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# Tachymetry survey POLITEKNIK MELAKA

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### Tachymetry survey POLITEKNIK MELAKA

1. 1. C 2005 / 2 / 1 ENGINEERING SURVEY 2 Article I. Article II. MODULE MALAYSIAN POLYTECHNICS MINISTRY OF EDUCATION UNIT 1UNIT 1
2. 2. ENGINEERING SURVEY 2 C 2005 / 1 / TACHYMETRY OBJECTIVES General Objective : To know and understand the basic concepts of distance measurement. Specific Objectives : At the end of the unit you should be able to :-  Explain the basic concepts of Optical Distance Measurement.  Discuss the system that has been use in tachymetry.  Calculate the distance by using the tachymetry system.  Explain the procedure to implement the field work  Explain the steps to process the observation data.  List errors in tachymetry survey. 2 U NI
3. 3. INPUTINPUT ENGINEERING SURVEY 2 C 2005 / 1 /  Explain the application of tachymetry in land surveying 1.1 INTRODUCTION The word tachymetry is derived from the Greek takhus metron meaning ‘swift measurement’. It is a branch of surveying where height and distances between ground marks are obtained by optical means only. An example of tachymetry method is the stadia method. This method employs rapid optical means of measuring distance using a telescope with cross hairs (Figure 1.1) and a stadia rod (one stadium = about 607 feet). The distance between marks can be obtained without using a tape. The tachymeter is any theodolite adapted, or fitted with an optical device to enable measurement to be made optically. Figure 1.1 Two Types of Stadia Hair 1.2 PRINCIPLES OF OPTICAL DISTANCE MEASUREMENT The tachymetry measurements are based on a common principle. Consider an isosceles triangle; the perpendicular bisector of the base is directly proportional to the length of this base. If the base length and paralactic angles are known, then the length of the perpendicular bisector can be calculated. (Figure 1.2) 3 Cross Hair reticle i = Stadia Interval
4. 4. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.2 Isosceles Triangle Geometry (Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed) Distance AB = ½ (Cd) x Cot α/2 If distance AB = D, distance Cd = S , so Whereby D = distance between two point S = base line α = paralactic angle 1.3 TACHYMETRY SYSTEM The alternatives of the tachymetry system are classified based on the basic principles, which are: a) Fixed angle: 1) The stadia system i) Incline Sights With The Staff Vertical ii) Incline Sights With The Staff Normal 4 D = ½ S Cot α/2
5. 5. ENGINEERING SURVEY 2 C 2005 / 1 / b) Variable angle 1) tangential system – vertical staff 2) subtence system – horizontal staff The theodolite is a standard instrument in each case. It is modified to suit the conditions. 1.3.1 The Stadia System The diaphragm in this system contains two additional horizontal lines known as stadia hairs. It is placed equidistant above and below the main horizontal cross hair (Figure 1.3). The distance between these stadia hair is called the stadia interval (Figure 1.1). This stadia interval is usually a constant, providing fixed-hair tachymetry. This interval may be altered on some instruments and the movement being measured on a micrometer. Figure 1.3 The View In The Telescope (Source: Ukur Kejuruteraan Asas, Abdul Hamid Mohamed) Observations are made on to a leveling staff which acts as the variable base. In the telescope’s field of view the stadia subtend a certain length of the staff or called staff intercept, which is greater the farther off the staff is held. The staff intercept is proportional to its distance from the instrument and so from this observed length of the staff the distance between it and the tachymeter can be obtained. 5
6. 6. ENGINEERING SURVEY 2 C 2005 / 1 / 1.3.1.1 The Stadia Formula The stadia method of providing the horizontal distance between instrument and staff is shown in Figure 1.4. This technique is always used in stadia tachymetry for engineering survey. The telescope consists of two centring tubes. The eyepiece and diaphragm are built at the end of tube. Move the object glass which is built at the other side when doing focusing. When the telescope is in focus, the image of the staff AB will be formed at ab in the plane of the diaphragm. Then a ray of light will emerge parallel to the optical axis similarly with the ray from B as shown. The rays here will form two similar triangles each with their apex at F, the base of the smaller triangle at the object glass being equal to the stadia interval i. Eyepiece Diaphragm Vertical axis Picket Figure 1.4 Stadia Principle (Source Land Surveying, Ramsay J.P. Wilson) f --- the focal length of the object glass F – the outer focal point of the object glass i --- the stadia interval ab I--- the distance from the outer focal point to the staff D---the horizontal distance required s--- the staff intercept AB c---the distance from object glass to instrument axis From these similar triangles: 6
7. 7. ENGINEERING SURVEY 2 C 2005 / 1 / i f s l = but l = D – (f + c), So, the stadia formula: i f s c)(f-D = + The term f / i is a constant in the stadia formula and is known as the stadia or multiplying constant and may be denoted by the letter K. The term ( f + c) partly of the constant f and partly of the variable c, which varies as the object lens is moved in focusing. However the variation in c is small, especially for sights greater than 10m, and for all practical purposes may also be considered a constant. The term ( f + c), usually about 300 to 450mm in this telescope, is known as the additive constant and may be denoted by the letter C. This reduces the stadia formula to the simple linear equation: CKsD += 1.3.1.2 The Analactic Lens Do you know who J. Porro is? He is the man who invented the analactic lens in 1840. In order to save the labour of multiplying the staff intercept each time and the adding the constant for the particular instrument, it would obviously be simpler if K were to be 100 and C zero. This would provide a stadia formula of D = 100s and calculation would merely consist of moving the decimal point of the staff intercept reading two places to the right. Most of the vernier instruments still in use today do not have an accurate K value of 100, but most modern tachymeters generally do. In 1840, the elimination of the additive constant was achieved by an Italian, J. Porro, when he invented the analactic lens. The inclusion of a second convex lens fixed in relation to the object glass had the effect of bringing the apex of the measuring triangle, the analactic 7 s i f cfD =+− )( )( cfs i f D ++=
