Theodolite surveying

Contains notes on Theodolite Surveying

1. Royal University of Bhutan
Jigme Namgyel Engineering College
Department of Civil Engineering & Surveying
Tutor : Phurba Tamang
Designation: Associate lecturer
Department of Civil Engineering and Surveying
Theodolite Surveying
MODULE: SURVEYING
Spring Semester 2020

2. MID-TERM REVIEW, 2017
UNIT 4: Theodolite Surveying
IntroductiontoTheodolite
Theodoliteisaveryusefulinstrumentforengineers.Itisusedprimarilyformeasuringhorizontal
andverticalangles.Howevertheinstrumentcanbeusedforotherpurposeslikeprolongingaline,
measuringdistancesindirectlyandlevelling.
Theodolitethesedaysarealltransittheodolites.Herethelineofsightcanberotatedinavertical
planethrough180degreesaboutitshorizontalaxis.Thisisknownastransitingandthusthename
‘transit’isderived.
TheodoliteTraversing
Traversingisthattypeofsurveyinwhichmemberofconnectedsurveylineformtheframework
andthedirectionandlengthsofthesurveylinesaremeasuredwithhelpofananglemeasuring
instrument.Theodolitetraversingisamethodofestablishingcontrolpointsandtheirposition
beingdeterminedbymeasuringdistancebetweenthetraversestationandanglesubtendedatthe
variousstationbytheiradjacentstations.Aftermeasuringangle,lengthanddirection,Galestable
ispreparedtocalculatefinaltraversing.Galestablehelpsincalculatingerrorsandgetting
accuratebearings,anglesandlengths

3. MID-TERM REVIEW, 2017
Components of Theodolite

4. MID-TERM REVIEW, 2017
List of Equipment for Theodolite Traversing
•Theodolite
•Rangingrod
•TripodStand
•Plumbbob
•MeasuringTape
•Pegs
Plumb bob
Theodolite
with tripod
stand
Ranging Rods
Wooden Pegs
Measuring Tape

5. MID-TERM REVIEW, 2017
Theodolite Adjustments
A.Setting,LevellingandCenteringTheodolite
•Releasetheclampscrewoftheinstrument
•Holdtheinstrumentintherighthandandfixitonthetripodbyturningroundonlythelowerpartwiththe
lefthand.
•Screwtheinstrumentfirmlyandbringallthefootscrewstothecenterofitsrun.
•Spreadthetripodlegswellapartandfixanytwolegsfirmlyintothegroundbypressingthemwiththe
hand.
•Movethethirdlegtoupordownuntilthemainbubbleisapproximatelyinthecenter.
•Thenmovethethirdleginoroutuntilthebubblesofthecross-levelisapproximatelyinthecenter.
•Fixthethirdlegfirmlywhenthebubblesareapproximatelyinthecentersoftheirrun.
•Placethetelescopeparalleltoapairoffootscrews.
•Bringthebubbletothecenterofitsrunbyturningthefootscrewsequallyeitherbothinwardsandboth
outwards.
•Turnthetelescopethrough90degrees,sothatitliesoverthethirdfootscrew.
•Turnthisthirdfootscrewsothatthebubblecomestothecenterofitsrun.
•Turnthetelescopethroughandcheckwhetherthebubbleremainscentral
•Lookingthroughtheopticalplummet,focusthecenteringindexmark.Slidethetheodoliteonthetripod
headuntilthereferencemarkiscenteredintheopticalplummet.
•Fullytightenthecenteringscrew.Lookthroughtheopticalplummetagainandadjustthetheodolitefoot
screwsforalignmentwiththereferencemark.

