unit no-03 trignometric levelling setting out work

Unit No-03Trignometric levelling

&Setting out work

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Mr. .R.S. Sonawane DCE,BE(Civill),ME(Geotechnical Engg) ,AMIEI

Trignometric Levelling

A method in which the relative elevation ofdifferent station are found out from themeasured vertical angle and known planehorizontal distance or geodetic distance, iscalled as the trignometric levelling.

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Terrestrial refraction

The effect of refraction is to make the object appearhigher than they really are .

In plan surveying where a graduated staff isobserved either horizontal line of sight or inclinedline of sight ,the effect of refraction is to decreasethe staff reading and the correction is appliedlinearly to the observed staff reading

In trignometric levelling employed the elevation ofwidely distributed points ,the correction is appliedto the observed angles.

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Let ,P and Q are the two points the difference in elevation between

these being required .Let,O = centre of the earth PO’ = tangent to the level line through P = horizontal line at PQO’ = horizontal line at Q<P’PO’ = α1 = observed angle of elevation from P to Q.<Q’QQ2 = β1 = observed angle of depression from Q to P.r = angle of refraction or angular correction for refraction= <P’PQ

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PP’ = tangent at P to the curve line of sight PQ=apparent sight .

QQ’ = tangent at Q to the curved line of sight QP =parent sight

d = horizontal distance between P and QR = mean radius of the earth = 6370 kmm = Co-efficient of refraction Ɵ = angle subtended at the centre by distance PP1 over

which the observation are made .

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The actual line of sight between P and Q should have been along the straight line PQ but due to the effect of the terrestrial refraction ,the actual line of sight curved concave towards the ground surface .PP’ is, therefore the apparent sight from Q to P and QQ’ is the apparent from Q to P.sincethe angle are measured on the circle of a theodolite ,they are measured in the horizontal plane .

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The angle measured at P towards Q is ,therefore ,the angle between the apparent sight P’P and the horizontal line PQ’

hence <P’PO’ = observed angle α1 ,the true angle elevation ,in the absence of refraction is

<P’PQ.calling this as r ,the correction is evidently subtractive

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Thus ,correct angle = <QPO’ = < P’PQ = α1- rSimilarly,the angle measured at Q towards P is <Q’QQ= β1 .the true angle of depression . In the absence

of refraction is <PQQ2.Hence the correction for refraction is <PQQ’ and be added

to the observed angle to get the correct angle.Thus the correct angle = < PQQ2 = < Q’QQ2 + <Q’QP =

β1 +r.Thus the correction for the refraction is subtractive to the

angle of elevation and additive for the angle of depression.

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Coefficient of refraction:-The co-efficient of refraction (m) is the ratio

of the angle of refraction and the angle subtended at the centre of the earth by the distance over which observation are tken

Thus, m = r/Ɵ or r = m. ƟThe value of ‘m’ varies roughly between 0.06

to 0.08

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Determination of correction refraction :-In order to determine the angle of refraction r.Case:-I : Distance ‘d’ small and ‘H’ large :-

r = ( Ɵ/2) – {(β1 – α 1) /2}It assumed that the refraction error ‘r’ is the same at both

station Writing r = m.Ɵand rearrange ,we get

2m Ɵ = Ɵ – (β1 - α 1 )β1 = α 1 + Ɵ( 1- 2m)

Thus the observed angle of depression Always exceed the angle of elevation by the amount Ɵ( 1- 2m)

Case :- II Distance ‘d’ large and ‘H’small:-In this case ,both α 1 and β1 are the angle of

depression Changing the sign of α 1 in equation r = ( Ɵ/2) – {(β1 – α 1) /2}.We get r = ( Ɵ/2) – {(β1 + α 1) /2}Which is reduce to: (β1 + α 1) = Ɵ (1 – 2m)

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Correction for curvature:-The correction for the curvature is

+Ɵ/2 for angle of elevation and - Ɵ/2 for angle of depression.Combined correction:-

Now,<O’PP1 = Ɵ/2 = d/ 2R radians =( d/ 2R sin1”) secondsAngular correction of refraction = mƟ

= (m.d/ R sin1”) seconds

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Hence , Combined angular correction =[ (d/ 2R sin1”) –(m.d/ R sin1”) ]= ( 1 – 2m ) d Seconds

2R sin1” The combined correction is positive for angles

of elevation and negative for angle of depression

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Axis signal correction ( Eye & Object Correction):-

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In order to observe the points from the theodolite station .signals of appropriate heights are erected at points to be observed .The signal may or may not be same height as that of the instrument .

If height of the signal is not same as that of the height of instrument axis above the station , a correction known as the axis signal correction or eye and object correction is to be applied.

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Let ,h1= height of instrument at P,for observation to

Q.h2 = height of instrument at Q,for bservation to

P.s1 = height of instrument at P ,instrument being Q.s2 = height of instrument at Q ,instrument being P.

d=horizontal distance between P and Qα = Observed angle of elevation uncorrected for

the axis signal

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β=Observed angle of depression uncorrected for the axis signal.