8. 8. ENGINEERING SURVEY 2 C 2005 / 1 / point, into exact coincidence with the vertical axis of the instrument, as illustrated in Figure 1.5. Figure 1.5 The analectic Telescope (Source : Land Surveying, Ramsay J.P. Wilson) The term f / i = 1/100 become K = 100. Distance for f and c become similar but in the opposite side. Therefore C = 0. The stadia formula would now become KsD = , the additive constants are eliminated. This externally focusing telescope is known as an analactic telescope. 1.3.1.3 Evaluation of Stadia Constants In most modern surveying telescopes the stadia constant is designed to be 100 and the additive constant 0. To confirm the value of these constants or to establish the stadia of an old or a new instrument, the following fieldwork should be carried out (Figure 1.6) Figure 1.6 Evaluation of Stadia Constants, K and C (Source: Asas Ukur Kejuruteraan, Abdul Hamid Mohamed) 8 Object glass Focus point Analactic point Analactic lens Diaphragm
9. 9. ENGINEERING SURVEY 2 C 2005 / 1 / a) Choose a fairly level ground b) Set out four pegs A, B, C, and D on that ground. AB is 100m, AC is 40m and AD is 90m. c) Set up the tachymeter over the peg at A and observe to a staff that held at C. d) Not the staff intercepts. e) Transfer the staff to D and note the staff intercepts. Distance Stadia Reading Staff Intercept 40 90 1.620, 1.420,1.220 1.871,1.421,0.971 0.400 0.900 Table 1 Obsevation Data ( Source: Asas Ukur Kejuruteraan, Abdul Hamid Mohamed) The observation data is shown in table 1. K and C can be calculate by using the stadia formula, D = Ks + C. D is the distance between staff and the tachymeter, s stands for staff intercept. 40 = 0.4 K + C ------------------------------ (1) 90 = 0.9 K + C ------------------------------ (2) Now, we can solve the problem by using simultaneous equation. (2) – (1) 90 – 40 = 0.9 K – 0.4 K 50 = 0.5 K K 0.5 50 = K = 100 Replace K =100 in (1) 40 = 0.4 ( 100) + C C = 40 -40 C = 0 1.3.1.4 Inclined Sight 9
10. 10. ENGINEERING SURVEY 2 C 2005 / 1 / As height differences between staff positions and instrument increase, it will become impossible to use the horizontal line of sight which so far has only been considered. In such case a tachymeter must be used to provide an inclined line of sight and the angle of elevation or depression must be recorded. The stadia formula must now reflect the angle of inclination of the line of sight and two such cases arise: a) where the staff is held vertically at the far station b) where the staff is held to the line of sight from the instrument 1.3.1.4.1 Incline Sights With The Staff Vertical Figure 1.7 shows that an observation of an inclined sight to a staff held vertically. A, X and B are the readings on the staff and A’, X and B’ are those which would have been taken had the staff been swung about X to position it at right-angles or normal to the line of sight. In figure 1.7, s = the staff intercept AB h = the length of the centre hair reading from the staff base V = the vertical component XY, the height of the centre hair reading above (or below) the instrument axis D = the length of the line of sight IX H = the horizontal distance required. H I = instrument height Figure 1.7 Incline Sight With The Staff Vertical 10
11. 11. ENGINEERING SURVEY 2 C 2005 / 1 / (Source : Land Surveying, Ramsay J.P. Wilson) From the stadia formula D = Ks + C, it can be seen that the term s in this case is the distance A’B’ normal to the line of sight. However, the observed value of s is the length AB, so A’B’ actually equal s, cos θ almost exactly. Therefore the length of the inclined sight D = Ks + C , but H, the horizontal distance actually required, obviously equals D= cosθ , therefore the stadia formula now becomes: H = Ks cos2 θ + C cos θ From the right angled triangle IXY can been seen that: V = D sin θ But D = Ks cos θ + C V = Ks cos θ sin θ + C sin θ But cos θ sin θ = ½ sin 2θ ∴V = ½ Ks sin 2θ + C sin θ In instruments where the additive constant is zero and K = 100, these formulae are simplified as follows: H = 100s cos2 θ V = (100/2) s sin 2θ To obtain the reduced level at the staff position where the reduced level of the instrument station is known, the height difference between the points is applied as follows: Difference in height, dH = H. I. ± V –h Where H.I = the height of instrument (always positive) V = the vertical component (positive for angles of the elevation, negative for angles depression) h = the centre hair reading (always negative) The reduced level of the instrument position I plus the difference in height equal the reduced level of the staff position S. Therefore: R.L.s = R.L.I + H.I ± V – h 11
12. 12. ENGINEERING SURVEY 2 C 2005 / 1 / Example 1: In this example, the value of hi cannot be seen on the rod due to some obstruction. Here, a rod reading of 2.72 with a vertical angle of -6º 37’ was booked, along with the h of 1.72 and a rod interval of 0.241., Calculate the horizontal distance and the vertical distance. Then find the elevation for station 3. Figure 1 Solution: H = 100s cos2 θ = 100 x 0.241 x cos2 6º 37’ = 23.8m V = (100/2) s sin 2θ = 100 s cos θ sin θ = 100 x 0.241 x cos 6º 37’x sin 6º 37’ = -2.76m R.L.3 = R.L.2 + H.I ± V – h = 185.16 + 1.72 +- 2.76 – 2.72 = 181.40 12
13. 13. ENGINEERING SURVEY 2 C 2005 / 1 / So, the elevation for station 3 is 181.40, 1.3.1.4.2 Incline Sights With The Staff Normal Figure 1.8 shows the observation on a staff held normal to an inclined line of sight. The same notation applies as in figure 1.7. Figure 1.8 Incline Sight With The Staff Normal to The Line of Sight (Source: Land Surveying, Ramsay J.P. Wilson) This time the staff reading normal to the line of sight is the actual reading and does not have to be reduced as in the previous case. Therefore D = Ks + C But H= D kos θ ± (the distance from point X to the vertical through the staff base) H = (Ks + C) cos θ ± h sin θ As before V = D sin θ, therefore: V = (Ks + C) sin θ In instruments where the additive constant C is zero, K = 100 and the value of θ is less than 10º (the assumption is generally made that the term h sin θ is zero), these formulae can be simplified as : 13