6. MID-TERM REVIEW, 2017
Theodolite Adjustments
B.InstructionsforTheodoliteTraversing
•Theareatobesurveyedisfirstthoroughlyexaminedtodecidethebestpossiblewayofstartingthework.
•ConsideraclosedtraversewithstationsnamelyABCDEFA.Thetraversestationsaremarkedontheground
bywoodenpegswithnailsontop.
•SetuptheinstrumentatA.Completetheadjustments(levellingandcentering)beforesightingthestationB.
•Oncelevellingandcenteringhavebeenachieved,turnonthetheodolitebypressingthepowerkey.
•Placethecompassoverthetheodoliteandrotatetofindthedirectionofmeridian(North).
•Oncethedirectionhasbeset,presstheHOLDkeytwicetolockthereferencedirection.
•PresstheL/Rkeyforhorizontalangleoptions.Rmodeisusedwhenthetraversingiscarriedoutin
clockwisedirectionandLmodeincounter-clockwisedirection.
•PressV/%keytoseetheinclinationangleofopticaltelescope.Theanglehastobein000’0’’(zero
inclination)tomaintainthelineofcollimationthroughouttheprocessoftraversing.
•Afterzero-inclinationhasbeenmaintained,usetheverticalclampscrewtorestricttherotationof
telescope.
•Now,sightthelevellingstafforrangingrodplacedatBandrecordtheforebearing.
•StationtheinstrumentatStationBandsightatStationAandrecorditsbackbearing.
•Similarly,completetheprocessforentiretraversestations.
•Findtheincludedangles∠A,∠B,∠C,∠D,∠E,and∠F.Measurethelengthoftraverselegsconnectingallthe
stations.
•Plotthetraverseandfindtheclosingerrorgraphically(DraftinAutoCAD)

7. MID-TERM REVIEW, 2017
Theodolite Adjustments
•Checktheadjustmentofinterioranglesusingthecondition,i.e.thesumoftheincludedanglesshouldbe
(2n±4)x90degrees,wherenisthenumberofsidesofclosedtraverse.
•PerformallessentialchecksusingtheGale’sTraverseTable.
•Re-plotthecorrectedtraverse.

8. MID-TERM REVIEW, 2017
Traverse Survey and Computations
Introduction
•Atraverseisaseriesofconnectedlineswhoselengthsanddirectionsaremeasuredinthefield.The
surveyingperformedtoevaluatesuchfieldmeasurementsisknownastraversing.
•Atraverseisoftwotypes,openandclosed.
OpenTraverse:Anopentraverseisonethatdoesnotreturntothestartingpoint.Itconsistofaseriesoflines
expandinginthesamedirection.Anopentraversecannotbecheckedandadjustedaccurately.Itisemployed
forsurveyinglongnarrowstripsofcountry,e.g.thepathofahighway,railway,canal,pipeline,transmission
lines,etc.
•ClosedTraverse:Atraverseissaidtobeaclosedoneifitreturnstothestartingpoint,therebyforminga
closedpolygon.Inaddition,atraversewhichbeginsandendsatthepointswhosepositionsontheplanis
knownarealsoreferredtoasaclosedtraverse.Aclosedtraverseisemployedforlocatingtheboundaries
oflakesandwoods,forareadetermination,controlformappinggandforsurveyingmoderatelylarge
areas.
Commonusesoftraversing
•Todetermineexistingboundarylines,tocalculateareawithintheboundary,toestablishcontrolpointsfor
mappingandalsoforphotogrammetricwork,toestablishcontrolpointsforcalculatingearthwork
quantities,forlocatingcontrolpointsforrailroadshighwaysandotherconstructionwork.

9. S
Traverse Survey and Computations
CA
B D
E
F
( Open Traverse ) An open traverse terminates at a point of unknown position
A
B D
E
F
C
F ’
( Known Point )
A ’ ( Known Point )
A
D
C
B
E
F
( Closed Traverse ) A closed traverse terminates at a point of known location.

10. Measurement of Traverse Angles
1.InteriorAngles
Interioranglesofaclosedtraverseshouldbemeasuredeitherclockwiseoranticlockwise.
Clockwisemeasurementofanglesisalwaysrecommended.
A
B
E
C
D

11. Measurement of Traverse Angles
2.DeflectionAngles
Opentraverse,e.g.routesurveysareusuallyrunbydeflectionanglesoranglestotheright.A
deflectionangleisformedatatraversestationbyanextensionofthepreviouslineandthe
succeedingone.ThenumericalvalueofadeflectionanglemustbealwaysfollowedbyRorLto
indicatewhetheritwasturnedrightorleftfromtheprevioustraverselineextended.
A
B
C
D
E
F
R R
L L

12. Measurement of Traverse Angles
3.AnglestotheRight
Anglesmeasuredclockwisefromabacksightonthepreviouslinearecalledanglestotherightor
azimuthsfromthebackline.Thiscanbeusedinbothopenorclosedtraverse.Rotationshould
alwaysbeclockwisefromthebacksight.
A
B
C
D
E

