C2005 Ukur Kejuruteraan 2

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AREA AND VOLUME

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

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

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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 lines

b) 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 rectilinear area. When areas are

calculated from coordinates, the calculation must be repeated another way

to prove the result.

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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 :

AB - 63.0 m

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BC - 45.0 m CD - 60.0 m DA - 78.0 m BD - 93.3 m AC - 76.0 m

SOLUTION

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

Chainage AD

Offset

A 0.0 0.0 16.0 6.0 33.0 7.0 40.0 0.0 49.0 7.0 61.0 7.0 68.0 0.0 B 78.0 11.0 89.0 5.0 93.0 9.0

Chainage CD

Offset

C 0.0 0.0 10.0 4.2 20.0 6.4 30.0 8.1 40.0 10.3 50.0 11.3 D 60.0 13.2

AB and BC are straight boundaries. Offsets to the

irregular boundaries are as follows :

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

= 2126.3 m2

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

388.5 140.0

- 140.0

248.5 m2 (total plus area on AD)

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2.5 CALCULATING AREAS FROM COORDINATES

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 :

Station Easting Northing

E + E Double Longitude

ΔN

A 150 100 B 95.2 164.3 245.2 64.3 15 766.36 C 127.9 210.7 223.1 46.4 10 351.84 D 176.3 239.8 304.2 29.1 8 852.22 E 219.4 222.4 395.7 -17.4 6 885.18

F 237.5 163.8 456.9 -58.6 26

774.34

A 150 100 387.5 -63.8 24

722.50

34 970.42 58

382.02

34

970.42

2A = 23

411.60

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Area = 11 705.8 m2

= 1.1706 ha

2.6 AREAS OF IRREGULAR FIGURES

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:

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Area = (width of internal) [½ (first + last ordinate) + sum of remaining

ordinates]

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.

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:

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

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2.6.4 Simpson’s rule

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.

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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

a) The trapezoidal rule

b) The mid-ordinate rule

c) 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.

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Feedback 2a

2.1)

FIGURE 2.5 : Graph of y against x

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

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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:

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'.

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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.

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)


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)


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

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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. 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

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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 :

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)

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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 :

= 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 Question

Calculate, 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

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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 :

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}

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Specimen Question

Calculate, 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

Solution

V = 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.

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.

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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 :

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

= (2.5 / 3 ) (9822)

= 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 2

V = 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 Question

Picture 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 n A 12.16 1 12.16 B 12.48 2 24.96 C 13.01 1 13.01 D 12.56 2 25.12 E 12.87 4 51.48 F 13.53 2 27.06 G 12.94 1 12.94 H 13.27 2 26.54 J 13.84 1 13.84

∑n = 16 207.11

Mean level = 207.11 / 16

= 12.944 m

Depth of excavation = 12.944 – 8.00

= 4.944

Volume = Total area x Depth

= 30 x 20 x 4.944

= 2966.4 m3

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(b) Calculation from triangles

It 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 Product

R.L. is used = n (R.L.) x n

A 12.16 1 12.16

B 12.48 3 37.44

C 13.01 2 26.02

D 12.56 3 37.44

E 12.87 7 90.09

F 13.53 2 27.06

G 12.94 2 25.88

H 13.27 2 26.54

J 13.84 2 27.68

∑n = 24 310.55

Mean level = 310.55 / 24

= 12.940 m

Depth 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.

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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.

Calculate

a). Volume by mean areas

b). Volume by end areas

c). Volume by prismoidal rule

2.3 Given the previous example but with the centre line turned through 90º,

calculate volume

a) By mean areas b) By end areas c) By prismoidal

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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 areas

V = w ( A1 + 2A2 + A3 ) / 2

= 30.00 (54.00 + 189.00 + 144.00) / 2

= 5805.0 m3

c). By Prismoidal Rule

V = 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 m² A2 = [ ( 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

26

Volume a). By mean areas V = 60.00 ( 54.00 + 95.64 + 145.64 ) / 3 = 5920.2 m³ b). By end areas V = 30.00 ( 54.73 + 191.28 + 145.64 ) / 2 = 5874.8 m³ c). By prismoidal rule V = 30.00 ( 54.73 + 382.56 + 145.64 ) / 3 = 5829.3 m³

Self Assessment

Calculate the volumes in Figure 1 and Figure 2.

Figure 1

Figure 2

27

Feedback to Self Assessment

Figure 1

A1 = [ ( 0.75 + 4.75 ) / 2 ] x 7 = 19.25 ft²

A2 = [ ( 0.75 + 3.75 ) / 2 ] x 5 = 11.25 ft²

A3 = [ ( 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

= 7.29 yrd³

Figure 2

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

= 27.4 / 3 ( 13.5 x 13.5 )

= 1664.6 m³

28

MASS HAUL DIAGRAM

3.1 INTRODUCTION

Mass-haul diagrams (MHD) are used to compare the economics of the various

methods of earthwork distribution on road or railway construction schemes. With

the combined use of the MHD plotted directly below the longitudinal section of

the survey centre-line, one can find :

i. The distances over which ‘cut and fill’ will balance.

ii. Quantities of materials to be moved and the direction of movement.

iii. Areas where earth may have to be borrowed or wasted and the amounts

involved.

iv. The best policy to adopt to obtain the most economic use of plan.

3.2 DEFINITION AND IMPORTANT PHRASES

Bulking An increase in volume of earthwork after excavation

Shrinkage A decrease in volume earthwork after deposition and

compaction.

Haul distance (d) The distance from the working face of the excavation to

the tipping point.

Average haul distance

(D)

The distance from the centre of gravity of the cutting to

that of the filling.

Freehaul Distance The distance, given in the Bill of Quantities, included in

the price of excavation per cubic metre.

Overhaul Distance The extra distance of transport of earthwork volumes

beyond the freehaul distance.

Haul

The sum of the product of each load by its haul

distance. This must equal the total volume of excavation

multiplied by the average haul distance, i.e. ∑ vd = VD

Overhaul The product of volumes by their respective overhaul

distance. Excess payment will depend upon overhaul.

Station Metre A unit of overhaul, viz. 1 m3 x 100 m.

Borrow The volume of material brought into a section due to a

deficiency.

Waste The volume of material taken from a section due to

excess

29

Figure 3.1 : Mass-haul diagram (Source : Land Survey, Ramsay)

Figure 3.2 : Freehaul and Overhaul (Source : Land Survey, Ramsay)

30

3.3 CONSTRUCTION OF THE MASS-HAUL DIAGRAM

Volumes of cut and fill along a length of proposed road are as follows :

Volume

Chainage Cut Fill 0

100 290 200 760 300 1680 400 620 480 120 500 20 600 110 700 350 800 600 900 780

1000 690 1100 400 1200 120

Draw a mass-haul diagram and exclude the surplus excavated material along

this length. Determine the overhaul if the freehaul distance is 300 m.

Volume Aggregate Chainage Cut Fill volume

0 100 290 + 290 200 760 + 1050 300 1680 + 2730 400 620 + 3350 480 120 + 3470 500 20 + 3450 600 110 + 3340 700 350 + 2990 800 600 + 2390 900 780 + 1610

1000 690 + 920 1100 400 + 520 1200 120 + 400

3470 3070 3070

check 400

31

Figure 3.3 : Mass-haul diagram (Source : Land Survey, Ramsay)

Graphical Method (figure 3.3)

i. As the surplus of 400 m³ is to be neglected, the balancing line is drawn

from the end of the mass-haul curve, parallel to the base line, to form a

new balancing line ab.

ii. As the freehaul distance is 300 m, this is drawn as a balancing line cd.

iii. From c and d, draw ordinates cutting the new base line at c1d1.

iv. To find the overhaul :

a) Bisect cc1, to give c2 and draw a line through c2 parallel to the base

line cutting the curve at e and f, which now represent the centroids

of the masses acc1 and dbd1.

b) The average haul distance is the centroids of the masses acc1 and

dbd1.

c) The overhaul distance = the haul distance – the free haul distance

Planimetric method

Distance to centroid = haul / volume

= (area x horizontal scale x vertical scale) / volume ordinate

from area acc1

area scaled from mass-haul curve = 0.9375 cm²

horizontal scale = 1 cm = 200 cm

vertical scale = 1 cm = 1600 cm³

Therefore

32

haul = 0.9375 x 200 x 1600 = 300 000

volume (ordinate acc1 ) = 2750

distance to centroid = 300 000 / 2750 = 109.1 m

chainage of centroid = 350 – 109.1 = 240.9 m

for area dbd1

area scaled = 1.9688 cm²

Therefore

haul = 1.9688 x 320 000 = 630 016

volume (ordinate dd1 ) = 2750

distance to centroid = 229.1 m

chainage of centroid = 650 + 229.1 = 879.1 m

average haul distance = 879.1 -240.9 = 638.2 m

overhaul distance = 638.2 – 300 = 338.2 m

Therefore

overhaul = 338.2 x 2750 = 9300 station metres

Instead of the above calculations, the overhaul can be obtained direct, as the

sum of the two mass-haul curve areas acc1 and dbd1 is:

Area acc1 = (302 950) / 100 station metre

Area dbd1 = (634 950) / 100 station metre

Total area = overhaul = (936 900) / 9369 station

metre

Proof :

Take any cutoff by a balancing line, Figure 5.4. Let a small increment of area δ A

= (say) 1 m³ and length of haul be L. Then

δA = 1 m³ x L / 100 station metre

Therefore

A = n x 1 m³ x ( ∑L / n )

= total volume x average haul distance

Therefore, Area = total haul

33

Activity 3a

3.1 Below are the definitions used in this unit. Fill in the blank with the

appropriate terms.

an increase in volume of earthwork after

excavation

a decrease in volume earthwork after

deposition and compaction.

the distance from the working face of the

excavation to the tipping point.

the distance from the centre of gravity of

the cutting to that of the filling.

the distance, given in the Bill of quantities,

included in the price of excavation per

cubic metre.

the extra distance of transport of earthwork

volumes beyond the freehaul distance.

the sum of the product od each load by its

haul distance. This must equal the total

volume of excavation multiplied by the

average haul distance, i.e. ∑ vd = VD

the product of volumes by their respective

overhaul distance. Excess payment will

depend upon overhaul.

a unit of overhaul, viz. 1 m3 x 100 m.

the volume of material brought into a

section due to a deficiency

the volume of material taken from a section

due to excess

34

Feedback 3a

3.1).

an increase in volume of earthwork after

excavation

a decrease in volume earthwork after deposition

and compaction.

the distance from the working face of the

excavation to the tipping point.

the distance from the centre of gravity of the

cutting to that of the filling.

the distance, given in the Bill of quantities,

included in the price of excavation per cubic

metre.

the extra distance of transport of earthwork

volumes beyond the freehaul distance.

the sum of the product of each load by its haul

distance. this must equal the total volume of

excavation multiplied by the average haul

distance, i.e. ∑ vd = VD

the product of volumes by their respective

overhaul distance. excess payment will depend

upon overhaul.

a unit of overhaul, viz. 1 m3 x 100 m.

the volume of material brought into a section

due to a deficiency

the volume of material taken from a section due

to excess

Station Metre

Borrow

Waste

Bulking

Shrinkage

Haul Distance (d)

Overhaul

Haul

Overhaul Distance

Freehaul Distance

Average Haul

Distance

35

Self Assessment

The table shows the stations and the surface levels along the centre-line, the formation level being at an elevation above datum of 43.5 m at chainage 70 and thence rising uniformly on a gradient of 1.2%. The volumes are recorded in m³, the cuts are plus and fills minus.

Chn Surface Vol Chn Surface Vol Chn Surface Vol Level Level Level

70 52.8 74 44.7 78 49.5 +1860 -1080 -237

71 57.3 75 39.7 79 54.3 +1525 -2025 +362

72 53.4 76 37.5 80 60.9 +547 -2110 +724

73 47.1 77 41.5 81 62.1 -238 -1120 +430

74 44.7 78 49.5 82 78.5

1) Plot the longitudinal section using a horizontal scale of 1 : 1200 and a vertical scale of 1 : 240.

2) Assuming a correction factor of 0.8 applicable to fills, plot the MHD to a

vertical scale of 1000 m3 to 20 mm.

3) Calculate total haul in stn. m and indicate the haul limits on the curve and section.

4) State which of the following estimates you would recommend.

a) No freehaul at 35 p per m³ for excavating, hauling and filling b) A freehaul distance of 300 m at 30 p per m³ plus 2 p per stn m for

overhaul.

36


1) The volume at chainage 70 is zero 2) The mass ordinates are always plotted at the station and not between

them. 3) The mass ordinates are now plotted to the same horizontal scale as

the longitudinal section and directly below it. 4) Check that maximum and minimum points on the MHD are directly

below grade points on the section. 5) Using the datum line as a balancing line indicates a balancing out of

the volumes from chainage 70 to XY and from XY to chainage 82.

Total haul (taking each loop separately) = total volume x total haul distance. The total haul distance is from the centroid of the total cut to that of the total fill and is found by bisecting AB and A’B’, to give the distances CD and C’ D’. Total haul = ( AB x CD ) / 100 + ( A’ B’ x C’D’ ) / 100 = ( 3932 x 450 ) / 100 + ( 1516 x 320 ) / 100 = 22 545 stn m

37

a) If there is no freehaul, then all the volume is moved regardless of distance for 35 p per m³.

Estimate costs : ( AB + A’ B’ ) x 35 p = 5448 x 35 = 190 680 p

b) The purpose of plotting the free haul distance on the curve is to assess the overhaul.

From MHD : Cost of freehaul = (AB + A’B’) x 30 p per m³ = 163 440 p Cost of overhaul = [ EG ( JK – EF ) / 100 + E’G’ (J’K’ – E’F’) / 100 ] x 2p = 13 628 p Total cost = 163 440 + 13 628 = 177 068 p The second estimate is cheaper by 13 612 p = £136.12 All the dimensions in the above solution are scaled from the MHD

38

CURVES

4.1 INTRODUCTION

In the geometric design of motorways, railways and pipelines, the design and setting out of

curves is an important aspect of an engineer’s work. The initial design is usually based on a

series of straight sections whose positions are defined largely by the topography of the area. The

intersections of pairs of straights are then connected by horizontal curves. In the vertical design,

intersecting gradients are connected by curves in the vertical plane. Curves can be listed under

three main headings as follows:

1. Circular curve of constant radius

2. Transition curves of varying curves (spirals)

3. Vertical curves.

4.2 CIRCULAR CURVES

Horizontal, circular or simple curves are curves of constant radius required to connect two

straights set out on the ground. Such curves are required for roads, railways, kerb lines, pipe

lines and may be set out in several ways, depending on their length and radius. Figure 4.1

illustrates how two tangents are joined by a circular curve and shows some related circular curve

terminology. The point at which the alignment changes from straight to circular is known as the

BC (beginning of curve).The BC is located at a distance T (sub tangent) from PI (Point of tangent

intersection).

