Surveying Test 3

Trigonometric levelling is the process of determining the differences of elevations of stations from observed vertical angles and known distances.

In Trigonometric levelling:

1.  The vertical angles are measured by means of Theodolite.

 2. The horizontal distances by instrument

3. Relative heights are calculated using trigonometric functions.

The trigonometric Leveling can be done in two ways:

1. Observations taken for the height and distances: In this, we can measure the horizontal distance between the given points if it is accessible and take the observation of the vertical angles and then compute the distances using them.

2. Geodetic Observations: In this, the distances between the two points are geodetic distances and the principles of the plane surveying are not applicable here. The corrections for the curvature and refraction are applied to the angles directly.

Concept:

In cartography, contour lines are an imaginary line of a terrain that joins the point of equal elevation above a given level, such as mean sea level or benchmark.

Contour interval on map sheet denotes vertical distance between two successive contour lines.

Following are the characteristics of contour lines in surveying for reading contour maps:

• Two contour lines do not intersect each other except in the case of an overhanging cliff and a cave penetrating a hill.
• All lines close themselves within the map boundaries or outside it.
• If the contour lines are very close to each other, this indicates a steep slope.
• If the contour lines are at a very large distance to each other, this indicates a gentle slope.
• If the closed lines have higher elevation in the center then it represents the hill or mountain.
• If the closed line have increasing elevation as we move away, then it represents a pond or a depression in the ground profile.
• The direction of the steepest slope is along the shortest between the contours.

 The contour interval depends upon the following factors:

• Scale of the Map: If the scale is small, the contour interval is kept large and vice versa so as to avoid overcrowding of contours.
• Purpose of Map: The contour interval selected should be small so that the map serves the intended purpose.
• Nature of Ground: For a flat ground the interval is kept small and for a steep slope the interval is kept large.
• Time and Funds: Contour interval is kept large when time and funds are less.

Concept:

Tacheometry is one of the methods used in surveying to accurately measured the horizontal distance between two inaccessible points on the ground.

Horizontal distance from instrument station and ground point is given by:

D = ks + c

Where,

k = multiplying constant

i = stadia interval

f = focal length of the objective

k = f/i

c = additive constant

c = f + d

Calculation:

Given:

k = Multiplying constant = 100

c = Additive constant = 0.15

∵ D = kS + c

⇒ D = 100 × (1.62 – 0.90) + 0.15 = 72.15 m

Now, R.L of instrument station = 100.1 m

Height of Trunnion axis = 1.60 m

∴ R.L of line of collimation = 100.10 + 1.60 = 101.70

R.L of staff station = 101.70 - 1.20 = 100.50 m

Concept:

i) Mid-Ordinate Rule

Area of plot = h1 × d + h2 × d + … + hn × d = d (h1 + h2 + … hn)

∴ Area = common distance × sum of mid-ordinates

ii) Average-Ordinate Rule

\(\text{Area}=\frac{{{\text{O}}_{1}}+{{\text{O}}_{2}}+\ldots +{{\text{O}}_{\text{n}}}}{{{\text{O}}_{\text{n}+1}}}\times \text{l}\)

\(\text{i}.\text{e}.\text{ }\!\!~\!\!\text{ Area}=\frac{\text{Sum }\;\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ ordinates}}{\text{No}.\;\text{ }\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ ordinates}}\times \text{length }\;\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ base }\;\!\!~\!\!\text{ line}\)

(iii) Trapezoidal Rule

\(\text{Total }{area}=\frac{\text{d}}{2}\left\{ {{\text{O}}_{1}}+2{{\text{O}}_{1}}+2{{\text{O}}_{2}}+\ldots +2{{\text{O}}_{\text{n}-1}}+{{\text{O}}_{\text{n}}} \right\}\)

\(\text{Total }{ Area}=\frac{\text{Common }\;\!\!~\!\!\text{ distance}}{2}\times \{\left( 1\text{st }\!\!~\!\!\text{ ordinate}+\text{last }\!\!~\!\!\text{ ordinate} \right)+2\left( \text{sum }\!\!~\!\!\text{ of }\!\!~\!\!\text{ other }\!\!~\!\!\text{ ordinate} \right))\}\)

(iv) Simpson’s Rule

\(\text{Total }{ area}=\frac{\text{d}}{3}\left( {{\text{O}}_{1}}+4{{\text{O}}_{2}}+2{{\text{O}}_{3}}+4{{\text{O}}_{4}}+\ldots +{{\text{O}}_{\text{n}}} \right)\)

\(\text{Total }{ Area}=\frac{\text{Common }\;\!\!~\!\!\text{ Distance}}{3}\times \left\{ \left( 1\text{st }\!\!~\!\!\text{ ordinate}+\text{last }\!\!~\!\!\text{ ordinate} \right)+4\left( \text{sum }\!\!~\!\!\text{ of }\!\!~\!\!\text{ even }\!\!~\!\!\text{ ordinates} \right)+2\left( \text{sum }\!\!~\!\!\text{ of }\!\!~\!\!\text{ remaining }\!\!~\!\!\text{ odd }\!\!~\!\!\text{ ordinates} \right) \right\}\)

Calculation:

By using Trapezoidal rule:

d = 3m, 0₁ = 2m, 0₂ = 3.15 m …

\(\therefore A = d\left[ {\frac{{{0_1} + {0_n}}}{2} + {0_2} + {0_3} + \ldots } \right]\) 

\( = 3\left[ {\frac{{2 + 3.14}}{2} + 4.3 + 3.60 + 7.50 + 6.45 + 4.65} \right]\)= 87.21 m²

Using Simpson’s Rule:

\(A = \frac{d}{3}\left[ {{O_1} + {O_n} + 4\left( {{O_2} + {O_4} + {O_6} + \ldots {O_{n - 1}}} \right) + 2\left( {{O_3} + {O_5}} \right)} \right]\) 

\(= \frac{3}{3}\left[ {\left( {2 + .314} \right) + 4\left( {3.15 + 3.60 + 6.45} \right) + 2\left( {4.30 + 7.50 + 4.65} \right)} \right]\) = 90.84 m²