13. Measurement of Traverse Angles
4.AzimuthAngles
Atraversecanberunbyreadingazimuthanglesdirectly.Azimuthsaremeasuredclockwisefrom
thenorthendofthemeridianthroughtheanglepoints.Ateachstationthetransitistobeoriented
bysightingthepreviousstationwiththebackazimuthofthelineasthescalereadings.
N
N
N N
A
B
C
Suppose the Azimuth of AB is . The azimuth of BA is 160
0
29′20′′ 340
0
29′20′′

14. Latitude and Departure
LatitudeofalineisthedistancemeasuredparalleltotheNorthSouthLine.
DepartureofalineisthedistancemeasuredparalleltotheEastWestLine.
N
EW
S
θ
θ
L sin θ
L cos θ
L sin θ( Departure )
L cos θ( Latitude )
Note: Theodolite traverse is not plotted according to interior angles or bearings. It is plotted by computing
the latitudes and departures of the points and then finding the independent coordinates of the points.

15. Latitude and Departure
Thelatitudeanddepartureoflinesarealsoexpressedinthefollowingways:
•Northing=LatitudetowardsNorth=+L
•Southing=LatitudetowardsSouth=-L
•Easting=DeparturetowardsEast=+D
•Westing=DeparturetowardsWest=-D
Check for Closed Traverse:
1.The algebraic sum of latitudes must be equal to zero
2.The algebraic sum of departures mush be equal to zero
Line Length ( L )Reduced Bearing (θ) Latitude ( L cos θ)Departure ( L sin θ)
AB L N θE + L cos θ + L sin θ
BC L S θE -L cos θ + L sin θ
CD L S θW -L cos θ -L sin θ
DA L N θW + L cos θ -L sin θ

16. Latitude and Departure
Check for Closed Traverse:
1.Sum of Northings = Sum of Southings
2.Sum of eastings = Sum of Westings
Line Length
( L )
Reduced Bearing
(θ)
Northing ( + )Southing ( -)Easting ( + )Westing ( -)
AB L N θE L cos θ L sin θ
BC L S θE L cos θ L sin θ
CD L S θW L sin θ L sin θ
DA L N θW L cos θ L sin θ

17. Balancing of Traverses
Incaseofaclosedtravers,thealgebraicsumoflatitudesmustbeequaltozeroandthatof
departuresmustalsobeequaltozerointhedialconditions.Inotherwords,thesumofnorthings
mustbeequaltothatofthesouthings,andthesumoftheeastingsmustbetheasthatofthe
westings.Inactualpracticesomeclosingerrorisalwaysfoundtoexistwhilecomputingthe
latitudeanddeparturesofthetraversestations.Thetotalerrorsinlatitudeanddepartureare
determined.Theseerrorsarethendistributedamongthetraversestationsproportionately.
ClosingErrors:
Theerrorsinfieldmeasurementsofanglesandlengthssometimesresultsinimproperclosureof
thetraverse(Endpointdoesnotcoincidewiththestartingpoint).Thedistancebywhichatraverse
failstocloseisknownasclosingerrororerrorofclosure.
ClosingError=σ??????
2
+σ??????
2
where,L=LatitudeandD=Departure
RelativeClosingError=ClosingError/Perimeteroftraverse.
PermissibleAngularError=leastcountx??????,whereN=Numberofsides
tan??????=
σ??????
σ??????
=whereθindicatesthedirectionofclosingerror.

18. S
Balancing of Traverses
Traverse for Permissible Angular ErrorPermissible relative closing error
1. Land, roads and railway surveys 1’ x ?????? 1 in 3000
2. City survey, important foundry survey 30’’ x ?????? 1 in 5000
3. Very Important Survey 15’’ x ?????? 1 in 10000
A
B
D
C
E
F
Closing Error
σ??????
σ??????