The length of a circular curve (L) is dependent on the central angle (∆) and the value of R

(radius). The tangent deflection angle (∆) is equal to the curve’s central angle (Figure 4.2). The

point at which the alignment changes from circular back to tangent is known as the EC (end of

curve). Since the curve is symmetrical about the PI, the EC is also located at distance T from the

PI. From a study of geometry, we recall that the radius of a circle is perpendicular to the tangent

at the point of tangency. Therefore, the radius is perpendicular to the back tangent at the BC and

to the forward tangent at the EC. The terms BC and EC are also referred to by some agencies as

PC (point of curve) and PT (point of tangency) and by others as TC(tangent to curve) and

CT(curve to tangent).

39

Figure 4.1 Circular Curve Terminologies

(Source: Surveying With Construction Application, B.F. Kavanagh)

4.2.1 Circular Curve Geometry

Most curve problems are calculated from field measurements (∆ and the chainage of PI) and from

design parameters(R). Given R (which is dependent on the design speed) and ∆, all other curve

components can be computed. An analysis of Figure 4.2 will show that the curve deflection angle

(PI, BC, EC) is ∆/2 and that the central angle at O is equal to ∆, the tangent deflection. The line

(O-PI), joining the centre of the curve to the PI, effectively bisects all related lines and angles.

a) Tangent:

In Triangle BC, O, PI,

2tan

2tan

RT

R

T

b) Chord :

In triangle BC, O, B

40

2sin2

2sin2

1

RC

R

C

c) Mid- ordinate:

2cos

2cos

ROB

R

OB

but OB = R- M

2cos1

2 cos R M - R

RM

d) External:

In triangle BC, O, PI

O to PI = R + E

12

sec

1

2cos

1

2cos

R

RE

ER

R

41

Figure 4.2 Geometry Of The Circle.


e) Arc:(Figure 4.3)

3602

3602

RL

R

L

where is expressed in degrees and decimals of a degree.

Figure 4.3 Relationship Between The Degree Of Curve (D) And The Circle.


The sharpness of the curves is determined by the choice of the radius R; large

42

radius curves are relatively flat, whereas small radius curves are relatively sharp.

D is defined to be that central angle subtended by 100 ft of arc. (in railway

designs, D is defined to be that central angle subtended by 100 ft of chord.)

From Figure 4.3, D and R:

RD

R

D

58.5729

2

100

360

Arc:

DL

D

L

100

100

f) Deflection angle

Figure 4.4 Deflection angle.

(Source: Land Surveying, Ramsay J.P. Wilson)

In ∆ T1AO, curve T1A = R x 21

Curve T1A = Chord T1A

1(rad) = Curve T1A / 2R

= Chord T1A

1(minutes) = (Curve T1A x 180 x 60) / 2R

= (1718.9 x chord T1A) / R

EXAMPLE 4a

43

Refer to Figure 4.5, Given ∆ = 16 ° 38’

R = 1000 ft and PI at 6 + 26.57, calculate the

station of the BC and EC. Calculate also

lengths C, M and E.

SOLUTION:

ft

RT

18.146

'198tan1000

2tan

ft

RL

31.290

360

6333.1610002

3602

PI at 6 + 26.57

-T 1 46.18

BC = 4 + 80.39

+L 290.31

EC = 7 + 70.70

Figure 4.5 (Source: Surveying With

Construction Application, B.F.

Kavanagh)

44

ft

RE

ft

RM

ft

RC

63.10

)1'198(sec1000

12

sec

52.10

)'198cos1(1000

2cos1

29.289

'198sin10002

2sin2

4.2.2 Compound Circular Curves

A compound circular curves are curves formed when of two (usually) or more circular arcs

between two main tangents turn in the same direction and join at common tangent points. Figure

6.4 shows a compound curve consisting of two circular arcs joined at a point of compound curve

(PCC). The lower chainage curve is number 1, whereas the higher chainage curve is number 2.

The parameters are R1, R2, ∆1, ∆2 (∆1 + ∆2 = ∆), T1 and T2. If four of these six or seven

parameters are known, the others can be solved. Under normal circumstances, ∆1, ∆2, or ∆, are

measured in the field, and R1 and R2 are given by design considerations, with minimum values

governed by design speed.

Although compound curves can be manipulated to provide practically any vehicle

path desired by the designer, they are not employed where simple or spiral

curves can be used to achieve the same desired effect. Practically, compound

curves are reserved for those applications where design constraints (topographic

or cost of land) preclude the use of simple or spiral curves, and they are now

usually found chiefly in the design of interchange loops and ramps. Smooth

driving characteristics required that the larger radius be more than 1-1/3 times

larger than the smaller radius (this ratio increases to 1-1/2 when dealing with

interchange curves).

Solutions to compound curve problems vary, as several possibilities exist as to which of the data

are known in any one given problem. All problems can be solved by use of the sine law or cosine

law or by the omitted measurement traverse technique. If the omitted measurement traverse

technique is used, the problem becomes a five-sided traverse (Figure 4.6) with sides R1, T1, R2

45

and (R1- R2) and with angles 90°, 180° - ∆° + 90°, 180°+ ∆2° and ∆1°. An assumed azimuth that

will simplify the computations can be chosen.

Figure 4.6 Compound Circular Curves


4.3 Reverse Curves

Reverse curves are seldom used in highway or railway alignment. The instantaneous change in

direction occurring at the point reverse curve (PRC) would cause discomfort and safety problems

for all but the slowest of speeds. Additionally, since the change in curvature is instantaneous,

there is no room to provide super elevation transition from cross-slope right to cross-slope left.

However, reverse curves can be used to advantage where the instantaneous change in direction

poses no threat to safety or comfort.

The reverse curve is particularly pleasing to the eye and is used with great success on park

roads, form paths, waterway channels, and the like. The curve can be encountered in both

situations illustrated in Figure 4.7 a. and b. the parallel tangent application is particularly common

(R1 is often equal to R2). As with compound curves, reverse curves have six independent

parameters ( R1, ∆1, T1, R2, ∆2, T2); the solution technique depends on which parameters are

unknown, and the techniques noted for compound curves will also provide the solution to reverse

curve problems.

46

Figure 4.7 Reverse Curves (a-Non parallel curve, b- Parallel tangents)


4.4 Transition Curves

The centrifugal force acting on a vehicle as it moves along a curve increases as

the radius of the curve decreases. A vehicle moving from the straight with no

centrifugal force acting upon it, into a curve would suddenly receive the

maximum amount of centrifugal force for that radius of curve. To prevent this

sudden lateral shock on passengers in the vehicle, a transition curve is inserted

between the straight circular curve

(Figure 4.8). The transition curve is a curve of constantly changing radius. The

radius (R) of transition curves varies from infinity at its tangent with the straight to

a minimum at its tangent point with the circular curve. The centrifugal force thus

builds up gradually to its maximum amount.

47

Figure 4.8 The Transition Curves.


The purpose of a transition curve then is to achieve a gradual change of direction from the

straight (radius ∞) to the curve (radius R) and permit the gradual application of super-elevation to

counteract centrifugal force.

The central fugal force tending to thrust a vehicle sideways on a curve is resisted by the friction

between the wheels and the surface. If the outer edge of the surface is raised or super elevated,

the resultant forces tend to reduce the frictional force necessary to hold the vehicle on the

surface. At a particular slope the frictional force necessary can be eliminated by the formula

below:

2

tangR

v

where v is the velocity and g is the acceleration due to gravity. As vehicle speeds vary, the

fractional resistance is always necessary and a vehicle may stop on the curve. The super

elevation must not be too great.

4.4.1 Spiral Curve and Composite Curve

A spiral is a curve with a uniformly changing radius. Spirals are used in highway

and railroad alignment changes from tangent to circular curves, and vice versa.

The length of the spiral curve is also used for transition from normally crowned

pavement to fully superelevated pavement.

S = shift

48

Figure 4.9 shows how the spiral curve is inserted between tangent and

circular curve alignment. It can be seen that at the beginning of the spiral

(T.S. = tangent to spiral) the radius of the spiral is the radius of the tangent

line (infinitely large) and that the radius of the spiral curve decreases at a

uniform rate until, at the point where the circular curve begins (S.C = spiral

to curve) the radius of the spiral equals the radius of the circular curve.

The spiral curve, used in horizontal alignment, has a uniform rate of

change of radius (curvature). This property permits the driver to leave a

tangent section of highway at a relatively high rate of speed without

experiencing problems with safety or comfort.

A composite curve is a curve that forms by combination of two transition curves or through

combination of two transition curves and a circular curve.

Figure 4.9 Spiral Curves


4.5 Vertical Curves

Vertical curves are used in highway and street vertical alignments to provide a

gradual change between two adjacent grade lines. Some highway and municipal

agencies introduce vertical curves at every change in grade-line slope, whereas

other agencies introduce vertical curves into alignment only when the net change

in slope direction exceeds a specific value (for example 1.5% or 2%).

In Figure 4.10, g1 is the slope of the lower chainage grade line, g2 is the slope of

the higher chainage grade line, BVC is the beginning of the vertical curve, EVC is

the end of the vertical line, and PVI is the point of intersection of the two adjacent

49

grade lines. The length of vertical curve (L) is the projection of the curve onto a

horizontal surface and, as such, corresponds to plan distances.

The algebraic change in slope direction is A, where A = g2 – g1.

Example 4b:

g1 = +1.5% and g2= -3.2%

A = g2 – g1

= -3.2-1.5

= -4.7

The geometric curve used in vertical alignment designs is the vertical axis parabola. The parabola

has the desirable characteristics of

(1) a constant rate of change of slope, which contributes to smooth alignment

transition,

(2) ease of computation of vertical offsets, which permits easily computed curve

elevations

Figure 4.10 Vertical Curves (Profile View Shown)


The origin of the axes is placed at the BVC (Figure 4.11), the general equation becomes y = ax2 +

bx, and because the slope at the origin is g1, the expression for slope of the curve at point

becomes

slopedx

dy

= 2ax + g1

The general equation can finally be written as y = ax2 + g1x

50

Figure 4.11 Types of Vertical Curve


Activity 4a

4.1 Fill in the blanks with related circular curve terminology.

51

Figure 1

4.2 Solve the puzzle by using the clues as shown below.

Horizontal:

1) The __________ curve is a curve of constantly changing radius.

2) __________ curves are used in highway and street vertical alignment to provide a gradual change

between two adjacent grade lines.

3) __________ curves are curves of constant radius required to connect two straights set out on the

ground.

4) Circular curve is also known as ____________ curves.

Vertical:

5) The ________curves can be encountered in both situations which are a non parallel curve and

parallel tangents.

1 5

7

2

6

3

4

A

B

C

D B

B

E

H B

B

G B

B F B

B

I B

B J B

B

52

6) A _______ curve of two (usually) or more circular arcs between two main

tangents turning in the same direction and joining at common tangent points.

7) A ________ is a curve with a uniformly changing radius.

Feedback 4a

4.1

Figure 1

A – Back tangent

B – Point of intersection

C – Deflection angle

D – Radius

E – Mid ordinate

F – Long chord

G – Sub tangent

H – End of Curve

I – External

J – Length of curve

4.2

53

4.6 SETTING OUT CURVES

This is the process of establishing the centre-line of the curve on the ground by means of pegs at

10m to 30m intervals. In order to do this, the tangent and intersection points must be first fixed in

the ground in their correct positions.

The straights OI1, I1I2, I2I3,etc., will have been designed on the plan in the first instance(Figure

4.12). Using railway curves, appropriate curves will now be designed to connect the straights.

The tangent points of these curves will then be fixed making sure that the tangent lengths are

equal, i.e. T1 I1 = T2I1 and T3 I2 = T4I2. The coordinates of the origin, point O, and all the

intersection points will only now be carefully scaled from the plan. Using these coordinates, the

bearings of the straights are computed and using the tangent lengths on these bearings, the

coordinates of the tangent points are also computed. The difference of the bearings of the

straights provides the deflection angles(Δ) of the curves which, combined with the tangent length,

enables computation of the curve radius, through chainage and all-setting-out data. Now the

tangent and intersection points are set out from existing control survey stations and the curves

ranged between them using the methods detailed below.

Figure 4.12 Curve Setting Out

(Source: Engineering Surveying, W.Schofield)

1T

5R A N S I T I O N

7S E

P 2V E R T I

6C A L

I E O 3C I R C U L A R M

A S P

L E O

U

4H O R I Z O N T A L

D

54

4.6.1 Setting Out By Offsets With Tangent Angle Method The following methods of setting out curves is the most popular and it is called Rankine’s

deflection or tangential angle method, the latter term being more definitive.

In figure 4.13, the curve is established by a series of chords T1X, XY, etc. Thus, peg 1 at X is

fixed by sighting to I with the theodolite reading zero, turning off the angle 1 and measuring out

the chord length T1 X along this line. Setting the instrument to read the second deflection angle

gives the direction T1 Y, and peg 2 is fixed by measuring the chord length XY from until it

intersects at Y. The procedure is now continued, the angles being set out from T1 I and the

chords measured from the previous station. It is thus necessary to be able to calculate the setting

out angles as follows:

Assume OA bisects the chord T1 X at right-angles, then

Angle AT1 O =90°- 1 , but angle IT1 =90°

angle IT1A= 1

By radians arc length T1X= R21

1 rad = (arc T1X /2R) (Chord T1X / 2R)

1min = (chord T1X x 180º x 60) /2Rπ

= 1718.9(Chord / R)

or º = (Dº x Chord ) / 200 where degree of curve is used.

Figure 4.13 Tangent Angle Method (Source: Engineering Surveying, W.Schofield)

Example 4c:

The centre-line of two straights is projected forward to meet at I, the deflection angle being 30°. If

the straights are to be connected by a circular curve of radius 200 m, tabulate all the setting-out

data, assuming 20-m chords on a through chainage basis, the chainage of I being 2259.59 m.

