theodolite notes

Surveying II 10CV44

Select a point O in the plane of the mid cross section and join it to the vertices of the end cross
sections. The prismoid is thus divided into a number of pyramids with O as the apex and the
bases as side and end faces of the prismoid.

Volume of pyramid OA1B1C1D1 = (1/3) (L/2)A1 = A1L/6
Volume of pyramid OA2B2C2D2 = (1/3) (L/2)A2 = A2L/6
Volume of pyramid OA1B1A2B2 = (1/3) h L EF

= 1/ 3 L h EF

= 2/3 L(area EFO)

Similarly,

Volume of pyramid OC1D1C2D2 = 2/3 L(area GHO)

Volume of pyramid OA1D1A2D2 = 2/3 L(area EHO)

Volume of pyramid OB1C1B2C2 = 2/3 L(area FGO)

Total volume = A1L/6 + A2L/6 + 2/3 L(area EFO)

+ 2/3 L(area GHO) + 2/3 L(area EHO)

+ 2/3 L(area FGO)

= A1L/6 + A2L/6 + 2/3 L(area EFGH)
= A1L/6 + A2L/6 + 2/3 AnL
= 1/6(A1 + A2 + 4An)
Let A1, A2, ..., An be the areas of various cross-sections spaced at a uniform interval L.
Volume between first three sections constituting the first prismoid,
V1 = L/3 (A1 + 4A2 + A3)
Volume of the next prismoid = L/3 (A3 + 4A4 + A5) ... and so on
Total volume,

V = L/3 (A1 + 4A2 + A3 + A3 + 4A4 + A5 + ... + An-2 + 4An-1 + An)
V = L/3 [(A1 + An) + 4(A2 + A4 + ... + An-1) + 2(A2 + A5 + ... + An-2)]

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Surveying II 10CV44

Prismoidal Correction (CP)

As defined before, it is the difference between the volume computed by the end area formula and
the prismoidal formula. If the area is calculated by the end area formula, the prismoidal
correction is subtracted to obtsin the exact volume. The value of the correction for various cases
is as follows.

Level section, CP = Ls/6 (h – h')2

Two-level section, CP = L/6s (d – d') (d1 – d1')

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Surveying II 10CV44

Side-hill two-level section,
CP = L/12s (d – d') [(b + nh)/2 – (b + n'h')/2] for cut
CP = L/12s (d1 – d1') [(b/2 – nh) – (b/2 – n'h')] for hill

Three-level section, CP = d/12 (h – h') [(d + d1) (d' + d1')]

where d, d1, h and n refer to the cross section at one end and d', d1', h' and n' refer to the same at
the other end.

Curvature Correction (CC)

In all the formulae stated above, it was assumed that the c/l of the cutting or embankment is a
straight line. In practice, however, often the c/l is curved. In such cases the volume of earthwork
is calculated by assuming the c/l to be straight and effect of curvature is accounted for in the final
estimates. The curved volumes are calculated with the pappu's theorem which states that the
volume swept by a constant area rotating about a fixed axis is equal to the product of that area
and the length of the path traced by the centroid of the area.

Level Section No correction is required, since the area is symmetrical about the centre axis.

Two-level Section and Three-level Section
CC = ± L/6R (d2 – d12) (h + b/2s)

where R is the radius of the curve and CC is the curvature correction for a length L

Two-level Section The curvature correction to the area

CC = Ae/R per unit length

where e = dd1 (d + d1) / 3An

Side-hill Two-level Section

where CC = Ae/R per unit length for large area
e = 1/3 (d + b/2 – nh),

and e = 1/3 (d1 + b/2 + nh), for small area

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Surveying II 10CV44

Volume from spot levels

This method of estimating the earthwork quantities is also known as the unit area method. In this

method the area is divided into regular figures such as squares, rectangles or triangles and the

levels of corners of the figures are measured before and after the construction. Thus the depth of

the cut or fill at each corner is a known parameter. The corners of the figures may be at different

elevations but lie in the same plane. The surface of the ground within the figure, therefore lies in

an inclined plane.

a b
a1 b1

d d1 c1
c

If the depth of the cuts are a1, b1, c1, and d1, respectively, and A is the area of the figure abcd,
then the volume is given by

V = (a1 + b1 + c1 + d1)A/4

Mass Diagram

The execution of earthwork comprises of four operations, viz. Cutting, loading, hauling, and
filling. Except hauling, the cost of the remaining three operations depend upon the character of
the material. The haulage cost is a function of the weight of the material as well as the distance
from the place of excavation to the place of fill. Since overhaul is an added cost and as it affects
the economy considerably, it is desirable to deal with the problem carefully. This may be
accomplished graphically by means of a distribution diagram known as the mass diagram or
mass curve. It is a diagram, usually regular, whose abscissae represent distances and ordinates
the volume of earthwork.

Sources of Error

Some of the common errors are :

1. Approximating the end areas.

2. Failing to apply prismoidal and curvature corrections.

3. Computing the areas of cross sections beyond the limit justified by the field data.

4. Computing the volumes beyond the nearest cubic metre.

Mistakes

Some of the typical mistakes are:

1. Errors in airthmetic.

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Surveying II 10CV44

2. Using the prismoidal formula when end area volumes are sufficiently accurate.
3. Mixing cut and fill quantities.
4. Failing to consider transition sections when passing from cut to fill, or from fill to cut.

Dept of Civil Engineering, SJBIT Page 105