19. Balancing of Traverses
Inactualpracticesomeclosingerrorisalwaysfoundtoexistwhilecomputingthelatitudeand
departuresofthetraversestations.Thetotalerrorsinlatitudeanddeparturearedetermined.
Theseerrorsarethendistributedamongthetraversestationsproportionately.Thefollowingrules
areusedtodistributetheerrorsproportionately.
1.Bowditch’sRule
Bythisrule,thetotalerrorinlatitudeanddepartureisdistributedinproportiontothelengthsofthe
traverselegs.Thisisthemostcommonmethodoftraverseadjustment.
a)Correctiontolatitudeofanyside=
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b)Correctiontodepartureofanyside=
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20. Balancing of Traverses
2.TransitRule
a)Correctiontolatitudeofanyside=
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b)Correctiontodepartureofanyside=
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3.ThirdRule
a)Correctiontonorthingofanyside=
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×
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(??????��????????????�????????????�????????????�????????????�??????����)
b)Correctiontosouthingofanyside=
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×
�
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(??????��????????????�????????????�????????????�????????????�??????����)
c)Correctiontoeastingofanyside=
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×
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(??????��????????????�????????????�????????????����????????????��??????�)
d)Correctiontowestingofanyside=
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×
�
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(??????��????????????�????????????�????????????����????????????��??????�)
Note: If the error is positive, correction will be negative and vice vera

21. Balancing of Traverses ( Bowditch’s Rule )
Note: If the error is positive, correction will be negative and vice vera
Line Length
Consecutive Coordinates Correction
Corrected Consecutive
Coordinates
LatitudeDeparture LatitudeDeparture LatitudeDeparture
AB 70 +21.500 -65.450 +0.072 -0.064 +21.572 -65.514
BC 80 -80.755 -5.250 +0.083 -0.073 -80.672 -5.323
CD 43 -41.000 +13.550 +0.044 -0.039 -40.956 +13.511
DE 38 -14.250 +35.150 +0.038 -0.034 -14.212 +35.116
EA 115 +114.150 +22.315 +0.118 -0.105 +114.268 +22.210
Total 346 -0.355 +0.315 +0.315 -0.315 0 0
Perimeter Error Correction Adjusted

22. Balancing of Traverses ( Third Rule)
Note: Round up to 3 decimal places ( Ex: 3.1285 will be represented as 3.129 and 3.1284 will be 3.128 )
Line Length
Consecutive Coordinates Correction to Corrected Consecutive Coordinates
Northing
+
Southing
-
Easting
+
Westing
-
NorthingSouthingEastingWestingNorthing
+
Southing
-
Easting
+
Westing
-
AB 70 31.500 65.45+0.029 +0.14621.529 65.596
BC 80 80.755 5.250 -0.106 +0.012 80.650 5.262
CD 43 41.00013.550 -0.053-0.030 40.94713.520
DE 38 14.25035.150 -0.019-0.078 14.23135.072
EA 115114.150 22.315 +0.149 -0.049 114.299 22.266
Total346 135.650136.00571.01570.700+0.178-0.178-0.157+0.158135.828135.82870.85870.858
Perimeter = 346Error = -0.355Error = +0.315 Error = 0.000Error = 0.000

23. D
Determination of Included Angles from Bearings
Theincludedanglebetweentwolinesmayeitherbeinteriororexteriorangles.Whentraversingis
doneanticlockwise,theincludedanglesareinterior,whereasinthecaseofclockwisetraverse,
thesearetheexteriorones.Thesearealwaysmeasuredclockwisefromtheprecedinglinetothe
forwardline.
Example:DeterminethevalueofincludedanglesinaclosedtraverseABCDconductedinclockwise
direction,giventhefollowingforebearingsoftherespectivelines.
LINE Fore Bearing
AB 40°
BC 70°
CD 210°
DA 280°

24. D
Determination of Included Angles from Bearings

25. D
Determination of Included Angles from Bearings

26. D
Determination of Included Angles from Bearings

27. D
Determination of Included Angles from Bearings

28. D
Determination of Included Angles from Bearings
Since,thetraversingisforthisexampleiscarriedoutinclockwisedirection,theincludedanglesarethustaken
asexteriorangles.
Check:
TheoreticalSumofIncluded(Exteriorangles)=(2n+4)x90°=(2x4+4)x90=1080°
Also,sumofcalculatedincludedangles=∠A+∠B+∠C+∠D=300°+210°+320°+250°=1080°

29. D
Determination of Included Angles from Bearings
Alternative
Ifinterioranglesaretakenasincludeangles(Usual
Method),thenthefollowingcheckcanbeperformed
Check:
TheoreticalSumofIncluded(InteriorAngles)
=(2n-4)x90°=(2x4-4)x90=360°
Also,sumofcalculatedincludedangles
=∠A+∠B+∠C+∠D=60°+150°+40°+110°=360°