Solution:

55

Tangent length = R tan Δ/2

= 200 tan 15°

= 53.59 m

Chainage of T1 = 2255.59 - 53.59

= 2206 m

1st sub-chord = 14 m

Length of circular arc = RΔ = 200(30°) rad = 104.72 m

From which the number of chords may now be deduced

1st sub-chord = 14 m

2nd, 3rd, 4th, 5th chords = 20 m each

Final sub-chord = 10.72 m

Total = 104.72 m {Check}

Chainage of T2; = 2206 m + 104.72 m = 2310.72 m

Deflection angles:

For 1st sub-chord = 1718.9 (14/200) = 120.3 min = 2° 00' 19"

Standard chord = 1718.9 (20/200) = 171.9 min = 2° 51' 53"

Final sub-chord = 1718.9 (10.72 /200) = 92.1 min = 1° 32' 08"

Check: The sum of the deflection angles = Δ/2 = 14° 59' 59" 15°

The error of 1" is, in this case, due to the rounding-off of the angles to the nearest second and is negligible.

4.6.2 Setting Out By Offset From The Tangent

The position of the curve (in Figure 4.14) is located by right-angled offsets Y set out from

distances X, measured along each tangent, thereby fixing half the curve from each side. The

offsets may be calculated as follows for a given distance X. Consider offset Y3, for example.

In ΔABO,

AO2 = OB

2 - AB

2

(R-Y3)2= R

2 – X3

2

and Y3=R-(R2-X3

2)½

Chord number

Chord length (m)

Chainage (m)

Deflection angle o , „

Setting-out angle

o , „

Remarks

1 14 2220.00 2 00 19 2 00 19 peg 1

2 20 2240.00 2 51 53 4 52 12 peg 2

3 20 2260.00 2 51 53 7 44 05 peg 3

4 20 2280.00 2 51 53 10 35 58 peg 4

5 20 2300.00 2 51 53 13 27 51 peg 5

6 10.72 2310.72 1 32 08 14 59 59 peg 6

56

thus for any offset Yi, at distance Xi, along the tangent

Yi = R - (R2 - Xi

2) ½

Figure 4.14 Setting Out By Offset From Tangent (Source: Engineering Surveying, W.Schofield)

4.6.3 Setting Out By Offset With Sub-Chords

In Figure 4.15 assume T1A is a sub-chord of length x, from equation

Offset CA = (½ chord x chord) / Radius = (chord2)/2R, the offset CA = O1 == X

2 / 2R.

As the normal chord AB differs in length from T1A, the angle subtended at the centre will be 2θ

not 2. Thus, the offset DB will not in this case equal 2CA.

Construct a tangent through point A, then from the figure it is obvious that angle EAB = θ, and if

chord AB = y, then offset EB = y2 / 2R.

Angle DAE = , therefore offset DE will be directly proportional to the chord length, thus:

DE = (O1 / x) y = (x

2 y)2Rx = (xy)/ 2R

Thus the total offset DB = DE + EB

= ( y / 2R) (x +y)

= (Chord / 2R) (sub-chord + chord)

Thus having fixed B, the remaining offsets to T2; are calculated as y2/R and set out in the usual

way.

If the final chord is a sub-chord of length x1, however, then the offset will be

(x1/ 2R)((x1 + y)

A more practical approach to this problem is actually to establish the tangent through A in the

field. This is done by swinging an arc of radius equal to CA,

57

i.e. x2 / 2R from Ti. A line tangential to the arc and passing through peg A will then be the

required tangent from which offset EB, i.e. y2/lR, may be set off.

Figure 4.15 Setting Out By Offset With Sub-chords


4.6.4 Setting Out By Offset With Long-Chords

In this case (Figure 4.16) the right-angled offsets Y are set off from the long chord C, at distances

X to each side of the centre offset Y0.

An examination of Figure 6.16 shows the central offset Y0 equivalent to the distance T1A on

Figure 6.14, thus:

Yo = R –[R2 - (C/2)

2]1/2

Similarly, DB is equivalent to DB on Figure 6.14, thus:

DB=R-(R2 – X1

2) 1/2

and offset Y1, = Y0 - DB

:. Y1 = Y0- [R - (R2 - X1

2)'/

2]

and for any offset Yi, at distance Xi, each side of the mid-point of T1 T2,:

Y1, = Y0 - [R - (R2 - X

2)]

1/2

58

Figure 4.16 Setting Out By Offset With Long-chords


4.7 Setting Out A Circular Curve.

Circular curves may be set out in a variety of ways, depending on the accuracy required, its

radius of curvature and obstructions on site. Methods of setting out are as follows:

Using one theodolite and a tape by the tangent angle method. This method can be used

on all curves, but is necessary for long curves of radius unless they are set out by

coordinates.

Using two theodolites. This method can be used on smaller curves where the whole

length is visible from both tangent points and where two instruments are available.

Using tapes only by the method of offsets from the tangent. This method is used for

minor curves only.

Using tapes only by the method of offsets from the long chord. This method is used for

short radius curves.

Normally, a circular curve is set up by using a theodolite and a tape (tangent angle method).

Before the curve can be set out the tangent points must be located on the ground. For any

particular pair of straights there is only one point on each straight for a curve of given radius or

degree to leave the first tangentially in order to join the other tangentially. These tangent points

cannot be scaled off a plan with sufficient accuracy and they must be located by field

observations. The tangent points are represented in Figure 4.17 by the points A and B. The

method of locating these tangent points is summarised in (a) to (h) as follows:

a) Set up the theodolite near A and extend straight towards P.

b) Set up on the straight BF and produce it to meet the line at P.

c) Mark the intersection of tangents at P.

d) Measure angle EPF and obtain angle θ.

e) Calculate tangent length PA by using the formula T = R tan ´θ.

f) Place pegs at A and B on the lines. (From P measure the lengths PA and PB = T and line

in the points A and B on the straights with the theodolite still set up at P. Mark the pegs at

A and B distinctively representing tangent points. This can be done by painting the peg or

by placing three pegs, the centre peg representing the tangent point.)

g) Set up at A and measure angle PAB, which should equal ´θ.

59

h) Complete the chaining of the first straight to A.

NOTE :

(i) The chaining of the first straight is completed by measuring the distance from the last chain

peg and noting the actual chainage of the tangent point A.

ii) In many cases the intersection points P may have been previously marked on the ground. In

such cases the field work consists of pegging the straights and measuring θ without having to

locate the intersection by the method described above.

After the tangent points have been pegged as described above, the points on the curve must be

located. The interval between chainage pegs on the curve should be measured along the actual

arc. As chords are used in locating the pegs, the difference in length should be calculated strictly

as they are slightly shorter than the arc distances. This would be done in precise work, e.g.

underground railways. In most practical cases where R exceeds twenty times the chord length

this difference is negligible. The tangent point A will seldom fall exactly at a peg interval. Since

the chainage must be continuous, the chord AG to the first point on the curve may be shorter than

the regular chord length which is c, usually equal to the peg interval or half the peg interval if

additional pegs are needed to mark the curve clearly on the ground. There will generally also be a

sub-chord at the end of the curve. Let these

sub-chord lengths be denoted by c' and c".

The method of locating the points on the curve is summarised in (a) to (k} as follows:

a) Obtain the first sub-chord c' = c — EA.

Assuming E is the position of the last chainage peg on the straight, then EA + c’ =c and as EA

has been measured and c is known, the length of the sub-chord can be obtained.

b) Calculate for chord length c.

This can be calculated from sin = (c/2R) or = 1718.9 ( c/R) minutes.

c) Calculate ’ for the first sub-chord.

This can be calculated in the same way as for , but for flat curves it can be obtained with

sufficient accuracy from ’ = (c’ /c)

d) Calculate the final sub-chord and its ".

Calculate from θ and the radius the length of the curve (L= Rθ). Then the chainage of A + L =:

chainage of B. The amount, by which the chainage of B exceeds and exact number of peg

intervals, plus the initial sub-chord, is the length of the end sub-chord c".

e) Draw up a table of deflection angles to the various points.

i) This will take the following form:

1st deflection angle to G = (c’/c) = ”

60

2nd deflection angle to H = ’ +

3rd deflection angle to K = ’+ + .

ii) The final deflection angle to tangent point B must equal ´θ, allowance being made for

the sub-chords,

i.e. ´θ = ' + + + . . . + ".

f) From A set out ’for the line AG.

The instrument is set up at A and P is sighted at a reading of 0° 00' 00" and the horizontal circle is

clamped with the lower clamp. The first deflection angle ’ is set on the vernier or optical

micrometer using the upper clamp and tangent screw only, so that the line of sight is along AG.

g) Place G a distance of c' m from A on line AG.

The zero of the tape is held at A and the distances marked with a peg, which is then moved on to

the line AG as defined by the theodolite sighting.

h) From A set out ’+ for the line AH.

This is the second deflection angle PAH obtained from the table in (e).

i) Set out GH =c.

The zero of the tape is now held at Q and the chord length or peg interval along the tape is

marked with a peg which is moved on to the line AH as given by the theodolite.

j) Repeat the same process to set out the remaining pegs.

i) Continue until the last peg on the curve has been placed and measure the remaining distance

to B which should equal the calculated length c" of the final sub-chord. Also set out the final

deflection angle, which should pass through tangent point B, indicating no disturbance of the

instrument.

ii) As a final check on the accuracy, locate point B by the deflection angle and sub-chord c". If this

position does not coincide with the tangent point B, the distance between the two is the actual

error of tangency. If this is large, indicating an error, the whole process must be repeated. Where

calculations are inaccurate by a few millimetres in the final placing of the pegs, it is usual to adjust

the last few pegs to secure tangency. In accurate tunnel work the degree of precision must, of

course, be greater.

61

k) The first chainage peg on BF will be c — c" from -B.

Having calculated the distance of the first chainage peg F on the second straight, chaining may

be proceeded with after moving the instrument to B or some other convenient point on this

straight.

Figure 4.17 Setting out a circular curve


Example 4d:

Two straights AP and PB intersect with an angle of deflection of 12º 20' as illustrated in Figure

6.17. They are to be connected by a circular curve of radius 600 m. The chainage of the

intersection point is 12 + 73.16. Calculate the setting-out data required to peg the curve at a

continuous chainage with pegs at 25m intervals.

Solution :

a) Calculate the tangent lengths from T == R tan ´θ

T = 600 tan 6° 10'

= 600 x 0.108046

= 64.83 m.

b) Calculate, the arc length from L = Rθ

L = 600 x 12° 20' x (2π / 360)

= 600 x 0.21526

= 129 .16 m.

c) Calculate the chainages:

Chainage of P = 12 + 73.16

Less T = 64.83

Chainage of A = 12 + 08.33

Add L = 129.16

Chainage of B = 13 + 37.49 m.

62

d) Calculate the sub-chords.

The last peg on the straight is at chainage 12 + 00, therefore the next peg

must be at chainage 12 + 25. There is still 8-33 m on the straight to the

tangent point, so there will be 25.00 — 8.33 = 16.67 m along the curve to

the first peg on the curve. Thus 16.67 m is the length of the first sub-

chord.

As the curve length is 12 9.16 m and the first sub-chord is 16.67 m, there is 129.16 – 16.67 =

112.49 of arc left. Four 25m standard chords make up the next 100m, leaving a final sub-chord of

12.49 m. The measurement of the arc distance by these chords is sufficiently accurate for most

practical purposes although theoretically the measured distance is shorter than the arc distance.

e) Calculate the deflection angles from = 1718.9( c/R)

= 1718.9 x (25/600)

= 71.62

= 1° 11. 62’

= 1° 11' 37.2”.

i) The initial sub-chord is 16.67 m, so its deflection angle will be in the proportion of:

(16-67 /25 ) x (1º11.62’) = 47.76’

= 47' 45.6”.

ii) The final sub-chord is 12.49 m, so its deflection angle will be in the proportion of:

(12.49 / 25) x 1° ll.62’ = 35.78’.

= 35’ 46.8”

g) Tabulate the deflection angles. The deflection angles are tabulated as follows:

Instrument at A = 12 + 08.33

To peg at P Chord Length Bearing

° ’ ”

12 + 25

12 + 50

12 + 75

13

13 + 25

16.67

25

25

25

25

00 00 00

+ 47 45.6

00 47.45.6

+ 1 11 37.2

1 59 22.8

+ 1 11 37.2

3 11 00.0

+ 1 11 37.2

4 22 37.2

+ 1 11 37.2

5 34 14.4

63

B = 13 + 37.49

12.49

+ 35 46.8

6 10 01.2 = ´θ (Check)

Table 4.1 Calculation Of Setting Out Circular Curve


NOTE: There is always likely to be minor rounding off of errors such as the 1.2”, which is

negligible. To keep these errors to a minimum the calculation is always carried out to 0.1”, but the

observed bearings are rounded off to 1" for more accurate work and frequently to 10" or oven

20", depending on the theodolite being used for setting out.

4.8 Obstructions To Setting Out

Obstructions on site may prevent normal setting out in a variety of ways. Most problems of this

kind can easily be overcome if setting out is by means of coordinates, but two common problems

which often arise are the following:

a) Where the intersection point is inaccessible.

b) Where there are obstructions to sighting the deflection angle to every point on the curve from

the initial tangent point.

4.8.1 The Inaccessible Intersection Point.

It may not be possible to measure θ at the intersection point if it is inaccessible e.g. on mountain

roads. By setting out a line such as XY in figure 4.18, and by measuring its length and the angles

α and β the triangle XPY can be solved for the lengths PX and PY and θ can be deduced. The

tangent points can then be located from X and Y and the curve set out in the usual way.

Figure 4.18 : An inaccessible intersection-point


4.8.2 Obstructions To Sighting The Deflection Angles.

Where obstructions prevent the sighting to every peg on the curve, the following procedure must

be adopted as illustrated in Figure 4.19.

