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NTS STUDY

Nodal Theory of Structure : Every Node Matters, Every Structure Tells A Story.

Surveying Master class part 2: All topic Cover in one Place

Surveying Master class part 2: All topic Cover of Survey Engineering 

Surveying Master class part 2: All topic Cover of Survey Engineering


Fundamental Principles of Levelling

Levelling is used to determine the elevations of various points on the Earth's surface.

  • Level Line: A line parallel to the Earth's surface (Mean Sea Level).

  • Horizontal Line: A line at 90° to the plumb line and tangent to the level line.

  • Mean Sea Level (MSL): Determined by the average of sea fluctuations over a 19-year period.

  • Reduced Level (RL): The elevation of any point with respect to the MSL.

Key Definitions in Levelling

  • Reduced Level (RL): The height of any point measured with respect to Mean Sea Level (MSL).

  • Absolute Level: If the height of a point is measured from the center of the Earth, it is called an Absolute level.

  • Bench Mark (BM): A fixed point on Earth whose RL is already known. These can be Permanent or Temporary.

Types of Sights

Readings taken during levelling are divided into three types:

  • Back Sight (BS): The first reading taken on a Bench Mark after the instrument is set up.

  • Fore Sight (FS): The final reading taken from a single position of the instrument.

  • Intermediate Sight (IS): All other readings taken between the back sight and the fore sight.

Methods of Calculating RL

There are two main methods for determining the RL of points:

  1. Height of Instrument (HI) Method

  2. Rise & Fall Method

Height of Instrument (HI) Method Formulas

  • To find HI:

    $$HI = RL_{BM} + BS$$

  • To find the RL of a point:

    $$RL = HI - \text{Staff Reading (IS or FS)}$$

Rise and Fall Method

This is the second primary method for calculating Reduced Level (RL).

  • Principle: The difference between the previous reading and the next reading determines if the ground is rising (Rise) or falling (Fall).

  • Rule:

    • If the difference is positive (+ive), it is called a Rise.

    • If the difference is negative (-ive), it is called a Fall.

  • RL Calculation: The RL of the next point is found by adding the Rise or subtracting the Fall from the previous RL ($RL_{next} = RL_{previous} \pm \text{Rise/Fall}$).

Arithmetical Check

To check the accuracy of levelling calculations:

$$\Sigma BS - \Sigma FS = \text{Last RL} - \text{First RL}$$

(Where $\Sigma BS$ is the sum of all Back Sights and $\Sigma FS$ is the sum of all Fore Sights).

Numerical Examples

1. Finding RL using HI Method

Question: The RL of Bench Mark (A) is 100.5 m. The staff reading at A is 2.5 m and at B is 1.1 m. Find the RL of B.

Calculation:

  1. Find HI: $HI = RL_{BM} + BS = 100.5 + 2.5 = 103\text{ m}$.

  2. RL of B: $RL = HI - \text{Staff Reading at B} = 103 - 1.1 = \mathbf{101.9\text{ m}}$.

2. Inverted Staff Levelling

To find the RL of a high point (like a ceiling or the underside of a bridge), the staff is held inverted.

  • Identification: Inverted staff readings are always shown with a negative (-ive) sign.

    Question: The RL of a building's roof (top) is 510 m. The inverted staff reading at the roof is 2.2 m and the normal reading on the floor is 1.2 m. Find the RL of the floor.

    Calculation:

  1. Find HI: $RL_{top} - (\text{Inverted Reading}) = 510 - 2.2 = 507.8\text{ m}$.

  2. Floor RL: $HI - \text{Floor Reading} = 507.8 - 1.2 = \mathbf{506.6\text{ m}}$.

Earth Curvature and Refraction Corrections

When levelling over long distances ($d$), errors occur due to the Earth's curvature and atmospheric refraction.

1. Curvature Correction ($C_c$)

Due to the Earth's curvature, staff readings appear higher than they actually are.

  • Formula: $C_c = -0.0785 d^2$

  • Rule: This correction is always negative. (Note: $d$ is in kilometers).

2. Refraction Correction ($C_r$)

Light rays bend downward due to air density, making the reading appear slightly lower than actual.

  • Formula: $C_r = +0.0112 d^2$

  • Relationship: It is 1/7th of the curvature correction ($C_r = \frac{1}{7} |C_c|$).

  • Rule: This correction is always positive.

Distance of Visible Horizon

The maximum distance ($d$) that can be seen from a specific height ($h$).

