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R.L of F= R.L of B.M. + h + D tan z
Corrections for curvature and refraction

C =0.06735(D*D) so the true R.L is R.L of B.M. + h +D tanz + C

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If the both the angle of depression and elevation are given to us then we can directly find the height of the whole building.
Let us assume the angle of elevation is z1 and angle of depression is z2 and the object is accessible and the distance between instrument and foot of building is D
then,

Height of building= D tan z1 + D tan z2

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Case 2�Base of the object is not accessible

Depending upon the terrain, three cases may rise:

1. When the instrument axes at both stations P and Q are at the same level.

2. When the instrument axes at the stations P and Q are at different levels but the difference in level I small.

3. When then instrument axes at stations P and Q are at different levels and the difference in level is more.

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Case 3 �Base of the object is not accessible

The instrument stations and the elevated object not in the same vertical plane
This is the most practical case on field if we consider in comparison with other cases
In this case we use the sine law for finding the distances example D1 and D2

For example

(d sin z1)/sin z3 = D2