14. 14. ENGINEERING SURVEY 2 C 2005 / 1 / H = 100 s cos θ V = 100 s sin θ To obtain the reduced level at the staff station, then the height difference between the points is first reduced as follows: Difference in height, dH = H. I.± V – h cos θ 1.3.2.3 Comparison of Methods Conditions Staff Normal/Staff Vertical a) When holding staff ♣ Staff can be held vertically with greater ease than in the normal position. ♣ It’s simpler to plumb a staff with a staff bubble than hold the staff normal to a line of sight. ♣ For normal holding, it needs to be attached with a peep-sight perpendicular to the face of the staff, so the staff-man can sight towards the instrument. ♣ In bush the peep-sight may be obscured, preventing normal holding, while the upper part of the staff is still available for sighting in the vertically held position. ♣ The normal position may also be found by swinging the staff until the lowest possible reading of the centre cross hair is obtained. However, it is difficult to signal to the staff-man the correct position in the bush. Conditions Staff Normal/Staff Vertical b) Reduction of observation ♣ The vertical staff reduction formulae are simpler than the normal staff reduction formula when the h sin θ and h cos θ are included in the normal formulae. c) Careless staff holding ♣ Errors of distance and elevation are very much more marked when there is a deviation from the normal position especially on steep sights. The normal position may also be found by swinging the staff until the lowest possible reading of the centre cross hair is obtained. However, it is difficult to signal to the staff-man the correct position in the bush. 14
15. 15. ENGINEERING SURVEY 2 C 2005 / 1 / 1.3.2 The Tangential System In this system the paralactic angle subtended by a known length of staff is measured directly. Figure 1.9 shows the method where observations are taken to an ordinary levelling staff held vertically. Figure 1.9 The Tangential System of Tachymetry (Source: Land Surveying, Ramsay J.P. Wilson) The instrument is set up and the vertical circle is read on both faces to give the angle of elevation (or depression) to a whole staff graduation. This process is repeated to another whole graduation to give as large a staff intercept, s, as possible. From the staff intercept and the two observed vertical angles θ and φ, the horizontal distance H may be calculated as: AY = H tan θ BY = H tan φ AY –BY = s = H ( tan θ – tan φ) ∴ )tan(tan φθ − = s H or )tan(tan θφ − = s H ( for sight down hills) The difference in height between the instrument station and the staff station is found as follows : 15
16. 16. ENGINEERING SURVEY 2 C 2005 / 1 / Vertical component, V = BY = H tan φ Height difference, dH = H.I ± V – BX Check : AY=H tan θ ( when dH = H.I. ± V – AX) 1.3.3 The Substance Bar The substance system is a particular form of the tangential system where the measured base is held horizontally as illustrated in Figure 1.10 instead of vertically. The paralactic angle is measured with greater accuracy using the horizontal circle instead of the vertical circle The horizontally held base is especially made for this purpose and is known as a substance bar. The substance bar is a specially made instrument supported on a tripod with two sighting targets set a precise distance apart, usually 2m. The central target in the substance bar is placed midway between the end targets for traverse angle measurement and for use in sighting with the auxiliary base method. The sighting device is fixed at right-angles to the line of the bar so that it may be positioned at right angle to the line of sight from the theodolite. As temperature affects the bar length, the subtence bar targets are usually attached to invar rods or wires, which have a low coefficient of expansion, so that their nominal distance apart remains almost constant. The targets may be lit from behind for night observations, which have the advantage of a less disturbed atmosphere resulting in increased accuracy in the angular measurement. 16
17. 17. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.10 The Principles of Horizontal Distance Measurement ( Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed) From the figure above, it can be seen that the horizontal distance 2 cot 2 αs H = because half the bar length divided by the perpendicular sector of the isosceles triangle of half the measured angle α. Usually the bar is 2m long to simplify the calculation. So 2 cot α =H . As the paralactic angle is measured on the horizontal plane, the distance obtained is always the horizontal distance and no slope corrections are ever necessary however far above or below the theodolite the substance bar may be. If height differences between theodolite and bar stations are required then a vertical angle θ must be measured to the line of the bar and the vertical component calculated from the formula V = H tan θ (figure 1.11). The height of the theodolite above its station (Hi) and the height of the bar above its station (Hb) must be measured. Then the height difference between stations X and Y(dHXY) is shown as below: dH = Hi ±V – Hb where Hi = Height of the theodolite V = vertical component Hb = Height of substance bar So, the reduced level of the staff position Y, RL x = RL Y + dHXY 17
18. 18. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.11 The Height Difference Between Stations ( Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed) 18
19. 19. ENGINEERING SURVEY 2 C 2005 / 1 / Activity 1a 1.1 What is meant by the term ‘tachymetry’? 1.2 Explain the basic principles upon which tachymetric measurement are based? 1.3 A vertical staff is observed with a horizontal external focusing telescope at a distance of 112.489m. Measurements of the telescope are recorded as : Objective to diaphragm 230mm Objective to vertical axis 150mm If the readings taken to the staff were 1.073, 1.629 and 2.185, calculate a) the distance apart of the stadia lines (i) b) the multiplying constant (K) c) the additive constant (C) 1.4 What are the main differences between the stadia system and tangential system? 19
20. 20. ENGINEERING SURVEY 2 C 2005 / 1 / Feedback 1a 1.1 Tachymetry means swift measurement where height and distances between ground marks are obtained by optical means only. 1.2 The tachymetry measurements are based on the common principle of the isosceles triangle. The perpendicular bisector of the base is directly proportional to the length of this base. If the base length and paralactic angle are known, then the length of the perpendicular bisector can be calculated. 20 Try your best to answer these questions.
