Tacheometry pets ucis Cty + Tacheometry is defined as an optical distance measurement method. The other names given to Tacheometry are Tachymetry of Telemetry. ‘As compared to chaining on flat grounds, the accuracy of tacheometric distances is low, but on rough and steep grounds the accuracy is more. ¢ Itis a transit theodolite fitted with stadia diaphragm. The stadia diaphragm consists of two stadia hairs at equal distances, one above and the other below the horizontal hair of the cross- hairs. Important characteristics of Tacheometer 1, Value of the multiplying constant, k = 100. 2. Value of the additive constant, C = 0. 3. Telescope should be fitted with an anallactic lens. 4, Magnification power of the eyepiece is kept high. Exec * It is also called as vertical stave. * Itis a 5-15 m long rod, graduated in decimals of metre. ¢ For small distances up to 100 m, an ordinary leveling staff may be used but beyond this a stadia rod is used, since the graduations of an ordinary levelling staff become indistinct. = 4 : » ‘The staff can be held either vertical or normal to the line of sight.CIVIL ENGINEERING Tacheometry 221 Note: At can be judged when staff intercept is minimum: The staff is held normal to the line of Methods of Tacheometry . There are threé methods of measuring distances by optical mins Stadia Method In a tacheometer the various wires, in addition to the cross-wires on the diaphragm, are known as stadia wires and the vertical distance between these stadia wires is termed as stadia interval. When the parallactic angle a, defined with the help of stadia wires, is kept fixed and the staff intercept is varied, e.g., AB and AB, the method is known as fixed hair method. : Another way used to make the observation ist0__keep the staff intercept fixed, e.g., A B, and A" BY, and vary the parallactic anglepe.g. « aid a’. In this case the stadia wires will have to be Moyed and is accordingly called as the movable hair or subtense method. In both the above method a tacheometer and staff are used to take the observations. Tangential Method In this method observations are made for vertical angles and staff intercepts are obtained with the cross-wires only. % Stadia wires are not used at all. This method of tacheometry is quite similar to the method of trigonometrical levelling. Range Finding This method is used rine the horizontal distance and direction of a line without going to the far end of hie line, instrument used is called as range finder. A fixed base is measurement pute the ranges. The instruments and methods used are based on base angles or the angle of parallax. By this me} izontal distance can be measured but with the help of a level the vertical dist: a Also be measured. A great care sh e taken to observe the staff intercepts during tacheometric surveying. seta V My mecte tet MC ail | Let us assume that line of sight is horizontal and the telescope used is external focussing type. The staff is held vertically at Q.CIVIL_ENGINEERING 222 where, O = Optical center of objective N& Stadia wires A'B' = Stadia intervals C's S = AB = Staff intercept f = Focal length of objective d=Horizontal distance between optical center O and vertical axis of tacheometer. Horizontal cross wire f,f, = The conjugate focal lengths of the centre O and staff, and optical Tn triangles AOB and A'OB' pans A'B 8) From the lens formula Substituting for fa in gam) ftd= Lespsa p= (euro D=Ks+C re equation is Tacheometric Dis : =e : es cctivel ag Distance Equation, K and C are the multiplying and additiv® ‘plyi aCe multiplying constant K = (4) is also called as stadia interval factor. depends upon the stadia interval and the principal focal length of the objecti¥?