GEOMATIC - Areas & Volume

7/23/2019 GEOMATIC - Areas & Volume

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With Wisdom, We Explore

AREAS AND VOLUME

Mohd Efendi Daud (Dr. Sc)B.Surv (UTM, Malaysia) Msc (UTM, Malaysia), Dr. Sc (a!oya U"iv.,

#apa")

(Geomatic Division)

acu!t" o# $ivi! % Environmenta! En&ineerin&'Universiti un ussein Onn Ma!a"sia' *+,-- atu /ahat'

0ohor' MALA1S2A./hone 3 4+-5,6757+78 4+-9:5*675,-8 a; 3

4+-5,675-+-E<mai! 3 efendi=uthm.edu.m"

>e?3 htt@3BBB.#Cass.uthm.edu.m"

mailto:[email protected]

http://www.fkass.uthm.edu.my/





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2NRODU$2ON

• Area is often required in the context ofdesign, e.g., we might need the

surface area of a lake, the area of atract of land or of a cross-sectionalarea of road cutting.

• Volumes are often calculated by

integrating the area at regularintervals e.g., along a road centerline,or by using regularly spaced contours.



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• e need to know the volume ofmaterials in virtually all facets of civil

engineering especially in design andconstruction e.g., earthworkcomputations, amount of cut or !ll,amount of concrete, and etc.

• "here are two cases for calculating theareas#$ Areas of %traight &oundary

$ Area of 'rregular &oundary

2NRODU$2ON



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AREAS OSRA2G

OUNDAR1 • "he determination of areas depend

upon the topography of the terrain

and the accuracy required.• "he area of land portion, may be

determined by the following

methods#$ from the !eld notes, and

$ from the plotted plan or map



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$om@utation o# Areas #rom ie!dNotes

• henever the area of a plot of land is to bedetermined directly from the !eld notes, itshould be ensured that the survey lines includethe whole area and the land is divided into

geometrical !gures such as#-$ "riangles,

$ %quares,

$ (ectangles and etc.

AREAS OSRA2G

OUNDAR1



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AREAS OSRA2G

OUNDAR1 $om@utation o# Areas #rom ie!d Notes

( ) ( ) ( )c sb sa s s −−−

)ase *#

( )cba s ++=

2

1Area of triangle A&) is given aswhere,

= 0.5*ab*Sin C

= 0.5*ac*Sin B

= 0.5*bc*Sin A



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AREAS OSRA2G


)ase +#

bh2

1=∆



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)ase #

AREAS OSRA2G


ba×= "he area of a rectangle

S O



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AREAS OSRA2G


)ase #

( ) d ba ×+=

2

1 "he area of a trapeium

AREAS O



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$om@utation o# Areas #rom /!ans

9. Gra@hica! method

AREAS OSRA2G

OUNDAR1

AREAS O



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$om@utation o# Areas #rom /!ans

. 2nstrumenta! method

AREAS OSRA2G

OUNDAR1

he /!animeter



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AREA 1$OORD2NAES

• "he area of shape A&)/0A can becalculated in the following

manner1111



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AREA O



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AREA O2RREGULAR

OUNDAR1 9. he ra@eoida! Ru!e

where,2*, 2+, 3 # 24set5 # 'nterval between o4set 6constant

A*A+ A A A8

AREA O



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9. he ra@eoida! Ru!e

AREA O2RREGULAR

OUNDAR1

"rapeoidal formula

A* 9 : 62* ; 2+7 5A+ 9 : 62+ ; 27 5A 9 : 62 ; 27 5 and,

A8 9 : 628 ; 2<7 5

F L (O9 4 O+ 4 (O 4O7 4 O,4 O6)H

AREA O



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. he Sim@sonIs Ru!e

2n Sim@sonIs Ru!e' it is assumed that the

irre&u!ar?oundar" is com@rised o# @ara?o!ic arcs.

AREA O2RREGULAR

OUNDAR1

Parabola arc

A F /7 (O9 4 O5 4 ,(O 4 O, 4 O+) 4 (O7 4O6)H

A F /7 9st

4 !ast ordinates) 4 ,

(even ordinates)4 (odd ordinates)H

AREA O



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. he Sim@sonIs Ru!e

AREA O2RREGULAR

OUNDAR1

Parabola arc

't is important to note that %impson=s (ule requires aeven num?er o# divisions o# the area, i.e. the tnum?er o# ordinates must ?e odd.



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VOLUMES

Mohd Efendi Daud (Dr. Sc)B.Surv (UTM, Malaysia) Msc (UTM, Malaysia), Dr. Sc (a!oya U"iv.,

#apa")

(Geomatic Division)

acu!t" o# $ivi! % Environmenta! En&ineerin&'Universiti un ussein Onn Ma!a"sia' *+,-- atu /ahat'

0ohor' MALA1S2A./hone 3 4+-5,6757+78 4+-9:5*675,-8 a; 3

4+-5,675-+-E<mai! 3 efendi=uthm.edu.m"

>e?3 htt@3BBB.#Cass.uthm.edu.m"







www.uthm.edu.my


2NRODU$2ON

• 'n civil engineering calculation of volumesof earthwork is required.