21. 21. ENGINEERING SURVEY 2 C 2005 / 1 / Distance AB = ½ (Cd) x Cot α/2 If distance AB = D, distance Cd = S , so D = ½ S Cot α/2 Whereby D = distance between two point S = base line α = paralactic angle 1.3 From equation D = Ks + C = )( cfs i f ++ )( cfD fs i +− = )150.0230.0(489.112 )073.1185.2(230 +− − = 380.048.112 )100.1(230 − = = 2.3 mm Therefore, 100 3.2 230 === i f K 21
22. 22. INPUTINPUT ENGINEERING SURVEY 2 C 2005 / 1 / C = f +c = 230+150 = 380mm 1.4 In the stadia system, the apex angle of the measuring triangle is defined by the stadia hairs on the telescope diaphragm. The base length is obtained by observing the intersection of the stadia hairs on the image of the measuring staff seen in the telescope’s field or view. The tangential system in which the apex angle subtended by a basic of known length is accurately measured, usually with the single-second theodolite. In order to obtain the distance between instrument and base, the tangent of the angle or angles observed must be used in the calculation. Well done. You have done a good job!! 1.4 STADIA FIELD PRACTICE Stadia tachymetry is mainly used in surveying details in selected areas. Adequate horizontal and vertical control, supplied by traversing and leveling is required to orientate the survey and to provide station levels. It is best suited to open ground where few hard levels are required. 22
23. 23. ENGINEERING SURVEY 2 C 2005 / 1 / In field practice, the transit is set on a point for which the horizontal location and elevation have been determined. If necessary, the elevation of the transit station can be determined after setup by sighting on a point of known elevation and working backward through equation elevation station (Rod) = elevation station (instrument) + Hi - h ± V. Hi is instrument height, V is the vertical component and h is the centre hair reading 1.4.1 PROCEDURE OF FIELD WORK Figure 1.12 shows an area which needs topographic survey. There are some object illustrated in that figure, such as station (A), building(B), road(C), fence(D) and drainage(E). The procedures below show the way to implement the stadia field works. a) Establish 4 control stations (station 1, station 2, station 3 and station 4) by using wooden pegs. b) Implement horizontal control networks on each station in order to obtain the coordinate for every station. Record the data in a field book. c) After that, implement the leveling process to get the elevation of each station. Enter the observations in the field book. d) Now, use either stadia tachymetry method or stadia electronic method to set the theodolite over station 2. e) Measure the height of the theodolite at station 2 as Hi2 with a steel tape. f) Set the horizontal circle to zero. g) Sight the reference station (Station 1) at 0º00’. h) Sight the stadia point to Station 3 by loosening the clamp (clamp is tight). i) Sight the main horizontal hair roughly on the value of h, then move the lower hair to the closest even foot (decimeter) mark. j) Read the upper hair, determine the rod interval, and enter the value in the notes. k) Sight the main horizontal hair precisely on the h value. l) Wave off the rod holder on point a, point b and point c. m) Read and book the horizontal angle and the vertical angle from station 2 to points a, b and c. Try to take as many details as possible. n) Check the zero setting for the horizontal angle before moving the instrument to station 3. o) Repeat step d to m for observation at station 3(3-d, 3-e,3-f) , station 4 (4-g,4- h,4-i) and station1(1-j). 23
24. 24. ENGINEERING SURVEY 2 C 2005 / 1 / p) Finally reduce the notes (compute horizontal distances and elevation) after field hours and check the reductions. Figure 1.12 Stadia Field Works. (Source: Ukur Kejuruteraan 1 , Baharin Mohammad) 24
25. 25. C 2005 / 2 / ENGINEERING SURVEY 2 1.4.2 Recording of Observation Stadia tachymetry is best booked in tabulated form as below. Station and Instrument Height Horizontal Angle (α) Vertical angle Middle Stadia reading Stadia reading (a –upper reading) b- lower reading) Horizontal Length H = Ks Cos2 θ-C Vertical difference V=(Ks Sin 2θ)/2 - sinθ Difference in Height ΔH = Hi ±V-h Reduced level of station Reduced level of point Remarks Station 2 1.542m 50 23 00 +88 31 0.6m a- 0.890 b- 0.310 57.961m 1.501m 2.443m 100 102.443m Station 1- control station 343 25 00 -92 32 0.5m a- 0.551 b- 0.449 10.068m -1.153m -0.111m 99.889m a- beside drainage 342 57 00 - 96 36 0.4m a-0.454 b-0.346 10.657m -1.233m -0.091m 99.909m b- beside drainage 357 00 00 -96 20 0.8m a-0.837 b- 0.763 7.340m -0.811m -0.069m 99.931m c- road side 305 31 00 -94 28 1.2m a-1.242 b-1.548 8.353m -0.652m -0.310m 99.690m d-tree (radius- 2.7m) 214 16 00 -94 37 1.2m a- 1.230 b- 1.170 5.961m -0.481m -0.139m 99.861m e- beside drainage 220 37 00 -94 05 1.3m a- 1.326 b- 1.274 5.174m -0.369m -0.127m 99.873m f- beside drainage 250 36 00 -94 06 1.3m a- 1.334 b- 1.266 6.765m -0.485m -0.243m 99.757m g- lamp post 255 26 00 -94 23 1.3 a- 1.323 b- 1.277 4.573m --0.351m -0.109m 99.891m h- road side Table1.1 Tachymetry Stadia Method Booking (Source: Ukur Kejuruteraan 1 , Baharin Mohammad) 25
26. 26. Explanation of the booking Column 1 : Station number and height of instrument Column 2 : The bearing of the ray oriented on the control points Column 3: Vertical angle (θ) or the zenith angle. Z = (θ = 90-Z) For example point 1 Z = 93°, then θ = 90°- 93°= 3° 00' Column 4: Middle stadia reading. Column 5 : Upper and lower stadia readings Column 6 : H, the horizontal length = KsCos2θ - C ( Usually 100sCos2θ) by using the data from column 3 and 5. Column 7 : Vertical difference = H tan θ or (usually 50 sin 2θ - C) by using the data from column 3 and 5. Column 8: Difference in height, which is calculated from formula Hi ±V-h Column 9 : Axis level of the station Column 10 : Reduced level of the point : axis level ± (V-h) Column 11 : Remarks amplification of diagram. 1.5 ACCURACY OF STADIA OBSERVATION a) Accuracy of distance measurement Under ideal conditions, it should be possible to obtain an accuracy of 0.01% in distance measurement, but this is seldom achieved in practice. Using an ordinary levelling staff with 10 mm divisions practical accuracies approximate to the following: Distance 20m 100m 150m Accuracy ±100mm ±200mm ±300mm b) Accuracy of height measurement Provided that the staff is held vertically with reasonable care and angles of sighting are less than 10˚, then heights should be accurate to within 0.01 per cent of the sighting distance. 1.5.1 Errors in horizontal distances • The error of careless staff holding can be resolved by using staff bubbles when implementing field observation. • Error in reading the stadia intercept, which is immediately multiplied by 100(K1), thereby making it significant. This source of error will increase C Ks − 2 2sin θ