64

(a) Pegs 2, 3 and 4 have been placed turning off deflection angle δ each time. Peg 5 cannot

be placed from peg 1 owing to an obstruction.

i) Triangle 1X4 is isosceles, therefore angle X14 = angle 14 X = 3.

ii) The angle between the chord 1-4 produced and the tangent X4 produced is also 3

and the angle is required to be turned off this tangent to locate peg 5.

iii) An angle of 180° + 4 is required to be turned off line 4-1 in order to locate the

direction 4-5.

b) Set up the theodolite at peg 4, sight peg 1 at a zero setting and turn of at an angle equal

to 180° + (5 - 1) = 180° + 4; i.e. must be multiplied by the number of standard chord

lengths between the two points being sighted to.

NOTE:

i) The longest possible backsight should always be used to orient the theodolite.

ii) If a sub-chord exists between the instrument and the point sighted to, the

angle to be turned off will be 180° + ( x Number of standard chords between the

pegs sighted) + ', the deflection angle of the sub-chord.

iii) The rule for obtaining the angle applies between any two pegs on any one circular curve.

Figure 4.19 : Obstructed Deflection Angles


4.9 COMPUTING AND SETTING OUT A TRANSITION CURVE

4.9.1 Introduction

It is normal to set out a transition curve using deflection angles from the tangent point or by

deflection distances for short transitions, in the same way as for circular curves. The deflection

angles for transitions are not equal as are those for circular curves. The chord length used is

often half that used on the circular curve. In practice the setting-out data are usually extracted

from tables which relate to various design speeds. The only calculations needed are for the

tangent lengths using the observed deflection angle .

65

Figure 4.20 : Transition Curve Detail


Once the tangent points have been established, the transitions are set out from both tangent

points to T1 and T2, the limits of the circular curve. Then from T1 or T2 the direction of the tangent

to the circular curve is obtained by turning off 2/3' (Figure 4.20) from the chord to the transition

and the circular curve deflection angles are set out as before.

4.9.2 Setting-out Calculations.

If the tabulated data are not available, the length of the transition must first be obtained from the

formula below:

The rate of change of radial acceleration

This forms part of highway design and is dependent on traffic speed, available space and the

radius to be adopted. The setting out surveyor will be provided with the transition length and the

radius or degree of curve. With this information and the observed deflection angle , the following

calculations are needed before setting out the pegs:

(a) Shift. This is calculated from:

L2

S=

24R

The radius R = 5729.58 /D, if only the degree of curve is given.

(b) Tangent lengths. This is calculated from:

This distance will be taped back from the intersection point and the two tangents T0, and T3

pegged.

(c) Deflection angles. The deflection angles for 10-, 15- or 20-m chords are calculated from:

where l is the continuous chainage along the transition. For small angles of deflection the

summation of the chord lengths may be taken to equal the lengths l.

The final deflection angle to locate T1 is 572.958 minutes, and ' is three times this value.

(d) Length of circular curve. This is obtained from R' in the usual way, where:

' = ( — 2') and

2'' = 57.2958 degrees.

RL

v3

2tan)( 2

1L

SRTP o

utesRL

ld min958.572

2

R

L

R

L

66

NOTE: If 2' is greater than , it is not possible for the transitions to be contained within the straights. Longer transitions leading to a curve of smaller radius will have to be used.

Example 4e.

Calculate the setting-out data for a 75m transition curve to connect an 8° circular curve joining two straights with an angle of deflection of 20° 00', using 15m chords.

Solution:

(a) To calculate the radius and shift:

(b) To

calcula

te the

tangen

t

length

s:

(c) To calculate and tabulate the deflection angles and deflection distances:

Chord L L2 x

15 15 225 2'24" 0.010

15 30 900 9'36" 0.084

15 45 2025 0.283

15 60 3600 38'24"

15 75 5625 60'00" 1.309

m

R

LShiftS

m

DRRadius

327.0

20.71624

75

24

20.716

8

58.5729

28.5729,

2

'

2

m

LsRTTangent

84.163

5.3710tan)33.020.716(

2tan)(, 2

1

mRL

landx

6

3

2

2

2

01067 . 0

75 20 . 716

958 . 572

958 . 572

l

l

Minutes

RL

l

67

Table 4.2 Deflection Angle And Deflection Distances.


NOTE : Either the deflection angles or the deflection distances are calculated and used for setting

out. In many cases, where the distances are short they form the more convenient setting out

method.

(d) To check the final deflection angle and calculate ':

Final 8 = 572.958 minutes

= 572.958

= 60'

= 1° 00", which checks the final deflection angle calculated and tabulated in Table 6.2.

'' =

= 3° 00'

2' = 6° 00"

but 2'' = 57.2958 degrees

= 6° 00", which checks the above value.

(e) To calculate the length of the circular curve:

consumed by both transitions, = 2' = 6°.

L curve = 716.20 x (20° - 6°) x (2/360)

= 716.20 X 0.24435

= 175.00 m.

4.9.3 Setting out curves containing a transition.

To illustrate this procedure the following describes the process of setting out the curve, the details

of which are found in the question above:

(a) Setting out the transitions.

The first transition is set out from To. The other tangent point T3 is set out along the second

straight the same distance from P as T0, equal to 163.84 m. The final transition is then set out

between T3 and T2 as before.

(b) Setting out the circular curve.

R

L

20.716

75

R

L

radiansR

L

2

68

Assuming 15m chords are to be used, there will be eleven 15m chords and one 10m sub-chord to

make up the arc of 175 m. The standard deflection angle will be:

set up at T1 for the sub-chord:

Set up at T1 (Figure 4.20), sight T0 and swing through 180° +2/3'+ to sight to the first peg on the

curve, completing the setting out to T2 in the usual way for circular curves. This initial angle is

180° + 2° + 36' = 182° 36' 00".

4.10 PROCEDURE FOR COMPUTING A VERTICAL CURVE

1. Compute the algebraic difference in grades: A = g2-g1.

2. Compute the chainage of the BVC and EVC. If the chainage of the PVI is known, ½ L is

simply subtracted and added to the PVI chainage.

3. Compute the distance from the BVC to the high or low point (if applicable):

and determine the station of the high/low point.

4. Compute the tangent grade-line elevation of the BVC and the EVC.

5. Compute the tangent grade-line elevation for each required station.

6. Compute the midpoint of chord elevation:

7. Compute the tangent offset (d) at the PVI (i.e., distance VM in Figure 4.21):

8. Compute the tangent offset for each individual station (see line ax2 in Figure 4.21):

where x is the distance from the BVC or EVC (whichever is closer) to the required station.

9. Compute the elevation on the curve at each required station by combining the tangent

offsets with the appropriate tangent grade-line elevations—add for sag curves and

subtract for crest curves.

Ax

g1L

2

EVC ofelevation BVC ofElevation

2

chord ofmidpoint and PVI ofelevation in differenced

2

2

2

4

2

tan xL

dor

L

xdgentoffset

'364100

15of

'244100

10of

69

Figure 4.21 Geometric

Properties Of The Parabola

(Source: Surveying With Construction

Application, B.F. Kavanagh)

EXAMPLE 4f

The techniques used in vertical curve computations are illustrated in this example.

Given that L= 300 ft, g1 = -3.2%, g2 = + 1.8%, PVI at 30 + 30, and elevation =465.92, determine

the location of the low point and elevations on the curve at even stations as well as at the low

point.

Solution :

1. A = 1.8 - (-3.2) = 5.0

2. PVI - ½L = BVC

BVC at (30 +- 30) - 150 = 28 + 80.00

PVI + ½L = EVC

EVC (30 + 30) + 150 = 31 + 80.00

EVC - BVC = L

(31 + 80) - (28 + 80) = 300 Check

3. Elevation of PVI = 465.92

150 ft at 3.29r = 4.80 (see Figure 6.20)

Elevation BVC = 470.72

Elevation PVI = 465.92

150 ft at 1.8% = 2.70

Elevation EVC = 468.62

4. Location of low point

5. Tangent grade-line computations are entered in Table 4.3.

Example:

)(00.192

5

3002.3

1

fromtheBVCft

A

Lgx

470.08

0.64 - 470.72

20) (0.032 - 470.72 00 29at Elevation

70

Figure 4.22: Sketch For Example 4


Station Tangent Elevation + Tangent Offset = Curve Elevation

BVC 28 + 80 470.72 (0/150)2 x 1.8.75 =0 470.72

29 + 00 470.08 (20/150)2 x 1.875 = 0.03 470.11

30 + 00 466.88 (120/150)2 x 1.875 = 1.20 468.08

PVI 30 + 00 465.92 (150/150)2 x 1.875 = 1.875 467.80

LOW30 + 72 466.68 (108/150)2 x 1.875 = 0.97 467.65

31 + 00 467.18 (80/150)2 x 1.875 = 0.53 467.71

EVC 31 + 80 468.62 (0/150)2 x 1.875 = 0 468.62

30 + 62 466.50 (118 / 150) 2 x 1.875 = 1.16 467.66

30 + 72 466.68 (108 / 150) 2 x 1.875 = 0.97 467.65

30 + 82 466.86 (98 / 150) 2 x 1.875 = 0.80 467.66

*Where x is distance from BVC or EVC, whichever is closer.

Table 4.3 Parabolic Curve Elevation by Tangent Offset

6. Mid-chord elevation:

7. Tangent offsets are computed by multiplying the distance ratio squared, [x/(L/2)]2, by the

maximum tangent offset (d). See Table 4.3.

8. The computed tangent offsets are added (in this example) to the tangent elevation in

order to determine the curve elevation.

Activity 4b

4.3) Match the formulae used to calculate the following

97.4692

(EVC) 468.62 (BVC) 470.72

71

4.4) State the methods of setting out a circular curve without using a theodolite.

4.5) How is the problem of an inaccessible intersection point overcome?

4.6) The tangent length of a simple curve was 202.12m and the deflection angle for a

30m chord 2 18'. Calculate the radius, the total deflection angle, the length of curve

and the final deflection angle.

Feedback 4b

4.3) The formulae used to calculate the following:

Transition deflection

angle

Shift

Tangent length


distance

Length of the

circular curve

utesRL

ld min958.572

2

R

Ls

24

2

2tan)( 2

1L

SRTP o

RL

lx

6

3

reesR

Ldeg2958.57'2


angle

Shift

2tan)( 2

1L

SRTP o

RL

lx

6

3

72

4.4) a) Setting Out By Offset With Sub-Chords.

b) Setting Out By Offset With Long-Chords.

c) Setting Out By Offsets From Tangent.

4.5) If the intersection point is inaccessible, this problem can be solved by setting out

a line such as XY in figure 1. By measuring its length and the angles α and β the

triangle XPY can be solved for the lengths PX and PY and θ can be deduced.

The tangent points can then be located from X and Y and the curve set out in the

usual way.

Figure 1.

4.6) 2 18' = 138'

138'= 1718.9( 30/R )

R = 373.67

202.12 = R tan /2

202.12 = 373.67 tan /2

= 56 49' 06"

Length of curve = R rad = 373.67 x 0.991667 rad = 370.56m

Using 30m chords, the final sub-chord = 10.56m

final deflection angle = (138' x 10.56)/30

= 48.58'

Tangent length


distance

Length of the

circular curve

utesRL

ld min958.572

2

R

Ls

24

2

reesR

Ldeg2958.57'2

73

Self Assessment

= 0 48' 35"

1) Refer to the figure below, given = 12 51’, R = 400m , PI at 0 + 241.782. Calculate the station

of the BC and EC.

Figure 2

2) What is transition curve? Why is it important in engineering survey?

3) A downgrade of 1 in 20 = 5 in 100= -5%

An upgrade of 1 in 25 = 4 in 100 = +4%

Calculate the algebraic difference of the two gradients.

4) The straight lines ABI and CDI are tangents to a proposed circular curve

of radius 1600m. The length AB and CD are each 1200m. The intersection

point is inaccessible so that it is not possible directly to measure the

deflection angle; but the angles at B and D are measured as ABD = 123

48', BDC = 126 12' and the length BD is 1485m. Calculate the

distances from A and C of the tangent points on their respective straights

and the deflection angles for setting out 30m chords from one of the

tangent points. (Figure 3)

Figure 3

74


1) T = R tan (/2)

= 400 tan 025’ 30” = 45.044m

L = (2R )/360

= 2 x 400 x (12.850/360) = 89.170m

PI at 0+241.782 -T 45.044 BC = 0+196.738 +L 89.710 EC = 0+286.448

2) The transition curve is a curve of constantly changing radius. It is inserted between the

straight circular curves to prevent this sudden lateral shock on passengers in the vehicle.

In engineering survey, the purpose of a transition curves then is to achieve a gradual

change of direction from the straight (radius ∞) to the curve (radius R) and permit the

gradual application of super-elevation to counteract centrifugal force.

3) g2 = -5% and g1= +4%

A = g2 – g1

= -5%-4%

= -9%

4) 1 = 180 -123 48'

= 56 12'

2 = 180 -126 12'

= 53 48'

=1 + 2 = 110

= 180- = 70

Tangents length IT1 and IT2 = R tan /2

= 1600 tan 55 = 2285m

By sine rule in triangle BID:

m

BDBI

2.1275

70sin

'4853sin1485

sin

sin 2

m

BDID

1314

70sin

'1556sin1485

sin

sin 1

75

Thus AI = AB + BI = 1200 + 1275.2 =2475.2m CI = CD + ID = 1200 + 1314 = 2514m

AT1 = AI -IT1 = 2475.2 -2285 = 190.2m CT2 = CI -IT2 = 2514 -2285 = 229m

Deflection angle for 30m chord = 1718.9 x (30/1600) = 32.23'

= 0 32' 14"

76

SETTING OUT

5.1 INTRODUCTION

In surveying, existing features and levels are located on the ground and then

plotted at a reduced scale on plan. Setting-out is the reverse process, where the

position and levels of new works already recorded on a working plan are

transferred to the ground.

The accuracy of measurement required in setting out depends on the type of

building work that is to take place. Where prefabricated frames and components

are to be used, the setting out must be sufficiently precise for such components

to fit within the tolerances laid down. Measurements must therefore be made with

great care. As in surveying, the setting out must be arranged so that the work is

checked. Every peg placed must be proved to be in its correct position within

allowable limits.

5.2 PURPOSE AND FIELD WORK

The controlled process of setting out covers three aspects of positioning new

works.

a) Horizontal Control, in which the true relative positions of points are fixed

on the horizontal plane and marked by pegs in the ground.

b) Vertical Control, in which pegs defining different levels of construction are

suitably placed.

c) Works Control, in which the construction processes are controlled, e.g. the

vertical alignment of buildings during construction and the control of

embankment slopes and excavations.

