C2005 Ukur Kejuruteraan 2 283l5b

1

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

2 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.

3

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

4 BC CD DA BD AC -

45.0 m 60.0 m 78.0 m 93.3 m 76.0 m

AB and BC are straight boundaries. Offsets to the irregular boundaries are as follows : Chainage AD

Chainage CD

Offset A

B

0.0 16.0 33.0 40.0 49.0 61.0 68.0 78.0 89.0 93.0

Offset 0.0 6.0 7.0 0.0 7.0 7.0 0.0 11.0 5.0 9.0

C

D

0.0 10.0 20.0 30.0 40.0 50.0 60.0

0.0 4.2 6.4 8.1 10.3 11.3 13.2

SOLUTION The rectilinear area from A

= √ ((s – a) (s – b) (s – c))

5 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)

6 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 A 150 B 95.2 C 127.9 D 176.3 E 219.4

E+E Double Northing Longitude 100 164.3 245.2 210.7 223.1 239.8 304.2 222.4 395.7

ΔN

64.3 46.4 29.1 -17.4

F

237.5

163.8

456.9

-58.6

A

150

100

387.5

-63.8

15 766.36 10 351.84 8 852.22

34 970.42

2A

=

6 885.18 26 774.34 24 722.50 58 382.02 34 970.42 23 411.60

7 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

Technicians, S. Adam)

for

8 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:

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

2.6.3 Mid-ordinate rule

FIGURE 2.3 : Midordinate 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 midordinate 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)

10

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.

11

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 y

0 2

1 5

2 8

3 11

4 14

5 17

6 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.

12

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

13

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'.

14 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) = 158.2 square units c) Simpson’s rule Area = 2/3 [ (0 + 0) + 4(6.0 + 9.15 + 10.00 + 9.15 + 6.00) + 2(8.0 + 9.8 + 8.0)] = 2/3 [0 + 4(40.3) + 2(35.6)] = 2/3 (161.2 + 71.2) = 2/3 (232.4) = 154.9 square units The true area is given by π r² / 2, i.e π (10)² / 2 = 157.1 square units

15 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) ii) iii)

2.8

by cross-sections, generally used for long, narrow works such as roads, railways, pipelines, etc. by contours, generally used for larger areas such as reservoirs, landscapes, redevelopment sites, etc. by spot height, generally used for small areas such as underground tanks, basements, building sites, etc. 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) ii)

by calculating from the formula or from first principles the standard crosssections of constant formation widths and side slopes. 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

16 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 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) ii) iii)

Prism, in which the end polygons are equal and the side faces are parallelograms. Wedge, in which one end is a line, the other end a parallelogram, and the sides are triangles and parallelograms. Pyramids, in which one end is a point, the other end a polygon and the side faces are triangles.

The Prismoidal Formula Let D = A1 and A2 = M = V = a1, a2, m, v =

the perpendicular distance between the parallel end planes the areas of these end planes the mid-area, the area of the plane parallel to the end planes and midway between them, the volume of the prismoid and the equivalent for any prism, wedge or pyramid forming the prismoid

then in a prism and in a wedge and in a pyramid Prism volume v Wedge volume v Pyramid volume v

a1, = a2, = m a2 = 0 and m = 1/2 a1 a2 = 0 and m = 1/4 a1 = D . a1 = D/6 (6 . a1 ) = D/6 (a1, + 4m + a2) = ½ D . a1 = D/6 (3 . a1 ) = D/6 (a1, + 4m + a2) = 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 , 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)

17 Note : A. M does not represent the mean of the end areas A 1 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) Area (m2)

0 11

15 42

30 64

45 72

60 160

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

75 180

90 220

18 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 midsectional 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 midsectional 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}

19 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) Area (m2)

0 11

15 42

30 64

45 72

60 160

75 180

90 220

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.

20

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) Area (m2)

50 12

52.5 135

55 660

Using the end areas method : V = 2.5 { [ (12 + 1950)/2] + 135 + 660 + 1500 } = 8190 m3 Using Simpson’s rule from the prismoidal formula :

57.5 1500

60 1950

21

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.

22

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

Station A B C D E F G H J

R.L. 12.16 12.48 13.01 12.56 12.87 13.53 12.94 13.27 13.84

Number of times the R.L. is used = n 1 2 1 2 4 2 1 2 1 ∑n = 16

Product (R.L.) x n 12.16 24.96 13.01 25.12 51.48 27.06 12.94 26.54 13.84 207.11

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

23 (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.

