area and earthwork volume calculation methods for surveyors

7/25/2019 area and earthwork volume calculation methods for surveyors

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Area

Based on feldmeasurements

Considering theentire area

By osets to thebaes line

By latitudes&departures

By coordinates

Based on

measurementsscaled rom a

map

Instrumentalmethod

Usually by aplanimeter

Based on feld measurements considering the entire area

In this method, the area is divided into number of geometrical figures such as triangles,

rectangles, squares and trapeziums and then the area can be calculated according to one of

following method. Computing the area can be achieved in two steps.

1st step

Area of a triangle:

Area=s (sa) (sb ) ( sc )

Where a, b and c are sides and s = (a + b + c ! "

Area of rectangle

Area=a b

Where a and b are sides.


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#$

#""

Area of square

Area=a2

Where a is the side of the square.

Area of the trapezium

Area=1

2( a+b ) d

Where a and b are the parallel sides and d is the perpendicular distance between them.

2nd step

%he area along the boundaries is calculated as follows.

01

,02=ordinates

x1

, x2=chainages

Area of the shaded portion=01+0

2

2 (x2x1 )

Total area=area of the geomatrical shape+area of the boundary


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Areas rom osets to a baseline osets at regular intervals

%his method is suitable for long narrow strips of land. %he offsets are measured from the

boundar& to the base line or a surve& line at regular intervals. %his method can also be

applied to a plotted plan from which the offsets to a line can be scaled off. %here are '

methods in order to calculate the area.

id)ordinate rule

*verage)ordinate rule

%rapezoidal rule

impsons rule

id)ordinate rule

In order to appl& this method, we assume that the boundaries between the e-tremities of the

ordinates are straight lines.

%he base line is divided into a number of divisions and the ordinates are measured at the mid

points of each division.

*rea=common distancesum of mid ordinates

id ordinate=o$+o"!"=h$

*rea=d (h$+h"+/

*verage)ordinate rule

In this method also we assume that the boundaries between the e-tremities of the ordinates

are straight lines. %he offsets are measured to each of the points of the divisions of the base

line.

*rea=sum of ordinate!no.of ordinates length of base line

*rea=o$+o"+/+on!o (n+$l

%rapezoidal rule

%his rule is based on the assumption that the figures are trapeziums. %his rule is more

accurate than the previous two rules which are appro-imate versions of this rule. %his rule

can be applied for an& number of ordinates.

*rea enclosed b& one trapezium=o0+o$!"d

%otal area is the sum of each separate trapezium.

%otal area= o0+o$!"d + /


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%otal area=d!"(o$+"o"+/+"on)$+on

%otal area=common distance!"($stordinate + last ordinate+"1sum of other ordinate2

Add the average of the end offsets to the sum of the intermediate offsets. Multiply the total

sum thus obtained by the common distance between the ordinates to get the required area.

impsons rule

In this rule, we assume that the short lengths of boundar& between the ordinates are parabolic

arcs. %his method is more useful when the boundar& line departs considerabl& from the

straight line.

The area is equal to the sum of the two end ordinates plus four times the sum of the even

intermediate ordinates plus twice the sum of the odd intermediate ordinates, the whole

multiplied by one-third the common interval between them.

3ven though this method gives more accurate results out of other three methods, this is onl&

applicable when the number of divisions is even.

Areas rom osets to a baseline osets at irregular

intervals

%here are two methods to find the area when the ordinates distances are irregular.

$stmethod

In this method, the area of each trapezoid is calculated separatel& and then added together to

compute the total area.

"ndmethod

*rea b& coordinates

Latitudes and departures double meridian distances

%his method is one of the most frequentl& used for computing the area of a closed traverse.


5/16

In order to calculate the area b& this method, the latitudes and departures of each line of the

traverse are calculated. %hen the traverse is balanced.

* reference meridian is then assumed to pass through the most westerl& station of the traverse

4 the double meridian distances of the lines are computed.

eridian distances

%he meridian distance of an& point in a traverse is the distance of that point to the reference

meridian, measured at right angles to the meridian.

The meridian distance of any line is equal to the meridian distance of the preceding line plus

half the departure of the preceding line plus half the departure of the line itself.

Area by latitudes and meridian distances

3ast)west lines drawn from each station to the reference meridian, thus getting triangles and

trapeziums.

5ne side of each triangle or trapezium will be one of the lines, the base of the triangle or

trapezium will be the latitude of the line, and the height of the triangle or trapezium will be

the meridian distance of that line.

*rea of triangle or trapezium=latitude of the linemeridian distance of the line

Double meridian distance

%he double meridian distance of a line is equal to the sum of the meridian distances of the

two lines.

The D.M.D. of any line is equal to the D.M.D. of the preceding line plus the departure of the

preceding line plus the departure of the line itself.

*rea b& latitudes and double meridian distances

*=area of d6cC+area of Ccb7)area of d6*)area of *7b

*rea from departures and total latitudes

Area by double parallel distances and departures* parallel distance of an& line of a traverse is the perpendicular distance from the middle

point of that line to a reference line (chosen to pass through most southerl& station at right

angles to the meridian.

The DPD of any line is the sum of the parallel distances of its ends.

%he principle of finding area b& 6..6 method 4 6.8.6. method are identical.

Area by coordinates

%his method can be applied when the offset intervals are irregular.