-CIVIL _ENGINEERING Tacheometry 223 For ease of calculation of distances, the stadia wires are spaced such that the multiplying constant K = 100. s Theadditive constant C pecause the value of d vari telescope on different object (f+ d), Practically speaking, it is a constant value for a tacheometer, es by only a small and negligible amount when focussing the Value of C ranges from 0.25 to 0.35, If an anallactic lens is used into the telescope, it is so plaeed that ab the observations are reduced to the centre of the instrument and the constant O'beéomes zero and the Tacheometric distance equation is simplified as D = Ks, Hence, if the staff intercept is known, the horizontal digtance @an be readily obtained by multiplying it by 100. n Formular for Inclined Sights yet Cel 1. Staff Vertical As the staff is held vertical, the staff intercepiRB i is not normal to the line of sight OC. Draw through C and perpendiculaii#,0C, cutting OA at A’ and OB at Bi -pendicular to OC) ibtend an angle a, (90° - o/2) + al2 We know that the value of o/2 (its value being 17/11" for K = 100) is very small. Hence, the triangles AA’C and BB’C may be assumed to be right angled triangles. AB = A’'C + BIC = AC cos 6 + BC cos ® = (AC + BC) cos 0 =s cos 0224 Surveying CIVIL_ENGINEERINg ee re Inclined distance OC, ‘ L=K.A‘B’ + L= Ks cos 0 + © f D= L cos 0 = (Ks cos 6 + C) cos 0 = Ks cos? 0 + C cos 0 = FC = L sin 0 =SHIL+V-h Elevation of staff station for angle of depression 0 SHI. -V, _ 2, Staff Normal ¢ As the staff at Kylie he al to the line of sight AC therefo, ercept AB is normal to the line of . Case I: Lin in angle of elevation. Where, AB= 6 = intercept, Ci= h = central hair reading, 0= angle of elevation, and OC=L = inclined distance, _ Draw a perpendicular CF” to OF, i: L= Ks +C iCIVIL ENGINEERING Tacheometry 225 OF'= (Ks + C) cos0 D=OF +r RF = (Ks + ©) cos0 +h sind Elevation of the staff station, V = OC sind = L sino = (Ks + C) sing Elevation of staff station = HI. + V—h cos0 L=Ks+C OF'=L cos0 = (Ks + C) D= OF - FF’ = OF" = (Ks + C) cos hs Elevation of staff station, distance of the staff from the instrument if the stadia hairs are movable, and the staff whereas intercept ‘s’ is kept fixed, the tacheometer angle f changes with the staff position. : ‘The diaphragm has an arrangement for the measurement of the stadia interval i Se Each hair of the stadia diaphragm can be moved independently by a separate sliding frame.226 Bury © Each sliding frame is actuated by 4 micrometer wren Wits © When both the stadia hairs coincide with the contra mark On horizontal plane of the line of sight and reading on should be zero. © When an observation is made, the upper head ie rotate’ ti upper target. Similarly, the lower head is rotated till target. The fractions of the turn are read on the graduated determined. The movable hair method is also known as the ver ured with the helj i the micrometer ore ec ee ao hn ‘Sight is inclined and staff held is vertical, thenCIVIL_ENGINEERING Tacheometry pat KS 5in20+Csin 2m If the line of sight is inclined and staff held is normal, then D= (+c }cos0+ sine ve (B-c}sino \ Note : If there is an index error e in the micrometer scrél Method Advantages As compared to the stadia method (fixed hi oe sights which can be taken with great accuracyé 3 , this method is more accurate for long mly targets are to be bisected. Disadvantages 1. The method is slow. 2. It is very difficult to meas) ‘interval accurately. 3. Computations are tedious m comes in the denominator. The tangential: method of tacheométy is generally used when the diaphragm does not have stadia hairs, when the staff is too far from the instrument and it becomes difficult to read the staff. In this method, a st: With two big targets (vanes) spaced at a fixed vertical distance (8) of 2m or 3m ii é measured by sighting the two targets. he vertical intercept V are computed from the values of s, 8, and 0). the angles 0, and 0, are the angles of elevation or depression, thearetio CIVIL_ENGINEERING 228 cence ewveyng? — CCIVEENGINEE RWG Ss aad tan, V =Dtan6, Case 2: Both angles of Depression s *D= tan6, -tand, he V=D tané, Case 2: One angle of elevation and o depression inferior to the statia method. The method should be used only if the © stadia hairs. This method has the following disadvantages. have to be measured, it takes more time in comparison the stadia ‘oceur if the instrument gets disturbed between the two observations. be changes in atmospheric refraction in the Period between the two observations | Cause error. not easily reduced to the horizontal distance and vertical intercept.GIVIL_ENGINEERING _____Taeheometry P __ 228 ¢ following readings were taken with a tachometer with the line of sight horizontal on 4 held vertieal. 