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TACHEOMETRIC SURVEYING
2
UNIT I Tacheometric Surveying
CONTENTS
• Tacheometric Surveying• Tangential, Stadia and sub-tense methods• Stadia Systems• Horizontal and inclined sights• Vertical and Normal Staffing• Fixed and movable hairs• Stadia constants• Anallactic lens• Subtense bar
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Introduction
• Tacheometry – Greek word means quick measure• Measuring horizontal and vertical distance of a points on the earth
surface relatively to one another are determined without using a chain or tape or a separate levelling instrument.
• Preparation of contoured maps or plans with higher accuracy and also, it provides a check on distances measured with the tape.
• Need of Tacheometry : steep and broken ground, deep revines, stretches of water or swamp etc., where chaining is difficult or impossible
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Uses of Tacheometry
Measuring horizontal distances and differences in elevations.
1. Preparation of topographic maps which require both elevations and horizontal distances
2. Survey work in difficult terrain where direct methods are inconvenient
3. Detail filling
4. Reconnaissance surveys for highways, railways, etc.
5. Checking of already measured distances
6. Hydrographic surveys
7. Establishing secondary controlSivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Instrument- Transit theodolite fitted with a stadia diaphragm
In addition to it, convex lens (anallatic lens)is provided between the eye-piece and the object glass at a fixed distance.
Various pattern of stadia diaphragm
-Levelling Staff-Stadia Rod
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Instrument
- Transit theodolite fitted with a stadia diaphragm
-The stadia diaphragm essentially consists of one stadia hair above and the other an equal distance below the horizontal cross-hair, the stadia hairs being mounted in the ring and on the same vertical plane as the horizontal and vertical cross-hairs.
(1) The simple external-focusing telescope
(2) The external-focusing anallactic telescope (Possor`s telescope)
(3) The internal-focusing telescope.
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Tangential System
1. Diaphragm of the tacheometer is not provided with stadia hair
2. Single Horizontal Hair is used
3. Staff consist of two vanes at known distances
4. Two points are required to measure staff intercept
5. Angles, elevations or depressions are measured
6. Tangents are used to measure horizontal distances and elevations
(not generally used)
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Stadia SystemPrinciple: Tacheometric angle is constant
Staff intercept varies with distance between staff and instruments, which forms base
1. Diaphragm of tacheometer is provided with two stadia hair (upper and lower)2. Telescope is directed towards the staff held at a point whose distance from instruments is
to be found3. Difference in these readings gives staff intercept4. Horizontal distance is obtained by multiplying staff intercept by multiplying constant
Two Methods1.Fixed hair 2.Movable Hair Method
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Subtense Method1. Reverse of stadia method2. Staff intercept forms fixed base3. Tacheometric angle according with staff position
Fixed Base : fixed distance between two tangents or vanesInterval between the stadia wires is changed till lines of sight coincide with tangents and the subtended angle is noted
Base may be vertical or Horizontal
Base Vertical : Movable Hair method or Vertical base subtense method
Vertical angle is measured with the help of special diaphragm – high accuracy
Base Horizontal : Horizontal base subtense method Horizontal angle is measured by method of repetition using Transient theodolite
example: Subtense bar method
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Principle of TacheometerBased on Isoscales Triangle
Ratio of
k = ½ cot β/2
f – focal lengthi- stadia interceptSivapriya Vijayasimhan
A
B
QPO
S3S2S1
R
D1D2
D3
A’
B’O’
i β
UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan 11
Tacheometer must essentially incorporate the following features:(i) The multiplying constant should have a nominal value of 100 and the error contained in
this value should not exceed 1 in 1000. (ii) The axial horizontal line should be exactly midway between the other two lines. (iii) The telescope should be truly anallactic. (iv) The telescope should be powerful having a magnification of 20 to 30 diameters.
• The aperture of the objective should be 35 to 45 mm in diameter to have a sufficiently bright image.
• For small distances (say upto 100 meters), ordinary levelling staff may be used. For greater distances a stadia rod may be used.
