Chapter 3 Triangulation 3.1 Introduction To prevent accumulation of errors, it is necessary to provide a number of control points all over the area, which will form a control frame work on which entire survey surve y is to be based. Provision of such control points can be made either by one or a combination of both the following methods: 1) Theodolite Traverse 2) Triangulation Methods of triangulation is more accurate than that by the theodolite traverse, as accumulation of error in triangulation is less as compared to that of theodolite traversing
3.2 Triangulation The process of measuring the angles of a chain network of triangles formed by a number of stations marked on the surface of the earth, is called triangulation.
3.3 Principle of Triangulation If all the three angles and the length of one of its sides of a triangle are known, the lengths of the remaining sides of the triangle can be calculated calc ulated trigonometrically. Again, if the coordinates of any vertex of the triangle and azimuth of any side, are also known, then, the co-ordinates of the remaining vertices may be commuted. The side of the first triangle, whose length is predetermined is called c alled the “Base line” and the vertices of the individual i ndividual triangles are ar e known as “Triangulation Stations”. To minimize accumulation accumulatio n of errors in lengths, subsidiary bases at suitable intervals, are usually measured. To control the error in azimuth of stations, astronomical observations are made at intermediate stations. The triangulation sta tions at which astronomical observations for for azimuth are made, are called “Laplace Stations”.
3.4 Purpose of Triangulation Surveys Triangulation surveys are carried out for the following purposes: 1) Establishment of accurate control points for plane and geodetic surveys of large areas, by ground method. 2) Establishment of accurate control points for photogrammetric surveys of large areas. 3) Accurate location of engineering works i.e. a) Fixing the centre line, terminal points and shafts for long tunnels. b) Fixing control line, piers and abutments of long bridges over rivers. c) Transferring the control points across wide sea channels, large water bodies, etc. d) Finding the direction of the movement of clouds.
3.5 Classification of Triangulations On the basis of quality, accuracy and purpose, triangulations are clas sified as under:
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1) Primary triangulation or First order triangulation. 2) Secondary triangulation or Second order triangulation. 3) Tertiary triangulation or Third order triangulation. (1) Primary Triangulation: The highest grade of triangulation system which is carried out either for the determination of the shape and siz e of the earth’s surface or for providing precise plainmetric control points for subsidiary triangulations, is called primary triangulation. The stations of the first order triangulation are generally selected 16 km to 150 km apart. Every possible precaution is taken in making linear, angular and astronomical observations, and also in their computation (Fig. 3.1). (2) Secondary Triangulation: The triangulation system which is employed to connect two primary series and thus to t o provide control points closer together than those of primary triangulation, is called secondary triangulation. If any triangulation series which is carried out as primary triangulation does not attain the standard of the accuracy of that class, due to unfavourable conditions, may also be classified as s econdary triangulation or triangulation of second order (Fig. 3.2). (3) Tertiary Triangulation: The triangulation system which is emplo yed to provide control points between stations of primary and second order series of triangulation is called Tertiary triangulation. In the department of the Survey of India, tertiary triangulations, known as topo triangulations, form the immediate control for topo-graphical surveys of various scales (Fig. 3.3). Specifications of different triangulations:
Length of the base lines Length of the sides Average triangulation error (after correcting the spherical excess) Maximum station closure Actual error of the base Probable error of the base Discripancy between measures Probable error of distance
Primary Triangulation 8-12 km 16-150 km Less than
1′′
Not more than
3′′
1:50000 1:1000000 two
5
√
1 in 50000 to 1 in 250000
Probable error in astronomical observation
0′′. 5
Secondary Tertiary Triangulation Triangulation 2-5 km 100-500 m 10-25 km 2-10 km
3′′ 8′′
12′′ 15′′
1 in 25000 1 in 500000
1 in 10000 1 in 250000
10 mm
√
1 in 20000 to 1 in 50000
5′′
25 mm
√
1 in 5000 to 1 in 20000
10′′
3.6 Layout of Triangulation The arrangement of the various triangles of a triangulation series, is known as the layout of triangulation. A series of triangulation may consist of the following (1) Simple triangles in chain Page 2 of 27
(2) Braced quadrilaterals in chain (3) Centred triangles and polygons
Fig. 3.1 Simple Triangles
Fig. 3.2 Braced Quadrilaterals
Fig. 3.3. Centred Triangles and Polygon 1) Simple Triangles in Chain: This layout of triangulation is generally used when control points are provided in a narrow strip of terrain such as a valley between two ridges. This system is rapid and economical due to its simplicity of sighting only four stations and observations of long diagonals are avoided. On other hand, simple triangles of a triangulation do not provide any check on the accurac y of observations as there is only one route through which distances can be computed. To avoid excessive accumulated error, check base lines and astronomical observations for azimuth at frequent intervals are therefore very necessary in this kind of layout (Fig. 3.4).
