Area and Volume Survey Engineering (RZ)

ENGINEERING SURVEY C 2005 / 2 /

AREA AND VOLUME

OBJECTIVES

General Objective : To know and understand the basic concepts of Area and Volume Calculation

Specific Objectives : At the end of the unit you should be able to :-

Explain the basic concept of Area and Volume Method.

Define the usage of Area And Volume Calculation.

Describe the methods that have been used in Area and Volume Calculation .

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UNIT 2

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2.1 INTRODUCTION

Estimation of area and volume is basic to most engineering schemes such as route alignment, reservoirs, construction of tunnels, etc. The excavation and hauling of material on such schemes is the most significant and costly aspect of the work, on which profit or loss may depend. Area may be required in connection with the purchase or sale of land, with the division of land or with the grading of land. Earthwork volumes must be estimated :

to enable route alignment to be located at such lines and levels that cut and fill are balanced as far as practical.

to enable contract estimates of time and cost to be made for proposed work. to form the basis of payment for work carried out.

It is frequently necessary as part of engineering surveying projects to determine the area enclosed by the boundaries of a site or the volume of earthwork required to be moved. Many of the figures involve accepted mensuration formulae (see 1.6 ) but it is more common to meet irregular shapes and these require special attention.

2.2 PLAN AREAS

The basic unit of area in SI units is the square metre (m²) but for large areas the hectare is a derived unit.

1 hectare (ha) = 10 000 m² = 2.471 05 acres

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2.2.1 Conversion Of Plannimetric Area Into Actual Area

Let the scale of the plan be 1 in H (or as representative fraction 1/H). Then 1 mm is equivalent to H mm and 1 mm² is equivalent to H² mm² is equivalent to H mm², i.e. H² x 10-6 m²

2.3 AREA CALCULATION

Areas of ground may be obtained from the plotted plan but results are only as accurate as it is possible to scale off the drawings. Accuracy is greatly increased by using the measurements taken in the field. In most surveys the area is divisible into two parts :

a) The rectilinear areas enclosed by the survey linesb) The irregular areas of the strips between these lines and the boundary

In order to calculate the area of the whole, each of these areas must be evaluated separately because each is defined by a different form of geometrical figure.

2.3.1 Rectilinear Areas

The method of evaluating the rectilinear area enclosed by survey lines depends on the method of survey.

a) If chain surveying is used, the areas of the triangles forming the survey network are calculated from the field dimensions from the formula :

Area = √ s(s – a) (s – b) (s – c)

Where a, b and c = the lengths of the triangles sides and

s = (a + b + c) / 2

b) If traversing is used and the survey stations are coordinated, the computed coordinated are used in the area calculation.

Whichever calculation method is used, checks must be applied to prove the area calculations. In a chain survey network the work must be arranged so that two different sets of the triangles forming the rectilinear figure are used in evaluating the total area, which is thus twice calculated. These two results will not normally agree precisely because the network will not be geometrically perfect. Owing to observational errors, the two results are meaned to produce the final

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rectilinear area. When areas are calculated from coordinates, the calculation must be repeated another way to prove the result.

2.3.2 Irregular Areas

Unless boundaries are straight and the corner points coordinated there are usually irregular strips of ground between the survey lines and the property boundaries. The area of the irregular strips are either positive or negative to the rectilinear area and since they are divided up by offsets between which the boundary is supposed to run straight, they are computed as a series of trapezoids. The mean of each pair of offsets is taken and multiplied by the chainage between them. Where the offsets are taken at regular intervals, the trapezoidal rule or Simpson’s rule for areas is used, (see section 2.6).

NOTE

a. The field work should be arranged to overcome difficulties with corners. This is usually achieved by extending the survey line to the boundary, allowing for the triangular shape which may occur.

b. In order to check the irregular area the calculations should be repeated by another person, or a check against gross error may be made taking out a planimeter area of the plot.