A B C D E F G H J

12.16 12.48 13.01 12.56 12.87 13.53 12.94 13.27 13.84

Number of times the R.L. is used = n 1 3 2 3 7 2 2 2 2 ∑n = 24

Product (R.L.) x n 12.16 37.44 26.02 37.44 90.09 27.06 25.88 26.54 27.68 310.55

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.

24

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

25

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

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. ii. iii. iv. 3.2

The distances over which ‘cut and fill’ will balance. Quantities of materials to be moved and the direction of movement. Areas where earth may have to be borrowed or wasted and the amounts involved. The best policy to adopt to obtain the most economic use of plan. DEFINITION AND IMPORTANT PHRASES

Bulking Shrinkage Haul distance (d) Average haul distance (D) Freehaul Distance Overhaul Distance

Haul

Overhaul Station Metre Borrow Waste

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

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 volume (ordinate acc1 ) distance to centroid chainage of centroid

for area dbd1 area scaled Therefore haul =

= 300 000 = 2750 = 300 000 / 2750 = 109.1 m = 350 – 109.1 = 240.9 m

= 1.9688 cm²

1.9688 x 320 000 volume (ordinate dd1 ) distance to centroid chainage of centroid average haul distance overhaul distance

= 630 016 = 2750 = 229.1 m = 650 + 229.1 = 879.1 -240.9 = 638.2 – 300

= 879.1 m = 638.2 m = 338.2 m

= 338.2 x 2750

= 9300 station metres

Therefore overhaul

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 Therefore, Area = total haul

= n x 1 m³ x ( ∑L / n ) = total volume x average haul distance

33

Activity 3a

3.1

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

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

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.

Bulking

Shrinkage

Haul Distance (d)

Average Haul Distance Freehaul Distance

Overhaul Distance

Haul

Overhaul

Station Metre

the volume of material brought into a section due to a deficiency the volume of material taken from a section due to excess

Borrow

Waste

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 70

Surface Level 52.8

Vol

Chn 74

Surface Level 44.7

+1860 71

57.3

75

53.4

44.7

54.3 +362

80

60.9

-2110 77

41.5

+724 81

62.1

82

78.5

-1120 78

49.5

Vol

-237 79

37.5

-238 74

Surface Level 49.5

-2025 76

47.1

78

39.7

+547 73

Chn

-1080

+1525 72

Vol

+430

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

to Self Assessment

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

The second estimate is cheaper by 13 612 p

= 177 068 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 ed 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 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), ing the centre of the curve to the PI, effectively bisects all related lines and angles. a) Tangent: In Triangle BC, O, PI,

T   tan R 2  T  R tan 2

b) Chord : In triangle BC, O, B

40

1 C 2  sin  R 2 C  2 R sin

 2

c) Mid- ordinate:

OB   cos R 2 OB  R cos but

 2

OB = R- M

 2   M  R1  cos  2 

R - M  R cos

d) External: In triangle BC, O, PI O to PI = R + E

R   cos RE 2  1  E  R  1  cos   2       R sec  1 2  

41

Figure 4.2 Geometry Of The Circle. (Source: Surveying With Construction Application, B.F. Kavanagh) e) Arc:(Figure 4.3)

L   2R 360  L  2R 360 where

 is expressed in degrees and decimals of a degree.

Figure 4.3 Relationship Between The Degree Of Curve (D) And The Circle. (Source: Surveying With Construction Application, B.F. Kavanagh)

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:

D 100  360 2R 5729.58 D R Arc:

L   100 D  L  100 D f) Deflection angle

Figure 4.4 Deflection angle. (Source: Land Surveying, Ramsay J.P. Wilson) In ∆ T1AO, curve T1A = R x 21 Curve T1A = Chord T1A 1(rad) = Curve T1A / 2R = Chord T1A 1(minutes) = (Curve T1A x 180 x 60) / 2R = (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. Figure 4.5 (Source: Surveying With Construction Application, B.F. Kavanagh)

SOLUTION:

 T  R tan 2  1000 tan 819'  146.18 ft

PI at -T BC = +L EC =

6 + 26.57 1 46.18 4 + 80.39 290.31 7 + 70.70

L  2R

 360

 2  1000   290.31 ft

16.6333 360

44  2  2  1000  sin 819'

C  2 R sin

 289.29 ft   M  R1  cos  2   1000(1  cos 819' )  10.52 ft    E  R sec  1 2    1000(sec 819'1)  10.63 ft 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 at common tangent points. Figure 6.4 shows a compound curve consisting of two circular arcs ed 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 (Source: Surveying With Construction Application, B.F. Kavanagh) 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) (Source: Surveying With Construction Application, B.F. Kavanagh) 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 engers 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

S = shift

Figure 4.8 The Transition Curves. (Source: Land Surveying, Ramsay J.P. Wilson) 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

v t an   gR 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.