27. 27. ENGINEERING SURVEY 2 C 2005 / 1 / with the length of sight. The obvious solution is to limit the length of sight to ensure a good resolution of the graduations. • Error in the determination of the instrument constants K1 and K2, resulting in an error in distance directly proportional to the error in the constant K1 and directly as the error in K2. • Effect of differential refraction on the stadia intercepts. This is minimized by keeping the lower reading 1 to 1.5m above the ground. • Random error in the measurement of the vertical angle. This has a negligible effect on the staff intercept and consequently on the horizontal distance. In addition to the above sources of error, there are many others resulting from instrumental errors, failure to eliminate parallax, and natural errors due to high winds and summer heat. The lack of statistical evidence makes it rather difficult to quote standards of accuracy; however, the usual treatment for small errors will give some basis for assessment. 1.5.2 Errors in elevations The main sources of error in elevation are errors in vertical angles and additional errors rising from errors in the computed distance. Figure 1.13 clearly shows that whilst the error resulting from errors in vertical angles remains fairly constant, the results from additional errors rising from errors in the computed distance increases with increased elevation. θtanDH = ( ) ( )[ ] ( ) ( )[ ] 046.0 "1sin"205sec2005tan48.0 sectan sec tan 222 2/1222 2 ±= ×°+°±= +±=∴ = =∴ θδθθδδ θδθδ θδδ DDH DH DH This result indicates that elevation need be quoted only to the nearest 10mm. Accuracies of 1 in 1000 may still be achieved in tachymetry traversing, due to the compensating effect of accidental errors, reciprocal observation of the lines and a general increase in care. 27
28. 28. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.13 Errors In Elevation. (Source : Engineering Surveying, W. Schofield) 1.6 PLOTTING After the field observations, the collected data must now be processed. The traverse closure is calculated and then all adjusted values for northing, easting and elevations are computed manually or by computer programs. After that, the data is processed by using software such as TRPS and Autocad. All the details can be plotted the following way. a) Plotting Control Station.(Figure 1.14) ♪ Place the control station on a grid paper. ♪ The grid paper is printed with grid lines at 1mm intervals. ♪ When plotting on grid paper, the stations are defined by using the coordinate system. ♪ Station 1 is assumed as the origin whereby coordinate x is 1000m and coordinate y 1000m. The position of station is plotted starting from the lower left corner of the grid paper. ♪ Scaling along the x-axis from coordinate x station 1, plot the coordinate of station 2, X2. Using the same way also plot coordinate Y2. Repeat this step for station 3 and station 4 28
29. 29. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.14 Determination Of Station Position On Grid Paper. (Source: Ukur Kejuruteraan 1 , Baharin Mohammad) b) Detailing (Figure 1.15) ♪ Instead of using coordinates, plotting can be done by scaling the bearing and distances of a detail. ♪ Normally a protractor and a scale ruler are needed in plotting. ♪ Place the circular protractor with its centre station 2 and the zero lined up with the reference station 2-1. ♪ Mark the bearing of θa on the paper against the protractor edge. ♪ Remove the protractor and draw the direction of the line 2-a. Scale the distance and plot the position of a. ♪ Repeat the same steps when marking off point c-j. ♪ Finally, join the points to form the detail. 29
30. 30. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.15 Details Plotting On Grid Paper. (Source: Ukur Kejuruteraan 1 , Baharin Mohammad) c) Contours ♪ After marking all the details, the contours need to be plotted. ♪ Contours are lines on a map representing a line joining points of equal height on the ground. This method is most commonly adopted for larger areas. ♪ Reduced level is placed beside details and the height of each point is spotted. ♪ Finally, contour lines are plotted by using the interpolation method. d) Preparation of Title Block on Tracing Paper. ♪ All topography and engineering drawings have title blocks. ♪ Usually the title block is placed in the right corner of the plan, which also has the logo client, project name, date of project and other details.(Figure 1.16). 30
31. 31. ENGINEERING SURVEY 2 C 2005 / 1 / ♪ Revisions to the plan are usually referenced immediately above the title block, showing the date and a brief description of the revision. ♪ The title block is often of standard size and has a format similar to that in figure 2.4. ♪ All the drawings on tracing paper are done manually by using the technical pens with Indian ink or by using AutoCad software. ♪ The size of the technical pen is determined based on texts and lines required in a drawing. Figure 1.16 Title Block (Source: Ukur Kejuruteraan 1 , Baharin Mohammad) e) Final drawing. ♪ After completing the title block, transfer all the drawings from the grid paper to tracing paper. ♪ Then, write the additional text or draw lines and symbols in that drawing. ♪ The texts refer to the name of building, road, and values of height and contour intervals. ♪ Plot the text horizontally for all values except the value of height. ♪ Every detail has lines of different types and sizes. (For example, the root line of hedge is shown in black and the outline in green) ♪ Finally, plot the details by using a plotter. (Figure 1.17). 31 Scale Direction Grid value Logo client Plan number Plan title Datum explanation Legend Explanation of observation, Land Survey Firm, Name of surveyor, date, plan reference number and others.