• A stadia rod is usually of one piece, having 3 – 5 meters length. • A stadia rod graduated in 5 mm (i.e. 0.005 m) for smaller distances and while for longer
distances, the rod may be graduated in 1 cm (i.e. 0.01 m).
UNIT I Tacheometric Surveying
10' 7"
Common Patterns of Stadia Rods
LC of the stadia rods are less than the LC of ordinary Levelling Staff
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UNIT I Tacheometric Surveying
Stadia Tacheometry
Sivapriya Vijayasimhan
A
Object glass O
B
CB’C’A’
}}F Vi
f
d
v u
A, B, and C -the points cut by the three lines of sight corresponding to three wires A’, B’ and C’ - top, bottom and axial hairs of the diaphragm i = interval b/w the stadia hairs (stadia interval) – length of imageAB = S = staff interceptF= focusf = focal length of the objectiveV = vertical axis of the instrumentv=distance between optical centre and imageu= distance between optical centre and staffd= distance between optical centre and vertical axis of instrument
S
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UNIT I Tacheometric Surveying
Similar triangle B’O’A’ and BO’A i/S = v/u v = iu/SProperties of Lens , 1/v + 1/u = 1/f Sub values of v, 1/(iu/S) + 1/u = 1/f S/iu + i/u = 1/f
1 S + 1 = 1u i f
u = S + 1 f i
D = u + d = S + 1 f + d i
D = (f/i) x S + (f+d) f/i – multiplying : f+d – additive constant
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Horizontal Sight
• A, C, and B = the points cut by the three
Sivapriya Vijayasimhan
O is the optical centre of the objective of an external focusing telescopeA, B and C = the points cut by the three lines of sight corresponding to three wires a, b and c = bottom, top and hairs of the diaphragmab = i = interval b/w the stadia hairs (stadia interval)AB = s = staff intercept; f = focal length of the objective
UNIT I Tacheometric Surveying
f1 = horizontal distance of the staff from the optical centre of the objectivef2 = horizontal distance of the cross-wires from O.d = distance of the vertical axis of the instrument from O.D = horizontal distance of the staff from the vertical axis of the instruments. M = centre of the instrument, corresponding to the vertical axis.
Since the rays BOb and AOa pass through the optical centre, they are straight so that AOB and aOb are similar. Hence,
f1 sf2 i
=
Again, since f1 and f2 are conjugate focal distances, we have from lens formula,
1 1 1f f1 f2
+=
Multiplying throughout by ff1, we get f1 = f + ff1 f2Substituting the values of in the above, we get
f1 sf2 i
=
f1 = f + fsi
Horizontal distance between the axis and the staff is D = f1 + d
D = s + (f + d) = k . s + Cfi
UNIT I Tacheometric Surveying
Equation is known as the distance equation. In order to get the horizontal distance, therefore, the staff intercept s is to be found by subtracting the staff readings corresponding to the top and bottom stadia hairs.
Determination of constant K and C
1st method:
In this method, the additive constant C = (f + d) is measured from the instrument while the multiplying constant k is computed from field observations:
1. Focus the instrument to a distant object and measure along the telescope the distance between the objective and cross-hairs,
2. The distance d between the instrument axis and the objective is variable in the case of external focusing telescope, being greater for short sights and smaller for long sights. It should, therefore be measured for average sight. Thus, the additive constant (f + d) is known.
1 1 1f f1 f2
+=
UNIT I Tacheometric Surveying3. To calculate the multiplying constant k, measure a known distance D1 and take the intercept s1 on the staff kept at that point, the line of sight being horizontal. Using the equation,
D1 = ks1 + C or k = For average value, staff intercepts, s2, s3 etc., can be measured
corresponding to distance D2, D3 etc., and mean value can be calculated. Note: In case of some external focusing instruments, the eye-piece-diaphragm unit moves during focusing. For such instruments d is constant and does not vary while focusing.
