Question

Consider the traverse ABCDE as shown below. AB is the datum/reference direction of the traverse with Known coordinates for A and B as given below: A (E=327682.209, N=5813380.144, Z=20.467) and B(E: 327882.209, N:5813116.199, Z=23.779) where Z is the elevation or RL value of the point. Internal angles and horizontal distances are measured using a Total Station as tabulated below: Angle @ Direction Measured (unadjusted) distances (m) A | АВ Measured (unadjusted) Angles 152° 54'30" 111° 11'15" 57° 55'55" 163° 32' 35" 53° 26' 10" 435.160 486.920 361.860 354.580 DE EA Not into scale 1. Work out the direction (whole circle bearing), length and grade of AB using the coordinates of A and B https:l/en.wikipedia og/wik Grade (slope) 2. Using the computed direction for AB and measured internal angles as tabulated above, work out the bearings of all other directions (i.e. BC, CD, DE, EA and again AB to check). (Hint: refer to the bearings of the next direction in Lecture 3). 3. Close the traverse using Bowditch adjustment and compute adjusted bearings and distances for all traverse sides. 4. Rotate the traverse so that the direction of the reference line AB is unchanged before and after Bowditch adjustment. 5. Work out the coordinates of traverse corner points after rotation in Q4. Note: When possible and suitable tabulate the data and computations.

Answer 1

SOLUTION:-

Part 1: Calculation of Distance and direction of AB:

Given Data:

a) Coordinates of A (E,N) = (327682.209, 5813380.144)

b) Coordinates of B (E,N) = (327882.209, 5813116.199)

Distance of a line = (Different in E Coordinates)2 + (Different in N Coordinates)

AB = V(327882.209 - 327682.209)2 + (5813116.199 - 5813380.144)2 = 331.160 m

Difference in E Coordinates(de) Azimuth of a line = tan-1 (Difference in E Coordinates(d) )

200 327882.209 - 327682.209 Azimuth of AB = tan-(5813116.199 - 5813380.144) -263.945) = -(37099)

Since easting is positive and northing is negative, line is in SE direction.

Final Azimuth of AB = 108o - 37o9'9" = 142o50'51"

Part 2: Calculation of direction of other lines:

Step 1: Balance interior angles:

Sum of interior angles of a closed traverse is given by below formula:

Sum of the Measure of Interior Angles = (5 - 2)180o

Where n = no. of sides in the traverse.

For given traverse no. of sides = 5

Sum of all interior angles = (5 - 2) $\times$ 180o = 540o

Total Angular Error = Sum of angles measured in field

Adjustment for each angle = -1 $\times$ total angular Error / no of angles

Adjusted angles are presented in below table:

Step 2: Calculating preliminary aximuths:

Station Total Angular Error Deg Deg 153 А | 30 25 11 15 B с Measured Angles Min Sec 152 54 111 57 55 163 35 53 10 539 25 Correction in each angle 11'55" 11'55" 11'55" 11'55" 11'55" Corrected Angles Min Sec 6 111 23 10 50 163 44 30 53 38 5 540 0 | 55 -59'35" 58 32 E 26 Total 0

Given Azimuth of AB = 142o50'51"

Our traverse is clockwise, so we will use formula A to calculate the results. Results are computed in below table:

Formula: A) To calculate azimuths clockwise around a traverse: Azimuth of preceding line = Azimuth of back line-Angle between lines + 1800 B) To calculate azimuths counter-clockwise around a traverse: Azimuth of preceding line = Azimuth of back line +Angle between lines - 1800

Part 3 & 4: Close traverse by bowditch method:

Station Line Corrected Angles Deg Min Sec 153 Given Azimuth Min Preliminiary Azimuth Deg Min Sec Deg Sec L25 AB 142 50 51 111 23 BC 211 58 333 19 163 44 DE 349 35 53 38 FA 115 57 153 Check AB 142 50 51

Step 1: Calculating unadjusted departure, latitudes, error of closure:

Accordingly the results are calculated in table below:

Formula: Departure is the horizontal displacement of a line on orthographic coordinate plane and latitude is the vertical displacement. These are given by below formula: A) Latitude of a line = L cosa B) Departure of a line = L sina Where, L = Length of line and a = Azimuth

For a closed traverse, sum of latitudes and departures is always zero, traverse is not perfectly closed.

Line Azimuth Min Deg AB 142 50 BC 27 211 333 Length (m) Latitude (m) Departure (m) Sec 51 331.160 -263.945 200.0000 41 435.160 -371.188 -227.1200 51 486.920 435.118 -218.5480 21 361.860 355.903 -65.3900 16 354.580 -155.184 318.8180 1,969.680 0.704 7.760 CD 19 349 35 DE EA 57 115 Total

Error of closure for latitudes = 0.704 m

Error of closure for departures = 7.760 m

Step 2: Adjusted Latitudes and departures

Since Line AB Length and directions are fixed, total error will be adjusted among balance 4 lines. Adjusted Latitude and departures are calculated in below table:

Compass Rule: Adjustment in departures = Total departure error x length of line/perimeter of traverse Adjustment in latitudes = -Total latitude error x length of line/perimeter of traverse

Step 3: Final Adjusted lengths and final directions

Line Length of line (m) Preliminary Latitude Preliminary Correction in Correction in Departure latitude departure Adjusted Latitude Adjusted Departure G = C+E H = D+F A AB -263.945 0.000 -263.945 200.000 200.000 -227.120 -371.188 -2.061 -371.375 -229.181 FIXED 435.160 486.920 361.860 435.118 -218.548 -2.306 434.909 -220.854 0.000 -0.187 -0.209 -0.156 -0.152 -0.704 355.903 -65.390 -1.714 355.747 -67.104 354.580 -155.184 318.818 -155.336 317.139 -1.679 -7.760 Total 1,638.520 0.704 7.760 0.000 0.000

Adjusted Length = (Adjusted latitude? + Adjusted departure)

Adjusted lengths are computed in below table:

Adjusted azimuths are computed in below table:

Line Adjusted Latitude Adjusted Departure Adjusted Length -263.945 200.000 331.160 -371.375 -229.181 436.398 434.909 -220.854 487.773 355.747 -67.104 362.021 -155.336 317.139 353.138

Azimuth of a line = tan-1 $\left [ \frac{Adjusted/ departure}{Adjusted / latitude} \right ]$

Line Adjusted Latitude Azimuth Adjusted Departure Departure/ Latitude AB -263.945 200.000 -0.75773 142°50'51" BC -371.375 -229.181 0.61711 21104046" 434.909 -220.854 -0.50782 3330440" 355.747 -67.104 -0.18863 349°19'4" -155.336 317.139 -2.04163 116°5'45"

Part 5: Calculations of Coordinates:

Given Coordinates of A (E<N) = (327,682.209, 5,813,380.144)

Northing of a station = Northing of previous station + Latitude

Easting of a station = Easting of previous station + Departure

Accordingly results are furnished below:

Line Station Adjusted Latitude Adjusted Departure Northing Easting 5 813 380.144 327 682.209 AB -263.945 200.000 5 813 116.199 327 882.209 BC -371.375 -229.181 5 812 744.824 327 653.028 434.909 -220.854 5 813 179.733 327 432.174 DE 355.747 -67.104 5 813 535.480 327 365.070 -155.336 317.139 Check 5 813 380.144 327 682.209

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