32. 32. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.17 Details That Plot On A 8-Pen Plotter. (Source: Surveying With Construction Application, B.F. Kavanagh) 1.7 APPLICATION This method is easy to apply in the field, but unless a direct-reading tachymeter is used, the resultant computation for many ‘spot-shots’ can be extremely tedious, even with the use of a computer program. The very low order of accuracy and its short range limit its application to detail surveys in rural areas or contouring. 1.7.1 Detail Survey The theodolite is set up at a control station A ( Figure 1.18) and oriented to any other control station (RO) with the horizontal circle set to 0º 00’. Thereafter the bearings (relative to A–RO) and horizontal length to each point of detail (P1, P2, P3, etc) are obtained by observing the stadia readings on a staff held there, the horizontal circle reading (φ1, φ2,φ3, etc) and the vertical angle. The cross hair-reading is also required to compute the reduced level of the point. The field data is booked as shown in table 2.1. Note that the angles are required to the nearest minute or arc only. It is worth noting that the staff-man should be the most experienced member of the survey party who would appreciate the error sources, the limit accuracy available and thus the best and most economic staff positions required. 32
33. 33. ENGINEERING SURVEY 2 C 2005 / 1 / At station A Grid ref E 400, N300 Weather Cloudy,cool Stn level (RL) 30.84m OD Ht of inst (hi) 1.42m Axis level (RL + hi) 31.90m(Ax) Survey Canbury Park Surveyor J. SMITH Date 12.12.83 Staff point Angles observed Staff readings Staff intercept Horizontal Distance Ks cos2 θ Vertical Angle K/2 s sin 2 θ Reduced level Remarks Hori- zontal Vertical circle Vertical angle ° ′ ° ′ ° ′ s D ± V Ax ± V-h RO 0 00 Station B P1 48º12’ 95º20’ -5º20’ 1.942 1.404 0.866 1.076 106.67 -9.96 20.54 Edge of pond P2 80º02’ 93º40’ -3º40’ 0.998 0.640 0.281 0.717 71.41 -4.58 26.68 Edge of pond P3 107º56’ 83º20’ +6º40’ 1.610 1.216 0.822 0.788 77.74 +9.09 39.77 Edge of pond Table 1.2 Booking Of Field Data (Source : Engineering Surveying, W. Schofield) Figure 1.18 Detail Survey (Source : Engineering Surveying, W. Schofield) 33
34. 34. ENGINEERING SURVEY 2 C 2005 / 1 / 1.7.2 Contouring Contouring is carried out exactly the same manner as above, but with many more spot shots along each radial arm (Figure 1.19). The arms are turned off at regular angular intervals, with the staff –man obtaining levels at regular paced intervals along each arm and at each distinct change in gradient. Subsequent computation of the field data will fix the position and level of each point along each arm, which may then be interpolated for contours. Figure 1.19 Contouring (Source : Engineering Surveying, W. Schofield) 1.8 FURTHER OPTICAL DISTANCE-MEASURING EQUIPMENT 1.8.1 Direct-Reading Tachymeters Direct-reading tachymeters or self reducing tachymeters as they are also called, have curved lines replacing the conventional stadia lines. Figure 1.20 illustrates one particular make, in which the outer lines are curves to the function cos2 θ and the inner curves are to the function sinθ cosθ. Thus the outer curve staff intercept is not just S but S cos2 θ. Hence, one need only multiply and intercept reading by K1=100 to obtain the horizontal distance. Similiarly, the inner curve staff intercept is S sinθ cosθ, and need only be multiplied by K1 to produce the vertical height H. The separation of the curves varies with variation in the vertical angle. 34
35. 35. ENGINEERING SURVEY 2 C 2005 / 1 / There are other makes of instruments which have different methods of deriving at the solution. However, the objective remains the same-to eliminate computation. It should be noted that there is no improvement in accuracy. Figure 1.19 Outer Lines of Direct-reading tachymeters (Source: Engineering Surveying, W. Schofield) 1.8.2 Vertical-Staff Precision Tachymeter. The vertical-staff tachymeter as produced by Kern and named the Kern DK-RV, has a moveable diaphragm which varies with the inclination of the telescope, the amount of variation being controlled by a gear-and-cam mechanism. It is used with a specially- graduated vertical staff giving horizontal distances to an accuracy of 1 in 5000 over a maximum range of 150m. Figure 1.20 illustrates a portion of the special staff as viewed through the instrument. By rotating the telescope in the vertical plane, the horizontal reticule A is made to bisect the zero wedge. Rotation of the instrument in azimuth is carried out until the sloping reticule B bisects a small circular dot on the left-hand scale. The instrument now reads as follows: Reticule B = 15.00m Vertical reticule C = 0.88m Horizontal distance = 15.88m The same comments apply to this instrument as to the horizontal-staff precision tachymeter. 35