Fig. 3.4. Simple Triangles in Chain Page 3 of 27
2) Braced Quadrilateral in Chain: A triangulation system which consists of figures containing four corner stations and observed diagonals, is known as a layout of braced quadrilaterals. This system is treated to be the best arrangement of triangles as it provides a means of computing the lengths of the sides using different combination of the sides and angles (Fig. 3.5).
Fig. 3.5. A Braced Quadrilateral 3) Centred Triangles and Polygon: A triangulation system which consists of figures containing centred polygons and centred triangles, is known as centred triangles and polygons (Fig. 3.3). This layout of triangulation is generally used when vast area extending in all directions is required to be covered. The centred figures generally are quadrilaterals, pentagons, or hexagons with central stations. Though this system provides proper check on the accuracy of the work, the progress of the work, is generally low due to the fact that more settings of the instrument are required.
3.7 Ideal Figures for Triangulation The undermentioned factors need be kept in mind while deciding and selecting a particular figure in any triangulation system. 1) 2) 3) 4)
Simple triangles should be preferably equilateral. Braced quadrilaterals should be preferably squares. Centred polygons should be regular. No angle of the figure, opposite a known side should be small, whichever end of the series is used for computation. 5) The angles of simple triangle should not be less than in case of quadrilaterals no angle should be less than . In case of centred polygons, no angle should be less than . 6) The sides of the figures should be of comparable length.
40
45
30
Note: It may be noted that if a very small angle of a triangle does not fall opposite the known side, it does not affect the accuracy of triangulation i.e. neither the sides nor the azimuth.
3.8 Size of Triangulation The length of the sides of the triangle varies according to the order of the triangulation and topography of the country. In order to cover the area quickly and to reduce the computation, the triangles should be as large as possible, provided sufficient number of intersected points can be observed simultaneously.
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3.9 Layout of Primary Triangulation for Large Countries In large countries, the frame work of triangulation may be provided by the following two methods: a. Grid Iron System b. Centred System a) Grid Iron System: In this system, the primary triangulation is laid in series of chains of triangles, which usually run roughly along the meridians (north-south) and along the perpendiculars to the meridians (east-west) throughout the country. The distance between two such chains, may vary from 150 km to 250 kms (Fig. 3.6). The area between the parallel and perpendicular series of primary triangulation are covered by the secondary and tertiary triangulation systems. Grid Iron System has been adopted in India and also in Austria, Spain and France.
Fig. 3.6. Grid Iron System of Triangulation b) Centered System: In this system, the whole area of the survey is covered by a network of primary triangulation extending outwards in all directions from the initial base line,
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which is generally laid at the centre of the country. This system is generally used for the survey of an area of moderate extent and has been adopted in United Kingdom (Fig. 3.7).
Fig. 3.7. Centred System of Triangulation
3.10 Well Conditioned Triangles of a Triangulation System The arrangement of triangles in the layout and the magnitude of the angles in individual triangles, affect the accuracy of a triangulation system. The shape of the triangle in which any error in angular measurements, has a minimum effect upon the lengths of the computed sides, is known as a “well-conditioned triangle”. In any triangle of a triangulation system, the length of one of its side is generally obtained from computation of the adjacent triangle. The error in the other two sides, if an y, will affect the side of triangle(s) whose computation is based upon their values. Due to accumulated errors, the entire triangulation system is thus affected thereafter. To ensure that two sides of a triangle are equally affected, these should therefore be equal in length. This condition suggests that all the triangles must be Isosceles.
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Let A, B and C be the three angles of an Isosceles triangle ABC whose sides are a, b and c respectively. AB be the known computed side whereas AC and BC are sides of equal length (Fig. 3.8).
Fig. 3.8. Well-Conditioned Triangle
= or = ∗ ----Equation (3.1) Now, assume that an error in the angle A introduces an error in the side ‘a’. ∗ ----Equation (3.2) Differentiate equation (3.1) partially, we get = = cot----Equation (3.3) Dividing equation (3.2) by equation (2.1), we get = Again assume that an error in the angle C introduces an error in the side ‘a’. − ----Equation (3.4) Differentiating equation (3.1) partially, we get = () ∗ = cot----Equation (3.5) Dividing equation (3.4) by equation (3.1), = If = = ± , the probable errors in the angles, then the probable fraction error in the side ‘a’, is given by the equation = ± ( cot) (cot) . But C = 180 = 180 2. = ± (cot) (cot2)----Equation (3.6) Therefore From equation (3.6), it is clear that will be minimum if the value of (cot) (cot2) is Applying the sine rule to the triangle ABC, we get
minimum.