2.4 CALCULATING AREA FROM A CHAIN SURVEY

The figure shows the rectilinear area ABCD, which is calculated first. Their regular strips between the chain lines and the boundary must be separately evaluated and either added or subtracted as necessary from the main rectilinear area calculation result. The following data were obtained from the chain survey of the site :

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AB - 63.0 mBC - 45.0 mCD - 60.0 mDA - 78.0 mBD - 93.3 mAC - 76.0 m

SOLUTION

The rectilinear area from A = √ ((s – a) (s – b) (s – c))

The area of triangle ACD = √(107(31) (47) (29))

= 2126.3 m2

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Chainage AD

Offset

A 0.0 0.016.0 6.033.0 7.040.0 0.049.0 7.061.0 7.068.0 0.0

B 78.0 11.089.0 5.0

Chainage CD

Offset

C 0.0 0.010.0 4.220.0 6.430.0 8.140.0 10.350.0 11.3

AB and BC are straight boundaries. Offsets to

the irregular boundaries are as follows :


The area of triangle ABC = √(92(29) (47) (16))

= 1416.4 m2

Area of ABCD = 2126.3 + 1416.4

= 3542.7 m2

Check :

The area of triangle ABD = √((117.15 (54.15) (39.15) (23.85))

= 2433.8 m2

The area of triangle ABD = √(( 99.15 (39.15) (54.15) (5.85)))

= 1108.9 m2

Area of ABCD = 2433.8 + 1108.9

= 3542.7 m2

Area of triangle ABD: Plus Minus

(0+6) x 2 x 16 = 48.0

(6+7) x 2 x 17 = 110.5

(7+0) x 2 x 7 = 24.5

(0+7) x 2 x 9 = 31.5

(7+7) x 2 x 12 = 84.0

(7+0) x 2 x 7 = 24.5

(0+11) x 2 x 10 = 55.0

(11+9) x 2 x 15 = 150.0

388.5140.0

- 140.0

248.5 m2

(total plus area on AD)2.5 CALCULATING AREAS

FROM COORDINATES

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A = Area

SPECIMEN QUESTION

Calculate the area of the figure ABCDEF of which the coordinates are listed

below.

SOLUTION

The calculation is tabulated as shown :

Product

Station Easting Northing

E + E Double Longitude

ΔN

A 150 100B 95.2 164.3 245.2 64.3 15 766.36C 127.9 210.7 223.1 46.4 10 351.84D 176.3 239.8 304.2 29.1 8 852.22E 219.4 222.4 395.7 -17.4 6 885.18F 237.5 163.8 456.9 -58.6 26 774.34A 150 100 387.5 -63.8 24 722.50

34 970.42 58 382.0234 970.42

2A = 23 411.60

Area = 11 705.8 m2

= 1.1706 ha

2.6 AREAS OF IRREGULAR FIGURES

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There are several practical situations where it is necessary to estimate the area of irregular figures. Examples include estimation of areas of plots of land by surveyors, areas of indicator diagrams of steam engines by engineers and areas of water planes and transverse sections of a ship by naval architects. There are many methods whereby the area of an irregular plane surface may be found and these include:

(a) Use of a planimeter, (b) Trapezoidal rule, (c) Mid-ordinate rule and (d) Simpson’s rule.

2.6.1 The planimeter

A planimeter is an instrument for directly measuring areas bounded by an irregular curve. There are many different types of the instrument but all consist basically of two rods AB and BC, hinged at B (see Fig. 2.1). The end labelled A is fixed, preferably outside of the irregular area being measured. Rod BC carries at B a wheel whose plane is at right angles to the plane formed by ABC. Point C, called the tracer, is guided round the boundary of the figure to be measured. The wheel is geared to a dial which records the area directly. If the length BC is adjustable, the scale can be altered and readings obtained in mm2, cm2, m2 and so on.

FIGURES 2.1 : Planimeter

(Source : Mathematics for

Technicians, S. Adam)

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2.6.2 Trapezoidal rule

To find the area ABCD in Fig. 2.2, the base AD is divided into a number of equal intervals of width d. This can be any number; the greater the number the more accurate the result. The ordinates y1, y2, y3, etc. are accurately measured. The approximation used in this rule is to assume that each strip is equal to the area of a trapezium.

FIGURE

2.2 : Trapezoidal rule (Source : Mathematics for Technicians, S. Adam)

The area of a trapezium = ½ (sum of the parallel sides) (perpendicular distance between the parallel sides).