36. 36. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.20 Portion of Vertical-Staff Precision Tachymeter. (Source: Engineering Surveying, W. Schofield) 1.8.3 Total Station When electronic theodolites are combined with interfaced EDMIs and an electronic data collector, they become electronic tachymeter instruments- Total Stations. The total stations can read and record horizontal and vertical angles together with the slope distances. The microprocessors in the total stations can perform a variety of mathematical operations, for example, averaging multiple angle measurement, averaging multiple distances measurement, determining X, Y, Z coordinates and others. The data collected can be handled by a device connected by cable to the tachymeter but many instruments come with the data collector built into the instrument. Data are stored on board internal memory about (1300-points) and on memory cards (about 2000 points per card). The data can be directly transferred to the computer from the total station via cable, or the data transferred from the data storage cards first to a card reader-writer and from there to the computer. This section will be discussed further in Unit 6. 36
37. 37. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1.21 Total Station (Source: Surveying With Construction Application, B.F. Kavanagh) 37
38. 38. ENGINEERING SURVEY 2 C 2005 / 1 / Activity 1b 1.5 List down 5 steps that are needed to produce a topographic map. 1.6 The following observations were taken with a tachymeter, having constants of 100 and zero, from point A to B and C. The distance BC was measured as 157m. Assuming the ground to be a plane within the triangle ABC, calculate the horizontal distance and vertical distance for AB. At To Staff readings (m) Vertical Angle A B 1.48, 2.73, 3.98 + 7° 36’ C 2.08, 2.82, 3.56 -5° 24’ 1.7 In tachymetry survey, the accuracy of stadia observation is affected by several sources of error. Describe 3 of these errors. 1.8 Describe the procedure to implement the stadia field work in tachymetry survey. 38 Look for the solutions now.
39. 39. ENGINEERING SURVEY 2 C 2005 / 1 / Feedback 1b 1.5 There are 5 steps to produce a topographic map. a) Plotting Control Station b) Details plotting c) Contours d) Preparation of title block on tracing paper. e) Final drawing 1.6 Horizontal distance AB = 100 x S cos2 θ = 100 x (3.98 – 1.48) cos2 7° 36’ = 246 m Vertical distance AB = 246 tan 7° 36’ = +32.8m 1.7 There are three kind of errors: a) Careless staff holding. This is minimized by using staff bubbles. b) Error in reading the stadia intercepts. This source of error will increase with the length of sight. The obvious solution is to limit the length of sight to ensure good resolution of the graduations. c) Effect of differential refraction on the stadia intercepts. This is minimized by keeping the lower reading 1 to 1.5m above the ground. 1.8 Establish 4 control stations (station 1, station 2, station 3 and station 4) by using wooden pegs. • Implement horizontal control networks on each. Record the data in a field book. • After that, implement the levelling process to get the elevation of each station. Record the observations in the field book. • Now, use either stadia tachymetry method or stadia electronic method to set the theodolite over station 2. • Measure the height of the theodolite at station 2 as Hi2 with a steel tape. • Set the horizontal circle to zero. • Sight the reference station (Station 1) at 0º00’. • Sight the stadia point to Station 3 by loosening the clamp (clamp is tight). • Sight the main horizontal hair roughly on the value of h, then move the lower hair to the closest even foot (decimeter) mark. • Read the upper hair, determine the rod interval, and enter the value in the notes. • Sight the main horizontal hair precisely on the h value. 39
40. 40. ENGINEERING SURVEY 2 C 2005 / 1 / • Wave off the rod holder on point a, point b and point c. • Read and book the horizontal angle and the vertical angle from station 2 to points a, b and c. Try to take as many details as possible. • Check the zero setting for the horizontal angle before moving the instrument to station 3. • Repeat step d to m for observation at station 3(3-d, 3-e,3-f) , station 4 (4-g,4-h,4-i) and station1(1-j). • Finally reduce the notes (compute horizontal distances and elevation) after field hours and check the reductions. Figure 2 40 I got it !!!
