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C HAPTER Four
TRAVERSING PRINCIPLES
4.1. Introduction
A traverse consists of a series of straight lines connecting successive points. The points defining the ends of the traverse lines are called traverse stations or traverse points.
Distance along the line between successive traverse points is determined either by direct measurement using a tape or electronic distance measuring (EDM) equipment, or by indirect measurement using tachometric methods. At each point where the traverse changes direction, an angular measurement is taken using a theodolite.
Traverse party : it usually consists of an instrument operator, a head tape man and rare tape
man.
Equipments for the traverse party :-The equipments for the traverse party are the
theodolite, tapes, hand level, leveling staff, ranging pole & plumb bobs, EDM & reflector, stakes & hubs, tacks, marking crayon, points, walkie talkies, & hammer etc.
Purpose of traverse : It is a convenient, rapid method for establishing horizontal control
particularly when the lines of sights are short due to heavily built up areas where triangulation and trilateration are not applicable. The purpose includes:
- Property surveys to locate or establish boundaries;
- Supplementary horizontal control for topographic mapping surveys;
- Location and construction layout surveys for high ways, railway, and other private and public works;
- Ground control surveys for photogrammetric mapping.
A^ C
B D
4.2 Traversing by compass and theodolite
4.2.1. Types of traverse
1. Open traverse : It starts at a point of known position and terminates at a point of
unknown position.
- It is not possible to check the consistence of angles and distance measurement.
- To minimize errors, distances can be measured twice, angles turned by repetition, magnetic bearings observed on all lines and astronomic observations made periodically (not done in engineering works).
2. Closed traverse:- It originates at a point of known position and close on another point
of known horizontal position.
This type of traverse is preferable to all others since computational checks are possible which allow detection of systematic errors in both distance and direction.
Traverses also categorized on the method of observing angles.
X
A C
B D
D
B
Closed link traverse
X
A C Closed loop traverse
X C
A B
D
4.2.3 Interior angle traverse:
Interior angle traverse is the one that is employed for closed loop traverse. Successive stations occupied and back sight is taken to the preceding station with horizontal circle set zero. The instrument is then turned on its upper motion until the next station is bisected/sighted and the interior angle is observed. The horizontal circle reading gives the interior angle in the clockwise direction. Horizontal distances are determined by stadia and angles should be observed twice by double sighting.
Azimuth of a line =360-(back azimuth of preceding line + Clockwise interior angle).
In closed figure (^)
n
i 1 i ( n^2 )^180 n is the number of stations The error of closure can be distributed to all angles equally assuming that all observations are made with equal precision.
Example:
A clockwise interior angle in a closed traverse is as follows A= 84^0 58’, B=157^0 38’, C=24^0 37’ D= 153^0 14’ , E=103^0 54’, F= 139’ 06’ G= 236^0 49’ Compute the error of closure and adjust the interior angle.
A 3
1 2
4
Solution:
Station Observed Interior angle Correction Adjusted interior Angle A 84 0 58’ -0^0 2’ 84 0 56’ B 157 0 38’ -0 0 2’ 157 0 36’ C 24 0 37’ -0 0 2’ 24 35’ D 153 0 14’ -0 0 2’ 153 12’ D 103 0 54’ -0 0 2’ 103 52’ F 139 0 06’ -0 0 2’ 139 04’ G 236 0 47’ -0 0 2’ 236 45’ Sum 900 0 14’ 00” 9000 00’00’ (n-2)180^0 900 0 00 ‘ 00” Error of closure
Exercise: Calculate azimuth and bearing of all lines for Azimuth of AB Az AB=315^0 12’
4.2.4 Deflection angle traverses
This method of running traverses is widely employed than the other especially on open traverses. It is mostly common in location of routes, canals, roads, highways, pipe lines, etc.
Successive traverse stations are occupied with a theodolite with horizontal circle set at zero and back sight taken to the preceding station with a telescope reverse. The telescope is then plunged and the line of sight is directed to the next station, by turning the instrument about the vertical axis on its upper motion and the deflection angle is observed. Angles have to be observed by double sighting.
Azimuth of line =Azimuth of preceding line + R
Azimuth of line =Azimuth of preceding line - L
Solution: A 1 =A 2 (for closed loop) For closed loop traverse
Ri ^ Li ^3600
For the given traverse
^ L =370^0 18’^ ^ R =10^0 11”
10 0 11’-370^0 18’=-360^0 07’
360 0 07’-360^0 00’=0.07’
Correction per angle= (^001) ' 00 " 7
0 007 ' 0
This correction angle is added to deflection to the right and subtracted to deflection to the left. Station From/To Circle observed
Deflection angle
Correction Corrected deflection A G 0 0 00’ B 85 0 20’ 850 20’L -0^0 01’ 850 19’ B A 0 0 00’ C 10 0 11’ 100 11’R +0 0 01’ 100 12’ C B 0 0 00’ D D C E E D F F E G G F A
Ri ^ ^ Li ^3600
Exercise: Azimuth of line AB is given as Az AB= 85^0 24’.Calculate the azimuth and bearing of all other lines.