0, 1.285, 1.620 m mine ov horizontal distance from the instrument station to the staff station if k = 100 1B Hi: letermaine the RL: of the staff station if the RL, of the instrument station is 101.580 id the height of the trunnion axis is 1.460 in. iontal distance D = ks + 6 6 =0.16m 100(1,620 = 0.956) + 6.15 = 67.165 ma instrument station = 101.58 mi i of trunnion axis = 1.460 m ; RL. of line of collimation = 101.58 + 1,46 = 108.04 m ALL: of staff station = 108.04 = 1.985 = 101.755 m ne the distance between the instrument station P and the staff station Q from the (ba: : pee es striument = 1.400 m (staff vertical) = 6.645) 6.808 1.861 m ‘mine the Ras of Q if that of Bis 900.410 i. keseas’ 6+0.0 100(1,851 = 0.645) eos?*(4°80) = 70.165 m issn +00 a 100 x (1.861 =6.645) sin 9° = 5.588 mi hi. af Q = RL, of P + HL. + V = Staff reading (900.410 + 1.400) + 5.598 = 6.99)Surveying CIVIL ENGINEERINg 230 Determine the distance following data: staff statior between the instrument station P and the staff station Q from the RL. of the line of colimation = 200.150 m Vertical angle = —3°45’ Staff readin -450; 0.900; 0.350 m Also determine the R.L. of Q Take k = 100 and C = 0.0 Sol. Horizontal distance D = ks cos*6 100 x (1.45-0.35) x cos?(3°45' 109.529 m Veritcal distance V = sin (28) = 24100x(1.45-0.35)sin(7°30) = 7.179 m 2 RL. of Q= RL. of line of colimation — V — staff reading = 200.150 ~ 7.179 - 0.9 = 192.071 m Example 4. The following observations were taken with a tacheometer normal to the line of sight. If the staff readings are 1. inclination is 29°30’, determine the horizontal distan at the station P to a staff at Q held 71, 2.64 and 3.57 m, and the angle af| ce between P and Q. Also determine the elevation of Q if the line of colimation is at RL. of 200.00 m. Take k = 100} and C = 0.50. ¥ Sol. Here, r= 2.64m Hoxizontal distance D = (ks + ©) cos@+rsing = [100 (3.57 = 163.621 m Vertical distance V = (Ks + C) sing 1.71) + 0.5) cos (29°30) + 2.64 sin (29°30 = (100 x 1.86 + 0.5) sin (29°30 ') = 91.3837 m RL. of Q= RL. of P+ HL + V — State Teading x cog = RL. of Line collimation + V— Stat reading x cos I = 200.00 + 91.837 — 2.298 = 289.539 m = following observations w. I fo the line of sight ‘re taken with a tacheometer atthe sation Pio a walrat@ melCIVIL_ ENGINEERING Tacheometry 231 Gtaff readings = 1.450; 1.916 _ Angle of depression = 15°30" RL. of P = 201.45 m Height of trunnion axis above the peg at P = 1.315 m 380 m Determine the horizontal distance between P and Q, and the R.L. of Q, Take k = 100 and C = 0.0. Sol. : r= 1.915 m Horizontal distance D = (ks + C)cos@-rsind ee * = [100 x (2.38-1.45) +0) cos (15°30') — 1.915 sin (15°30') i = 89.106 m ‘Vertical distance V = (ks + C)sind = [100 x (2.88 — 1.45)] sin (15°30’) RL. of Q= RL. of P+ HI. — V— reos@ = 201.45 + 1.815 — 24.858 — 1.916 cos (15° 80’) = 176.067 m 4.853 m sean A subtense theodolite was used to determine the horizontal distance of a point from the | instrument station. The micrometer readings of the drum of the diaphragm were respectively 8.425 and 3.930 when the staff intercept was 3 m. The micrometer screw had 100 threads to em, The focal length of the object glass was 225 mm. The distancé of the instrument axis ‘from the centre of the object glass was measured as 200 mm, -P, Pi ae P, Pitch of the screw = 755m = 0.01 cm Ke fip = —-— = 2250 0.01 m= 8,425 + 3.930 = 7.355 C= f + d = 22.5 + 20.0 = 42.50 cm = 0.425 m — _ -2250x3 ¥ a +0.425 = a a ‘ ee 918.17 m eet Cig gs observed with a percentage theodolite corresporiding to angles of elevation of 4% , 1,525 and 2.925, respectively. If the the vertical angle on sighting the staff readingSeeseveas CIVIL ENGINEERING | €gual to the height of the trunnion ais above ground was 4.5%, calculate FO Gi Re sees etarce between instrament and staff @) the elevation of staff station if that of the instrument station was 493.700 From the formula | “Horizontal Gaance V= Dene, V= 100x004 =560m between point B and C, where C is the point at a height that ff the trunnion axis above the ground level. = Horivntal distance a 045-004) 0.005 140 * 0.005 = 0.70 m = 1525 + 0.70 = 2.225 m BL of Q= (493.70 + 2.295) + 5.60 — 1.525CIVIL_ ENGINEERING Tacheometry 233 Vertical distacne 2 i Vea= pkesin(20) Se Va $x100x0. 