D1 – C s
2nd method:
In this method, both the constants are determined by field observations as under:
1. Measure a line, about 200m long, on fairly level ground and drive pegs at some interval, say 50 meters.
2. Keep the staff on the pegs and observe the corresponding staff intercepts with horizontal sight.
3. Knowing the values of D and s for different points, a number of simultaneous equations can be formed by substituting the values of D and s in equation D = k.s + C. The simultaneous solution of successive pairs will give the values of k and C, and the average of these can be found.
UNIT I Tacheometric Surveying
For example, if s1 is the staff intercept corresponding to distance D1 and s2 corresponding to D2 we have,
D1 = k.s1 + C . . . . . (i) and D2 = k. s2 + C . . . . . (ii)
Subtracting (i) from (ii), we get
k = D2 – D1
s2 – s1
. . . . . . . . . (1)
Substituting the values of k in (i), we get
C = D1 - s1D2 – D1
s2 – s1
=D1s2 – D2s1
s2 – s1
. . . . . . . . . (2)
Thus equation (1) and (2) give the values of k and C.
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UNIT I Tacheometric Surveying
FIXED HAIR METHOD OF STADIA SYSTEM- Distance between the stadia hair is fixed- Distance between the station and staff = staff intercept x stadia constants
Methods to find Stadia Constants1. Line of sight is horizontal and staff vertical2. Line of sight inclined upwards and staff vertical3. Line of Sight inclined upwards and staff normal4. Line of Sight inclined downwards with staff vertical5. Line of sight inclined downwards and staff normal
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
1. Line of Sight Horizontal and Staff Vertical
General Tacheometric equation : =100 & = 0RL of Staff station, P = Hi – hWhere as Hi = RL of BM + BS
BS = Back Sighth = central hair readingSivapriya Vijayasimhan
BS
D
O
O’ S
h
Height of Instrument (Hi)
BMP
UNIT I Tacheometric Surveying
C
A
O’
DO
A`
B
C`
P
P’
h
V
Ө
α
L
2.Line of sight inclined upwards and staff vertical
α
S
Line of axis
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UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan
Ois the optical centre of the objective of an external focusing telescopeA, B and C = the positions of staff corresponding to the cut points of the stadia and central hairsS= AC = staff intercepth=central hair readingV=vertical distance between instrument axis central hairD=horizontal distance between instrument and staffL=inclined distance between instrument axis and Bθ = angle of elevationα = angle made by outer and inner rays with central ray
A’C’ is drawn perpendicular to central ray, O’BL = +
Consider triangle ABA’ and CBC’ ABA’ = CBC’ = AA’B = 90o + α BC’C = 90o –α hence α is very small, it is taken as zero
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UNIT I Tacheometric Surveying
AA’B and BC’C = 900
So, AC’ = AC cos = S cos
Then
x +
Sivapriya Vijayasimhan
UNIT I Tacheometric Surveying
C
A
O’
DO
A`
B
C`
P
P’
h
V
Ө
α
L
3.Line of sight inclined upwards and staff normal
α
S
Line of axis
h c
os
θ
h sin θ
L cos θ
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UNIT I Tacheometric Surveying
Vertical height of central hair = h cos θHorizontal distance between O and B = L cos θHorizontal distance, PP’ = h sinθ
Since staff is normal to line of collimation,
Horizontal distance, +h sin + +
Vertical Distance,
Sivapriya Vijayasimhan
RL of staff station, P = (RL of instrument axis) +V – h cos
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UNIT I Tacheometric Surveying4.Case IV: Line of Sight Inclined Downwards with staff vertical
Sivapriya Vijayasimhan
D Line of Axis
A B C’ C
V
h
P’
P
O
θO’
sin)(2
2Sin Sx
cos)(cosx 2
dfi
fV
dfSi
fD
RL of Staff P = (RL of axis of instrument) – V- h
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UNIT I Tacheometric Surveying5.Case V: Line of Sight Inclined Downwards with staff normal
Sivapriya Vijayasimhan
Line of Axis
A B C
V
h cosθh
P P1
O
θO’
RL of Staff P = (RL of axis of instrument) – V- h cos θ