α1= angle of elevation corrected for axis signalβ1=angle of depression corrected for axis signal.In figure.PA = horizontal line at P Q= Point observed BQ= difference in the height of signal at Q and

the height of instrument at P

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= (s2-s1)<BPA= α = angle observed from P to Q<BPQ= δ1 = axis signal correction ( angular )at

P.At B ,draw BC perpendicular to BP , to meet PQ

produce in C,

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For triangle PBO<BPO = <BPA + <APO = α + 900

<POB = Ɵ<PBO = 180 – ( 90 + α ) – Ɵ

= 90 – (α + Ɵ)<QBC = 90 – [ 90- (α + Ɵ)]

= (α + Ɵ)The angle δ1 is usually very small and hence <BCQ can

be approximately taken equal to 900.

BC = BQ cos (α + Ɵ) very nearly = (s2-h1)cos (α + Ɵ)] ………………………(1)

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For triangle PP1B,<BPP1= α + Ɵ/2<PBP1= 90 – (α + Ɵ)<PP1B = 180 –[ 90 - (α + Ɵ)]-(α + Ɵ/2)

= (90 + Ɵ/2)Now PB = PP1

sin PP1B sin PBP1PB = [(d.sin (90 + Ɵ/2)]/[sin(90 – (α + Ɵ)] = d.[ (cos Ɵ/2 ) / cos (α + Ɵ)]…………………..(2)

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For triangle PBC,Tanδ1 = BC

PBSubstituting the value of BC from (1) & (2)We get , Tanδ1 = (s2-h1)cos (α + Ɵ)

d. (cos Ɵ/2 )cos (α + Ɵ)

Tanδ1 = (s2-h1) cos2 (α + Ɵ)d .cos Ɵ/2 ………exactly

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Usually Ɵ is small in comparison to α and may be ignored

Tanδ1 = (s2-h1) cos2 α ……(a)d

The correction for signal evidently substrctivefor this case.

Similarly , if the if the observation are taken from Q towards P ,it can be proved that

Tanδ2 = (s2-h1) cos2 β ( additive)d

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The correction of axis signal is negative for angles of elevation and positive for angle of depression .

If , however angle α or β is very small ,we can take ,with sufficient accuracy

Tanδ1 = δ1 =( s2-h1) / d sin 1” seconds………….(b)Tanδ2 = δ2 =( s2-h2) / d sin 1” ………………………(c)By considering PB= PQ= PP1=d nd taking the arc with radius

equal to d. then

δ1 = [BQ/ d ]radians= [( s2-h1) / d ]= [( s2-h1) / d sin 1”] seconds

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This expression gives the sufficiently accurate result when the vertical angle is small , the difference is large and the difference in height of the signal and that of the instrument is small .after having calculated δ1 & δ2 , The angle corrected for the axis signal are given by

α1 ( elevation )= α - δ1 β1 ( Depression ) = β + δ2

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DETERMINATION OF DIFFRENCE IN ELEVATION:

The difference in elevation between the two points P & Q can be found out by two method(a) By single observation(b) By reciprocal observation.(a) By single observation:The following correction will have to applied:(1) Correction for Curvature.(2) Correction for refraction. (3) Correction for axis signal.

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Since the sign of these correction will depend upon the sign of the angle observed,

We shall consider following cases:(i) When the observed angle is the angle of

elevation.(ii) When the observed angle is the angle of

depression .

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For angle of Elevation

Mr. .R.S. Sonawane DCE,BE(Civill),ME(Geotechnical Engg) ,AMIEI 29

α = observed angle of elevation to Qα1 = observed angle corrected for axis signal = (α - δ1 )= [α – (s2 –h1)/d sin 1”] seconds.Therefore QP1= d.sin α1+(m.d/Rsin1”) +(d/ 2R sin1”)

Cos α1+(m.d/Rsin1”) +(d/ 2R sin1”)

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QP1= d.sin α1+(1-2m) (d/ 2R sin1”)Cos α1+(1 – m) (d/ R sin1”)

Where the quantities (1-2m) (d/ 2R sin1”) and (1 – m) (d/ R sin1”) are in seconds.Approximate Expressions:˂PP1Q= 90,Ѳ is very smallQP1 =H = PP1 tan QPP1= d tan [α1 - m Ѳ + Ѳ/2]= d tan [α1 - (1-2m) (d/ 2R sin1”) ]

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QP1= d.sin ᵝ1- (1-2m) (d/ 2R sin1”)Cos ᵝ1 - (1 – m) - (d/ R sin1”)

Approximate Expression˂ PQ1Q to be equal to 90 when Ѳ is very small

.ThenQ1P =H=QQ1 tan PQQ1=

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For angle of depression

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ᵝ = observed angle of depression to P

ᵝ1 = observed angle corrected for axis signal

= ᵝ + δ2= [ᵝ – (s1 –h2)/d sin 1”] secondsd = horizontal distance = arc QQ1 = chord QQ1= QB

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ᵝ = observed angle of depression to P

ᵝ1 = observed angle corrected for axis signal

= ᵝ + δ2= [ᵝ – (s1 –h2)/d sin 1”] secondsd = horizontal distance = arc QQ1 = chord QQ1= QB= d tan [ᵝ1 - (1-2m) (d/ 2R sin1”) ]

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Difference in elevation by Reciprocal :-

Reciprocal observation are generally made to eliminate the effect of refraction. in this method ,observation are made simultaneously from both station ( i.e P and Q )

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Both α1 and β1 are the angle of depression ,the expression for H can be obtained by changing the sign of α1 .

H = d sin[ (β1 - α1 )/2]cos [(β1 - α1 )/2 +Ɵ/2]

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unit no-03 trignometric levelling setting out work

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