Hence for the first strip, shown in Fig. 2.2, the approximate area is ½ (y1 + y2)d. For the second strip area is ½ (y1 + y2)d and so on. Hence the approximate area of

ABCD = ½ (y1 + y2)d + ½ (y3 + y4)d + ½ (y3 + y4)d + ½ (y4 + y5)d+ ½ (y5 + y6)d + ½ (y6 + y7)d

= ½ y1 d + ½ y2 d + ½ y2 d + ½ y3 d + ½ y3 d + ½ y4 d + ½ y4 d + ½ y5 d + ½ y5 d + ½ y6 d + ½ y6 d + ½ y7 d

= ½ y1 d + ½ y2 d + ½ y3 d + ½ y4 d + ½ y5 d + ½ y6 d + ½ y7 d= d [ ( y1 + + y7 ) / 2 + y2 + y3 + y4 + y5 + y6 ]

Generally, the trapezoidal rule states that the area of an irregular figure is given by:Area = (width of internal) [½ (first + last ordinate) + sum of remaining ordinates]

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2.6.3 Mid-ordinate rule

FIGURE 2.3 : Mid-ordinate rule method

(Source : Mathematics for Technicians, S. Adam)

To find the area of ABCD in Figure 2.3 the base AD is divided into any number of equal strips of width d. (As with the trapezoidal rule, the greater the number of intervals used the more accurate the result.) If each strip is assumed to be a trapezium, then the average length of the two parallel sides will be given by the length of a mid-ordinate, i.e. an ordinate erected in the middle of each trapezium. This is the approximation used in the mid-ordinate rule.

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The mid-ordinates are labelled y1, y2, y3, etc. as in Fig. 18.3 and each is then accurately measured. Hence the approximate area of ABCD

= y1 d + y2 d + y3 d + y4 d + y5 d + y6 d= d (y1 + y2 + y3 + y4 + y5 + y6 )

where d = ( length of AD / number of mid-ordinates )

Generally, the mid-ordinate rule states that the area of an irregular figure is given by:

2.6.4 Simpson’s rule

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Area = (width of interval) (sum of mid-ordinates)


FIGURE 2.4 : Simpson’s rule (Source : Mathematics for Technicians, S. Adam)

To find the circa A BCD in Figure 2.4 the base AD must be divided into an even number of strips of equal width d. Thus producing an odd number of ordinates. The length of each ordinate, y1, y2, y3, etc., is accurately measured. Simpson's rule states that (the area of the irregular area ABCD is given by;

Area of ABCD = d / 3 [(y1 + y7 ) + 4(y2 + y4 + y6) + 2(y3 + y5)]

More generally, the calculation of the area of:

Area = 1/3 (width of interval) [(first and last ordinates) + 4( sum of even ordinates)

+ 2 (sum of remaining odd ordinates)]

When estimating areas of irregular figures, Simpson's rule is generally regarded as the most accurate of the approximate methods available.

Activity 2a

2.1 The values of the y ordinates of a curve and their distance x from the origin are given in the table below. Plot the graph and find the area under the curve by :

x 0 1 2 3 4 5 6

y 2 5 8 11 14 17 20

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a) The trapezoidal ruleb) The mid-ordinate rulec) Simpson’s rule

2.2 Sketch a semicircle of radius 10cm. Erect ordinates at intervals of 2 cm and determine the lengths of the ordinates and mid-ordinates. Determine the area of the semicircle using the three approximate methods. Calculate the true area of the

semicircle.

Feedback 2a

2.1)

FIGURE 2.5 : Graph of y against x

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a) Trapezoidal rule

Using 7 ordinates with interval width of 1 the area under the curve is:

Area = 1 [ ½ (2 + 20) + 5 + 8 + 11 + 14 + 17 ] = [ 11 + 5 + 8 + 11 + 14 + 17 ] = 66 square units

b) Mid-ordinate rule

Using 6 intervals of width 1 the mid-ordinates of the 6 strips are measured. The area under the curve is:

Area = 1 (3.5 + 6.5 + 9.5 + 12.5 + 15.5 + 18.5)= 66 square unit

c) Simpson’s rule

Using 7 ordinates, given an even number of strips, i.e. 6, each of width 1, thus the area under the curve is:

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Area = 1 / 3 [ (2 + 20) + 4(5 + 11 + 17) + 2 (8 + 14) ]= 1 / 3 [ 22 + 4(33) + 2(22)]= 1 / 3 [ 22 + 132 + 44 ]= 198 / 3= 66 square units

The area under the curve is a trapezium and may be calculated using the formula ½(a+b)h, where a and b are the lengths of the parallel sides and h the perpendicular distance between the parallel sides.