(cot) (cot2) 4(cos) 2(cos) 1 = 0 Differentiating
with respect to ‘A’ and equating to zero, we get
---- Equation (3.7)
On solving the equation (3.7) for Cos A, we get A=
5614′
approximately.
5614′
Hence, the best shape of a triangle is an isosceles triangle whose base angles are each. But, for all particular purposes, an equilateral triangle may be treated as a well-conditioned triangle. The triangles, whose angles are less than or more than , should be avoided in the chain of triangles.
30
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120
3.11 Strength of Triangulation Figures The strength of the figures in triangulation, plays an important role to decide its layout and to ensure a desired degree of accuracy. Computation involves use of the angles of triangles and the length of a known side. The other sides are computed by the law of sines. For a given change in the angles, the sine of small angles changes more rapidly than those of large angles. This suggests that smaller angles than should not be used in the computation of triangulation. If, due to, unavoidable circumstances, angles less than are used, then it must be ensured that this is not opposite the side whose length is required to be computed for carrying forward the triangular series. In any triangulation system, more than one routes are possible for various stations. Hence, it becomes necessary to find some criterion for the selection of the most appropriate route so that best shaped triangulati on net route may be adopted.
30
30
The strength of the figure is decided by an index number which is denoted by R. The strength of the figure depends on the following factors: 1) Number of observed directions 2) Number of trigonometric conditions 3) Magnitude of the distances and angles If D = number of directions observed excluding the known sides of the figure.
, = − ∗ ∑( . )
C = number of the geometric conditions = tabular difference for 1 second in the logarithmic sine of angles A, B and C in the sixth place of decimal. Then
----Equation (3.8)
The value of C in equation (3.8) may be computed from the formula, C=
(′ ′ 1) ( 2 3)
----Equation (3.9)
Where n = total number of the lines including the fixed li ne. n’= number of lines observed in both directions. s = total number of stations. s’ = total number of occupied stations.
∑( . )
For the computation of the quantity used in the equation (3.8), Table (3.1) may be used. The required value is obtained in the column headed by the smaller angle and on the horizontal line containing the larger angle. If the computed value of R is less, the strength of the figure is more and vice versa.
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Table 3.1: Table for the Distance Angle Error Values
3.12 Accuracy of Triangulation Inspite of all precautions in every triangulation series, the error gets accumulated. It is therefore very essential to know then accuracy achieved in the triangulation network so that no appreciable error in plotting, is introduced. For calculating the accuracy, the following formula may be used
= ∑
----Equation (3.10)
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Where m = root mean square error of un-adjustable horizontal angles (in seconds) as obtained from the triangular errors.
∑
= sum of the squares of all the triangulation errors in triangulation series.
n = Total number of triangles in series. Note: The following points may be important considerations 1) All the triangles must be included in the computation. 2) All the four triangles of a braced quadrilateral must be included in the computation. 3) If the average triangular error of the series is , probable error in latitudes and departures after a distance of 100 km is approximately 8 meters.
8′′
3.13 Signals and Towers To define the exact position of triangulation stations during observations from other stations, signals are used. Various types of signals are centred vertically over the station marks and observations are made to these signals. It is very necessary to ensure that the signals whenever used, are truly vertical and centered over the station marks. The accuracy of triangulation is entirely dependent on the degree of accuracy of centering the signals. Greatest care of c entering the transit over the station mark will be useless, unless same degree of care of centering the signals, is impressed upon.
3.14 Classification of Signals The signals may be classified as under: a. luminous signals b. opaque signals a) Luminous signals: Luminous signals are further divided in to two categories i.e., sun signals and night signals. Sun signals: Those signals which reflect the rays of sun towards the station of observation, are known as heliotropes. Apparently, such signals can only be used in clear weather. Heliotropes: A heliotropes consists of a circular plate mirror with a small hole at its center. It reflects the sun’s rays and a sight vane with an aperture carrying a cross hairs. The circular mirror can be rotated horizontally as well as verticall y through . The heliotrope is centred over the station mark and the line of sight is directed towards the station of observation. The heliotrope man looks through the mirror hole and adjusts the sight vane till the flashes given from the station of observation fall at the center of cross hairs. Once this is done heliotrope is not disturbed. Now the heliotrope attendant rotates the frame carrying the mirror so that the black shadow of the small central hole of the plane mirror falls exactly at the cross of the sight vane. The reflected beam of rays will now be seen at the station of observation. Due to motion of the sun, this small shadow also moves and hence the attendant should remain on constant watch to ensure that the shadow always remains at the cross.