• "here are several methods available forthe determination of the volume of threedimensional !gures and earthworks.

• >enerally the methods are based on using

!eld survey techniques to acquire datathat represents the shape of the land sothat volume calculations can proceed.

• "he methods are as follow#



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VOLUMES

$ross<Sections

• A cross-section of the natural surface

indicates the existing ground pro!le normalto a !xed direction at a selection point.

• "he !xed direction is usually of signi!cancesuch as the center line of a road.

• 0xample of cross sections are described asfollow#



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$ross<Sections

• ?or a cut or !ll on horiontal ground

VOLUMES

$rea % h(&' "h)where h is the average cutting depth, " is the side slopeand ' is the formation width



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VOLUMES

$ross<Sections

• ?or a cut and !ll on sloping ground

Assuming a cut such as the ?ig.the cross-sectional area is found!rstly by calculating 5 @ >.

W L = S(b + nh)/(S + n)W G = S(b + nh)/(S-n)

Thus Area = (h '*")(W + W ) - '& *"

1: S & 1: n



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• * 9 g 6b+ ; hs7 6g-s7

• + 9 g 6b+ - hs7 6g-s7

• 5uas Bemotongan 9 6b+ ; gh7C 6g - s7D



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VOLUMES

Vo!umes #rom $ontours

contour *E+

contour *E ４

contour *E<

contour *EE

contour *FG

POINTS CONTOUR (m) AREAS (m )

A1 182 3150

A2 184 2460

A3 186 1630

A4 188 840

A5 190 210



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Vo!umes #rom $ontours < so!utions

VOLUMES

Trapezoidal formula

I = d [ (A1 + An) +A2 + A3 + A4 + .... + An -1

= d!2 [ (A1 + An) +2(A2 + A3 + A4 + .... + An-1)

= d!2 [ (A1 + A") +2(A2 + A3 + A4)

= 2!2 [ (31"# + 21#) + 2 ($4# + 1%3# + 24%#)

= 13, 220 m

Prizmoidal formula

I = d!3 [ (A1 + An) +4(A2 + A4 + &.. + An-1) + 2(A3 + A" )

= d!3 [ (A1 + A") +4(A2 + A4 ) + 2(A3)

= 2!3 [ (31"# + 21#) +4($4# + 24%# ) + 2(1%3#)

= 2!3 [ (33%#) +4(33## ) + 2(1%3#)

= 13,213.3 m



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Vo!umes #rom S@ot ei&hts

• "his method is particularly useful for large,open excavations such as tank, borrow pitsetc.

• "he area is divided into a grid, and levelsobtained at the intersection points.

• "he spacing of the grid depends on the terrain,accuracy required, and resources available.

VOLUMES



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Vo!umes #rom S@ot ei&hts

• "here are two methods$ (ectangular base, and

$ "riangular base

VOLUMES



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VOLUMES

(ectangular base - solution

Average 6/ug7 9 H.F*< 9 .<+Volume 9 Area x Average

9 6G x +87 x .<+ 9 <8.F m

+

Boints (educed 5evel/ug

6(educed 5ev/ug

5evel 6I72ccurance J

6J76J7 x 6I7

A *,.*8 *G.GG ,.*8 * ,.*8

& *,.HG *G.GG ,.HG + H.-G

) *-.,, *G.GG -.,, * -.,,

/ *,.F- *G.GG ,.F- + H.EE

0 *-.EG *G.GG -.EG - *F.+G

? *-.FH *G.GG -.FH + F.F-

> *8.*H *G.GG 8.*H * 8.*H

K *<.*G *G.GG <.*G + *+.+G

L *-.<H *G.GG -.<H * -.<H

*< H,.F-



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VOLUMES

"riangular base - solution

Boints (educed 5evel/ug

6(educed 5ev/ug

5evel 6I72ccurance J

6J76J7 x 6I7

A *.*8 *G.GG .*8 * .*8

& *.HG *G.GG .HG **.*G

) *. *G.GG . + E.<<

/ *.F *G.GG .F **.E+

0 *.EG *G.GG .EG < +E.EG

? *.FH *G.GG .FH *.F*

> *8.*H *G.GG 8.*H + *G.

K *<.*G *G.GG <.*G *E.G

L *.<H *G.GG .<H * .<H

+ ***.H8

Average 6/ug7 9 ***.H8+ 9 .<<Volume 9 Area x Average

9 6G x +87 x .<< 9 F+.+ m

+



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BheewwwM 0nd of BartV'

GEOMATIC - Areas & Volume

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