620sin8°40' = 4.671 m R.L. of A= RL. of line of colimination + Vo, “Let RL. of line of colimination i is 100.00 m _ RL, of A= 100 + 4.671 — 1.610 = 103.061 m _ Staff at B Horizontal distance ~ Hg= 100 x 0.62 cos*(0°10'40") Hey = 62.0(1 — sin?10'40") = 61.999 m = 100 + 0.192 ~ 1.410 = 98.782 m e of R.L. of A and B = 103.061 — 98.782 = 4.279 m A (61,3646)" + (61.999)" -(AB)* ca 3 "2x 61.646 61.999 OT apy 3800.23 + 3843.876 — AB? B = 87.52 m 4.279 = E poise Le 8Surveying _ CIE ENGINEER Horizontal Distance between B.M. and Q D, = 100(2.405 ~ 0.945) cos*(6"12) D, = 144,297 m Vertical distance 1 100 (2.406 - 0.946)ain(12"24') 2 V,= 15 RL, of BLM. = R.L.of Q + 1,600 + 16,676 — 1.675 421,625 = RL. of Q + 15,601 RL, of Q= 406,024 m ve D, between Qand P D, = 100(8.310 — 1,460) con"(4°12) = 186,002 m Vertical distance between P and Q. 6m Horizontal Dista: 100 V, a (9,310 — 1,450) ain (8° 94") " V,= 13,086 m Hol, of P= Rl, of Q 4 1,600 ~ 18,586 — 2.980 406,024 + 1,600 — 18,586 = 2.880 = $91,658 m Instrument at A Intercept, 1 (1.266 — 0,066) x 2 © 1.200 Vertical distance betwoon A & B® we *L2ain(l4a") © 14.618 m Hols, of Pe (Rul. of A+ 1,660) 14,615 0.685 ad AOL, OB Kil, of A+ 16.010 y Wl, of As 870,148 mCIVIL ENGINEERING Tacheometry 235 Objective Questions ‘The multiplying constant of a tacheometer is @ fi ) (fd) +i © (itd @ ftd The diaphragm of a stadia theodolite is normally fitted with two additional @ horizontal hair (©) vertical hair (© horizontal and two vertical hairs @ none of the above Which of the following represents a correct match? @) Movable hair method: The intercept of the staff is kept constant and stadia hair interval is variable (i) Fixed hair method: The intercept on staff is variable and stadia hair intervals is, fixed (ii) Tangential method: The stadia hait/are not used @ only (ii) is correct (©) only (i) and (ii) are correct (© all three statements are correct @ none is correct The stadia method ifi tacheometry is used to determine ( (@ horizontal angles (b) reduce effective length of the telescope © to eliminate multiplying constant @ make staff intercept proportional to its distance from the tacheometer. ‘The additive constant of a tacheometer is generally 10. ai 12. (@) a few cm (®) 100 cm © about 10m @)_ a dimensionless constant ‘The multiplying,constant of tacheometer is generally about (a) 200 (6) 100 ©) 50 @i1 For a tacheometer equipped with an anallactic lens, the additive and multiplying constants are, respectively (2) 0 100 @) 100 0 @o00 @ 100 100 If the spacing of cross-hairs in a stadia diaphragm of tacheometer is 12 mm, and the focal length of the object glass is 24 cm, then the multiplying constant of the tacheometer is @ 2 &) 100 © 0.005 @ 200 If the focal length of an object glass is 25 cm, stadia interval is 1.25 mm and the distance from object glass to the trunnion axis is 15 cm, the additive constant is fa) 01 (b) 0.4 ©) 1.66 @ 20 ‘The intercept of a staff (@) is maximum, if the staff is held truly normal to the line of sight >) is minimun, if the staff is held truly normal to the line of sight ©) decreases, if the staff is tilted away from normal @ decreases, if the staff is tilted towards normal Which of the following methods results in higher accuracy for measuring horizontal distance on rough grounds: (a) chaining (b) taping (© tacheometry @ allof the abovesurvevin GIVI ENGINEER, Additive constant lies im the range of 025 25m ) The multiplying constant is generelly taken as 100. If the staffintereept is kieva the horizontal distance ean be readily obtained by multiplying it by 100, 8 (@) Fur auallactic lens the additive constant Cis 0 and the multiplying constant hE 100, Hence the equation becomes, B=108 1®. (B) Additive constant C= fa = 025 +019 za MA: @) The intercept is minimum it the tal ® eld ay kal to the ie OF SEN