Most site operatives have little concept of the time, effort and expertise involved

in establishing setting out pegs. For this reason the pegs are frequently treated

with disdain and casually destroyed in the construction process. A typical

example of this is the centre-line pegs for route location which are the first to be

destroyed when earth-moving commences. It is important, therefore, that control

stations and BM should be protected in some way (usually as shown in figure

77

5.1) and site operatives, particularly earthwork personnel, impressed with the

importance of maintaining this protection.

Figure 5.1 : Control Station / BM Set In Concrete (Source : Land Surveying,

Ramsay)

Where the destruction of the pegs is inevitable, then referencing procedures

should be adopted to relocate their positions with the original accuracy of fixation.

Various configurations of reference pegs are used and the one thing that they

have in common is that they must be set well outside the area of construction

and have some form of protection, as in Figure 8.1.

A commonly-used method of referencing is from four pegs ( A, B, C & D )

established such that two strings stretched between them intersect to locate the

required position (figure 5.2). Distances AB, BC, CD, AD, AC and BD should all

be measured as check on the possible movement of the reference pegs, whilst

distances from the reference pegs to the setting-out peg will afford a check on

positioning. Intersecting lines of sight from theodolites at say A and B may be

used where ground conditions make string lining difficult.

Where ground conditions preclude taping, the setting-out peg may be referenced

by trisection from three reference pegs. The peg should be established to form

well-conditioned triangles of intersection ( Figure 5.3 ), the angles being

measured and set out on both faces of a 1” theodolite.

All information relating to the referencing of a point should be recorded on a

diagram of the layout involved.

78

Figure 5.2 : Locate The Required Position (Source : Land Surveying, Ramsay)

Figure 5.3 : Triangles Of Intersection (Source : Land Surveying, Ramsay)

5.3 BASIC SETTING-OUT PROCEDURES USING CO-ORDINATES

Plans are generally produced on a plane rectangular co-ordinate system, hence

salient points of the design may also be defined in terms of rectangular co-

ordinates on the same system. For instance, the centre-line of a proposed road

may be defined in terms of co-ordinates at, say, 30m intervals, or alternatively,

only the tangent and intersection points may be so defined. The basic methods of

locating position when using co-ordinates is by either polar co-ordinates, or

intersection.

79

5.3.1 By Polar Co-ordinates

In Figure 5.4, A, B and C are control station whose co-ordinates are known. It is

required to locate point IP whose design co-ordinates are also known. The

computation involved is as follows:

i. From the co-ordinates compute the bearing BA (this bearing may already

be known from the initial control survey computations).

ii. From the co-ordinates compute the horizontal length and bearing of B-IP.

iii. From the two bearings compute the setting-out angle AB(IP), i.e. β.

iv. Before proceeding into the field, draw a neat sketch of the situation

showing all the setting-out data. Check the data from the plan or by

independent computation.

Figure 5.4 : Control Station By Co-ordinates (Sources : Land Surveying,

Ramsay)

The field work involved is as follows :

i. Set up theodolite at B and backsight to A and note the horizontal circle

reading.

ii. Add the angle β to the circle reading BA to obtain the circle reading B – IP.

Set this reading on the theodolite to establish direction B – IP and

measure out the horizontal distance L.

If this distance is set out by steel tape, careful considerations must be given to all

the error sources such as standardization, slope, tension and possibly

temperature if the setting-out tolerances are very small. It should also be

carefully noted that the sign of the correction is reversed from that applied when

measuring a distance. For example, if a 30m tape was in fact 30.01m long, when

80

measuring a distance the recorded length would be 30m for a single tape length,

although the actual distance is 30.01m; hence a POSITIVE correction of 10mm is

applied to the recorded measurement. However, if it is required to set out 30m,

the actual distance set out would be 30.01m; thus this length would need to be

reduced by 10mm; i.e., a NEGATIVE correction.

The best field technique when using a steel tape is to align pegs carefully at X

and Y each side of the expected position of IP ( Figure 5.5 ). Now, carefully,

measure the distance BX and subtract it from the known distance to obtain

distance X – IP, which will be very small, possibly less than one metre. Stretch a

fine cord between X and Y and measure X – IP along this direction to fix point IP.

Figure 5.5 : Position Of IP (Source : Land Surveying, Ramsay)

Modern EDM, such as the Aga Geodimeter 122, displays horizontal distance, so

the length B – IP may be ranged direct to a reflector fixed to a setting-out pole.

The use of short range EDM equipment has made this method of setting out very

popular.

5.3.2 By Intersection

This technique, illustrated in Figure 5.6, does not require linear measurements;

hence, adverse ground conditions are immaterial and one does not have to

consider tape corrections.

The computation involved is as follows :

i. From the co-ordinates of A, B and IP compute the bearings AB, A – IP and

B – IP

ii. From the bearing compute the angles α and β

81

The relevant field work, assuming two theodolites are available, is as follows:

i. Set up a theodolite at A, backsight to B and turn off the angle α

ii. Set up a theodolite at B, backsight to A and turn off the angle β

Figure 5.6 : The Intersection (Source : Land Surveying, Ramsay)

The intersection of sight lines A – IP and B – IP locates the position of IP. The

angle δ is measured as a check on the setting out. If only one theodolite is

available, then two pegs locate position IP, as in Figure 5.2.

5.4 TECHNIQUE FOR SETTING OUT A DIRECTION

It can be seen that both the basic techniques of position fixing require the

turning-off of a given angle. To do this efficiently the following approach is

recommended.

In Figure 5.6, consider turning off the angle β equal to 20° 36' 20" using a Watts

No. 1

(20") theodolite (Figure 5.7(a)).

i. With theodolite set at B, backsight to A and read the horizontal circle - say,

02° 55'20".

ii. As the angle β is clockwise of BA the required reading on the theodolite

will be equal to ,02° 55' 20" + 20° 36' 20"), i.e. 23° 31' 40'.

As the minimum main scale division is equal to 20' anything less than this will

appear on the micrometer (Figure 5.7(a)). Thus, set the micrometer to read 11'

40”, then release the upper plate clamp and rotate theodolite until it reads

approximately 23° 20' on the main scale; using the upper plate slow-motion

82

screw, set the main scale to exactly 23° 20'. This process will not alter the

micrometer scale and so the total reading is 23° 31' 40", and the instrument has

been swung through the angle β = 20° 36' 20".

Figure 5.7 : (a) Watts No. 1 – 20” theodolite. (b) Wild T2 – 1” theodolite

(Source : Land Surveying, Ramsay)

If the Wild T.2 (Figure 5.7(b)) had been used, an examination of the main scale

shows its minimum division is equal to 10'. Thus, to set the reading to 23º 31' 40"

one would set only 01' 40" on the micrometer first before rotating the instrument

to read 23° 30' on the main scale.

Therefore, when setting out directions with any make of theodolite, the observer

should examine the reading system to find out its minimum main scale value,

anything less than which is put on the micrometer first.

Basically the micrometer works as shown in Figure 5.8, and, if applied to the

Watts theodolite, is explained as follows: .

Assuming the observer's line of sight passes at 90° through the parallel plate

glass, the reading is 23° 20' + S. The parallel plate is rotated using the

micrometer screw until an exact reading (23° 20') is obtained on the main scale,

as a result of the line of sight being refracted towards the normal and emerging

on a parallel path. The distance S, through which the viewer's image was

displaced, is recorded on the micrometer scale (11' 40") and is a function of the

rotation of the plate. Thus it can be seen that rotating the micrometer screw in no

way affects the pointing of the theodolite, but back-sets the reading so that

rotation of the theodolite is through the total angle of 20° 36' 20".

83

Figure 5.8 : Micrometer (Source : Land Surveying, Ramsay)

As practically all setting-out work involves the use of the theodolite and or level,

the user should be fully conversant with the various error sources and their

effects, as well as the methods of adjustment.

The use of co-ordinates is now universally applied to the setting out of pipelines,

motorways, general road works, power stations, offshore piling and jetty works,

housing and high-rise buildings, etc. Thus it can be seen that although the project

may vary enormously from site to site the actual setting out is completed using

the basic measurements of angle and distance.

There are many advantages to the use of co-ordinates, the main one being that the engineer can set out any part of the works as an individual item, rather than wait for the overall establishment of a setting-out grid.

5.5 USE OF GRIDS Many structures in civil engineering consist of steel or reinforced concrete

columns supporting floor slabs. As the disposition of these columns is inevitably

that they are at right-angles to each other, the use of a grid, where the grid

intersections define the position of the columns, greatly facilitates setting out. It is

possible to define several grids as follows,

(1) Survey grid: the rectangular co-ordinate system on which the original

topographic survey is carried out and plotted (Figure 5.9).

84

(2) Site grid: defines the position and direction of the main building lines of the

project, as shown in Figure 8.9. The best position for such a grid can be

determined by simply moving a tracing of the site grid over the original plan so

that its best position can be located in relation to the orientation of the major units

designed thereon.

(3) Structural grid: used to locate the position of the structural element0 within the

structure and is physically established usually on the concrete floor slab (Figure

5.9).

In order to set out the site grid, it may be convenient to translate the co-ordinates

of the site grid to those of the survey grid using the well-known transformation

formula.

E = ΔE + E1 cos θ – N1 sin θ

N = ΔN + N1 cos θ + E1 sin θ

where

ΔE, ΔN = difference in easting and northing of the respective grid origins

E1, N1 = the co-ordinates of the point on the site grid

θ = relative rotation of the two grids

E, N = the co-ordinates of the point transformed to the survey grid

Thus, selected points, say X and Y (Figure 5.9) may have their site-grid

coordinate values transformed to that of the survey grid and so set-out by polars

or intersection from the survey control. Now, using XY as a base-line, the site

grid may be set out using theodolite and steel tape, all angles being turned off on

both faces and grid intervals carefully fixed using the steel tape under standard

tension.

When the site grid has been established, each line of the grid should be carefully

referenced with marks fixed clear of the area of work. As an added precaution,

these marks could be further referenced to existing control or permanent, stable,

on-site detail.

85

Figure 5.9 : Plan With Grid Lines (Source : Land Surveying, Ramsay)

Activity 5a

5.1 What is the meaning of setting- out? 5.2 What are the three forms of control provided by setting-out processes?

5.3 What are the two procedures of setting-out by using coordinates?

5.4 What are the forms of using grid procedures?

Feedback 5a

5.1 Setting-out is the reverse process, where the position and levels of new

works already recorded on a working plan are transferred to the ground.

5.2 The three forms of control provided by setting-out processes is

a. Horizontal Control, in which the true relative positions of points are

fixed on the horizontal plane and marked by pegs in the ground.

86

b. Vertical Control, in which pegs defining different levels of

construction are suitably placed.

c. Works Control, in which the construction processes are controlled,

e.g. the vertical alignment of buildings during construction and the

control of embankment slopes and excavations.

5.3 The two procedures of setting-out by using a co-ordinates is

Polar coordinates

Intersection

5.4 The forms of using grid procedures is

Survey grid

Site grid

Structural grid

5.6 SETTING OUT BUILDINGS

For buildings with normal strip foundations the corners of the external walls are

established by pegs located direct from the survey control or by measurement

from the site grid. As these pegs would be disturbed in the initial excavations

their positions are transferred by theodolite on to profile boards set well clear of

the area of disturbance [Figure 5.10). Prior to this, their positions must be

checked by measuring the diagonals as shown in Figure 5.11.

The profile boards must be set horizontal with their top edge at some pre-

determined level such as damp proof course (DPC) or finished floor level (FFL).

Wall widths, foundation widths, etc. can be set out along the board with the aid of

a steel tape and their positions defined by saw-cuts. They are arranged around

the building as shown in Figure 5.11. Strings stretched between the appropriate

marks clearly define the line of construction.

In the case of buildings constructed with steel or concrete columns, a structural

grid must be established to an accuracy of about ±2 to 3 mm, otherwise the

prefabricated beams and steelwork will not fit together without some distortion.

The position of the concrete floor slab may be established in a manner already

described. Thereafter the structural grid is physically established by hi1ty nails or

small steel plates set into the concrete. Due to the accuracy required, a 1"

87

theodolite and standardized steel tape corrected for temperature and tension

should be used.

Once the bases for the steel columns have been established, the axes defining

the centre of each column should be marked on and, using a template oriented to

these axes, the positions of the hooding-down bolts defined (Figure 5.12). A

height mark should be established, using a level, at a set distance (say, 75 mm)

below the underside of the base-plate, and this should be constant throughout

the structure. It is important that the base-plate starts from a horizontal base to

ensure verticality of the column.

Figure 5.10 : Profile Board (Source : Land Surveying, Ramsay)

Figure 5.11: Diagonal Checks (Source : Land Surveying, Ramsay)

88

5.7 CONTROLLING VERTICALITY

5.7.1 Using a plumb bob

In low-rise construction a heavy plumb bob (5 to 10 kg) may be used as shown in

Figure 5.13. If the external wall was perfectly vertical, then when the plumb bob

coincides with the centre of the peg, distance d at the top level would equal the

offset distance of the peg at the base. This concept can be used internally as well

as externally provided that holes and openings are available.

Figure 5.12 : Hooding-down Bolts (Source : Land Surveying, Ramsay)

Figure 5.13 : Heavy Plumb Bob (5 to 10 kg) (Source : Land Surveying, Ramsay)

89

5.7.2 Using a theodolite

If two centre-lines at right-angles to each other are carried vertically up a

structure as it is being built, accurate measurement can be taken off these lines

and the structure as a whole will remain vertical. Where site conditions permit,

the stations defining the "base figure” (four per line) are placed in concrete well

clear of construction (Figure 5.14(a)). Lines stretched between marks fixed from

the pegs will allow offset measurements to locate the base of the structure. As

the structure rises, the marks can be transferred upon to the walls by theodolite,

as shown in Figure 5.14(b), and lines stretched between them. It is important that

the transfer is carried out on both faces of the instrument.

Where the structure is circular in plan the centre may be established as in Figure

5.14(a) and the radius swung out from a pipe fixed vertically at the centre. As the

structure rises, the central pipe is extended by adding more lengths. Its verticality

is checked by two theodolites (as in Figure 5.14(b)) and its rigidity ensured by

supports fixed to scaffolding. The vertical pipe may be replaced by laser beam or

autoplumb, but the laser would still need to be checked for verticality by

theodolites.