360
Note: 1) Heliotropes do not give better results as compared to the opaque signals.
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2) Heliotropes are found useful only if the signal station is in plains and the station of observation is on an elevated ground. Night signals: While making observations at night, night signals are used. Various types of night signals are:
1) Various forms of oil lamps with a reflector. These are used for sights less than 80 km. 2) Acetylene lamps designed by captain G.T. McCaw. These are used for sight more than 80 km. b) Opaque Signals: The opaque, or non-luminous signals used during day are of various forms and the ones most commonly used are the following:
1) Pole signal: It consists of a round pole painted black and white in alternate and is supported vertically over the station mark on a tripod. Pole signals are suitable up to a distance of 6 km (Fig. 3.9).
Fig. 3.9. A Pole Signal 2) Target signal: It consists of a pole carrying two squares or rectangular targets placed at right angles to each other. The targets are generally made of cloth stretched on wooden frames. Target signals are suitable up to a distance of 6 km (Fig. 3.10).
Fig. 3.10. A Target Signal
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3) Pole and Brush signals: It consists of a straight pole about 2.5 metre long with a bunch of long grass tied symmetrically round the top, making a cross. The signal is erected vertically over the station mark by heaping a pile of stones up to 1.7 metres round the pole. A rough coat of white wash is applied to make it more conspicuous to be seen against a black back ground (Fig. 3.11). This type of opaque signals is very useful. It should preferably be erected over every station of observation during reconnaissance.
Fig. 3.11. A Pole and Brush Signal 4) Stone cairn: It consists of stones built up to a height of 3 meters in a conical shape. This white washed opaque signal is very useful if the back-ground is dark (Fig. 3.12).
Fig. 3.12. A Stone Crain 5) Beacon: It consists of red and white cloth tied round the three straight poles. This can be easily centred over the station marks (Fig. 3.13). Beacons are useful when simultaneous signals are made at both the stations.
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Fig. 3.13. A Beacon Requirements of an Ideal Signal: An ideal signal should fulfil the following requirements:
1) It should be conspicuous i.e., it should be clearly visible from a distance against any back ground. 2) It should provide easy and accurate bisection by a telescope. 3) It should be capable of being accurately centred over the station mark. 4) It should exhibit very little phase error of bisection of the signal. Phase of a Signal:
When a cylindrical opaque signal which is partly illuminated by the sun and partly remains in shadow, is observed, the observer sees only illuminated portion from his station and thus bisects the centre of the illuminated portion. The error of bisection thus introduced, is known as phase of the signal. To reduce the measured angle to the centre of the signal, a phase correction is therefore necessary. Two cases may arise: 1) An observation is made on brightest portion 2) An observation is made on bright line 1) When th e br ight porti on i s bisected (F ig. 3.14)
Let O = observer’s station P = Signal’s centre AB = Illuminated portion of the signal C = Midpoint of illuminated portion AB OC = Line of Sight r = Radius of the signal
= Angles BOP and AOP respectively
= Angle between the sun and the line OP (produced) Page 13 of 27
D = OP horizontal distance of signal
= Phase correction = Angle COP
Fig. 3.14. Phase of Signal (Brightest Side is Observed)
= − ∆ tan = ⇒ = = ( −) = .
Therefore Phase Correction = From
, we get
---- Equation (3.11) .
Similarly
Substituting the values of
) ( in equation (3.11) we get = .
2) When the Observation is made on the Bright Line (Fig. 3.15)
Let the incident ray SC = reflected ray from the bright line.
= Phase correction Page 14 of 27
= Angle between the sun and line OP
Fig. 3.15. A Phase of Signal (Brightest Line is Observed)
∠ = 180 ( ) ∠ = 180 ∠ = 90 − ∴ ∠ = 180 ( ∠) = 180 90 − = 90 Assuming the rays of the sun to be parallel, we get SC ||
O
, ignoring
too small.
being
− ∴ tan = − − ⇒ = ----Equation (3.12) Note: The phase correction is applied algebraically to the observed angle according to the relative positions of the sun and the signal.