  • Combined Formula: $h = 0.0673 d^2$

  • To find distance: $d = 3.855 \sqrt{h}$

    (Where $h$ is height in meters and $d$ is distance in kilometers).

Reciprocal Levelling

Used when there is a large obstacle (like a river or valley) where direct surveying is not possible.

  • Principle: The instrument is placed alternately at two points (A and B), and readings are taken from both sides so that curvature and refraction errors cancel out.

  • Height Difference ($\Delta H$):

    $$\Delta H = \frac{(h_2 - h_1) + (h'_2 - h'_1)}{2}$$

Sensitivity of Bubble Tube

The sensitivity ($\alpha$) indicates the change in staff reading when the bubble moves by one division.

  • Main Formulas:

    1. $\alpha = \frac{S}{nD}$

    2. $\alpha = \frac{l}{R}$

    3. In seconds: $\alpha = \frac{S}{nD} \times 206265''$

  • $S$: Staff intercept.

  • $n$: Number of divisions.

  • $D$: Distance between instrument and staff.

  • $R$: Radius of bubble tube.

  • $l$: Length of one division.



Contours and Their Values

Contour Line: An imaginary line connecting points of equal elevation or equal Reduced Level (RL) on the Earth's surface.

Contour Interval (CI): The vertical distance between two consecutive contour lines. For a single map, the CI is kept constant to clearly distinguish various features of the area.

Horizontal Equivalent: The horizontal distance between two consecutive contour lines. This distance varies across the map depending on the terrain.

Identification of Slopes

The spacing and pattern of contour lines indicate the nature of the ground slope:

  • Uniform Slope: Indicated by contour lines that are parallel and equally spaced.

  • Steep Slope: Indicated by closely spaced contour lines.

  • Mild Slope: Indicated by widely spaced contour lines.

Identification of Hills and Valleys

Closed contour lines represent either a hill or a valley (depression):

  • Hill: If the values increase toward the center (e.g., 100, 110, 120, 130).

  • Valley: If the values decrease toward the center (e.g., 100, 90, 80, 70).

Special Conditions and Exceptions

Normally, contour lines of different elevations do not cross or meet. However, there are exceptions:

  • Vertical Cliff: Contour lines appear to touch or merge into a single line.

  • Overhanging Cliff or Cave: Contour lines cross or cut each other.

Methods of Area Calculation

Three primary methods are used to calculate the area of a plot:

1. Triangulation Method

The area is divided into small triangles.

  • Semi-perimeter ($s$): $s = \frac{a+b+c}{2}$

  • Heron's Formula: $Area = \sqrt{s(s-a)(s-b)(s-c)}$

2. Coordinate Method

The area is calculated using a central baseline and various offsets ($x_1, x_2...$). The total area is the sum of all individual parts.

3. Regular Offset Method

When offsets are taken at a uniform distance ($d$), the following rules are used:

  • Average Ordinate Rule: $A_T = \text{Base Length} \times \text{Average of Ordinates}$

  • Trapezoidal Rule: $A_T = \frac{d}{2} [ (h_1 + h_n) + 2(h_2 + h_3 + \dots + h_{n-1}) ]$

  • Simpson's Rule: (Most accurate as it assumes boundaries are curves)

    $$A_T = \frac{d}{3} [ (h_{first} + h_{last}) + 4(\Sigma \text{even offsets}) + 2(\Sigma \text{odd offsets}) ]$$

Trigonometric Levelling

This method determines the RL of points using vertical angles and horizontal distances.

Case 1: When Distance (D) is Measurable

  • Vertical Height ($V$): $V = D \tan \theta$

  • RL Calculation: $RL_{Point} = BM + BS + V$

Tacheometry

A Tacheometer is a specialized theodolite equipped with three horizontal cross-hairs (stadia hairs).

  • Main Advantage: Horizontal distance ($D$) can be measured directly without a tape.

  • Tacheometric Equation: $D = KS + C$

    • $K$ (Multiplying Constant) = $f/i$ (usually 100).

    • $C$ (Additive Constant) = $f+d$ (usually 0 for anallactic telescopes).

    • $S$ = Staff intercept (difference between top and bottom hair readings).

Curves

Used to change the direction of a road or railway line.

  • Simple Curve: A single circular arc with a constant radius ($R$).

  • Compound Curve: Two or more circular arcs of different radii turning in the same direction.

  • Reverse Curve: Two circular arcs of different radii turning in opposite directions.