Hence area = ½(2 + 20)(6) = 66 square units. This problem demonstrates the methods for finding areas under curves. Obviously the three 'approximate' methods would not normally be used for an area such as in this problem since it is not 'irregular'.

2.2). The semicircle is shown in Fig. 2.6 with the lengths of the ordinates and mid-ordinates marked, the dimensions being in centimetres.

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FIGURE 2.6 : Sketch a semicircle

a) Trapezoidal rule

Area = 2 [ ½ (0 + 0) + 6.0 + 8.0 + 9.15 + 9.80 + 10.0 + 9.80 + 9.15 + 8.0 + 6.0 ]= 2 (75.90)= 151.8 square units

b) Mid-ordinate rule

Area = 2 [ 4.3 + 7.1 + 8.65 + 9.55 + 9.95 + 9.95 + 9.95 + 8.65 + 7.1 + 4.3 ]= 2 (79.10)= 158.2 square units

c) Simpson’s rule

Area = 2/3 [ (0 + 0) + 4(6.0 + 9.15 + 10.00 + 9.15 + 6.00) + 2(8.0 + 9.8 + 8.0)]= 2/3 [0 + 4(40.3) + 2(35.6)]= 2/3 (161.2 + 71.2)= 2/3 (232.4)= 154.9 square units

The true area is given by π r² / 2, i.e π (10)² / 2 = 157.1 square units

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Well done! Keep it up!.

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2.7 VOLUME CALCULATION

In construction works, the excavation, loading, hauling and dumping of earth frequently forms a substantial part of the project. Payment must be made for the labour and plan needed for earthworks and this is based on the quantity or volume handled.

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These volumes must be calculated and depending on the shape of the site, this may be done in three ways :

i) by cross-sections, generally used for long, narrow works such as roads, railways, pipelines, etc.

ii) by contours, generally used for larger areas such as reservoirs, landscapes, redevelopment sites, etc.

iii) by spot height, generally used for small areas such as underground tanks, basements, building sites, etc.

2.8 CROSS SECTION VOLUME CALCULATION

Cross-sections are established at some convenient intervals along a centre line of the works. Volumes are calculated by relating the cross-sectional areas to the distances between them. In order to compute the volume it is first necessary to evaluate the cross-sectional areas, which may be obtained by the following methods:

i) by calculating from the formula or from first principles the standard cross-sections of constant formation widths and side slopes.

ii) by measuring graphically from plotted cross-sections drawn to scale, areas being obtained by plannimeter or division into triangles or square.

NOTE : The graphic measure of the cross-sectional area is most often used and provides a sufficiently accurate estimate of volume, but for railways, long embankments, breakwaters, etc., with fairly regular dimensions, the use of formulae may be easier and perhaps more accurate.

2.8.1 Prismoidal Method

In order to calculate the volume of a substance, its geometrical shape and size must be known. A mass of earth has no regular geometrical figure in most approaches. The prismoid is a solid, consisting of two ends which form plane, parallel figures, not necessarily of the same number of sides and which can be measured as cross-sections. The faces between the parallel ends are plane surfaces between straight lines which join all the corners of the two end faces. A prismoid can be considered to be made up of a series of prisms, wedges and pyramids, all having a length equal to the perpendicular distance between the parallel ends. The geometrical solids forming the prismoid are described as follows :

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i) Prism, in which the end polygons are equal and the side faces are parallelograms.

ii) Wedge, in which one end is a line, the other end a parallelogram, and the sides are triangles and parallelograms.

iii) Pyramids, in which one end is a point, the other end a polygon and the side faces are triangles.