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3.15 Base Line Measurement The accuracy of any order of triangulation depends upon the accuracy of the measurement of its base line. Hence, in triangulation a base line is of prime importance. The length of a base line depends upon the grade of the triangulation. In India, for the network of first order triangulation, baselines of lengths ranging from 1.7 mi les (2.83 km) to 7.8 miles (13 km) were used. For topographical triangulation, the length of the base line generally used is 88 yards (80.467 m). Site selection for a Base Line: The following factors are generally considered for the selection of the site for the base line of any triangulation system.
1) The ground at the site should be either fairly level, or uniformly sloping or gently undulating. 2) It should be free from obstructions throughout its length. 3) The ground should be firm and smooth. 4) The two extremities of the base line should be intervisible at ground level. 5) The site should be such that well shaped triangles can be obtained while connecting its end stations to the stations of the main triangulation.
3.16 Equipment for Base Line Measurement The measurement of base line forms the most important part of the triangulation operations. The base line is laid down with great accuracy of measurement and alignment as it forms the basis for the computations of triangulation system. The length of the base line depends upon the grades of the triangulation. Apart from main base line, several other check bases are also measured at some suitable intervals. In India, ten bases were used, the lengths of the nine bases vary from 6.4 to 7.8 miles and that of the tenth base is 1.7 miles. Standards of Length: The ultimate standard to which all modern national standards are referred is the international meter established by the Bureau International der Poids at Measures and kept at the Pavilion de Breteuil, Sevres, with copies allotted to various national surveys. The meter is marked on three platinum-iridium bars kept under standard that they are subject to very small secular change in their dimensions. Accordingly, the meter has now been standardized in terms of wavelength of cadmium light. The various national standards are as follows: (i) Great Britain: The legal unit in Great Britain is the Imperial yard of bronze bar with gold plugs kept at the Board of Trade in London.
1 Imperial Yard = 0.91439180 legal meters = 0.91439842 international meters. (ii) The United States: The standard is copy no. 27 of the international meter. Primary triangulation is computed in meters and is converted in feet by the statutory ratio:
1 meter = 39.37 inch. (iii) India: The triangulation is computed in terms of the old 10 feet bar ‘A’ as it was in 1840 70, having its length equal to 9.9999566 British feet at that time. The modern survey standards are however a nickel meter (of 1911) and a silica meter (of 1925) kept at Dehra Dun and standardized at the National Physical Laboratory. Page 16 of 27
Forms of Base Measuring apparatus:
There are two forms of base measuring apparatus: a) Rigid Bars b) Flexible Apparatus a) Rigid Bars:
Before the introduction of invar tapes, rigid bars were used for work of highest precision. The rigid bars may be divided into two classes: (i) Contact apparatus, in which the ends of the bars are brought into successive contacts. Example: The Eimbeck Duplex Apparatus. (ii) Optical apparatus, in which the effective lengths of the bars are engraved on them and observed by microscopes. Example: The Colby Apparatus and the Woodward Iced Bar Apparatus.
The rigid bars may also be divided into the following classes depending upon the way in which the uncertainties of temperature corrections are minimized: i. ii.
iii.
Compensating base bars, which are designed to maintain constant length under varying temperature by a combination of two more metals. Example: The Colby Apparatus. Bimetallic non-compensating base bars, in which two measuring bars act as a bimetallic thermometer. Example: The Eimbeck Duplex Apparatus (US Coast and Geodetic Survey), Borda’s Rod (French System) and Bessel’s Apparatus (German System). Monometallic base bars, in which the temperature is either kept constant at melting point of ice, or is otherwise ascertained. Example: The Woodward Iced Bar Apparatus, and Struve’s Bar (Russian System).
