area and volume - bar bar black sheep traversing: the traverse in which angular measurements are...

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1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 1

Chapter -7-

Traversing

Ishik University Sulaimani Civil Engineering Department Surveying II – CE 215


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)

Contents

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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.


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.

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.

A

B

C D

E

F

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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.


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.

LEFT HAND ANGLES

RIGHT HAND ANGLES

A

B

C

D

E

F B

C

D E

F

A

G

X

Y

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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


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

2. Traversing - Computations

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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 = (160 +180) + 100 = 440 - 360 80

Az34 = Az32 + angle at 3 = (80+180) +210 = 470 - 360 110

Az45 = Az43 + angle at 4 = (110+180) +190 = 480 - 360 120

1

2

3

4

5

100

210

190

Example 1.


Example 2.

If Az. AB =330 ˚00’00’’

Find the azimuths for

BC, CD, DE and EA

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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

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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).

Example 4.


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The maximum allowable error in the traverse which is given by

= kn

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.

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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

= 000'41"

E

N



The angular measurement for each angle should therefore be better than 000'41". 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

1/1000 0° 03 ' 26" 01 '

1/3000 0 ° 01 ' 09" 01 '

1/5000 0 ° 00 ' 41" 30"

1/7500 0 ° 00 ' 28" 20"

1/10000 0 ° 00 ' 21" 20"

1/20000 0 ° 00 ' 10" 10"

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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 : 10000.

In the example given, if the total distance measured along a traverse is 253.56m, and the linear

misclosure is 0.01m, then the proportional linear misclosure is

0.01/253.56 = 1/25356 or approximately 1 : 25000


relative precision = linear misclosure / traverse length

expressed as a number 1 / ?

Example 5:

linear misclosure = 0.08 ft.

traverse length = 2466.00 ft.

relative precision = 0.08/2466.00 = 1/30,000

Surveyor would expect 1-foot error for every 30,000 feet surveyed

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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 = ( E2 + N2)

E

N


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.

5. Balancing Angles

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Example 6.


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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.


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

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Signs of Departures and Latitudes

North

East West

South

Departure (+)

Latitude (+)

Departure (-)

Latitude (+)

Departure (+)

Latitude (-)

Departure (-)

Latitude (-)


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

7. Easting and Northing

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8. 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

8. Balancing The Traverse

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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.


The compass or Bowditch rule which was named after the distinguished American navigator

Nathaniel Bow ditch (1773-1838), 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.

(a) Bowditch’s method ( Compass rule);

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These correction are given by the following equations:

D= total closure in latitude or the algebraic sum of the north and south latitudes (NL + SL) L = total closure in departure or the algebraic sum of the east and west departures (ED + WD)


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.

(b) Transit rule ;

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(b) Transit rule

These corrections are given by the following equations


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

Adjust Linear Error

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Example 7.

1. Counter-clockwise approach solution

2. Clockwise approach solution


1. Counter-clockwise approach solution

2. Clockwise approach solution

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Compass Rule Adjustment for Traverse

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Homework 7

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Homework 8


Homework 9

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Homework 9 continue


Homework 10

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TRAVERSE AREA

COMPUTATION

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• The area of a closed traverse may be calculated from

1) The Coordinates Method (x and y)

2) Double Meridian Distance Method (DMD)

TRAVERSE AREA


1. The coordinates method

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Example 8. Compute the area by rectangular coordinate method for the following coordinates;


Solution ;

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Solution ;


Example 9.

Compute the area by rectangular coordinate method for the following coordinates;

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Solution ;


Solution ;

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2. Double Meridian Distance Method (DMD)


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Example 10.

The following table shows the balanced latitudes and departure, for the traverse example shown. Compute the area by DMD method.


Solution ;

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Solution ;


Example 11.

The following table shows the balanced latitudes and departure, for the traverse example shown. Compute the area by DMD method.

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Then,

Homework -11-

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area and volume - bar bar black sheep traversing: the traverse in which angular measurements are...

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