Steel and concrete columns may also be checked for verticality using the

theodolite. By string lining through the columns, positions A-A and B-B may be

established for the theodolite (Figure 5.15); alternatively, appropriate offsets from

the structural grid lines may be used.

Figure 5.14 : (a) Plan (b) Section (Source : Land Surveying, Ramsay)

90

Figure 5.15 : String Lining (Source : Land Surveying, Ramsay)

With the instruments set up at A, the outside face of all the uprights should be

visible. Now cut the outside edge of the upright at ground level with the vertical

hair of the theodolite. Repeat at the top of the column. Now depress the

telescope back to ground level and make a fine mark; the difference between the

mark and the outside edge of the column is the amount by which the column is

out of plumb. Repeat on the opposite face of the theodolite. The whole procedure

is now carried out at B. If the difference exceeds the specified tolerances, the

column will need to be corrected.

5.7.3 Using optical plumbing

For high-rise buildings the instrument most commonly used is an autoplumb

(Figure5.16). This instrument provides a vertical line of sight to an accuracy of ±

1 second of arc (1 mm in 200 m). Any deviation from the vertical can be

quantified and corrected by rotating the instrument through 90º and observing in

all four quadrants; the four marks obtained would give a square, the diagonals of

which would intersect at the correct centre point.

A base figure is established at ground level from which fixing measurements may

be taken. If this figure is carried vertically up the structure as work proceeds, then

identical fixing measurements from the figure at all levels will ensure verticality of

the structure (Figure 5.17).

91

Figure 8.16 : The optical system of the autoplumb (Source : Land Surveying, Ramsay)

A base figure is established at ground level from which fixing measurements may

be taken. If this figure is carried vertically up the structure as work proceeds, then

identical fixing measurements from the figure at all levels will ensure verticality of

the structure (Figure 5.17).

To fix any point of the base figure on an upper floor, a Perspex target is set over

the opening and the centre point fixed as above. Sometimes these targets have a

grid etched on them to facilitate positioning of the marks. The base figure can be

projected as high as the eighth floor, at which stage the finishing trades enter and

the openings are closed. In this case the uppermost figure is carefully

referenced, the openings filled, then the base figure re-established and projected

upwards as before. The shape of the base figure will depend upon the plan

shape of the building. In the case of a long rectangular structure, a simple base

line may suffice, but T shapes and Y shapes are also used.

92

Figure 5.17 : (a) Elevation (b) Plan (Source : Land Surveying, Ramsay)

5.8 CONTROLLING GRADING EXCAVATION This type of setting out generally occurs in drainage schemes where the trench,

bedding material and pipes have to be laid to a specified design gradient.

Manholes (MH) will need to be set out at every change of direction or at least

every 100 m on straight runs. The MH (or inspection chambers) are generally set

out first and the drainage courses set out to connect into them.

The centre peg of the MH is established in the usual way and referenced to four

pegs, as in Figure 5.2. Alternatively, profile boards may be set around the MH

and its dimensions marked on them. If the boards are set out at a known height

above formation level the depth of excavation can be controlled, as in Figure

5.18.

Figure 5.18 : Control Excavation (Source : Land Surveying, Ramsay)

93

5.8.1 Use of sight rails Sight rails (SR) are basically horizontal rails set a specific distance apart and to a

specific level such that a line of sight between them is at the required gradient.

Thus they are used to control trench excavation and pipe gradient without the

need for constant professional supervision.

Figure 5.19 illustrates SR being used in conjunction with a boning rod (or

traveller) to control trench excavation to a design gradient of 1 in 200 (rising).

Pegs A and B are offset a known distance from the centre-line of the trench and

levelled from a nearby TBM.

Assume peg A has a level of 40m and the formation level of the trench at this

point is to be 38 m. It is decided that a reasonable height for the SR above

ground would be 1.5 m. i.e. at a level of 41.5; thus the boning rod must be made

(41.5 — 38) == 3.5 m long, as its cross-head must be on level with the SR when

its toe is at formation level.

Figure 8.19 : Illustrates SR (Source : Land Surveying, Ramsay)

•

Consider now peg B, with a level of 40.8 m at a horizontal distance of 50 m from A. The proposed gradient is 1 in 200, which is 0.25 m in 50 m, thus the formation level at B is 38.25 m. If the boning rod is 3.5m, the SR level at B is

94

(38.25 + 3.5)= 41.75m and is set f41.75 — 40.8) = 0.95 m above peg B. The remaining SRs are established in this way and a line of sight or string stretched between them will establish the trench gradient 3.5 m above the required level. Thus, holding the boning rod vertically in the trench will indicate, relative to the sight rails, whether the trench is too high or too low. Where machine excavation is used, the SR are as in Figure 5.20, and offset to the side of the trench opposite to where the excavated soil is deposited.

Knowing the bedding thickness, the invert pipe level may be calculated and a

second cross-head added to the boning rod to control the pipe laying, as shown

in Figure 5.21. Due to excessive ground slopes, it may be necessary to use

double sight rails with various lengths of boning rods as shown in Figure 5.22.

Note:

1. SR offset far enough to allow the machine to pass.

2. Offset distance marked on pegs which support SR boards. 3. Length of traveller marked on both SR and traveler. 4. All crossheads should be leveled with a spirit leveled. 5. Colouring SR and traveller to match can overcome problems of using

wrong traveler. 6. SR must be offset square to the points to which they refer.

95

Figure 5.20 : Use of offset sight rails (SR) (Source : Land Surveying, Ramsay)

Figure 5.21 : Boning rod to control the pipe laying

(Source : Land Surveying, Ramsay)

96

Figure 5.22 : Double sight rails (Source : Land Surveying, Ramsay)

5.9 RESPONSIBILITY ON SITE Responsibility with regard to setting out is defined in Clause 17 of the ICE Conditions of Contract:

The contractor shall be responsible for the true and proper setting-out of the

works, and for the correctness of the position, levels, dimensions, and

alignment of all parts of the works, and for the provision of all necessary

instruments, appliances, and labour in connection therewith. If at any time

during the progress of the works, any error shall appear or arise in the

position, levels, dimensions, or alignment of any part of the works, the

contractor, on being required so to do by the engineer, shall, at his own cost,

rectify such errors to the satisfaction of the engineer, unless such errors are

based on incorrect data supplied in writing by the engineer or the engineer's

representative, in which case the cost of rectifying the same shall be borne by

the employer. The checking of any setting out, or of any line or level by the

engineer or the engineer's representative, shall not, in any way, relieve the

contractor of his responsibility for the correctness thereof, and the contractor

shall carefully protect and preserve all bench-marks- sight rails. pegs, and

other things used in setting-out the works.

The clause specifies three persons involved in the process, namely, the

employer, the engineer and the agent, whose roles are as follows:

The employer who may be a government department, local authority or private

individual, is required to carry out and finance a particular project. To this end, he

commissions an engineer to investigate and design the project, and to take

responsibility for the initial site investigation, surveys, plans, designs, working

97

drawings, and setting-out data. On satisfactory completion of his work he lets the

contract to a contractor whose duty it is to carry out the work. On site the

employer is represented by the engineer or his representative, referred to as the

resident engineer (RE), and the contractor's representative is called the agent.

The engineer has overall responsibility for the project and must protect the

employer's interest without bias to the contractor. The agent is responsible for the

actual construction of the project.

5.10 RESPONSIBILITY OF THE SETTING-OUT ENGINEER

The setting-out engineer should establish such a system of work on site that will

ensure the accurate setting out of the works well in advance of the

commencement of construction. To achieve this, the following factors should be

considered;

(1) A complete and thorough understanding of the plans, working drawings,

setting-out data, tolerances involved and the time scale of operations. Checks on

the setting-out data supplied should be immediately implemented.

(2) A complete and thorough knowledge of the site, plant and relevant personnel.

Communications between all individuals is vitally important. Field checks on the

survey control already established on site, possibly by contract surveyors, should

be carried out at the first opportunity.

(3) A complete and thorough knowledge of the survey instrumentation available

on site, including the effect of instrumental errors on setting-out observations. At

the first opportunity, a base should be established for the calibration of tapes,

EDM equipment, levels and theodolites.

(4) A complete and thorough knowledge of the stores available, to ensure an

adequate and continuing supply of pegs, pins, chalk, string, paint, timber, etc.

(5) Office procedure should be so organized as to ensure easy access to all

necessary information. Plans should be stored flat in plan drawers, and those

amended or superseded should be withdrawn from use and stored elsewhere.

Field and level books should be carefully referenced and properly filed. All

setting-out computations and procedures used should be clearly presented,

referenced and filed.

98

(6) Wherever possible, independent checks of the computation, abstraction, and

extrapolation of setting-out data and of the actual setting-out procedures should

be made.

It can be seen from this brief itinerary of the requirements of a setting-out

engineer that such work should never be allocated, without complete supervision,

to junior, inexperienced members of the site team.

Activity 5b

5.5) What are the three forms of control verticality? 5.6) Describe how some forms of earthwork of setting-out may be controlled by

means of sight rails.

Feedback 5b

5.5) The three forms of control verticality is

Using a Plum Bop

Using a Theodolite

Using optical-plumbing 5.6)

Sight rails (SR) are basically horizontal rails set a specific distance apart and to a

specific level such that a line of sight between them is at the required gradient.

Thus they are used to control trench excavation and pipe gradient without the

need for constant professional supervision.

Figure X illustrates SR being used in conjunction with a boning rod (or traveller)

to control trench excavation to a design gradient of 1 in 200 (rising). Pegs A and

B are offset at a known distance from the centre-line of the trench and levelled

from a nearby TBM.

99

Figure X : Illustrates SR

Assume peg A has a level of 40m and the formation level of the trench at this

point is to be 38 m. It is decided that a reasonable height for the SR above

ground would be 1.5 m. i.e. at a level of 41.5; thus the boning rod must be made

(41.5 — 38) = 3.5 m long, as its cross-head must be on level with the SR when

its toe is at formation level.

•

Consider now peg B, with a level of 40.8 m at a horizontal distance of 50 m from

A. The proposed gradient is 1 in 200, which is 0.25 m in 50 m, thus the formation

level at B is 38.25 m. If the boning rod is 3.5m, the SR level at B is (38.25 + 3.5)=

41.75m and is set f41.75 — 40.8) = 0.95 m above peg B. The remaining SRs are

established in this way and a line of sight or string stretched between them will

establish the trench gradient 3.5 m above the required level. Thus, holding the

boning rod vertically in the trench will indicate, relative to the sigh rails, whether

the trench is too high or too low. Where machine excavation is used the SR are

as in Figure Y, and offset to the side of the trench opposite to where the

excavated soil is deposited.

Knowing the bedding thickness, the invert pipe level may be calculated and a

second cross-head added to the boning rod to control the pipe laying, as shown

in Figure Z. Due to excessive ground slopes it may be necessary to use double

sight rails with various lengths of boning rods as shown in Figure K.

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Note:

1. SR offset far enough to allow the

machine to pass.

2. Offset distance marked on pegs which support SR boards. 3. Length of traveller marked on both SR and traveler. 4. All crossheads should be leveled with a spirit leveled. 5. Colouring SR and traveller to match can overcome problems of using wrong traveler. 6. SR must be offset square to the points to which they refer.

Figure Y : Use of offset sight rails (SR)

Figure Z : Bonning rod to control the pipe laying

Figure Z : Double sight rails

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Feedback of Self Assessment

Self Assessment

1) Describe the employer’s responsibility on the work site. 2) Describe the engineer’s responsibility of setting out.

1. Employer’s responsibility on the work site

The employer who may be a government department, local authority or private

individual is required to carry out and finance a particular project. To this end, he

commissions an engineer to investigate and design the project, and to take

responsibility for the :

initial site investigation

surveys

plans

designs

working drawings

settling out data

On satisfactory completion of his work, he lets the contract to a

contractor whose duly it is to carry out the work.

On site the employer is represented by the engineer or his

representative, referred to as the resident engineer (RE), and the

contractor's representative is called the agent.

The engineer has overall responsibility for the project and must

protect the employer's interest without bias to the contractor.

The agent is responsible for the actual construction of the project.

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2. Engineers responsibility of setting out.

A complete and thorough understanding of the plans, working drawings,

setting-out data, tolerances involved and the lime scale of operations.

Checks on the setting-out data supplied should be immediately

implemented.

A complete and thorough knowledge of the site. plant and relevant

personnel. Communications between all individuals is vitally important. Field

checks on the survey control already established on site, possibly by

contract surveyors, should be carried out at the first opportunity.

A complete and thorough knowledge of the survey instrumentation available

on site, including the effect of instrumental errors on setting-out

observations. At the first opportunity, a base should be established for the

calibration of tapes. EDM equipment, levels and theodolites.

A complete and thorough knowledge of the stores available, to ensure an

adequate and continuing supply of pegs, pins, chalk, string, paint, timber,

etc.

Office procedure should be so organized as to ensure easy access to all

necessary information. Plans should be stored flat in plan drawers, and

those amended or superseded should be withdrawn from use and stored

elsewhere. Field and level books should be carefully referenced and

properly filed. All setting-out computations and procedures used should be

clearly presented, referenced and filed.

Wherever possible, independent checks of the computation, abstraction,

and extrapolation of setting-out data and of the actual setting-out

procedures should be made.

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ELECTRONIC DISTANCE MEASUREMENT (EDM)

6.1.1 Electronic Distance Measurement

Electronic distance measurement (EDM), first introduced in the 1950s by the

Geodimeter Inc. founders, has since those early days undergone continual

refinement. The early instruments, which were capable of very precise

measurements over long distances, were large, heavy, complicated, and

expensive. Rapid advances in related technologies have provided lighter,

simpler, and less expensive instruments these EDM instruments (EDM Is) are

manufactured for use with theodolites and as modular components of Total

Station instruments. Technological advances in electronics continue at a rapid

rate as evidenced by recent market surveys that indicate that most new

electronic instruments have been on the market for less than two years.

Current EDMIs use infrared light, laser light, or microwaves. The microwave

systems use a receiver/transmitter at both ends of the measured line, whereas

infrared and laser systems utilize a transmitter at one end of the measured line

and a reflecting prism at the other end. Some laser EDMIs will measure short

distances (100 - 350 m) without a reflecting prism reflecting the light directly off

the feature (e.g. building wall) being measured, Microwave instruments are often

used in hydrographic surveys and have a usual upper measuring range of 50 km.