The Prismoidal Formula Let D = the perpendicular distance between the parallel end planes, A1 and A2 = the areas of these end planes, M = the mid-area, the area of the plane parallel to the end planes

and midway between them, V = the volume of the prismoid and a1, a2, m, v = the equivalent for any prism, wedge or pyramid forming the

prismoid

then in a prism a1, = a2, = m and in a wedge a2 = 0 and m = 1/2 a1

and in a pyramid a2 = 0 and m = 1/4 a1

Prism volume v = D . a1 = D/6 (6 . a1 ) = D/6 (a1, + 4m + a2) Wedge volume v = ½ D . a1 = D/6 (3 . a1 ) = D/6 (a1, + 4m + a2)Pyramid volume v = 1/3 D . a1 = D/6 (2 . a1 ) = D/6 (a1, + 4m + a2)

As the volume of each part can be expressed in the same terms, the volume of the whole can take the same form. Thus the prismoidal formula is expressed in the following way : V = D/6 (A1, + 4M + A2)

Note : A. M does not represent the mean of the end areas A1 and A2 except where the

prismoid is composed of prisms and wedges only. B. The formula gives the volume of one prismoid of which the end and mid-

sectional areas are known.

The prismoidal formula may be used to calculate volume if a series of cross-sectional areas, A1, A2, A3,…. An, have been established a distance d apart. Each alternate cross-section may be considered to be the mid-area M of a prismoid of length 2d.

Then the volume of the first prismoid of length 2d :

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= 2d / 6 (A1, + 4A2 + A3)and of the second = 2d / 6 (A3, + 4A4 + A5)and of the nth = 2d / 6 (An-2, + 4An-1 + An)

summing up the volumes of each prismoidal :

V = d / 3 (A1, + 4A2 + 2A3 + 4A4…… + 2An-2 + 4An-1 + An)Which is Simpson’s rule for volumes.

Specimen QuestionCalculate, using the prismoidal formula, the cubic contents of an embankment of which the cross-sectional areas at 15m intervals are as follows :

Distance (m) 0 15 30 45 60 75 90

Area (m2) 11 42 64 72 160 180 220

Solution,V = 15 / 3 (11 + 220 + 4 ( 42 + 72 + 180 ) + ( 64 + 160))V = 5 ( 231 + 1176 + 448 )V = 9275 m3

Note :A. The 15m interval is divided by 3, as the length of the individual prismoids

used is 30m, which in the prismoidal formula is divided by 6.B. A mass of earth, length double the usual cross-sectional interval of 15m,

20m or 25m, is considerably different from a true prismoid, so this method is not as accurate as it would be if the true mid-sectional area had been measured. This results in the use of prismoids of length equal to, instead of double, the interval between cross-sections.

2.8.2 End Areas Method

It is no more accurate to use the prismoidal formula where the mid-sectional areas have not been directly measured than it is to use the end areas formula, particularly as the earth solid is not exactly represented by a prismoid. Using the same symbols the volume may be expressed as :

v = d [ ( A1 + A2) / 2 ]

although this is only correct where the mid-area is the mean of the end areas :

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M = ( A1 + A2) / 2

However, in view of the inaccuracies that arise in assuming any geometric shape between cross-sections and because of bulking and settlement and the fact that the end areas calculation is simple to use, it is generally used for most estimating purposes.

Note :A. The summation of a series of cross-sectional areas by this method

provides a total volume :V = d{[( A1 + A2) / 2 ] + A2 + A3 + …… An-1}

Specimen QuestionCalculate, using the end areas method, the cubic contents of the embankment of which the cross-sectional areas at 15m intervals are as follows :

Distance (m) 0 15 30 45 60 75 90

Area (m2) 11 42 64 72 160 180 220

SolutionV = 15{[ (11 + 220) / 2 ] + 42 + 64 + 72 + 160 + 180 }V = 9502.5 m3

2.9 VOLUME CALCULATION FOR CONTOUR LINES

Contour lines may be used for volume calculations and theoretically this is the most accurate method. However, as the small contour interval necessary for accurate work is seldom provided due to cost, high accuracy is not often obtained. Unless the contour interval is less than 1m or 2m at the most, the assumption that there is an even slope between the contour is incorrect and volume calculation from contours become unreliable.