4.2.5 Angle to the right traverse
This method can be used in open, closed, or closed loop traverses. Successive theodolite stations are occupied and back sight is taken to the preceding station with the horizontal circle set zero. Then foresight is taken on the next station using the upper motion in the clockwise direction. The reading gives the angle to the right at the station and angles should be observed by double sighting.
Azimuth a line= angle to the right - Back azimuth of preceding line.
Error of closure
A 1 X 1 2 3 4 ( 4 1 ) 180 A 4 Y
The condition of closure can be expressed by
A 1 + 1 + 2 + -----n -(n-1)180-A 2 =
Where A1 & A 2 are Azimuths of the starting and closing stations. n=no of traverse stations (exclusive of fixed stations).
Any misclosure can be distributed equally to all angles assuming equal precision. Exercise: Try the above example in 5.2.4 (Deflection angle traverse.)
4
X
1 3
2 Y
4.3 Traverse Computations
Field operation for traverses yields angles or directions and distance for a set of lines connecting a series of traverse stations. Angles can be checked for error of closure and corrected so that preliminary corrected values can be computed. And observed distances can be reduced to equivalent horizontal distanced. The preliminary directions and reduced distances are suitable for use in traverse computations, which are performed in a plane rectangular coordinate system.
Computation with plane coordinates by considering the figure below.
Let the reduced horizontal distance of traverse lines ij and jk be dij and d (^) jk respectively, and Ai and Aj be the azimuths of ij and jk. Let Xij and Yij be the departure & latitude.
X (^) ij = dij sin Ai = departure Yij = dij cos Ai =latitude
If the coordinates of i are x (^) i and yi So, the coordinates of j are:
Y
X
d (^) jk
i k
yij
x (^) ij
d (^) ij
A (^) j
A (^) j j
x (^) ij
x (^) j
xk
yi
yi yk
yij
Xj=xi+xij ; yj= yi+yij Xk=xj+xjk ; yk=yj-yjk =xi+xij+xik ; =yi+yij-yjk xjk =djk sin Aj yjk=djk cos Aj Note: the signs of azimuth functions
If the coordinates for the two ends of a traverse line are given, distance between two ends can be determined as:
dij =[(xj-xi)^2 +(yj-yi)^2 ]1/
The azimuth of line ij from north and south is
yi yj Asij xi xj yj yi Aij xj xi
tan^1 tan^1
After coordinates for all the traverse points (all the departure and latitudes) for all lines have been computed, a check is necessary on the accuracy of the observations and the validity of calculations. In a closed traverse, the algebraic sum of the departures should equal the difference between the x- coordinates at the beginning and ending stations of the traverse. Similarly, the algebraic sum of the latitudes should equal the difference between the y coordinates at the beginning and ending stations.
In a closed loop traverse, the algebraic sum of the latitudes and the algebraic sum of the departures each must equal zero.
For a traverse containing n stations starting at i=1 and ending at station i=n, the foregoing conditions can be expressed as follows.
III Sin - Cos -
II sin+ Cos -
I Sin + Cos +
IV Sin - Cos +
W E
N
S
ii) Instrumental errors. Error in the adjustment of the theodolite. Always observe on both faces of the theodolite when measuring horizontal and vertical angles. The theodolite must be properly leveled before observations are made. So that ensure the plate bubble remains in the same position in its tube when the theodolite is rotated through 360. Ensure that the theodolite is stable with the legs firmly planted in solid ground and that the tripod adjusting screws are properly tightened. The theodolite must be properly centered over the station mark with an optical plummet or plumbing rod. If the horizontal circle is moved between observations the reduced angles will be in error. This can occur for any of the following reasons.
i) Screwing the theodolite too loosely to the tripod head. ii) Omitting to secure the movable head. iii) Omitting to clamp the lower plate. iv) Using the lower tangent screw instead of the upper tangent screw. v) Moving the orientation screw on single-axis theodolites.
4.5 Checking and adjusting traverse.
Traverse adjustment should be applied before the results of the traverse are usable for determining areas or coordinates for publishing the data, or for computing lines to be located from the traverse stations, to make the traverse mathematically consistent. The closure in latitudes and departures must be adjusted out.
4.5.1 The compass rule.
Consider a traverse station i, xi =correction to Xi yi = correction to Yi X (^) t =total closure correction of the traverse in the X coordinate.
Yt =total closure correction of the traverse in the Y coordinate. Li =distance from station i to the next station. L =total length of traverse dx (^) t= (X (^) n -X 1 )- departure dyt = (Yn -Y 1 )- latitude Then the corrections are
t (^) L dY t dX and yi Li L
xi Li
.
Alternatively, corrections may be applied to the departure and latitudes prior to calculating coordinates
xij dijL dxt and yij dijL dy t
xij and yij are respective corrections to the departure and latitude of line ij which has a length of dij.