30. D
Determination of Included Angles from Bearings
PracticeExample:
Followingarethebearingstakeninaclosedtraverse
LINE Fore Bearing Back Bearing
AB 142°30′ 322°30’
BC 223°15′ 44°15′
CD 287°00′ 107°45′
DE 12°45′ 193°15′
EA 60 °00 ‘ 239 °00 ‘
Solution:
∠A=263°30‘,∠B=260°45’,∠C=242°45‘,∠D=265°00‘,∠E=226°45‘

31. D
Gale’s Traverse Table
TraverseComputationsareusuallydoneinatabularform.OnesuchformisGale’sTraversetablewhichis
widelyusedbecauseofitssimplicity.
Station
Line
Length
Interior angles
Corrections
Corrected angles
WCB
RB
Quadrants
Consecutive
coordinates
Correction
(Bowditch Rule)
Corrected
Consecutive
Coordinates
Independent
Coordinates
Lat. Dep. Lat. Dep. Lat. Dep.
N (+) S (-)E (+) W (-)
N (+)S (-)E (+)W (-)(+)(-)(+)(-)N (+)S (-)E (+)W (-)
A
AB
B
BC
C
CD
D
DA
A
Total Error in
Latitude =
Total Error in
Departure =

32. D
Gale’s Traverse Table
ThefollowingstepsareinvolvedinTheodoliteTraversing
1.Inthecaseoftheodolitetraversing,theincludedanglesareadjustedtosatisfythegeometrical
conditions,i.e.thesumoftheincludedanglesshouldbe(2n±4)x90°,wherenisthe
numberofsidesoftheclosedtraverse.Theplussignisusedwhentheanglesareexterior
anglesandtheminussignwhentheyareinteriorangles.
2.Fromtheobservedbearingofaline,thewholecirclebearingsofallotherlinesarecalculated
andthenthesebearingsarereducedtothoseinthequadrantalsystem.
3.Fromthelengthsandcomputedreducedbearingsofthelines,theconsecutivecoordinatesi.e.
latitudesanddeparturesareworkedout.
4.Acheckisperformedtofindoutwhetherthealgebraicsumoflatitudesandthealgebraicsum
ofdeparturesarezero.Ifnotacorrectionisappliedusingthetransitrule.
5.Theindependentcoordinatesarethenworkedoutfromtheconsecutivecoordinates.Theorigin
issoselectedthattheentiretraverseliesinthenortheastquadrant.Thisisdonetofacilitate
plottingofthetraverseonasheetwiththelefthandbottomcornerofthesheetastheorigin.

33. D
Gale’s Traverse Table
Angle Observed Value Side Measured Length ( m )
DAB 97°41′ AB 22.11
ABC 99°53′ BC 58.34
BCD 72°23′ CD 39.97
CDA 89°59′ DA 52.10
Example:ThemeanobservedinternalanglesandmeasuredsidesofaclosedtraverseABCDA(in
anticlockwiseorder)areaasfollow:
Adjusttheangles,computethelatitudesanddeparturesassumingthatDisdueNorthofA,adjust
thetraversebytheBowditchmethod;andgivethecoordinatesofB,CandDrelativetoA.Assesthe
accuracyoftheseobservationsandjustifyyourassessment.

34. D
Gale’s Traverse Table
Solution:

35. D
Calculation of Traverse Area
Theareofaclosedtraversemaybecalculatedfrom
A.Thecoordinate(xandy)
B.Thedepartureandtotallatitudes.
A.CalculationofAreafromCoordinates
Thegivenconsecutivecoordinatesofatraverseare
convertedintoindependentcoordinateswith
referencetothecoordinatesofthemostwesterly
station.Thus,thewholetraverseistransferredto
thefirstquadrant.Fromthefigure,pointAisthe
mostwesterlystation.
Then,thecoordinatesarearrangedindeterminantformasfollows.