Although microwave systems can be used in poorer weather conditions (fog,

rain, etc.) than can infrared and laser systems, the uncertainties caused by

varying humidity conditions over the length of the measured line may result in

lower accuracy expectations. Hydrographic measuring and positioning

techniques have, in a few short years, been largely supplanted by Global

Positioning System (GPS) techniques.

Infrared and laser EDMIs come in long range (10 - 20 km), medium range (3 - 10

km) and short range (0.5 to 3 km). EDMIs can be mounted on the standards or

the telescope of most theodolites; additionally, they can be mounted directly in a

tribrach. When used with an electronic theodolite, the combined instruments can

provide both the horizontal and the vertical position of one point relative to

another. The slope distance provided by an add-on EDMI can be reduced to its

horizontal and vertical equivalents by utilizing the slope angle provided by the

theodolite. In Total Station instruments, this reduction is accomplished

automatically.

104

6.1.2 Electronic Angle Measurement

The electronic digital theodolite, first introduced in the late 1960s (Carl Zeiss

Inc.), set the stage for modern field data collection and processing. When the

electronic theodolite is used with a built-in EDMI, (e.g., Zeiss EIta, Figure 5.1) or

an add-on and interfaced EDMI (e.g.. Wild T-1000, Figure 6.2), the surveyor has

a very powerful instrument. Add that instrument to an on-board microprocessor

that automatically monitors the instrument's operating status and manages built-

in surveying programs and a data collector (built-in or interfaced) that stores and

processes measurements and attribute data, and you have what is known as a

Total Station.

FIGURE 6.1 Zeiss Total Stations. The EIca 45 and 55 have on-board data

storage (1.900 data lines), whereas the EIta 50 requires an

interfaced data collector. On board programs include coordinates,

free stationing, polar points, heights of objects, connecting distances

(between remote points), and setting-out angle accuracy from 3 to 5

seconds and EDM distances to 1.500 m (single prism). (Source :

Courtesy of Carl Zeiss Inc-.Thornwood. N.Y. in the Ramsay)

FIGURE 6.2 Wild T-1000 Electronic

Theodolite, shown with Dl 1000 Distomat EDM and the

GRE 3 data collector. (Source : Courtesy of Leica Co.

Inc..Toront in the Ramsay)

105

6.2 Principles of Electronic Distance Measurement (EDM)

Figure 6.3 shows a wave of wavelength λ. The wave is travelling along the

x axis with a velocity of 299, 792.5 ± 0.4 km/s (in vacuum). The frequency of the

wave is, the time taken for one complete wavelength.

FIGURE 6.3 : Light Wave (Source : Courtesy of Leica Co. Inc..Toronto in the

Ramsay)

λ = c / ƒ

where λ = wavelength in meters

c = velocity in km/s

ƒ = frequency in hertz (one cycle per second)

Figure 6.4 shows the modulated electromagnetic wave leaving the EDMI and

being reflected (light waves) or retransmitted (microwaves) back to the EDMI. It

can be seen that the double distance (2L) is equal to a whole number of

wavelengths (n λ). plus the partial wavelength (Φ) occurring at the EDMI.

L = (n λ + Φ) / 2 meters

The partial wavelength (Φ) is determined in the instrument by noting the phase

delay required to precisely match up the transmitted and reflected or

retransmitted waves. The instrument (e.g.. Wild Distomat) can count the number

of full wavelengths (n λ). or, instead, the instrument can send out a series (three

or four) of modulated waves at different frequencies. (The frequency is typically

reduced each time by a factor of 10, and of course, the wavelength is increased

106

each time also by a factor of 10.) By substituting the resulting values of λ and Φ

into Equation (6 - 2), the value of n can be found.

FIGURE 6.4 :

Principles of EDM

measurement

(Source : Courtesy of

Kern Ins. –Leica in

the Ramsay)

S Station

Z Target

E Reference plane within the distance

meter for phase comparison between

transmitted and received wave

R Reference plane for the reflection of

the wave transmitted by the distance

meter

a Addition constam

e Distance meter component of

addition constant

r Reflector component of addition

constant

λ Modulation wave length

Φ Fraction to be measured of a whole

wave length of modulation (Δ λ)

The addition constant a applies to

a measuring equipment consisting of

distance meter and reflector. The

components e and r are only auxilliary

quantities.

107

The instruments are designed to carry out this procedure in a matter of seconds

and then to display the value of L in digital form. The velocity of light (including

infrared) through the atmosphere can be affected by (1) temperature, (2)

atmospheric pressure, and (3) water vapor content. In practice, the corrections

for temperature and pressure can be performed manually by consulting

nomographs similar to that shown in Figure 6.5, or the corrections can be

performed automatically on some EDMIs by the on-board processor/calculator

after the values for temperature and pressure have been entered.

For short distances using lightwave EDMIs, atmospheric corrections have a

relatively small significance. For long distances using lightwave instruments and

especially microwave instruments, atmospheric corrections can become quite

important. The following chart shows the comparative effects of the atmosphere

on both lightwaves and microwaves.

ERROR parts per million

Parameter Error Light Wave Microwave

t, temperature +1º - 1 - 1.25

p, pressure +1 mm Hg + 0.4 + 0.4

e, partial water 1 mm Hg - 0.05 + 7 at 20ºC

vapor pressure + 17 at 45ºC

At this point, it is also worth noting that several studies of general EDM use show

that more than 90 percent of all distance determinations involve distances of

1000 m or less and that more than 95 percent of all layout measurements

involve distances of 400 m or less. The values in the preceding chart would

seem to indicate that, for the type of measurements normally encountered in the

construction and civil field, instrumental errors and centering errors hold much

more significance than do the atmosphere-related errors.

FIGURE 6.5 : Atmospheric correction

graph.(Source : Courtesy of

Sokkia Co. Ltd in the Ramsay)

108

6.3 EDMI CHARACTERISTICS

Following are the characteristics of recent models of add-on EDMIs. Generally

the more expensive instruments have longer distance ranges and higher

precision.

Distance range

800 m to 1 km (single prism with average atmospheric conditions) Short-range

EDMIs can be extended to 1300 m using 3 prisms, Long-range EDMIs can be

extended to 15 km using 11 prisms(Leica Co.)

Accuracy range

±(15 mm + 5ppm) for short-range EDMIs ±(3mm + 1 ppm) for long-range EDMIs

Measuring time

1.5 seconds for short-range EDMIs to 3.5 seconds for long-range EDMIs Both

accuracy and time are considerably reduced for tracking mode measurements.

Slope reduction

Manual or automatic on some models Average of repeated measurements:

available on some models Battery capability is 1400 to 4200 measurements,

depending on the size of the battery and the temperature

Temperature range

-20°C to +50°C. Nonprism measurements: available on some models with

distances from 100 to 350 m (3 to 5 km with prisms)

6.4 PRISMS

Prisms are used with electro-optical EDMIs (light, laser, and infrared) to reflect

the transmitted signals (see Figure 6.6). A single reflector is a cube corner prism

that has the characteristics of reflecting light rays back precisely in the same

direction as they are received. This retro-direct capability means that the prism

can be somewhat misaligned with respect to the EDMI and still be effective. A

cube corner prism is formed by cutting the corners off a solid glass cube. The

quality of the prism is determined by the flatness of the surfaces and the

perpendicularity of the 90° surfaces.

109

Prisms can be tribrach-mounted on a tripod, centred by optical plummet, or

attached to a prism pole held vertical on a point with the aid of a bull's-eye level.

However, prisms must be tribrach-mounted if a higher level of accuracy is

required.

In control surveys, tribrach-mounted prisms can be detached from their tribrachs

and then interchanged with a theodolite (and EDMI) similarly mounted at the

other end of the line being measured. This interchangeability of prism and

theodolite (also targets) speeds up the work, as the tribrach mounted on the

tripod is centred and levelled only one lime. Equipment that can be interchanged

and mounted on tribrachs already set up is known as forced-centring equipment.

Prisms mounted on adjustable-length prism poles are very portable and as such,

are particularly suited for stakeout surveys. Figure 6.7 shows the prism pole

being steadied with the aid of an additional target pole. The height of the prism is

normally set to equal the height of the instrument. It is particularly important that

prisms mounted on poles or tribrachs be permitted to tilt up/down so that they

can be perpendicular to infrared signals that are being sent from much higher or

lower positions.

FIGURE 6.6 : Various target and

reflector systems in tribach mounts.

F

FIGURE 6.7 : Steadying the EDM reflector with

the aid of a second target pole

110

6.5 EDMI ACCURACIES

EDMI accuracies are stated in terms of a constant instrumental error and a

measuring error proportional to the distance being measured.

Typically accuracy is claimed as ±[5 mm + 5 parts per million (ppm)] or ±(0.02 ft

+ 5 ppm). The ±5 mm (0.02 ft) is the instrument error that is independent of the

length of the measurement, whereas the 5 ppm (5 mm/km) denotes the

distance-related error.

Most instruments now on the market have claimed accuracies in the range of

±(3mm 4- 1 ppm) to ±(10 mm + 10 ppm). The proportional part error (ppm) is

insignificant for most work, and the constant part of the error assumes less

significance as the distances being measured lengthen. At 100 m, an error of ±5

mm represents 1/20,000 accuracy, whereas at 1,000 m the same instrumental

error represents 1/200,000 accuracy.

When one is dealing with accuracy, it should be noted that both the EDMI and

the prism reflectors must be corrected for off-centre characteristics. The

measurement being recorded goes from the electrical centre of the EDMI to the

back of the prism (allowing for refraction through glass) and then back to the

electrical centre of the EDMI. The difference between the electrical centre of the

EDMI and the plumb line through the tribrach centre is compensated for by the

EDMI manufacturer at the factory. The prism constant (30 to 40 mm) is

eliminated either by the EDMI manufacturer at the factory or in the field.

The EDMIs prism constant value can be field-checked in the following manner: A

long line (>1 Km) is laid out with end stations and an intermediate station (see

Figure 6.8). The overall distance AC is measured, along with partial lengths AB

and BC. The constant value will be present in all measurements; therefore,

AC — AB — BC = instrument/prism constant (6-3)

Alternatively, the constant can be determined by measuring a known baseline if

one can be conveniently accessed.

111

6.6 EDMI OPERATION

Figures 6.8 to 6.10 show a variety of first-generation short- to medium-range

EDMIs, The operation of all EDMIs involves the following basic steps: (1) set up

(2) aim (3) measure (4) record.

6.6.1 Set Up

Tribrach-mounted EDMIs are simply inserted into the tribrach (forced centring)

after the tribrach has been set over the point by means of the optical plummet.

Telescope or theodolite yoke-mounted EDMIs are simply attached to the

theodolite either before or after the theodolite has been set over the point.

Prisms are set over the remote station point either by inserting the prism into an

already setup tribrach (forced centring) or by holding the prism vertically over the

point on a prism pole. The EDMI is turned on and a quick check is made to

ensure that it is in good working order—for example, battery, display, and the

like. The height of the instrument (telescope axis) and the height of the prism

(centre) are measured and recorded; the prism is usually set to the height of the

theodolite when it is mounted on an adjustable prism pole.

FIGURE 6.8 : Method

of determining the

instrument-reflector

constant

FIGURE 6.9 Pentax PM 81 EDM mounted on a 6-second Pentax

theodolite and also shown as tribrach mounted. EDM

has a triple-prism

range of 2 km (6.600 ft) with SE =

+/—(5 mm + 5 ppm). (Source : Courtesy of

Pentax Corp.,Colo in the Ramsay.)

6.6.2 Aim

The EDMI is aimed at the prism by using either the built-in sighting devices on

the EDMI or the theodolite telescope. Telescope or yoke-mount EDMIs will have

the optical line of sight a bit lower than the electronic signal. Some electronic

A B C

112

tacheometer instruments (ETIs) have a sighting telescope mounted on top of the

instrument. In those cases, the optical line of sight will be a bit higher than the

electronic signal.

Most instrument manufacturers provide prism/target assemblies, which permit

fast optical sightings for both optical and electronic alignment (see Figure 6.6).

That is, when the crosshair is on target, the electronic signal will be maximized at

the centre of the prism.

The surveyor can (if necessary) set the electronic signal precisely on the prism

centre by adjusting the appropriate horizontal and vertical slow-motion screws

until a maximum signal intensity is indicated on the display (this display is not

available on all EDMIs). Some older EDMIs have an attenuator that must be

adjusted for varying distances the signal strength is reduced for short distances

so that the receiving electronics are not overloaded. Newer EDMIs have

automatic signal attenuation.

6.6.3 Measure The slope distance measurement is accomplished by simply pressing the

"measure" button and waiting a few seconds for the result to appear in the

display. The displays are either LCD (most) or LED. The measurement is shown

to two decimals of a foot or three decimals of a meter: a foot/meter switch readily

switches from one system to the other. If no measurement appears in the

display, the surveyor should check on the switch position ,battery status,

attenuation, and crosshair location (sometimes the stadia hair is mistakenly

centred).

EDMIs with built-in calculators or microprocessors can now be used to compute

horizontal and vertical distances, coordinates, atmospheric, curvature, and prism

constant corrections. The required input data (vertical angle, ppm, prism

constant, etc.) are entered via the keyboard.

Most EDMIs have a tracking mode (very useful in layout surveys, which permits

continuous distance updates as the prism is moved closer to its final layout

position. Handheld radios are useful for all EDM work, as the long distances put

a halt to normal voice communications. In layout work, clear communications are

essential if the points are to be properly located. All microwave EDMIs permit

voice communication—which is carried right on the measuring signal.

113

Figure 6.10 shows a remote device (Kem RD 10). which is attached to the prism.

The display on the EDMl is transmitted to the RD 10 so that the surveyor holding

the prism is immediately aware of the results. In tracking mode, the RD 10

display will show the remaining left/right and near/far (+/-) layout distances so

that the surveyor holding the prism can quickly proceed to the desired layout

point—even on high-noise construction sites.

FIGURE 5.10 Kern RD 10 remote

EDM display shown attached to

EPM reflecting prism. Slope.

horizontal, and vertical distances

(from the EDM to the prism) are

displayed on the RD 10, Maximum

range is 1,300 feet (400 m).

