earth quantities

8/13/2019 Earth Quantities

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H. Shamaoma SBE Dept of Urban and Regional Planning ESC340: Land Surveying

Earth quantities

Volumes and areas


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Introduction

Estimation of areas and volumes key component of

many engineering construction projects

Huge quantities of material have to be moved in order

to form the necessary embankments, cuttings,

foundations, basements, lakes and so on, that have

been specified in the design

The excavation and hauling of materials in

engineering projects is the most significant and

costly aspect of the work on which profit or loss maydepend


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Plotted areas

Area of any straight sided figure can be

calculated by splitting it into triangles

Sum of triangles gives the area of the figures

The area calculations can be done using one of the

following formulae

Area = [S(S - a) (S - b) (S - c)], where a, b, and c are sidesand S = (a+b+c)

Area = (base x height of the triangles)

Area = x ab sinC, where C is angle btn sides a and b


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Plotted irregular areas

These can be determined in one of the followingways:

Superimposing a gridded transplant sheet of material

With number of squares and scale, area can be estimated

Give and take technique

Give and take lineGive

Take


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Plotted irregular area

planimetreThis instrument measures the area of a plane figure as a pointer

is moved around the figure's edge


7/20H. Shamaoma SBE Dept of Urban and Regional Planning ESC340: Land Surveying

Computing irregular area

O1O8O7O6

O5O4O3O2

dd d d d dd

Survey line

O9

d

Two fundamental rules used in computing irregular

areas

(i) Trapezoid rule and (ii) Simpsons rule

Trapezoid rule

The survey line divided into a number of small equal intercepts

of length dand offsets O1, O2, , On,`



Trapezoid rule If d is short enough for the length of the boundary between

the offsets to be assumed straight, then the area is dividedinto a series of trapezoids

Area of trapezoid (1) =

Area of trapezoid (2) =

Summing up we get:

Area =

9321

.....222

OOOOd

dOO

*2

21

dOO

*2

32

O1O8O7O6

O5O4O3O2

dd d d d dd

Survey line

O9

d



Trapezoid rule

dxOO

Area2

)1( 21

dxOO

Area2

)2( 32

dx

OO

Area 2)7( 87

dxOO

Area 2)8( 98

O1 O8O7O6O5

O4O3O2

dd d d d dd

Survey line

O9

d

9321 .....222

OOOOd

Area In general case we get

1321 .....

2 n

n OOOOO

dArea


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Simpsons rule

O1

G

DA

C

B

O2 O3

d d

F

A1 A2

Area =

In general we get

Area =



Simpsons rule

Simpsons Rule states that: the area enclosed by a

curv i l inear f igure div ided into an even number of str ips

of equal width is equal to oneth i rd the w idth of a str ip,

mul t ip l ied by sum of the two extreme offsets, twice the

sum of the remaining odd o ffsets, and four t imes the

sum of the even offsets.

Area =



Area by coordinates

N

Q0 T P S ER

E

A

B

D

Consider closed traverse

ABCDEAC

Stn coords are: EA, NA;

EB, NB; EC, NC .etc.

Area ABCDEA = areas (ABPT + BCQP + CDRQDESREATS)



Area by coordinates

In general, area =

Note that the polygon should be labeled in a clockwise order*

=

=

=



Area by coordinatesN

Q0 T P S ER

E

A

B

D

Determine the area in

hectares enclosed by theline of a closed traverse

survey ABCDE from the

following data

C

Point

Id

Ni Ei+1 E i-1 N i(Ei+1-Ei-1)

A 418.11 806.71 652.08 64652.3493

B 523.16 942.17 613.26 172072.556

C 366.84 901.89 806.71 34915.8312

D 203.18 652.08 942.17 -58940.4862

E 259.26 613.26 901.89 -74830.2138

Sum 137870.0365

Point Id E (m) N (m)

A 613.26 418.11

B 806.71 523.16

C 942.17 366.84

D 901.89 203.18

E 652.08 259.26Area =

= 13.7870 ha

Area = 13.7870 ha *0.5

= 6.8935 ha



Area from cross sections

CL WW

bb

h

Original

ground

surface

1in n

Side slope 1

vertical in n

horizontal

L

W

bb

W

h

CFormation level

Embankment

Cutting

Plan width = 2W = 2(b + nh)

Area of cross-section =h(2b + nh)

Same formula used for cut and

embankment

Depth at centerline = hunits

Formation width = 2bunits

Side width = w

Plan width = 2w

Side slopesIinn



Volume of earthworks

2

(

........222

11

433

322

211

nn

ntotal

AA

d

AA

d

AA

d

AA

dV

2

2111

AAdV

If two cross-sectional areas A1and A2are horizontal

distance d1 apart volume between them V is given

by:

)......(22

1321 nn AAAAAd

Total volume =

End areas method



Volume of earthworks

Prismoidal formula

321 43

AAAd

543 43

AAAd

V3-5=V1-3=

34251 2)(43 AAAAAd

V =

The general formula for n cross-sections,

wherenmust be odd is:

areasoddremainingareasevenAAd

n 243

1V=



Volume from spot heights

h1 h2 h3 h4

h5 h6 h7 h8

h9 h10 h11

Method generally used for calculating

volumes of huge open excavations

such as:

Basements

Borrowpits

Underground tanks

Spoil heaps

etc

Typically a square or triangular grid is established on the

ground and spot levels are taken at each grid intersection

just as described in contouring by spot levels


20/20H Sh SBE D t f U b d R i l Pl i ESC340 L d S i

Volume from spot heightsh1 h2 h3 h4

h5 h6 h7 h8

h9h10 h11

4.76

3.21

5.14

4.77 5.82

6.72 8.10

6.07

1.98 2.31 3.55

The volume of grid square h1h2h6h5

Plan area of each Grid square = 100 m2

= (4.76 + 5.14 + 4.77 + 3.21)* 100

= 447 m3

Applying the formula to all the grids we obtain the general formula:

VdepthsquadrupledepthstripledepthsdoubledepthsnglesiA 4324

)]77.4(4)82.5(3)31.221.372.614.5(255.398.107.610.876.44

100

32394)08.1946.1776.3446.24(25 m

V =

V =

V =

V is outside the grids and is calculated separately

Volume = mean height x plan area

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