The Colby Apparatus
Fig. 3.16. The Colby Apparatus This is a compensating and optical type rigid bar apparatus designed by Maj-Gen. Colby to eliminate the effect of changes of temperature upon the measuring appliance. The apparatus was employed in the Ordinance Survey and the Indian Surveys. All the ten bases of G.T. of Survey of India were measured with the Colby apparatus. The apparatus (Fi g. 3.16) consists of Page 17 of 27
two bars, one of steel and the other of brass, each 10 ft. long and riveted together at the centre of their length. The ratio of coefficients of linear expansion of these metals having been determined as 3:5. Near each end of the compound bar, a metal tongue is supported by double conical pivots held in forked ends of the bars. The tongue projects on the side away from brass rod. On the extremities of these tongues, two minute marks a and are put, the distance between them being exactly equal to . The distance ab (or ) to the junction with the
′ ′ ′
10′0′′ ′ ′ ′
ℎ of the distance ac (or ) to the brass junction. Due to change in temperature, ′ if the distance of steel changes to by an amount x, the distance ′ of brass will change to ′ by an amount thus unfaltering the positions of dots a and ′ . The brass is steel is kept
coated with a special preparation in order to render it equally susceptible to change of temperature as the steel. The compound bar is held in the box at the middle of its length. A spirit level is also placed on the bar. In India, five compound bars were simultaneously employed in the field. The gap between the forward mark of one bar and the rear bar of the next was kept constant equal to by means of a framework based on the same principles as that of the compound bar. The framework consists of two microscopes, the distance between the cross-wires of which was kept exactly equal to . To start with, the cross-wires of the first microscope of the framework were brought into coincidence with the platinum dot, let into the centre of the mark of the one extremity of the base line. The platinum dot a of the first compound bar was brought into the coincidence with the cross-hairs of the second microscope. The cross-hairs of the first microscope of the second frame-work (consisting two microscopes apart) are then set over the end of the first rod. The work is thus continued till a length of ( = is measured at a time with the help of 5 bars and 2 frameworks. The work is thus continued till the end of the base is reached.
10′
6′′
6′′
6′′10′ ∗ 55∗6′′) 52′6′′ ′
b) Flexible Apparatus:
In recent years, the use of flexible instruments has increased due to the longer length that can be measured at a time without any loss in accuracy. The flexible apparatus consists of (a) steel or invar tapes, and (b) steel and brass wires. The flexible apparatus has the following advantages over the rigid bars: i. ii. iii.
Due to the greater length of the flexible apparatus, a wider choice of base sites is available since rough ground with wider water gap can be utilized. The speed of measurement is quicker, and thus less expensive. Longer bases can be used and more check bases can be introduced at closer intervals.
Steel Tapes
Steel tapes are semi-tempered bands of tough, flexible steel which has a thermal coefficient of expansion of very nearly 0.00000645 per degree Fahrenheit. The temperature of a steel tape cannot be measured with sufficient accuracy by mercurial thermometer in the day time. Accurate results can, however, be obtained if the measurements are made at night or on cloudy or even hazy days when there is little radiant heat. At these times the tape and air temperatures are nearly the same so that the temperature of the tape can be accurately determined and corrections applied.
Page 18 of 27
Invar Tapes and Wires
The research of Dr. Guillaume, of the French Bureau of Weights and Measures, led to the discovery of invar, the least expansible steel alloy containing about 36% nickel. The coefficient of thermal expansion is the lowest of all the known metals and alloys and seldom exceeds 0.0000005 per degree F. However, the temperature coefficient not only varies with the percentage of nickel, but also with the thermal and mechanical treatment given to each type. Every tape has its own coefficient which must be separately determined. Another peculiar thing with invar is that it undergoes some secular change in its length which increases slowly with time especially in the first few years. Due to this reason, invar can never be used for permanent standards. The instability, however, can be reduced by a process of artificial ageing, which consists in annealing them by exposure for several days to the temperature of boiling water. The coefficient of expansion of invar tapes also show slight variation with time, and should be determined both before and after a base line measurement. Invar is much softer than steel and must be handled very carefully. It should be wound upon a large reel or drum. The tensile strength of invar varies from 100000 to 125000 with an elastic modulus of 22 x . Invar tapes can be obtained in lengths of 100 ft. to 300 ft. in the ¼ in. width, and in the 6 mm width between 24 and 30 to 100 meters. The met ric tape, the ends are divided to 1/25 in. or 1/100 ft. Due to their high cost, they are not used for ordinary work.
⁄
⁄
10
Equipment for Baseline measurement:
i. ii. iii. iv. v.
Three Standardized tapes: out of the three tapes one is used for field measurements and the other two are used for standardizing the field tape at suitable intervals. Straining device, marking tripods or stakes, and supporting tripods or staking. A steel tape for spacing the tripods or stakes. Six thermometers, four for measuring the temperature of the field tape and two for standardizing the four thermometers. A sensitive and accurate spring balance.
The Field Work
The field work for the measurement of base line i s carried out by two parties: (1) The setting out party, consisting of two surveyors and a number of porters have the duty to place the measuring tripods in alignment in advance of the measurement, and at correct intervals. (2) The measuring party consisting of two observers, recorder, leveller and staffman, for actual measurements. The base line is cleared off the obstacles and is divided into suitable sections ½ to 1 kilometer in length and is accurately aligned with a transit. Whenever the alignment changes, stout posts are driven firmly in the ground. The setting out party then places measuring tripods in alignments in advance of the measurement which can be done by two methods: i. ii.