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%he procedure for this method is that from the given distances 4 offsets, a point is selected as

the origin.

%hen the coordinated of all other points are arranged with reference to the origin.

Instrumental method%he instrument used for computation of area from a plotted map is the planimeter. %he area

obtained from this instrument is more accurate than other graphical methods.

%he two t&pes of planimeter are9

*msler polar planimeter

:oller planimeter

8rocedure of finding the area with a planimeter

%he ;ernier of the inde- mar< is set to the e-act graduation mar


7/16

Volume

Fromcross

sections

Leelsection

!"o#leel

section

$ide hillt"o#leelsection

!hree leelsection

%ulti#leelsection

From spotleels

Fromcontours

By crosssections

By e&ualdepth

contours

Byhori'ontal

planes

;olume calculation from cross sections

%his is the most widel& used method.

%he total volume is divided into a series of solids b& the planes of cross sections.

Cross)sections are established at some convenient intervals along a center line of the wor


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%hree level sections

ulti) level section

?evel sectionIn this case the ground is level transversel&.

6epth of centre line or height of emban


9/16

*rea = $!"m 1(b!" + mh(w$+ w" @ b"!"2

ide hill two level sections

In this case, the ground slope crosses the formation level so that one portion of the area is in

cutting and the other in filling.

*rea of fill = A 1(b!" +


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%hree level sections

*rea = A m 1(w$+w" (mh+b!" @ b"!"

%he volumes of the prismoids between successive cross)sections are obtained either b&trapezoidal formula or b& prismoidal formula.

%he prismoidal formula

* prismoid is defined as a solid whose end faces lie in parallel and consist of an& two

pol&gons, not necessaril& of the same number of sides, the longitudinal faces being surface

e-tended between the end planes.

%he longitudinal faces ta


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*$=area of cross)section of one end plane.

*"= area of cross)section of other end plane.

*=the mid area

In order to calculate the volume of earth wor< between a number of sections having area a$,

a", aB, /, an spaced at a constant distance h apart.

%otal voume=h!B1(a$+an+'(a"+a'/an)$+"(aB+a/an)"2

%his is also


12/16

In order to calculate the volume of earth wor< between a number of sections having area a$,

a", aB, /, an spaced at a constant distance h apart.

V = d{[( A1+ A2) / 2 ] + A2+ A3+ An-1}

level section two levelsections

side hill twolevel sections

three levelsections

prismoidal

correct

ion

Cp=Ds

6 (h1h2 )

2

;olume from spot levels

In this method, the field wor< consists in dividing the area into a number fo squares,

rectangles or triangles and measuring the levels of their corners before and after the

construction.

%hus, the depth of e-cavation or height of filling at ever& corner is


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;olume of a group of rectangles or squares having the same area

Eh$= some of depths used once

Eh"= sum of depths used twice

EhB= sum of depths used thrice

Eh'= sum of depths used four times.

*=horizontal area of the cross)section of the prism

;olume of a group of triangles having the same area

;olume from contour plan

Contour lines ma& be used for volume calculations and theoreticall& this is the most accurate

method.

Dowever, as the small contour interval necessar& for accurate wor< is seldom provided due to

cost, high accurac& is not often obtained.

Fnless the contour interval is less than $m or "m at the most, the assumption that there is an

even slope between the contours is incorrect and volume calculation from contours become

unreliable.%here are four distinct methods depending upon the t&pe of wor


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7& equal depth contours

In this method, the contours of the finished or graded surface are drawn on the contour map at

the same interval as that of contours.

*t ever& point, where the contours of the finished surface intersect a contour of the e-istingsurface the cut or fill can be found b& simpl& subtracting the difference elevation between the

two contours.

7& Goining the points of equal cut or fill, a set of lines is obtained.

%hese lines are the horizontal proGections of the lines cut from the e-isting surface b& planes

parallel to the finished surface.

%he volume between an& two successive areas is determined b& multipl&ing the average of

the two areas b& the depth between them, or b& the prismoidal formula.

D=contour interval

;)total volume

;=sigmah!" (a$+*" b& trapezoidal

;=igma h!B(*$+'*"+*B b& prismoidal formula

7& horizontal planes

%his method consists in determining the volumes of earth to be moved between the horizontal

planes mar


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Chainag

e along

the

surve&

line

0.0 "0.0 '0.0 H0.0 0.0 $00.0 $B0.0 $H0.0 $J0.0

5rdinate

(m

.' J. K.B' H."B K.J K.B J.$ H.H '.

*ccording to the given data above we can plot a graph to determine the enclosed area of the

given surve& line.

( )( *( +( ,( -(( -)( -*( -+( -,( )((

(

)

*

+

,

-(

-)

,.*

/.0

1.2*

+.)2

1.,/

1.2

/.,-

+.+2

*.0

In this particular surve& line there are two distinct Chainage intervals of "0m and B0m.

%herefore when calculating the area it has to be separated into two parts.

%he method used to calculate the area is the trapezoidal rule since it is more accurate than the

mid)ordinate rule and average)ordinate rule and impsons rule cannot be applied because the

number of divisions is not even in the plotted plan.


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( )( *( +( ,( -(( -2( -+( -/((

)

*

+

,

-(

-)

Y-alues

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area and earthwork volume calculation methods for surveyors