41. 41. ENGINEERING SURVEY 2 C 2005 / 1 / Self Assessment 1) Tachymetry is used to determine the elevation of the instrument station B base on elevation of station A. Explain the tachymetry stadia formula below by using illustrations. R.L.B = R.L.A + H.I + V – h Where: R.L.B = elevation of the instrument station B R.L.A = elevation of the instrument station A H.I.= instrument height h = the length of the centre hair reading from the stsff base V = the vertical component XY, the height of the centre hair reading above the instrument axis 2) A line of third order levelling is run by theodolite, using tachymetry methods with a staff held vertically. The usual three staff readings of centre and both stadia hairs are recorded together with the vertical angle (VA). A second value of height difference is found by altering the telescope elevation and recording the new readings by the vertical circle and centre hair only. The two values of the height differences are then meaned. Compute the difference in height between the points A and B from the following data: The stadia constant are :multiplying constant =100; additive constant = 0. Backsights VA Staff Foresights VA Staff Remarks (all measurements in m) + 0º 02’ 00” 1.890 1.417 Point A 0.945 +0º 02’ 00” 1.908 -0º 18’ 00” 3.109 Point B 2.012 0.914 0º 00’ 00” 3.161 (height difference between the two ends of theodolite ray = 100s cos θ sin θ, where s= stadia intercept and θ = VA) 3) The rod reading made to coincide with the value of the hi, is typical of 90 percent of all stadia measurements. In figure 1, the vertical angle is + 1º 36’ and the rod 41
42. 42. ENGINEERING SURVEY 2 C 2005 / 1 / interval is 0.401. Both the rod hi and the rod reading (R.R.) are 1.72m. Calculate the horizontal distance and the vertical distance. Then find the elevation for station 2. . Figure 2 4) A theodolite has a tachymetry constant of 100 and an additive constant of zero. The centre of reading on a vertical staff held on a point B was 2.292m when sighted from A. If the vertical angle was +25° and the horizontal distance AB is 42
43. 43. Feedback to Self Assessment ENGINEERING SURVEY 2 C 2005 / 1 / 190.326m, calculate the other staff readings and thus show that the two intercept intervals are not equal. Using these values calculate the level of B if A was 37.95m and the height of the instrument 1.35m. 5) The table below shows a tachymetry stadia field booking. Complete the table below and calculate elevation of each point. Station and instru- ment height Hori- zontal Angle (α) Vertical angle Middle Stadia reading a –upper reading b- lower reading Horizont al Length H = 100s Cos2 θ-C Vertical difference V=100s cosθ sinθ Reduced level of station Reduced level of point Remarks Station 4 10.417 1.417 45° 51' -90° 18’ 3.100 3.301 2.900 Beside road 170°18’ -98° 48’ 1.120 1.252 1.000 Lamp post 120°21’ 87° 46’ 2.202 2.475 2.100 Centre line 43 How to answer this??? “Try your best to find the solution.”
44. 44. ENGINEERING SURVEY 2 C 2005 / 1 / 1) The figure above shows R.L.B = R.L.A + H.I + V – h Where: R.L.B = elevation of the instrument station B R.L.A = elevation of the instrument station A. H.I.= instrument height h = the length of the centre hair reading from the staff base V = the vertical component XY, the height of the centre hair reading above the instrument axis 2) V = 100s sin θ cos θ = 50s sin 2 θ To A, V = 50 (1.890 -0.945) sin 0º 04’ 00” = 0.055m Difference in level from instrument axis = 0.550 – 1.417 = -1.362 Check Reading V = 50 (0.945) sin 0º 40’ 00” = 0.550m Difference in level from instrument axis = 0.550 -1.980 44 Theodolite Staff
45. 45. ENGINEERING SURVEY 2 C 2005 / 1 / = -1.358 Mean = 1.360m To B, V = 50 (3.109-0.914) sin -0º 36’ 00” = -1.149m Difference in level from instrument axis = -1.149-2.012 = -3.161 Check level = -3.161 Mean = - 3.161m Difference in level AB = -3.161+1.360 = -1.801m 3) H= 100s cos2 θ = 100 x 0.401 x cos2 1º 36’ = 40.1m V = (100/2) s sin 2θ = 100 s cos θ sin θ = 100 x 0.401 x cos 1º 36’x sin 1º 36’ = +1.12m R.L.2 = R.L.1 + H.I ± V – h = 185.16 + 1.72 +1.12 – 1.72 = 186.28 So, the elevation for station 2 is 186.28. 4) 45
46. 46. ENGINEERING SURVEY 2 C 2005 / 1 / Figure 1 From basic equation, CD = 100s cos2 θ 190.326m = 100 s cos2 25° s = 2.316m From figure 1, HJ = s cos2 25° = 2.316 * cos2 25° = 2.1m Inclined distance CE = CD sec cos2 25° "23'340 210 1.2 2 °==∴ radα "11'170°=∴ α Now by reference to figure 1: DG = CD tan (25°- α ) = 190.326 tan (25° -0° 17’ 11”) = 87.594 DE = CD tan 25° = 190.326 tan 25° = 88.749 DF = CD tan (25° + α ) = 190.326 (25° + 0° 17’ 11”) = 89.910 It can be seen that the stadia intervals are: GE = DE – DG = S1 = 88.749 – 87.954 46
47. 47. ENGINEERING SURVEY 2 C 2005 / 1 / = 1.115 EF = DF – DE = S2 = 89.910 – 88.749 = 1.161 From which it is obvious that the a) Upper reading = (2.292 +1.161) = 3.453 b) Lower raeding = (2.292 – 1.155) = 1.137 Vertical Height DE = h = CD tan 25° = 190.326 tan 25° = 88.749( as above) ∴ Level of B = 37.95 + 1.35 + 88.749 2.292 = 125.757 m 5) 47 =2.316 (Check) CD = 100s cos2 θ…..(Bla bla bla)
48. 48. ENGINEERING SURVEY 2 C 2005 / 1 / Station and instru- ment height Hori- zontal Angle (α) Vertical angle Middle Stadia reading a –upper reading b- lower reading Horizont al Length H = 100s Cos2 θ-C Vertical difference V=100s cosθ sinθ Reduced level of station Reduced level of point Remarks Station 4 10.417 1.417 45° 51' -90° 18’ 3.100 3.301 2.900 40.099 0.210 8.524 a- Beside road 170°18’ -98° 48’ 1.120 1.252 1.000 24.610 3.810 6.904 b -Lamp post 120°21’ 87° 46’ 2.202 2.475 2.100 37.443 1.460 11.092 c-Center line Reduced level of point a = 10.417 +1.417 -0.210 – 3.100 = 8.524 Reduced level of point b = 10.417 +1.417 -3.810 – 1.120 = 6.904 Reduced level of point c = 10.417 +1.417 + 1.460 – 2.202 = 11.092 48 Congratulations, you can proceed to the next unit. IN P U T