Measurement on Wheeler’s methods by Wheeler’s base line apparatus Jaderin’s method
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i) Wheeler’s base line apparatus (Fig. 3.17)
Fig. 3.17. Wheeler ’s Base Line Apparatus The marking stakes are driven on the line with their tops about 50 cm above the surface of the ground, and at distance apart slightly less than the length of the tape. On the tops of the marking stakes, strips of zinc, 4 cm in width, are nailed for the purpose of scribing off the extremities of the tapes. Supporting stakes are also provided at intervals of 5 to 15 meters, with their faces in the line. Nails are driven in the sides of the support stakes to carry hooks to support the tape. The points of supports are set either on a uniform grade between the marking stakes or at the same level. A weight is attached to the other end of the straining tripod to apply a uniform pull. To measure the length, the rear end of the tape is connected to the straining pole and the forward end of the spring balance to the other end of which a weight is attached. The rear end of the tape is adjusted to coincide with the mark on the zinc strip at the top of the rear marking stake by means of the adjusting screw of the slide. The position of the forward end of the tape is marked on the zinc strip at the top of the forward marking stake after proper tension has been applied. The work is thus continued. The thermometers are also observed. ii) Jaderin’s method
In this method, introduced by Jaderin, the measuring tripods are aligned and set at a distance approximately equal to the length of the tape. The ends of the tapes are attached to the straining tripods to which weights are attached. The spring balance is used to measure the tension. The rear mark of the tape is adjusted to coincide with the mark on rear measuring the tension. The rear mark of the tape is adjusted to coincide with the mark on rear measuring tripod. The mark on the forward measuring tripod is then set at the forward mark of the tape. The tape is thus suspended freely and is subjected to constant tension. An aligning and levelling telescope is also sometimes fitted to the measuring tripod. The levelling observations are made by a level and light staff fitted with a rubber pad for contact with the tripod heads. The tension applied should not be less than 20 times the weight of the tape.
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Fig. 3.18. Jaderin’s Method Measurements by Steel and Brass wires: Principle of Bimetallic Thermometer
The method of measurement by steel and brass wires is based on Jaderin’s application of the principle of bimetallic thermometer to the flexible apparatus. The steel and brass wires are each 24 m long and 1.5 to 2.6 mm in diameter. The distance between the measuring tripods is measured first by the steel wire and then by the brass wire by Jaderin’s method as explained above (Fig. 3.18) with reference to invar tape or wire. Both the wires are nickel plated to ensure the same temperature conditions for both. From the measured lengths given by the steel and brass wires, the temperature effect is eliminated as given below: Let
= distance as computed from the absolute length of steel wire
= distance computed from the absolute length of the brass wire
= co-efficient of expansion of steel = co-efficient of expansion of brass
D= corrected distance
= mean temperature during measurement
= temperature at standardization
T=
= (1) = (1) ( ) = = −− = temperature increase
Now,
Therefore,
or
---- Equation (1)
---- Equation (2)
Substituting the value of T in (1) for steel wire, we get
= {1 ( )} Therefore, correction for steel wire = accuracy.
= (−−) ≈ (−−)
with sufficient
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Similarly, correction for brass wire =
= (−−)
.
The corrections can thus be applied without measuring the temperature in the field. The method has however been superseded by the employment of invar tapes or wires.
3.17 Satellite Station: Reduction to Centre In order to secure well-conditioned triangle or better visibility, objects such as church spires, steeples, flag poles, towers, etc. are some times selected as the triangulation stations. When the observations are to taken from such a station, it is impossible to setup instrument over it. In such a case, a subsidiary station, known as a satellite station or eccentric station or false station is selected as near to the main station as possible, and observation are taken to the other triangulation stations with the same precision as would have been used in the measurement of angles at the true station. These angles are later correcte d and reduced to what they would have been if the true station was occupied. The operation of applying the corrections due to the eccentricity of the station is generally known as ‘reduction of centre’. The distance between the true station and the satellite station is determined either by method of trigonometrical levelling or by triangulation. Satellite stations should be avoided as far as possible in primary triangulation.
Fig. 3.19. Satellite Station ‘S’ to the Left of Main Station ‘B’ In Fig. 3.18, let A, B, C = triangulation stations, S = Satellite station for B, d = BS = eccentric distance between B and S, determined by trigonometrical levelling or by triangulation.