DL cos θ
L
h sinθ
sin)(sin
sin
sincos)(cos
)(
dfSi
fV
LV
hdfSi
fD
dfiSi
fL
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UNIT I Tacheometric SurveyingMovable Hair method of Stadia SystemPrinciple- Distance between stadia wires varies: staff intercept is constant- Staff has two tangents at known distance and third target at middleInstrument- Theodolite +subtense diaphragm = Subtense Theodolite- Upper and lower stadia wires can moved in vertical plane by using micrometer screws
- Distance = Turns of micrometer screws- Complete turns is read on scale and fractional parts on top and bottom eye piece- Sum of micrometer readings = total distance moved by stadia wiresSivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Observation- Middle target is bisected by central fixed hair- Micrometer screws are operated to move stadia wire up and down- Upper and lower targets are bisected by top and bottom wires
1.Line of Sight is horizontal
Where,C – constant varying from 600 to 1000n – number of readings in micrometerS – staff intercept (distance b/w upper and lower targets)
2.Line of Sight is inclined
Sivapriya Vijayasimhan
d)(fx n
CSD
d)cos(f
cos2
n
CSD
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UNIT I Tacheometric Surveying
Subtense BarInstrument used to measure horizontal distance between instrument and a point on ground
Instrument• Theodolite – ordinary transit theodolite• Subtense bar made of metal of varying length 3 to 4 m• The bar can be locked in position by clamping screws • The bar can be levelled with the help of circular sprit level on the top. • At the mid point of the bar, a telescope arrangement or a sight rule with pair of vanes is
provided to align the bar perpendicular to the line of sight • Two targets are placed on the either ends of the bar such that they are equidistant from
the mid point • No staff is needed
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan
Alidade: line of sight perpendicular to the axis of the bar
Target
3 to 4 m
Spirit LevelTarget
Tripod
Alidade
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UNIT I Tacheometric Surveying
Procedure
BAC is measured by method of repetition , θAP is perpendicular to BC and bisects P
Note: –ve error in measurement of θ produce +ve error in D and vice-versa
Sivapriya Vijayasimhan
A P
C
B
S
D
2cot
2
122
tan
2/
2tan,
SD
SD
D
SBAC
θ
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UNIT I Tacheometric Surveying
If an error of δθ ( -ve) will cause an error of δD (+ve)
If an error of δθ ( +ve) will cause an error of δD (-ve)
Sivapriya Vijayasimhan
D
D
D
D
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UNIT I Tacheometric Surveying
Tangential System Of Tacheometry- No stadia hairs- Levelling staff with vanes or targets at known distance- Horizontal and vertical distances are measured by measuring the angles of
elevation or depression
Methods Case I : Both Angles of target are Angles of elevationCase II : Both angles of target are Angles of DepressionCase III : One angle is angle of elevation and the other is angle of depression
Sivapriya Vijayasimhan
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UNIT I Tacheometric Surveying
Case I : Both Angles of target are Angles of elevation
Sivapriya Vijayasimhan
O’
O
S
h
V
B
A
D
C1
C2
θ2θ1
O’ -Instrument axisO – Instrument stationC1 – Staff stationV – vertical distance between lower vane and axis of instrumentS – distance between the targetsθ1 - vertical angle by upper targetsθ2 - vertical angle lower targetsh – height of lower vane above the staff station
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UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan
21
2
21
21
2
1
tantan
tan
tantan
)tan(tan
tan
tan
SV
SD
DS
DV
DSV
21
2
21
21
2
1
tantan
tan
tantan
)tan(tan
tan
tan
SV
SD
DS
DV
DSV
RL of station C1 = RL of instrument axis + V - h
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UNIT I Tacheometric Surveying
Case II : Both Angles of target are Angles of depression
