Chapter -7- Traversing. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 1. Contents
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1 Ishik University Sulaimani Civil Engineering Department Surveying II CE 215 Chapter -7- Traversing 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 1 Contents 1. Traversing 2. Traversing Computations 3. Traversing Angular Error 4. Traversing Precision 5. Linear Misclosure 6. Balancing Angles 7. Computation of Latitude and Departure 8. Easting and Northing 9. Traverse Adjustment (Balancing Traverse) a. Bowditch s method ( Compass rule) b. Transit rule 10. Traverse Area 1. The Coordinate Method 2. Double Meridian Distance Method (DMD) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 2 1
2 1. Traversing A series of connected straight lines each joining two points on the ground, is called a traverse. End points are known as traverse stations & straight lines between two consecutive stations, are called traverse legs. A traverse survey is one in which the framework consists of a series of connected lines, the lengths and directions of which are measured with a chain or a tape, and with an angular instrument respectively. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 3 Traverses may be either a closed traverse or an open traverse: 1. Closed Traverse: (POLYGON or LOOP TRAVERSE) A traverse is said to be closed when a complete circuit is made, i.e. when it returns to the starting point forming a closed polygon or when it begins and ends at points whose positions on plan are known. The work may be checked and balanced. It is particularly suitable for locating the boundaries of lakes, woods, etc. and for the survey of moderately large areas. A F B E C D 2. Open Traverse: (LINK TRAVERSE) A traverse is said to be open or unclosed when it does not form a closed polygon. It consists of a series of lines extending in the same general direction and not returning to the starting point. Similarly, it does not start and end at the points whose positions on plan are known. It is most suitable for the survey of a long narrow strip of country e.g. the valley of a river, the coast line, a long meandering road, or railway, etc. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 4 2
3 Classification of traverses based on instruments used: 1. Chain Traversing: In chain traversing, the entire work is done by a chain or tape & no angular measuring instrument is needed. The angles computed by tie measurements are known as chain angles. 2. Compass Traversing: The traverse in which angular measurements are made with a surveying compass, is known as compass traversing. The traverse angle between two consecutive legs is computed by observing the bearings of the sides. 3. Plane Table Traversing: The traverse in which angular measurements between the traverse sides are plotted graphically on a plane table with the help of an alidade is known as plane table traversing. 4. Theodolite Traversing: The traverse in which angular measurements between traverse sides are made with a theodolite is known as theodolite traversing. 5. Tachometric Traversing: The traverse in which direct measurements of traverse sides by chaining is dispensed with & these are obtained by making observations with a tachometer is known as tachometer traversing. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 5 A F X B LEFT HAND ANGLES A C RIGHT HAND ANGLES B D E C D E Y F G a) is obviously closed b) must start and finish at points whose co-ordinates are known, and must also start and finish with angle observations to other known points. Working in the direction A to B to C etc is the FORWARD DIRECTION. This gives two possible angles at each station. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 6 3
4 2. Traversing - Computations Check on closed traverse: Sum of the measured interior angles (2n-4) x 90 Or (internal angles) = (n 2) 180 Sum of the measured exterior angles (2n+4) x 90 Or (external angles) = (n + 2) 180 The algebric sum of the deflection angles should be equal to 360. Right hand deflection is considered +ve, left hand deflection ve 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 7 2. Traversing - Computations The general formula that is used to compute the azimuths is: forward azimuth of line = back azimuth of previous line + clockwise (internal) angle The back azimuth of a line is computed from : back azimuth = forward azimuth 180 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 8 4
5 Example 1. For a traverse from points 1 to 2 to 3 to 4 to 5, if the angles measured at 2, 3 and 4 are 100, 210, and 190 respectively, and the azimuth of the line from 1 to 2 is given as 160, then Az23 = Az21 + angle at 2 = ( ) = Az34 = Az32 + angle at 3 = ( ) +210 = Az45 = Az43 + angle at 4 = ( ) +190 = /28/2018 Assistant Lecturer / Asmaa Abdulmajeed 9 Example 2. If Az. AB = Find the azimuths for BC, CD, DE and EA 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 10 5
6 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 11 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 12 6
7 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 13 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 14 7
8 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 15 Example 3. The same problem when the bearings of the sides are expressed in quadrantal system. Calculate; 1. W.C.B. angle 2. Azimuth angle 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 16 8
9 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 17 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 18 9
10 Example 4. Compute and tabulate the bearings of a regular hexagon given the starting bearing of side AB = S 50 10'E (Station C is easterly from B). 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 19 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 20 10
11 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Traversing Angular Error The maximum allowable error in the traverse which is given by = k n k therefore depends on the maximum allowable angular error as it relates to the least count of the instrument. For a 1/5000 traverse, the value of k = 30", so = 30" n. If a misclosure exists, then the figure computed is not mathematically closed. This can be clearly illustrated with a closed loop traverse. The co-ordinates of a traverse are therefore adjusted for the purpose of providing a mathematically closed figure while at the same time yielding the best estimates for the horizontal positions for all of the traverse stations. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 22 11
