Concept:
For an error-free plotted traverse:
ΣL = 0, ΣD = 0
ΣL = Algebraic sum of Latitudes of all the sides
ΣD = Algebraic sum of departure of all the sides
For a segment AB
Latitude = L cos θ & Departure = L sin θ
In whole circle bearing, Bearings are taken w.r.t North Direction
Calculation:
As Latitude is +ve & Departure is - ve
The line AB lies in 4^{ th} quadrant ( > 270 ° )
From the figure
\(\tan \theta = \frac{{\sum D}}{{\sum L}} = \frac{{-45.1}}{{78}}\)
∴ θ = - 30°
The whole circle bearing of the line AB is
α = - 30° + 360° = 330°Total closing error(c) = \(\sqrt {{{\left( {\sum L} \right)}^2} + {{\left( {\sum D} \right)}^2}}\)
Sum of latitude = 0.4 m
Sum of Departure = 0.3 m
\(C = \sqrt {{{\left( {0.4} \right)}^2} + {{\left( {0.3} \right)}^2}} \)
C = 0.5m
Relative error of closure or Relative precision of this traverse = \(\frac{c}{p}\)
p ⇒ Perimeter of traverse = 1 km = 1000 m
Relative precision = \(\frac{{0.5}}{{1000}} = 5 \times {10^{ - 4}} = \frac{1}{{2000}}\)
The departure is positive and latitude is negative
Concept:
L = Latitude is the projection on North-South meridian
Latitude is positive in north direction and negative in south direction
Northing → Positive, Southing → Negative
D = Departure is the projection on East-west meridian
Departure is positive in east direction and negative in west direction
Easting → Positive, Westing → Negative
θ = Bearing angle
l = Length of the line
L = l × Cos θ
D = l × Sin θ
Calculation:
Given,
The traverse line has W.C.B from 90° to 180°
Latitude lies in south direction, hence it will be negative.
Departure lies in east direction, hence it will be positive.
Explanation
Loose needle/Free needle method-
Fast needle method-
Included angle method-
Direct angle method-
Deflection angle method-
Concept:
For 20 m chain length,
Degree of curve (D) is given by:
\(D = \frac{{1146}}{R}\)
For 30 m chain length,
Degree of curve (D) is given by:
\(D = \frac{{1719}}{R}\)
Calculation:
Given,
L = 20 m
R = 573 m
\(D = \frac{{1146}}{{573}}\)
\(D = {2^o}\)
Explanation
Transit Rule:
Bowditch’s rule:
If the reduced bearing of line AB is N 30° E and length is 100 m, then the latitude and departure respectively of the line AB will be:
Concept:
L = Latitude is the projection on North-South meridian
D = Departure is the projection on East-west meridian
θ = Bearing angle
l = Length of the line
L = l × Cos θ
D = l × Sin θ
\({\rm{Line}}\;{\rm{closure = L}}{\rm{.C = }}\sqrt {{{\left( {{\rm{\Sigma L}}} \right)}^{\rm{2}}}{\rm{ + }}{{\left( {{\rm{\Sigma D}}} \right)}^{\rm{2}}}} \)
Calculation:
Given,
l = 100 m, θ = 30°
L = + 100 × cos 30° = + 86.60 m
D = + 100 × sin 30° = + 50 m
The latitude and departure respectively of the line AB are + 86.60 m and 50 m
Latitude of traverse line
Explanation:
Transit rule:
This method is developed for balancing a traverse in which angles are measured with a higher degree of precision than the lengths of the sides. It is based on the assumption that the error in departure or latitude of a traverse side is proportional to its departure or latitude. Thus, according to the transit rule, the corrections to the departure or latitude of a traverse, the side can be calculated by using: Correction for departure: \(\;{\bf{\delta }}{{\bf{d}}_1} = \frac{{{{\bf{D}}_1}}}{{{\bf{\Sigma D}}}} \times {{\bf{e}}_{\bf{D}}}\) Correction for latitude: \({\bf{\delta }}{{\bf{l}}_1} = \frac{{{{\bf{L}}_1}}}{{{\bf{\Sigma L}}}} \times {{\bf{e}}_{\bf{L}}}\) Where, δd1 = Correction in departure of side 1 D1 = Departure of traverse side 1 ΣD = Arithmetic sum of the departures of all the sides of the traverse eD = Total error in departure δl1 = Correction in latitude of side 1 L1 = Latitude of the traverse side 1 ΣL = Arithmetic sum of the latitudes of all the sides of the traverse eL = Total error in latitude |
A traverse consists of series of straight lines of known length related to one another by known angles between the lines.