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 (Source: Surveying With Construction Application, B.F. Kavanagh) 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) (2)

a constant rate of change of slope, which contributes to smooth alignment transition, ease of computation of vertical offsets, which permits easily computed curve elevations

Figure 4.10 Vertical Curves (Profile View Shown) (Source: Surveying With Construction Application, B.F. Kavanagh) 2

The origin of the axes is placed at the BVC (Figure 4.11), the general equation becomes y = ax + bx, and because the slope at the origin is g1, the expression for slope of the curve at point becomes

dy  slope dx = 2ax + g1 2

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

50

Figure 4.11 Types of Vertical Curve (Source: Surveying With Construction Application, B.F. Kavanagh)

Activity 4a 4.1 Fill in the blanks with related circular curve terminology.

51

B C I B

J

G

B

B

B

B

F E

B

H

B

B

B

B

A

D B B

Figure 1 4.2 Solve the puzzle by using the clues as shown below.

1

5

7 2

6

3

4

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.

52 6) A _______ curve of two (usually) or more circular arcs between two main tangents turning in the same direction and ing at common tangent points. 7) A ________ is a curve with a uniformly changing radius.

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

1

T

7

S

I

A

N

S

I

T

I

O

E

R

T

I

6

A

L

V

T

A

C

E

O

R

M

A

S

P

L

E

O

R

N

E

I C

R

2

P 3

5

C

U

L

A

U 4

H

O

R

I

Z

O

N

L

D 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)

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 T 1X, 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 T 1 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= R21 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°

Chord number

Chord length (m)

Chainage (m)

Deflection angle o,„

Settingout angle o,„

Remarks

1

14

2220.00

2 00 19

2 00 19

peg 1

2 3 4 5 6

20 20 20 20 10.72

2240.00 2260.00 2280.00 2300.00 2310.72

2 51 53 2 51 53 2 51 53 2 51 53 1 32 08

4 52 12 7 44 05 10 35 58 13 27 51 14 59 59

peg 2 peg 3 peg 4 peg 5 peg 6

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, 2 2 2 AO = OB - AB 2 2 2 (R-Y3) = R – X3 2 2 ½ and Y3=R-(R -X3 )

56 thus for any offset Yi, at distance Xi, along the tangent 2 2 ½ Yi = R - (R - Xi )

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 2 2 Offset CA = (½ chord x chord) / Radius = (chord )/2R, the offset CA = O1 == X / 2R. As the normal chord AB differs in length from T 1A, 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 2 chord AB = y, then offset EB = y / 2R. Angle DAE = , therefore offset DE will be directly proportional to the chord length, thus: 2 DE = (O1 / x) y = (x y)2Rx = (xy)/ 2R Thus the total offset DB = DE + EB = ( y / 2R) (x +y) = (Chord / 2R) (sub-chord + chord) 2

Thus having fixed B, the remaining offsets to T2; are calculated as y /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 2

i.e. x / 2R from Ti. A line tangential to the arc and ing through peg A will then be the 2 required tangent from which offset EB, i.e. y /lR, may be set off.

Figure 4.15 Setting Out By Offset With Sub-chords (Source: Engineering Surveying, W.Schofield) 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 T 1A on Figure 6.14, thus: 2

2 1 2

Yo = R –[R - (C/2) ] /

Similarly, DB is equivalent to DB on Figure 6.14, thus: 2

2 1 2

DB=R-(R – X1 ) / and offset Y1, = Y0 - DB 2

2

2

:. Y1 = Y0- [R - (R - X1 )'/ ] and for any offset Yi, at distance Xi, each side of the mid-point of T1 T2,: 2 2 1 2 Y1, = Y0 - [R - (R - X )] /

58

Figure 4.16 Setting Out By Offset With Long-chords (Source: Engineering Surveying, W.Schofield) 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 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) b) c) d) e) f)

g)