12 3. Traversing Angular Error The required accuracy of the survey in terms of its proportional linear misclosure also defines the equipment and allowable misclosure values. For example, for a traverse with an accuracy of better than 1/5000 would require a distance measurement technique better than 1/5000, and an angular error that is consistent with this figure. If the accuracy is restricted to 1/5000, then the maximum angular error is 1/5000 = tan E N = 0 00'41" 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Traversing Angular Error The angular measurement for each angle should therefore be better than 0 00'41". The general relationship between the linear and angular error is given by the following table Prop. Linear accuracy Maximum angular error Least count of instrument 1/ ' 26" 01 ' 1/ ' 09" 01 ' 1/ ' 41" 30" 1/ ' 28" 20" 1/ ' 21" 20" 1/ ' 10" 10" 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 24 12
13 4. Traversing Precision By itself the linear misclosure only gives a measure of how far the computed position is from the actual position (accuracy of the traverse measurements). Another parameter that is used to provide an indication of the relative accuracy of the traverse is the proportional linear misclosure. Here, the linear misclosure is divided by total distance measured, and this figure is expressed as a ratio e.g. 1 : In the example given, if the total distance measured along a traverse is m, and the linear misclosure is 0.01m, then the proportional linear misclosure is 0.01/ = 1/25356 or approximately 1 : /28/2018 Assistant Lecturer / Asmaa Abdulmajeed 25 relative precision = linear misclosure / traverse length expressed as a number 1 /? Example 5: linear misclosure = 0.08 ft. traverse length = ft. relative precision = 0.08/ = 1/30,000 Surveyor would expect 1-foot error for every 30,000 feet surveyed 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 26 13
14 5. Linear Misclosure These discrepancies represent the difference on the ground between the position of the point computed from the observations and the known position of the point. The easting and northing misclosures are combined to give the linear misclosure of the traverse, where linear misclosure = ( E 2 + N 2 ) N E 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Balancing Angles Method 1: Single Measurements: Apply an average correction to each angle where observing conditions were approximately the same at all stations. The correction is computed for each angle by dividing the total angular misclosure by the number of angles. Method 2: Single Measurements: Make larger corrections to angles where poor observing conditions were present. This method is seldom used. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 28 14
15 Example 6. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 29 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 30 15
16 6. Computation of Latitude and Departure The coordinates of points are defined as departure and latitude. The latitude is always measured parallel to the reference meridian. The departure perpendicular to the reference meridian. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 31 Latitude (L) The latitude of a line is its orthographic projection on the N-S axis representing the meridian. Thus, the latitude of a line is the distance measured parallel to the North- South line. Thus, Latitude(L)=l cosθ Departure (D) The departure of a line is its orthographic projection on the axis perpendicular to the meridian. The perpendicular axis is also known as the E-W axis. Thus, Departure(D)= l sinθ 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 32 16
17 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 33 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 34 17
18 Signs of Departures and Latitudes North Departure (-) Latitude (+) Departure (+) Latitude (+) West East Departure (-) Latitude (-) Departure (+) Latitude (-) South 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Easting and Northing The coordinates (X,Y) given by the perpendicular distances from the two main axes are the eastings and northings, respectively, as shown in Fig. The easting and northing for the points P and Q are (EP, NP,) and (EP, NP,), respectively. Thus the relative positions of the points are given by 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 36 18
19 8. Balancing The Traverse 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Balancing The Traverse A traverse is balanced by applying corrections to latitudes and departures. This is called balancing a traverse. The following are common methods of adjusting a traverse 1) Bowditch's rule 2) Transit rule 3) Third rule 4) Graphical construction method 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 38 19
20 The term balancing is generally applied to the operation of adjusting the closing error in a closed traverse by applying corrections to departures and latitudes The following methods are generally used for balancing a traverse: (a) Bowditch s method ( Compass rule); When the linear errors are proportional to l and angular errors are proportional to 1/ l, where l is the length of the line. This rule can also be applied graphically when the angular measurements are of inferior accuracy such as in compass surveying. In this method the total error in departure and latitude is distributed in proportion to the length of the traverse line. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 39 (a) Bowditch s method ( Compass rule); The compass or Bowditch rule which was named after the distinguished American navigator Nathaniel Bow ditch ( ), is a very popular rule for adjusting a closed traverse. The compass rule is based on the assumption that all lengths were measured with equal care and all angles taken with approximately the same precision. It is also assumed that the errors in the measurement are accidental and that the total error in any side of the traverse is directly proportional to the total length of the traverse. The compass rule may be stated as follows: The correction to be applied to the latitude (or departure) of any course is equal to the total closure in latitude (or departure) multiplied by the ratio of the length of the course to the total length or perimeter of the traverse. To determine the adjusted latitude of any course the latitude correction is either added to or subtracted from the computed latitude of the course. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 40 20