Example:
In a closed traverse the distance between traverse stations and the deflection angle are as hereunder. Compute the error of closure and adjust the traverse using compass rule.
Line Distance (m) Deflection angle Azimuth AB 225.94 (^) A 10234’L 67 0 50’ BC 143.39 B 85^0 55’R 153 0 45’ CD 188.47 C 150 47’L 20 58’ =557.80 D 78^0 20’R
A (^) XA =170 0 24’ A (^) DY =80 0 38’
Adjustment of coordinates by the compass rule
Station Distance Corrections Adjusted coordinates,m xi yi X^ Y A 100 100 B 225.94 0.296 -0.373 309.288 185. C 143.39 0.483 -0.609 373.642 56. D 188.47 0.730 -0.920 382.0 245.
Adjusted distance and Azimuth
DAB =[(X B -X A)^2 +(YB -YA)^2 ]1/
DAB =[(309.288-100)^2 +(185.483-100)^2 ]1/
=226. 073m AAB =tan- B A
B A Y Y
X X
=tan-
483 100
288 100
4.6 Computation of Area
Area computation is one of the primary objective of land survey. A closed traverse is run, in which the lines of the traverse are made to coincide with property lines as possible. The length and bearings of all straight boundary lines are determined either directly or by computation.
In ordinary land surveying, the area of a tract of land is taken as its projection up on a horizontal plane, and it is not the actual area of the surface of land. For precise determination of the area of a large tract, such as state or nation, the area is taken as the projection of the tract up on the earth’s spheroidal surface to mean sea level.
Station Adjusted Distance Azimuth A B 226.073 67 0 46’52.36” C 143.684 153 0 23’31.1” D 188.170 2 0 32’44.75”
Methods of determining area:
1. The area of the tract may be obtained by use of the planimeter from a map or plan. It
may also be calculated by dividing the tract in to triangles and rectangles, scaling the dimensions of these figures, and computing their areas mathematically.
2. Area by triangles.
It is computation of areas individually mathematically by dividing the track in to triangles. If length of two sides and included angle of any triangle are known,
area 21 ab sin c If lengths of the three sides of any triangle are given,
S a b c
area ss a s b s c
2
1
( )( )
3. Area by coordinates:
When the points defining the corners of a tract of land are coordinated with respect to some arbitrarily chosen coordinate axes or are given in a regional system, these coordinates are useful not only in finding the lengths and bearings of the boundaries but also in calculating the area of the tract. The calculation involves finding the areas of trapezoids formed by projecting the line up on a parallel at right angle to this. Considering the figure under here
c
a b
B
C
A
c
A reference meridian is assumed to pass through the most westerly point of the survey; the double meridian distance of the lines are computed; and double the areas of the trapezoids or triangles formed by orthographically projecting the several traverse lines up on the meridian are computed. The algebraic sum of these double areas is double the area within the traverse. The meridian distance of a point is the total departure or perpendicular distance from the reference meridian and the meridian distance of a straight line is the meridian distance of its mid point. The double meridian distance of a straight line is the sum of the meridian distances of the two extremities. N
b B
A
C
c
The length of the orthographic projection of a line up on the meridian is the latitude of the line. The double area of the triangle or trapezoidal formed by projecting a given line up on the meridian is: Double area=DMD* latitude
In computing double area algebraic signs should be taken in to account.
Example:
For a traverse 123456 the adjusted distance and azimuths are given as below. Coordinate of 1(0.0, 0.0)
Meridian distance of B M.D.B =bB Meridian distance of C M.D.C =cC Double meridian distance of AB=0+ bB Double meridian distance of BC= MD@B+MD@C =bB+cC DMD of line =DMD of preceding line + departure of Preceding line + departure of the line.
d D
Adjusted Adjusted Line Distance Azimuth Departure Latitudes 12 405.18 1060 19’45” +388.84 -113. 23 336.59 570 54’01” +285.13 +178. 34 325.18 3350 28’43” -134.96 +295. 45 212.92 2190 28’33” -135.41 -164. 56 252.21 2660 55’30” -251.85 -13. 61 237.69 2190 40’28” -151.75 -182.
Compute the area in the traverse by using all methods.
Solution: Computation of area
1. Area by triangle
m
D X X Y Y
93
75 388. 84 ( 182. 95 ) ( 113. 92 )
( ) ( ) 2 2
62 6 2 2 6 2 2
m
D
- 74
52 403.^4388.^842196.^48 (^113.^92 )^2
m
D
53 403.^4673.^972196.^4864.^942
Using the formula. Area s s a s b s c
2 S a b ^ c
Station Coordinate, m X Y 1 0 0 2 388.84 -113. 3 673.97 64. 4 539.01 306. 5 403.4 196. 6 151.75 182. 1 0 0
1
2
(^6 )
3
1 2
3 4