Set up the theodolite near A and extend straight towards P. Set up on the straight BF and produce it to meet the line at P. Mark the intersection of tangents at P. Measure angle EPF and obtain angle θ. Calculate tangent length PA by using the formula T = R tan ´θ. 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.) 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 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 (Source: Land Surveying, Ramsay J.P. Wilson) 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 subchord. 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

12 + 25

16.67

12 + 50

25

12 + 75

25

13

25

13 + 25

25

Bearing °’” 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 (Source: Land Surveying, Ramsay J.P. Wilson) 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. NOTE: There

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 (Source: Land Surveying, Ramsay J.P. Wilson) 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 (Source: Land Surveying, Ramsay J.P. Wilson) 4.9 4.9.1

COMPUTING AND SETTING OUT A TRANSITION CURVE 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 (Source: Land Surveying, Ramsay J.P. Wilson)

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 T 1 or T2 the direction of the tangent 2 to the circular curve is obtained by turning off /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:

v3  RL

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:

P  To  ( R  S ) tan 1 2  

L 2

This distance will be taped back from the intersection point and the two tangents T 0, and T3 pegged. (c) Deflection angles. The deflection angles for 10-, 15- or 20-m chords are calculated from:

l2 d  572.958 min utes RL 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.

L (d) Length of circular curve. This is obtained from R' in the R usual way, where: ' = ( — 2') and 2'' = 57.2958

L degrees. R

66 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. NOTE:

Example 4e. Calculate the setting-out data for a 75m transition curve to connect an 8° circular curve ing two straights with an angle of deflection of 20° 00', using 15m chords.

Solution: (a) To calculate the radius and shift: Chord 15 15 15 15 15

L2

L

5729.28 Radius 15 , R  225 D 30 900 5729.58 2025  45 8 60 3600  716 75.20m 5625 ShiftS 



x

2'24" 9'36"

0.010 0.084 0.283

38'24" 60'00"

2

L 24 R '

75 2  24  716.20  0.327 m s:

L 2   (716.20  0.33) tan10  37.5

Tangent , T  ( R  s ) tan 1 2    163.84m

(c) To calculate and tabulate the deflection angles and deflection distances: 2 l3 l   572.958 Minutes andx  m 6 RL RL 572.958 2  l 716.20 75 2  0.01067l

1.309

(b) To calcula te the tangen t length

67

Table 4.2 Deflection Angle And Deflection Distances. (Source: Land Surveying, Ramsay J.P. Wilson) 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 ':

L minutes

Final 8 = 572.958

R 75 716.20

= 572.958 = 60' = 1° 00", which checks the final deflection angle calculated and tabulated in Table 6.2. '' =

L radians 2R

= 3° 00' 2' = 6° 00" but 2'' = 57.2958

L degrees R

= 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 T o. The other tangent point T3 is set out along the second straight the same distance from P as T 0, equal to 163.84 m. The final transition is then set out between T3 and T2 as before. (b) Setting out the circular curve.

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:

15 of 4   36' 100

set up at T1 for the sub-chord:

10 of 4   24' 100 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. 2. 3.

Compute the algebraic difference in grades: A = g2-g1. 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. Compute the distance from the BVC to the high or low point (if applicable):

x 4. 5. 6.

 g1L A

and determine the station of the high/low point. Compute the tangent grade-line elevation of the BVC and the EVC. Compute the tangent grade-line elevation for each required station. Compute the midpoint of chord elevation:

Elevation of BVC  elevation of EVC 2 7.

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

d 8.

difference in elevation of PVI and midpoint of chord 2 2

Compute the tangent offset for each individual station (see line ax in Figure 4.21): 2

 x   or 4d x 2 tan gentoffset  d  L  L2  2

 

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.

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

 g1 L A 3.2  300  5  192.00 ft ( fromtheBVC ) x

5.

Tangent grade-line computations are entered in Table 4.3. Example:

Elevation at 29  00  470.72 - (0.032  20)  470.72 - 0.64  470.08

70

Figure 4.22: Sketch For Example 4 (Source: Surveying With Construction Application, B.F. Kavanagh) Station BVC 28 + 80 29 + 00 30 + 00 PVI 30 + 00 LOW30 + 72 31 + 00 EVC 31 + 80 30 + 62 30 + 72 30 + 82

Tangent Elevation + Tangent Offset 2 470.72 (0/150) x 1.8.75 =0 2 470.08 (20/150) x 1.875 = 0.03 2 466.88 (120/150) x 1.875 = 1.20 2 465.92 (150/150) x 1.875 = 1.875 2 466.68 (108/150) x 1.875 = 0.97 2 467.18 (80/150) x 1.875 = 0.53 2 468.62 (0/150) x 1.875 = 0 2 466.50 (118 / 150) x 1.875 = 1.16 2 466.68 (108 / 150) x 1.875 = 0.97 2 466.86 (98 / 150) x 1.875 = 0.80

= Curve Elevation 470.72 470.11 468.08 467.80 467.65 467.71 468.62 467.66 467.65 467.66

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

Table 4.3 Parabolic Curve Elevation by Tangent Offset 6.