∠ =
= Observed angle at a ‘S’,
= True angle at B, Page 22 of 27
∠ = ∠ = ∠ =
= Observed angle at ‘S’, ,
,
AC = b, AB = c and BC = a, O = Point of intersection of lines AB and CS. 1) The angles CAB and ACB are known by observations to B from A and C respectivel y. The length of the side AC is known by computations from the adjacent triangle. The sides AB and BC can then be calculated by applying sine rule to the triangle ABC. Thus,
= = = =
---- Equation (1) ---- Equation (2)
∠
In the above expressions, may be taken equal to instance to calculate the sides AB and BC.
180 ∠ ∠
, at the first
2) Knowing the sides AB and BC, and the eccentric distance SB, triangles ABS and CBS can be solved by sine rule to get the values of the angles and respectively.
(+) sin = = sin = =
Thus, from triangle ABS, And, from triangle CBS,
Since BS is very small in comparision to BA and BC, the angles and we may write.
= (+) () = ∗206264 = ∗206264 () = ∠ = = ∠ =
and
are extremely small,
---- Equation (3)
And
---- Equation (4)
3) After having calculated the angles B as follows:
and
, the observed angle at S is reduced to that at
---- Equation (5)
The above expression for the true angle does not cover all the four possible cases corresponding to the four positions of the satellite station S, as shown by in Fig. 3.20 (a).
= = ∠ = =
Case I: Position
The true angle
Case II: Position
The true angle
Case III: Position
The true angle
,,
to the left of B [Fig. 3.19 (a) and Fig. 3.18]
to the right of B [Fig. 3.19 (b)] ---- Equation (6)
between AC and B [Fig. 3.19 (c)] ---- Equation (7) Page 23 of 27
=
Case IV: Position
The true angle
[Fig. 3.19 (d)] ---- Equation (8)
Fig. 3.20. Different Other Possible Locations of Satellite Station
To ascertain the signs of corrections (i.e. and ) when a number of angles are observed from the satellite S, it is convenient to assume SB as an arbitrary meridian. The observed angles are then reduced to this meridian and the corrections are computed from the formula,
( ) =
---- Equation (9)
Where = observed angle reduced to the assumed meridian D = Distance from the true station to the observed station The sign of
will be the same as the sign of
sin
.
Thus, in Fig. 3.21 let S = Satellite Station and B = True station, Page 24 of 27
Fig. 3.21. Bearings measured from Assumed Meridian SB
,,,,. ,,, ,,,, ,,, ,,, ,,, ,,,, = = sin are observed stations,
, etc. are the angles to as the reference meridian.
. respectively, reduced to SB by taking SB
= correction corresponding to = distances of
Then
,
;
. from point B respectively.
, etc.
Since is positive, the correction is also positive. i.e., the bearing is also obtained by adding to the bearing (i.e. ). Similarly, since is in the second quadrant, is positive i.e. to get the bearing , will be added to . Knowing the bearings of and the angle will be computed. If and are in the fourth quadrant, and will be applied negatively.
3.18 Extension of Base: Base Net The base lines are usually much shorter than the average length of the triangle sides. This is mainly due to two reasons: i. ii.
It is often not possible to get a favourite site for a longer base; and It is difficult and expensive to measure long base lines.
Hence, in connecting the comparatively short base line to the main triangulation, badly conditioned figure must be avoided by expanding the base in a series of stages. The group of triangles means for extending the base is known as the base net.
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There are a great variety of the extension layouts, but the following important points should be kept in mind in selecting the one: i. ii. iii.
Small angles opposite the known side must be avoided. The net should have sufficient redundant lines to provide three or four side equations within the figure. Subject to the above, it should provide the quickset extension with the fervent stations.
Fig. 3.22. Extension of Base Fig. 3.22 represents such a base net in which it is required to extend the base AB. The following steps are necessary: 1) Select two stations E and F to the either side AB such that AEB and AFB are wellconditioned triangles. 2) In the line AB, prolonged very accurately with the help of a theodolite, choose a favourable position C from which E and F are both visible, and which forms wellshaped triangles AEC and AFC. Thus if the angles at A, E, F and C are about , and those at base about , both sets of the triangles will be well-conditioned. 3) In the triangle AEB, AB is measured, and hence EB can be calculated by measuring all the three angles, A, E and B. From triangle EBC, BC can be computed from known side EB and the measured angles at E, B and C. 4) Similarly, BC can also be calculated from triangles AFB and FBC. Thus, two values of BC are obtained. 5) Two more values of BC = is extended to AC. Similar procedure can be adopted if further extensions to D etc. are required.
45
90
Fig. 3.23 shows various typical forms of base extensions.
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Fig. 3.23. Base Extension (Other forms)
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