The formula used for volume calculation is the end areas formula of Simpson’s rule for volumes, the distance d in the formula being contour interval. The area enclosed by each contour line is measured, usually by plannimeter, and these areas A1, A2, etc., are used in the formula as before ( see the end areas method). If the prismoidal method is used, each alternate contour line is assumed to enclose a mid-area or the outline of the mid-area can be interpolated between the existing contour intervals.

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This illustration shows an area contoured at 5m intervals and how the contours of proposed works, in this case a dam wall with an access road through a cutting, shown as packed lines, define the plan outline of the works. This also allows the volume of the earthworks to be calculated using the positions of the contour lines.

Picture 4.1 : Intersection Of Contoured Surfaces (Source : Land Survey, Ramsay)

The volume of the dam wall and the amount of cut may be obtained from the contour lines by calculating the volume of ground within the working area down to a common level surface and then calculating the new volume from the formation contour lines, the difference being the change in volume due to the works. This volume calculation is more usually carried out by using the cross-sectional method. The use of contours is a practical method of calculating volumes in several cases, one of which being the calculation of water at various levels in a reservoir. For example, in picture 4.1 the volume of water which could be contained up to the level of the 60m, contours could be calculated as follows from these data :

Contour above datum (m) 50 52.5 55 57.5 60

Area (m2) 12 135 660 1500 1950

Using the end areas method :

V = 2.5 { [ (12 + 1950)/2] + 135 + 660 + 1500 } = 8190 m3

Using Simpson’s rule from the prismoidal formula :

V = ( 2.5 / 3 ) [ 12 + 1950 + 4(135 + 1500) + 2 (660) ] = (2.5 / 3 ) (9822)

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= 8185 m3

Note :The small volume of water below 50m (not included in the above calculation) would be estimated from the interpolated depth of 2m at the deepest point, using the end areas formula, the lowest end area being 0, thus :

V = [ ( 12 + 0 ) / 2 ] x 2V = 12m3

This would then be added to either of the results above.

2.10 VOLUME CALCULATION FROM SPOT HEIGHT

This is a method of volume calculation frequently used on excavations where there are vertical sides covering a fairly large area, although it can be used for excavation with sloping sides. The site is divided into squares or rectangles, and if they are of equal size the calculations are simplified. The volumes are calculated from the product of the mean length of the sides of each vertical truncated prism ( a prism in which the base planes are not parallel ) and the cross-sectional area. The sizes of the rectangles is dependent on the degree of accuracy required. The aim is to produce areas such that the ground surface within each can be assumed to be plane.

Specimen QuestionPicture 4.2 shows the reduced levels of a rectangular plot which is to be excavated to a uniform depth of 8m above datum. Calculate the mean level of the ground and the volume of earth to be excavated.

Note :A. The mean or average level of the

ground is that level of ground which would be achieved by smoothing the ground off level, assuming that no bulking would take place.

B. The mean level of the ground is the mean of the mean height of each prism. It is not the mean of all the spot heights.

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Picture 4.2 : Calculating volume from spot height on a levelling grid. (Source : Land Survey, Ramsay)Solution(a) Calculation from rectangles :

Station R.L.

Number of times the Product

R.L. is used = n (R.L.) x nA 12.16 1 12.16B 12.48 2 24.96C 13.01 1 13.01D 12.56 2 25.12E 12.87 4 51.48F 13.53 2 27.06G 12.94 1 12.94H 13.27 2 26.54J 13.84 1 13.84

∑n = 16 207.11

Mean level = 207.11 / 16 = 12.944 mDepth of excavation = 12.944 – 8.00 = 4.944 Volume = Total area x Depth = 30 x 20 x 4.944 = 2966.4 m3

(b) Calculation from trianglesIt is usually more accurate to calculate from triangles as the upper base of the triangular prism is more

likely to correspond with the ground plane than the larger rectangle. The mean level of each prism is then

the mean of the three height enclosing the triangle instead of four as before.