Sivapriya Vijayasimhan
V
S
h
θ1 θ2
C2
A
B
C1
O’
O
D
12
2
12
12
1
2
tantan
tan
tantan
)tan(tan
tan
tan
SV
SD
DS
DSV
DV
RL of station A = RL of instrument axis - V - h
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UNIT I Tacheometric Surveying
Case III : One angle is angle of elevation and the other is angle of depression
Sivapriya Vijayasimhan
SV
h
θ1θ2
C2
C1
O’
O
D
21
2
21
1
2
tantan
tan
tantan
tan
tan
SV
SD
DVS
DV
RL of station A = RL of instrument axis - V - h
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UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan
Anallatic Lens- Convex lens between the object glass and diaphragm to make additive constant as zero- Reduces the brilliance of image- Distance = difference in stadia hair x multiplying constant (100)
DK
d
S
mf2 f1
V P
b2
a2
A
b
i’
a
b3
a3
f'a1
b1
N
N’
A
B
Oi
O – optical centre of object glass: A – optical centre of anallatic lensP – principal focus of anallatic lensK – distance between object glass and anallatic lensD – distance between vertical axis of instrument and staffS - staff interceptv – vertical axis of the instrument
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UNIT I Tacheometric Surveyingf - focal length of the objectivef1 and f2 – conjugate focal lengths of object glassf' – focal lengths of anallatic lensd - distance between optical centre and vertical axis of instrumentm - distance between optical centre and real image , abi – length of image a3b3 when anallatic lens is not providedi ‘– length of image a3b3 when anallatic lens is provided
Ray of light from AB along AN and BN’, meets at PP –principal focus of anallatic lensDiverging ray from P emerge direction parallel to axis of telescope after passing through
anallatic lensReal image “ab” is formed
Without inclusion of Anallatic lens, law of lenses
Sivapriya Vijayasimhan
2
1
21
111
f
f
i
S
fff
1
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UNIT I Tacheometric Surveying
With inclusion of Anallatic lens, law of lenses
With inclusion of anallatic lens, imaginary object a2b2 is seen.
Final image “ab” is formed in stadia hairs
Eliminate m,f2 and I, from equation 1 and 2;
Sivapriya Vijayasimhan
Km
Kf
i
i
KmKff
2
1'
'
111
1''
)('
1
1
1
2
2
1
12
11
12
i
i
f
Kf
Kfi
i
KmSubstitute
ff
fff
ff
Sf
ff
Si
if
S
f
in equation 2
2
3
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UNIT I Tacheometric Surveying
Sivapriya Vijayasimhan
Substitute values of i and f2 from equation 3 , we get
df
fdfK
Kff
fKfd
dKff
fKf
Kffi
SffD
dfDNow
Kff
fKf
Kffi
Sfff
'
0)'(
)'(
)'(
)'(
)'('
'
,
)'(
)'(
)'('
'
1
1
D is proportional to S
By adopting suitable values for f,f’,K and i; K’ is derived as follows
100)'('
''
Kffi
ffK
44
UNIT I Tacheometric Surveying
Errors and Precautions in Tacheometric Surveying1. Errors of observation2. Instrument Errors3. Errors due to natural causes
1.Errors of observation
Sivapriya Vijayasimhan
Error PrecautionsIncorrect centering and levelling Proper centering and levelling of plate bubble and altitude
should be taken careVerticality of staff Plumb bob is used to check the verticalityParallax error Proper focusing before starting of workDistance of station beyond the scope of telescope
Graduations on staff are clearly and distinctly seen
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UNIT I Tacheometric Surveying
2. Instrument Errors
3. Errors due to natural causes
Sivapriya Vijayasimhan
Error PrecautionsHigh wind: staff and instrument subject to variation
Work should be suspended or temporary barear used to some extended
Hot weather: Tacheometer subjected to expansion
Readings taken under some shade
Hot weather: Poor visibility of staff
No direct sunlight on object glass
Error PrecautionsAdjustments in Tacheometer Checked and rectified before starting Graduation of staff or stadia hair Checked and corrected or replacedMultiplying constant ≠0 Field test should be conducted inorder to avoid constant errors
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