21 These correction are given by the following equations: D= total closure in latitude or the algebraic sum of the north and south latitudes ( N L + S L ) L = total closure in departure or the algebraic sum of the east and west departures ( E D + W D ) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 41 (b) Transit rule ; The method of adjusting a traverse by the transit is similar to the method using the compass rule. The main difference is that with the transit rule the latitude and departure corrections depend on the length of the latitude and departure of the course respectively instead of both depending on the length of the course. The transit rule has no sound theoretical foundation since it is purely empirical. The rule is based on the assumption that the angular measurements are more precise than the linear measurements and that the errors in traversing are accidental. The transit rule may be stated as follows: The correction to be applied to the latitude (or departure) of any course is equal to the latitude (or departure) of the course multiplied by the ratio of the total closure in latitude (or departure) to the arithmetical sum of all the latitudes (or departures) of the traverse. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 42 21
22 (b) Transit rule These corrections are given by the following equations 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 43 Adjust Linear Error Transit rule When angles are more accurate than distances Proportion L error based on total N-S distance Proportion Dep error based on total E-W distance Compass Rule more common Assumes angles are as accurate as distances Proportion both errors based on total distance 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 44 22
23 Example Counter-clockwise approach solution 2. Clockwise approach solution 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Counterclockwise approach solution 2. Clockwise approach solution 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 46 23
24 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 47 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 48 24
25 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 49 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 50 25
26 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 51 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 52 26
27 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 53 Compass Rule Adjustment for Traverse 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 54 27
28 Compass Rule Adjustment for Traverse 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 55 Compass Rule Adjustment for Traverse 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 56 28
29 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 57 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 58 29
30 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 59 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 60 30
31 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 61 Homework 7 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 62 31
32 Homework 8 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 63 Homework 9 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 64 32
33 Homework 9 continue 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 65 Homework 10 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 66 33
34 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 67 TRAVERSE AREA COMPUTATION 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 68 34
35 TRAVERSE AREA The area of a closed traverse may be calculated from 1) The Coordinates Method (x and y) 2) Double Meridian Distance Method (DMD) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed The coordinates method 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 70 35
36 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 71 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 72 36
37 Example 8. Compute the area by rectangular coordinate method for the following coordinates; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 73 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 74 37
38 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 75 Example 9. Compute the area by rectangular coordinate method for the following coordinates; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 76 38
39 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 77 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 78 39
40 2. Double Meridian Distance Method (DMD) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 79 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 80 40
41 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 81 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 82 41
42 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 83 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 84 42
43 Example 10. The following table shows the balanced latitudes and departure, for the traverse example shown. Compute the area by DMD method. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 85 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 86 43
44 Solution ; 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 87 Example 11. The following table shows the balanced latitudes and departure, for the traverse example shown. Compute the area by DMD method. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 88 44
45 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 89 Homework -11- Then, 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 90 45
46 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 91 46
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