Traverse survey is a method of establishing control points, their position being determined by measuring the distances between the traverse stations which serve as control points and the angles subtended at the various stations by their adjacent stations.
A traverse is developed by measuring the distance and angles between points that found the boundary of a site.In any closed traverse, if the survey work is error free, then
1. The algebraic sum of all the latitudes should be equal to zero.
2. The algebraic sum of all the departures should be equal to zero.
3. The sum of the northings should be equal to the sum of the southings.
Which of the above statements are correct?
Concepts
The latitude of a line is its perpendicular projection in the N-S direction. It is positive for northing and negative for southing.
The departure of a line is its perpendicular projection in the E-W direction. It is positive for Easting and negative for Westing.
Error in any traverse survey is given by the summation of all the latitude or departures i.e. \(\sum L\) or \(\sum D.\)
A traverse is said to be closed, it is error-free.
Therefore, for a closed traverse: \(\sum {\bf{L}} = 0\;{\bf{and}}\;\sum {\bf{D}} = 0\)
Confusion Points
For the network to be error-free only the first two conditions, are necessary,
e.i. \(\sum {\bf{L}} = 0\;{\bf{and}}\;\sum {\bf{D}} = 0\)
The third one is correct but not required as it can be an inference drawn from condition two. So, the appropriate answer is option A.
Bowditch Rule is used when length and bearing both are equally precise. In this method, total error in Latitude and Departure is distributed as per the ratio of length of lines i.e.
Correction in Latitude is:
\({C_L} = \;\frac{l}{{{\rm{\Sigma }}l}} \times {\rm{\Sigma }}L\)
Correction in Departure is:
\({C_D} = \;\frac{l}{{{\rm{\Sigma }}l}} \times {\rm{\Sigma }}D\)
Where ΣL and ΣD are total error in latitude and departure respectively.
Σl is the summation of length of all lines involved in closed traverse.
Note:
Transit rule is used when bearings are more precise then length of line.
Concept:
Balancing of errors: The process of adjusting the latitudes and departures to make the algebraic sum of latitudes or departures to zero is called the balancing of errors.
The two types of balancing rules to eliminate errors in traverse surveying are as follows:
1) Bowditch rule (Compass rule)/Compass rule:
It is most commonly adopted when angular measurement and linear measurement both are nearly of same precision.
The correction is considered directly proportional to the length of the side.
By Bowditch rule Correction to a particular line is given by
\( {C_L} = \;l \times \frac{{{\bf{\Sigma }}L}}{{{\bf{\Sigma }}l}}\)
\({C_D} = \;l \times \frac{{{\rm{\Sigma }}D}}{{{\rm{\Sigma }}l}}\)
Where CL, CD is corrections in latitude and longitude for a line.
Correction for Latitude/Departure = Total error in Latitude/Departure × \(\frac{{{\bf{length}}\;{\bf{of}}\;{\bf{the}}\;{\bf{side}}}}{{{\bf{perimeter}}\;{\bf{of}}\;{\bf{the}}\;{\bf{traverse}}}}\)
2) Transit rule:
When angular measurements are more precise than linear measurements, the transit method is adopted.