Mid-chord elevation:

470.72 (BVC)  468.62 (EVC)  469.97 2 2

7.

Tangent offsets are computed by multiplying the distance ratio squared, [x/(L/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

71

Transition deflection angle

P  To  ( R  S ) tan 1 2  

x

Shift

Tangent length

l3 6 RL

2 '  57.2958

Transition deflection distance

L deg rees R

d  572.958

Length of the circular curve

s

l2 min utes RL

L2 24 R

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.

4b

4.3)

The formulae used to calculate the following:

Transition deflection angle

Shift

P  To  ( R  S ) tan 1 2   x

L 2

l3 6 RL

L 2

72

Tangent length

2 '  57.2958

Transition deflection distance

d  572.958

Length of the circular curve

4.4)

L deg rees R

s

l2 min utes RL

L2 24 R

a) Setting Out By Offset With Sub-Chords. b) Setting Out By Offset With Long-Chords. c) Setting Out By Offsets From Tangent. 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.

4.5)

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'

73 = 0 48' 35"

Self Assessment

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

to Self Assessment

1)

T = R tan (/2) = 400 tan 025’ 30” = 45.044m L = (2R )/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 engers 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%

4) 1

A = g2 – g1 = -5%-4% = -9% = 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:

BI 

BD sin  2 sin 

1485 sin 53 48'  sin 70   1275.2m

ID 

BD sin  1 sin 

1485 sin 56 15' sin 70   1314m 

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 Ramsay)

(Source : Land Surveying,

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 of rectangular coordinates on the same system. For instance, the centre-line of a proposed road may be defined in 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. ii. iii. iv.

From the co-ordinates compute the bearing BA (this bearing may already be known from the initial control survey computations). From the co-ordinates compute the horizontal length and bearing of B-IP. From the two bearings compute the setting-out angle AB(IP), i.e. β. 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 es 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 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 ing 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?

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 predetermined 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 s 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 90 º 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 . 2. Offset distance marked on pegs which 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 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.

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.

100 Note: 1. SR offset far enough to allow the machine to . 2. Offset distance marked on pegs which 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

101

Self Assessment

1) Describe the employer’s responsibility on the work site. 2) Describe the engineer’s responsibility of setting out.

of Self Assessment 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.

102

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.

103 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 builtin 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 Z E

R

a e r λ Φ

Station Target Reference plane within the distance meter for phase comparison between transmitted and received wave Reference plane for the reflection of the wave transmitted by the distance meter Addition constam Distance meter component of addition constant Reflector component of addition constant Modulation wave length 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.

Parameter t, temperature p, pressure e, partial water vapor pressure

ERROR parts per million Error Light Wave +1º -1 +1 mm Hg + 0.4 1 mm Hg - 0.05

Microwave - 1.25 + 0.4 + 7 at 20ºC + 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 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.

A

B

C

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

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

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 ed to a computer via an RS 232 interface. The 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 ed 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 ing 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 ing 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 ed 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.

121

Activity 6b

6.2 What is a Total Station? 6.3 Describe typical Total Station programs.

6b

6.2

6.3

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

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 ing the sighted points.

122 



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 ing 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

123 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.

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.)

±(15 mm + 5ppm) for short-range EDMIs ±(3mm + 1 ppm) for long-range EDMIs

Accuracy range

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

Distance range

Measuring time

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

IN P U T

124

-20°C to +50°C Nonprism measurements: available on some models; distances from 100 to 350 m (3 to 5 km with prisms)

Temperature range

to Self Assessment

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.)

±(15 mm + 5ppm) for short-range EDMIs ±(3mm + 1 ppm) for longrange EDMIs

Accuracy range

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

Distance range

Measuring time

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

-20°C to +50°C Nonprism measurements: available on some models; distances from 100 to 350 m (3 to 5 km with prisms)

Temperature range

125