Station R.L.Number of times the ProductR.L. is used = n (R.L.) x n

A 12.16 1 12.16B 12.48 3 37.44C 13.01 2 26.02D 12.56 3 37.44E 12.87 7 90.09F 13.53 2 27.06G 12.94 2 25.88H 13.27 2 26.54J 13.84 2 27.68

∑n = 24 310.55

Mean level = 310.55 / 24

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= 12.940 mDepth of excavation = 4.960 Volume = 30 x 20 x 4.944 = 2966.4 m3

Note :The diagonal forming the triangles would be noted in the field book on the grid layout to conform most suitably with the ground planes.

Activity 2b

2.3 An embankment is to be formed with its centre line on the surface (in the form of a plane) on full dip of 1 in 20. If the formation width is 12.00m and the formation heights are 3.00m, 4.50m and 6.00m at intervals of 30.00m, with side slopes 1 in 2, calculate the volume between the end sections.

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Calculate a). Volume by mean areasb). Volume by end areasc). Volume by prismoidal rule

2.4 Given the previous example but with the centre line turned through 90º, calculate volume

a) By mean areasb) By end areasc) By prismoidal

Feedback 2b

2.3 Area (1) = h1 (w + mh1) = 3.00 [ 12.00 + (2 x 3.00) ] = 54.00 m2

Area (2) = 4.50 [ 12.00 + (2 x 4.50) ] = 94.50 m2

Area (3) = 6.00 [ 12.00 + (2 x 6.00) ] = 144.00 m2

Volume :

a). By mean areas V = W(A/n) = 60.00 ( 54.00 + 94.50 + 144.00 ) / 3 = 5850.0 m3

b). By end areasV = w ( A1 + 2A2 + A3 ) / 2

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Try your best to answer this question.


= 30.00 (54.00 + 189.00 + 144.00) / 2 = 5805.0 m3

c). By Prismoidal RuleV = w ( A1 + 4A2 + A3 ) / 3 = 30.00 (54.00 + 378.00 + 144.00)/3 = 5760.0 m3

2.4 A = m ( h²0 k² + w² / 4 + wh0 m) + wh0

( k² - m² )

Cross-sectional areas

A1 = 2 [ (3.00² x 20² ) + ( 0.25 x 12.00²) + ( 12.00 x 3.00 x 2 ) + (12 x 3.00)

( 20² - 2² )

= [ (3600.00 + 36.00 + 72.00) / 198 ] + 36.00 = 54.73 m²A2 = [ ( 8100.00 + 36.00 + 108.00 ) / 198 ] + 54.00 = 95.64 m²

A3 = [ ( 14400.00 + 36.00 + 144.00 ) / 198 ] + 72.00 = 145.64 m²

Volume

a). By mean areasV = 60.00 ( 54.00 + 95.64 + 145.64 ) / 3 = 5920.2 m³

b). By end areasV = 30.00 ( 54.73 + 191.28 + 145.64 ) / 2 = 5874.8 m³

c). By prismoidal ruleV = 30.00 ( 54.73 + 382.56 + 145.64 ) / 3 = 5829.3 m³

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Self Assessment

Calculate the volumes in Figure 1 and Figure 2.

Figure 1

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Feedback to Self Assessment


Figure 2

Figure 1

A1 = [ ( 0.75 + 4.75 ) / 2 ] x 7 = 19.25 ft²A2 = [ ( 0.75 + 3.75 ) / 2 ] x 5 = 11.25 ft²A3 = [ ( 0.75 + 2.75 ) / 2 ] x 3 = 5.25 ft²

Volume, V = L / 6 ( A1 + 4Am + A2 ) = 17 / 6 (19.25 + 4 x 11.25 + 5.25) = 17 / 6 ( 19.25 + 45.00 + 5.25 ) = 17 / 6 ( 69.50 ) = 1181.50 / 6 = 196.92 ft³

29

29

How ???


= 7.29 yrd³

Figure 2 Volume, V = h / 3 ( area of base )

= 27.4 / 3 ( 13.5 x 13.5 )= 1664.6 m³

30

30

Congratulations you can can proceed to the next unit.

INPUT

Area and Volume Survey Engineering (RZ)

Education

area calculations

actual area

usage of area

calculating area

planimeter area

total area

final rectilinear area

rectilinear area abcd