(Source : Courtesy of Pentax

Corp.,Colo. In the Ramsay)

6.6.4 Record

The measured data can be recorded conventionally in HOLD note format, or they

can be manually entered into an electronic data collector. The distance data must

be accompanied by all relevant atmospheric and instrumental correction factors.

Total Station instruments, which have automatic data acquisition capabilities, are

discussed in Section 6.8.

6.7 GEOMETRY OF ELECTRONIC DISTANCE MEASUREMENT

Figure 6.11 illustrates the use of EDM when the optical target and the reflecting

prism are at the same height (see Figure 6.6—single prism assembly). The

slope distance (S) is measured by the EDMI, and the slope angle (a) is

measured by the accompanying theodolite. The heights of the EDMI and

theodolite (hi) are measured with a steel tape or by a graduated tripod centring

rod; the height of the reflector/target is measured in a similar fashion. As noted

earlier, adjustable-length prism poles permit the surveyor to set the height of the

114

prism (HR) equal the height of the instrument (hi), thus simplifying the

computations. From Figure 6.11, if the elevation of station A is known and the

elevation of station B is required:

Elev. STA. B = elev. STA. A + hi ± V - HR (6-4)

FIGURE 6.10 : Geometry of an EDM calculation (Source : Land Surveying,

Ramsay)

When the EDMI is mounted on the theodolite and the target is located beneath

the prism pole, the geometric relationship can be as shown in Figure 6.12. The

additional problem encountered in the situation depicted in Figure 6.12 is the

computation of the correction to the vertical angle (Δα) that occurs when Δhi and

ΔHR are different. The precise size of the vertical angle is important, as it is used

in conjunction with the measured slope distance to compute the horizontal and

vertical distances.

In Figure 6.12 the difference between ΔHR and Δhi is X (i.e., ΔHR - Δhi = X). The

small triangle formed by extending 5' [see Figure 6.12(b)] has the hypotenuse

equal to X and an angle of α. This permits computation of the side X cos α, which

can be used together with S to determine Δα :

( X cos α ) / S = sin Δα

115

Activity 6a

6.1) An EDM slope distance AB is determined to be 561.276 m. The EDMI is 1.820 m above its station (A), and the prism is 1.986 m above its station (5). The EDMI is mounted on a theodolite whose optical centre is 1.720 m above the station. The theodolite was used to measure the vertical angle (+60 2’ 38") to a target on the prism pole; the target is 1.810 m above station B. Compute both the horizontal distance AB and the elevation of station B, given that the elevation of station A = 186.275 m.

116

Feedback 6a

6.1) Solution

The given data are shown in Figure 5-15(a) and the resultant figure is shown.

The X value introduced in Figure 5-14(b) is, in this case, determined as follows :

X = (1.986- 1.810)-(1.820- 1.720)

= 0.176 - 0.100

= 0.076 m

If H had been computed by using the field vertical angle of 6°21'38", the result

would have been 557.82m not a significant difference in this example.

Elevation B = elev. A + 1.820 + 561.276 sin 60 22’ 06" - 1.986

= 186.275 + 1.820 + 62.257 - 1.986

= 248.336 m

If V had been computed by using 6°21'38", the result would have been 62-181 m

instead of 62.257m a more significant discrepancy.

117

6.8 TOTAL STATION

When these instruments are combined with interfaced EDMIs and electronic data

collectors, they become electronic tacheometer instruments (ETIs), also known

as Total Stations. Figure 6.11 to 6.15 illustrate some additional Total Stations

now in use.

These Total Stations can read and record horizontal and vertical angles together

with slope distances. The microprocessors in the Total Stations can perform a

variety of mathematical operations: for example, averaging multiple angle

measurements: averaging multiple distance measurements; determining X, Y, Z

coordinates, remote object elevations (i.e., heights of sighted features), and

distances between remote points; and making atmospheric and instrumental

corrections. The data collector can be a handheld device connected by cable to

the tacheometer (see Figure 6.11 and 6.15), but many instruments come with the

data collector built into the instrument. Figure 6.11 shows a Sokkia Total Station

Set 3, a series of instruments that have angle accuracies from 0.5 to 5 seconds,

distance ranges (one prism) from 1600 m to 2400 m. dual axis compensation, a

wide variety of built-in programs, and a rapid battery charger, which can charge

the battery in 70 minutes. Data are stored on-board in internal memory (about

1300 points) and/or on memory cards (about 2000 points per card). The data can

be directly transferred to the computer from the Total Station via an RS-232

cable, or the data can be transferred from the data storage cards first to a card

reader-writer and from there to the computer.

FIGURE 6.11 Sokkia Total Station Set 3 with cable-connected SDR2 electronic field book- Also shown is a two-way radio (2-mile range) with push-to-talk headset.

Many data collectors are really handheld computers, very sophisticated and very

expensive in excess of $2500. If the Total Station is being used alone, the

capability of performing all survey computations including closures and

adjustments is highly desirable. However, if the Total Station is being used as a

part of a system (field data collection/data processing/digital plotting), then the

computational capacity of the data collectors becomes less important. If the Total

118

Station is being used as part of a system, the data collector then can be

designed to collect only the basic information that is, slope distance, horizontal

angle, vertical angle, or coordinates and attribute data, such as point number,

point type, and operation code. Computations and adjustments are then

performed by one of the many coordinate geometry programs now available for

surveyors and engineers.

Most early models and some current models use the absolute method for reading

angles. These instruments are essentially optical coincidence instruments with

photo-electronic sensors to scan and read the circles, which are divided into

preassigned values from 0 to 360 degrees (or 0 to 400 grad or gon).

Some later models employ an incremental method of angle measurement. These

instruments have a circle divided into many graduations, with both sides of the

circle being scanned simultaneously; a portion of the circle is slightly magnified

and superimposed on the opposite side of the circle. As a result, a pattern is

developed that can be analyzed (with the aid of photodiodes) to read the circles.

Both systems enable the surveyor to conveniently assign zero degrees (or any

other value) to an instrument sighting after the instrument has been sighted in.

Most Total Stations have coaxial electronic and optical systems, which permit

one sighting for both electronic and optical orientation. Other Total Stations have

the telescope mounted a bit below or above the CDMI. These instruments

employ a specific target/prism assembly similar to that shown in Figure 6.6 (left

side). The assembly is designed so that when the crosshairs are centred on the

target, the EDMI measuring beam is exactly on the prism.

The Total Station has an on-board microprocessor that monitors the instrument

status (e.g., level and plumb orientation, battery status, return signal strength)

and makes corrections to measured data for the first of these conditions, when

warranted. In addition, the microprocessor controls the acquisition of angles and

distances and then computes horizontal distances, vertical distances,

coordinates, and the like.

Many Total Stations arc designed so that the data stored in the data collector can

be automatically downloaded to a computer via an RS 232 interface. The

download program is usually supplied by the manufacturer and a second

program is required to translate the raw data into a format that is compatible with

119

the surveyor's coordinate geometry (i.e.. processing) programs. The system

computer could be a mainframe, a mini, or a desktop, although lower costs and

increased capabilities have recently made the desktop computer the choice of

many surveyors and engineers.

Also, most Total Stations enable the surveyor to capture the slope distance and

the horizontal and vertical angles to a point by simply pressing one button. The

point number and point description for that point can then be recorded. In

addition, a wise surveyor will prepare a sketch showing the overall detail and the

individual point locations. This sketch will help keep track of the completeness of

the work and will be invaluable at a later date when the plot file is prepared.

Total Stations and/or their attached data collectors have been programmed to

perform a wide variety of surveying functions. Some programs require that the

proposed instrument station's coordinates and elevation as well as the

coordinates and elevations for proposed reference stations, be uploaded into the

Total Station prior to the field work.

After setup, the instrument station must be identified as such, and the hi and

prism heights must be measured and entered. Typical Total Station programs

include:

northing, easting, and elevation — determination

missing line measurement — This program enables the surveyor to

determine the horizontal and slope distances between any two sighted

points as well as the directions of the lines joining the sighted points.

resection — this technique permits the surveyor to set up the Total Station

at any convenient position and then determine the coordinates and

elevation of that position by sighting previously coordinated reference

stations. When sighting two points of known position, it is necessary to

measure both the distances and angles between the reference points;

when sighting several points (three or more) of known position, it is only

necessary to measure the angles between the points. It is important to

stress that most surveyors take more readings than are minimally

necessary to obtain a solution. These redundant measurements give the

surveyor increased precision and a check on the accuracy of the results.

120

azimuth — the azimuth of the line joining a sighted point from the

instrument station is readily displayed.

remote object elevation — the surveyor can determine the heights of

inaccessible points (e.g., electricity conductors, bridge components, etc.)

by simply sighting the pole-mounted prism as it is being held directly under

the object. When the object itself is sighted, the object height can be

promptly displayed (the prism height must first be entered into the Total

Station; it is often set at the value of the instrument hi).

offset measurements (a) distance offsets — When an object is hidden

from the Total Station, a measurement can be taken to the prism held out

in view of the Total Station and then the offset distance is measured. The

angle (usually 90°) to the hidden object along with the measured distance

is entered into the Total Station, enabling it to compute the position of the

hidden object, (b) angle offsets—the prism is held to the left or right of the

object being located (e.g., a concrete column). The prism is centred and

then an angle is measured to the predetermined centre of the object. The

program will compute the coordinates of the centre of the object (the

concrete column, in this case).

layout or setting-out positions — After the coordinates and elevations of

the layout points have been uploaded into the Total Station, the

layout/setting-out software will enable the Total Station to display the

left/right, forward/back, and up/down movements needed to place the

prism in each of the desired positions. This capability is a great aid in

property and construction layouts.

building face pickup — This program permits the surveyor to define the

vertical face of a building, including all cut-outs (doors and windows) by

simply turning angles to each feature.

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Activity 6b

6.2 What is a Total Station?

6.3 Describe typical Total Station programs.

Feedback 6b

6.2 Total Station

A combination of interfaced EDMIs and electronic data collectors, they become electronic tacheometer instruments (ETIs).

These Total Stations can read and record horizontal and vertical

angles together with slope distances.

The microprocessors in the Total Stations can perform a variety of

mathematical operations: for example, averaging multiple angle measurements: averaging multiple distance measurements; determining X, Y, Z coordinates, remote object elevations (i.e., heights of sighted features), and distances between remote points; and making atmospheric and instrumental corrections.

The data collector can be a handheld device connected by cable to

the tacheometer but many instruments come with the data collector built into the instrument

6.3 Typical Total Station programs include :

northing, casting, and elevation — determination

missing line measurement — This program enables the surveyor to

determine the horizontal and slope distances between any two

sighted points as well as the directions of the lines joining the sighted

points.

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resection — This technique permits the surveyor to set up the Tof'1

Station at any convenient position and then determine the

coordinates and elevation of that position by sighting previously

coordinated reference stations. When sighting two points of known

position, it is necessary to measure both the distances and angle

between the reference points; when sighting several points (three or

mere) of known position, it is only necessary to measure the angles

between the points. It is important to stress that most surveyors take

more readings than are minimally necessary to obtain a solution.

These redundant measurements give the surveyor increased

precision and a check on the accuracy of the results.

azimuth — the azimuth of the line joining a sighted point from the

instrument station is readily displayed.

remote object elevation — the surveyor can determine the heights of

inaccessible points (e.g., electricity conductors, bridge components,

etc.) by simply sighting the pole-mounted prism as it is being held

directly under the object. When the object itself is sighted, the object

height can be promptly displayed (the prism height must first be

entered into the Total Station; it is often set at the value of the

instrument hi).

offset measurements (a) distance offsets — When an object is

hidden from the Total Station, a measurement can be taken to the

prism held out in view of the Total Station and then the offset

distance is measured. The angle (usually 90°) to the hidden object

along with the measured distance are entered into the Total Station,

enabling it to compute the position of the hidden object, (b) angle

offsets—the prism is held to the left ur right of me object being

located (e.g., a concrete column). The prism is centered and then an

angle is measured to the predetermined center of the object. The

program will compute the coordinates of the center of the object (the

concrete column, in this case).

layout or setting-out positions — After the coordinates and elevations

of the layout points have been up-loaded into the Total Station, the

layout/setting-out software will enable the Total Station to display the

left/right, forward/back, and up/down movements needed to place the

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prism in each of the desired positions. This capability is a great aid in

property and construction layouts.

building face pickup — This program permits the surveyor to define

the vertical face of a building, including all cut-outs (doors and

windows) by simply turning angles to each feature.

Self Assessment

1. Below are the definitions used in this unit. Fill in the blanks with the

appropriate term.

Distance range

800 m to 1 km (single prism with average

atmospheric conditions) Short-range

EDMIs can be extended to 1300 m using

3 prisms Long-range EDM/s can be

extended to 15 km using 11 prisms(Leica

Co.)

Accuracy range

±(15 mm + 5ppm) for short-range EDMIs

±(3mm + 1 ppm) for long-range EDMIs

Measuring time

1.5 seconds for short-range EDMIs to

3.5 seconds for long-range EDMIs Both

accuracy and time are considerably

reduced for tracking mode

measurements

Slope reduction

Manual or automatic on some models

Average of repeated measurements:

available on some models Battery

capability: 1400 to 4200

measurements, depending on the size

of the battery and the temperature

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Temperature range

-20°C to +50°C Nonprism

measurements: available on some

models; distances from 100 to 350 m

(3 to 5 km with prisms)

Distance range

800 m to 1 km (single prism with average

atmospheric conditions) Short-range

EDMIs can be extended to 1300 m using

3 prisms Long-range EDM/s can be

extended to 15 km using 11 prisms(Leica

Co.)

Accuracy range

±(15 mm + 5ppm) for short-range

EDMIs ±(3mm + 1 ppm) for long-

range EDMIs

Measuring time

1.5 seconds for short-range EDMIs to

3.5 seconds for long-range EDMIs Both

accuracy and time are considerably

reduced for tracking mode

measurements

Slope reduction

manual or automatic on some models

Average of repeated measurements:

available on some models Battery capability:

1400 to 4200 measurements, depending on

the size of the battery and the temperature

Temperature range

-20°C to +50°C Nonprism

measurements: available on some

models; distances from 100 to 350 m

(3 to 5 km with prisms)

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C2005 Ukur Kejuruteraan 2

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