36. D
Calculation of Traverse Area
Thesumoftheproductsofcoordinatesjoinedbysolidlines,
Thesumoftheproductsofcoordinatesjoinedbydottedlines,

37. D
Calculation of Traverse Area
Example:Findtheareaoftheclosedtraverseusingthecoordinatemethod.
Side Latitude Departure
AB +225.5 +120.5
BC -245.0 +210.0
CD -150.5 -110.5
DA +170.0 -220.0
Solution:
Station Side Consecutive Coordinates Independent Coordinates
Latitude ( y )Departure ( x )Latitude ( y )Departure ( x )
A +200.00 +100.00
B AB +225.5 +120.5 +425.50 +220.50
C BC -245.0 +210.0 +180.50 +430.50
D CD -150.5 -110.5 +30.00 +320.00
A DA +170.0 -220.0 +200.00 +100.00
TheindependentcoordinatesofthemostwesterlystationAareassumedtobe+200.00,+100.00

38. D
Calculation of Traverse Area
Station Side Consecutive Coordinates Independent Coordinates
Latitude ( y )Departure ( x )Latitude ( y )Departure ( x )
A +200.00, y1 +100.00, x1
B AB +225.5 +120.5 +425.50, y2 +220.50, x2
C BC -245.0 +210.0 +180.50, y3 +430.50, x3
D CD -150.5 -110.5 +30.00, y4 +320.00, x4
A DA +170.0 -220.0 +200.00, y5 +100.00, x5
Theindependentcoordinatesarearrangedinadeterminantformasfollows:

39. D
Calculation of Traverse Area
Sumofproductsofcoordinatesjoinedbysolidlines:
ƩP=(200.00x220.5+425.50x430.50+180.50x320.00+30.00x100.00)=288037.75
Sumofproductsofcoordinatesjoinedbydottedlines:
ƩQ=(100.00x425.50+220.50x180.50+430.50x30.00+320.00x200.00)=159265.25
RequiredArea,A=0.5x(ƩP–ƩQ)=0.5x(288037.75–159265.25)
=64386.25m
2

40. D
Calculation of Traverse Area
B.CalculationofAreafromDepartureandTotalLatitude
Consideringthefollowingfigure,pointAisthemostwesterlystation,andthereferencemeridianis
assumetopassthroughit.

41. D
Calculation of Traverse Area
ProcedureforCalculationArea:
1.Thetotallatitude(thelatitudewithrespecttothereferencepoint)ofeachstationofthe
traverseisfoundout.
2.Thealgebraicsumofdeparturesofthetwolinesmeetingatastationisdetermined.
3.Thetotallatitudeismultipliedbythealgebraicsumofdeparture,foreachindividualpoint.
4.Thealgebraicsumofthisproductgivestwicethearea.
5.Halfofthissumgivestherequiredarea.

42. D
Calculation of Traverse Area
Example:Findtheareaoftheclosedtraverseusingthedepartureandtotallatitudemethod.
Side Latitude Departure
AB +225.5 +120.5
BC -245.0 +210.0
CD -150.5 -110.5
DA +170.0 -220.0
ThelatitudesofthestationsarecalculatedwithreferencetostationA.
TotalLatitudeofB=+225.50
TotalLatitudeofC=+225.5-245.0=-19.5
TotalLatitudeofD=+225.50–245.0–150.5=-170.0
TotalLatitudeofA=+225.5–245.0–150.5+170.0=0.0

43. D
Calculation of Traverse Area
AlgebraicsumofdeparturesatB=AB+BC=120.5+210.0=+330.5
AlgebraicsumofdeparturesatC=BC+CD=+210.0–110.5=+99.5
AlgebraicsumofdeparturesatD=CD+DA=-110.5–220.0=-330.5
AlgebraicsumofdeparturesatA=DA+AB=-220.0+120.5=-99.5
Theresultistabulatedasfollows:
Side LatitudeDeparture Station Total
Latitude
Algebraic
Sum of
adjoining
Departure
Double Area
Column 5 x Column 6
+ -
1 2 3 4 5 6 7 8
AB +225.5 +120.5 B + 225.50 + 330.50 74527.75
BC -245.0 +210.0 C -19.50 + 99.50 _____ 1940.25
CD -150.5 -110.5 D -170.00 -330.50 56185.00 _____
DA +170.0 -220.0 A 0.00 -99.50 _____ 0.00
Total +130712.75–1940.25
AlgebraicSum=+128772.50
TwiceArea=AlgebraicSumofColumn7and8=+128772.50
RequiredArea=0.5x128772.50=64386.25??????
2