  • Transition Curve: A curve with a varying radius (from infinity to a fixed $R$).

Photogrammetry and Scale

  • Scale of Vertical Photograph: $Scale = \frac{f}{H - h_{avg}}$

  • Relief Displacement ($d$): The shift in the image position of an object due to its elevation.

    • Formula: $d = \frac{r \cdot h}{H}$

Plane Table Surveying

A method where observation and plotting are done simultaneously in the field. Its main principle is Parallelism.

Methods of Plane Table Surveying:

  1. Radiation: Points are located by drawing rays from a single station. Best for small areas.

  2. Traversing: The table is moved through a sequence of stations. Used for narrow strips like roads.

  3. Intersection: Points are located by the intersection of rays drawn from two different stations. Useful for hilly or inaccessible terrain.

  4. Resection: Used to locate the position of the plane table on the map using already plotted points (e.g., Two-point and Three-point problems).




Scale of a Vertical Photograph

The scale of a vertical photograph is the ratio between the distance on the photograph and the actual distance on the ground. It is calculated using the following formula:

Formula:

$$Scale = \frac{f}{H - h_{avg}}$$

  • $f$: Focal length of the camera.

  • $H$: Flying height of the camera above Mean Sea Level (MSL).

  • $h_{avg}$: Average elevation (height) of the ground.

Relief Displacement ($d$)

Relief displacement is the distance between the image of the top point and the image of the bottom point of an object, measured from the principal point (camera axis) of the photograph.

Primary Formula:

$$d = r - r'$$

  • $r$: Distance from the principal point to the image of the top of the object.

  • $r'$: Distance from the principal point to the image of the base of the object.

Alternative Formula:

$$d = \frac{r \cdot h}{H}$$

(Where $h$ is the actual height of the object)

Numerical Example: Calculating Tower Height

Given Data:

  • Actual length of line AB on ground $= 300\text{ m}$.

  • Length of line AB measured on map $= 102.4\text{ mm}$.

  • Average ground elevation ($h_{avg}$) $= 553\text{ m}$.

  • Focal length ($f$) $= 152.4\text{ mm}$.

  • Distance of top point from center ($r$) $= 8\text{ cm}$.

  • Distance of bottom point from center ($r'$) $= 7\text{ cm}$ (therefore, $d = 1\text{ cm}$).

Calculation:

  1. First, the flying height ($H$) is determined using the scale and the measured lengths, resulting in $999.48\text{ m}$.

  2. Next, the height of the tower is calculated using the relief displacement formula adjusted for ground elevation:

    $$d = \frac{r \cdot h}{H - h_{avg}}$$

Result:

The actual height of the tower is calculated to be $55.8\text{ m}$.



Read in details now:


🏗️ Surveying: Complete Study Guide & Index

📔 Part 1: Fundamentals of Surveying

📏 Part 2: Linear Measurement & Chain Survey

🧭 Part 3: Angular & Instrumental Survey

🏔️ Part 4: Levelling & Elevation

🛰️ Part 5: Modern Technologies

📝 Part 6: Practice & Quizzes (MCQs)

📚 Quick Revision Resources

Surveying IS Codes with Latest Revision Years

1. General Surveying & Instruments

2. Chain and Tape Surveying

  • IS 1492:1970 – Specification for Metric Surveying Chains.

  • IS 1269 (Part 1):1997 – Material and Construction of Steel Tapes.

  • IS 1269 (Part 2):1997 – Woven Metallic and Glass Fibre Tapes.

  • IS 1659:2006 – Specification for Invar Tapes for High Precision Measurement.

3. Theodolite and Tacheometry

  • IS 8002:1976 – Specification for Surveying Chain Vertical Vernier Theodolite.

  • IS 8330:1976 – Specification for Tilting Levels (Optical).

  • IS 8636:1977 – Specification for Tacheometers.

4. Leveling and Contouring

  • IS 9128:1992 – Specification for Tilting Levels.

  • IS 9573:1980 – Specification for Automatic Levels.

  • IS 1779:1961 – Specification for 4-metre Leveling Staff (Folding Type).

5. Modern Surveying (Total Station & GPS)

  • IS 16481:2016 – Guidelines for Accuracy and Testing of Total Stations.

  • IS 14855:2000 – Terminology and Concepts for GIS and Remote Sensing.

6. Measurement of Building Works

  • IS 1200 (Part 27):1992 – Method of Measurement of Building and Civil Engineering Works (Earthwork & Surveying).


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