Correction to latitude/departure of any side = Total error in latitude/departure × \(\frac{{{\bf{Latitude/Departure}}\;{\bf{of}}\;{\bf{the}}\;{\bf{side}}}}{{{\bf{Arthmetical}}\;{\bf{sum}}\;{\bf{of}}\;{\bf{all}}\;{\bf{latitudes/departure}}}}\)
1. When the adjustment is made by the Bowditch rule, the length of the sides becomes less and the angle becomes more than that when the adjustment is made by the transit rule.
2. In Bowditch rule, it is assumed that the errors in linear measurements are proportional to \(\sqrt l \) and angular measurements are proportional to \(\frac{1}{{\sqrt l }}\)
Explanation:
For an open traverse with 'n' numbers of station:
Total no of fore bearing = n - 1, and total no back bearing = n - 1.
The stations can be primary classified as first station, intermediate station, and last station.
For every intermediate stations: One fore bearing and back bearing is required each.
For first station: Only one fore bearing required, No back bearing required
For end station: Only one back bearing required, No fore bearing required
Concept:
The equation for closing error and angle fo misclosure is given by,
\({\rm{e}} = \sqrt {{{\left( {{\rm{Σ L}}} \right)}^2} + {{\left( {{\rm{Σ D}}} \right)}^2}} \)
\({\rm{\theta }} = {\tan ^{ - 1}}\left[ {\frac{{{\rm{Σ D}}}}{{{\rm{Σ L}}}}} \right]\)
Where,
e = closing error, θ = angle of misclosure
ΣL = Algebraic sum of latitudes of all lines
ΣD = Algebraic sum of departures of all lines
The relative error of closure (r)
\({\rm{r}} = \frac{{{\rm{Closing\;error\;of\;a\;traverse}}}}{{{\rm{perimeter\;of\;a\;traverse}}}} = \frac{{\rm{e}}}{{\rm{P}}}\)
Where, P =perimeter of a traverse = Total length of a traverse
The degree of accuracy or the relative precision
\({\rm{Relative\;precision}} = \frac{1}{{\frac{{\rm{P}}}{{\rm{e}}}}} \)
Calculation:
Given,
ΣL = 0.4m, ΣD = 0.3 m
P = 1 km = 1000 m
we know that
\({\rm{e}} = \sqrt {{{\left( {{\rm{Σ L}}} \right)}^2} + {{\left( {{\rm{Σ D}}} \right)}^2}} \)
\({\rm{e}} = \sqrt {{{\left( {{\rm{0.4}}} \right)}^2} + {{\left( {{\rm{0.3}}} \right)}^2}} \)
e = 0.5 m
\({\rm{Relative\;precision}} = \frac{1}{{\frac{{\rm{P}}}{{\rm{e}}}}}=\frac{1}{{\frac{{\rm{1000}}}{{\rm{0.5}}}}} =\frac{5}{{10000}} = \frac{1}{{2000}}\)
Explanation
Loose needle/Free needle method-
Fast needle method-
Included angle method-
Direct angle method-
Deflection angle method-
A triangulation station selected close to the main station for avoiding intervening obstruction is called
Explanation:
Triangulation:
Pivot point:
The station where observations are not made, but the angles at the station are used in the triangulation series, is known as the Pivot station.
Satellite station:
Tie station:
Concept:
Total Station:
Uses of Total Station:
From figure.
Internal angle ∠B = ∠OBA + ∠OBC ---- (I)
i) ∠OBA = ∠PAB = 60°30’ [Parallel lines AP, OQ]
∠OBC = 180° - 122° = 58°
Substitute in (I)
∠B = 60°30’ + 58°
= 118°30’
∴ The correct option is 118° 30’.
Explanation:
In general, the closing error in traverse survey is given as:
\(e = \sqrt {{{\left( {{\rm{\Sigma }}L} \right)}^2} + {{\left( {{\rm{\Sigma }}D} \right)}^2}} \)
Where
ΣL = Summation of Latitudes of all lines involved in traverse or also called error in Latitude
ΣD = Summation of Departure of all lines involved in traverse or also called error in Departure
However, in a closed traverse in which bearings are observed, the closing error in bearing may be determined by observing the bearing of the last line and correction in bearing of last line is the closing error.