Surveying (CE-303) - Unit 5 Short Notes

UNIT 5: SURVEYING - EXAM-FOCUSED SHORT NOTES

1.0 FUNDAMENTALS & CLASSIFICATIONS

Surveying is the art of determining relative positions of points on, above, or below the Earth's surface by measuring distances, angles, and elevations.

Primary Classification:

• Plane Surveying: Assumes Earth is flat. Suitable for small areas (< 250 km²). Angles are plane angles.

• Geodetic Surveying: Accounts for Earth's curvature. For large areas. Angles are spherical.

Secondary Classification (by purpose): Topographic, Hydrographic, Engineering, City, Route, Cadastral.

Meridians:

Bearings:

• Whole Circle Bearing (WCB): Measured clockwise from North (0° to 360°).

• Reduced Circle Bearing (RCB)/Quadrantal Bearing: Measured from N/S towards E/W (e.g., N 30° E).

• Fore Bearing (FB): Direction of survey line in forward direction.

• Back Bearing (BB): Direction opposite to FB. Relationship: BB = FB ± 180°.

  • WCB: BB = FB + 180° (if FB < 180°), BB = FB - 180° (if FB > 180°).

  • RCB: Reverse N/S and E/W (e.g., N 30° E ↔ S 30° W).

Local Attraction: Deflection of magnetic needle due to nearby magnetic materials.

• Detection: Compare FB and BB of a line. If FB ≠ BB ± 180°, local attraction exists.

• Correction (Compass Surveying):

  1. Identify correct bearing from a line free from attraction (or known true bearing).

  2. Compute correction for each station.

  3. Apply correction to observed bearings to get correct bearings.

[!TIP] Exam Focus: Converting WCB ↔ RCB and correcting bearings in closed traverses with local attraction are very frequent.

2.0 CHAIN SURVEYING & OBSTACLES

Tapes: Steel (most common), Chain (Gunter's, Engineer's), Invar (for high precision).

Sources of Error:

• Personal: Non-uniform tension, incorrect alignment, incorrect reading, improper plumbing.

• Instrumental: Incorrect length (calibration error), bent tape, loose joints.

• Natural: Temperature, tension (sag), slope, wind.

Precautions: Use standardized pull, level tape, proper alignment, avoid temperature extremes, measure slope correction.

Obstacles:

• To Ranging: Hill/curved ground. Use reciprocal ranging.

• To Chaining: River, building, pond.

  • Direct: For small obstacles (perpendicular offsets, traversing around).

  • Indirect: For large obstacles.

    • Triangulation/Similar Triangles: Set perpendiculars by optical square/right angle prism.

    • Tacheometry/Subtense Bar.

    • Reciprocal Observation: From opposite banks.

[!TIP] Exam Focus: Sketches for indirect methods (especially perpendicular offsets and traversing) are often asked. Tape correction formula (temperature & pull) is crucial.

Tape Correction Formula:

For a tape of standard length L₀ at temperature T₀ and standard pull P₀:

$$L = L_0 \left[ 1 + \alpha (T - T_0) + \frac{(P - P_0)}{A E} \right]$$

Where, α = coeff. of thermal expansion, A = cross-sectional area, E = modulus of elasticity.

3.0 COMPASS SURVEYING

Temporary Adjustments:

1. Centring: Tripod over station.

2. Leveling: Using foot screws (spirit level parallel to pair of legs).

3. Focusing: Sliding eyepiece for clear sight & graduated ring.

4. Sighting & Reading: Direct object, rotate compass, read through prism.

Included Angle in Closed Traverse (Compass Rule):

For a traverse with n sides, sum of interior angles = (2n - 4) × 90°. Correction per angle: δ = (Error in sum) / n.

Correct each angle by ±δ (alternatively).

4.0 THEODOLITE SURVEYING

Fundamental Lines (with sketch):

1. Vertical Axis (VV): Axis of rotation of telescope & vernier plate.

2. Horizontal Axis (HH): Axis of rotation of telescope.

3. Line of Collimation (LC): Imaginary line through cross-hair intersection & optical center of objective.

4. Axis of Plate Levels (BB): Axis of bubble tube on vernier plate.

5. Axis of Altitude Bubble (CC): Axis of bubble on telescope.

Condition for Perfect Adjustment: VV ⊥ HH, HH ⊥ LC, BB ⊥ VV, CC ⊥ HH.

Temporary Adjustments:

1. Setting up: Tripod over station, approximate centring.

2. Centring: Optical plummet or plumb bob for precise centring.

3. Leveling: Rough (tripod legs), fine (plate level bubble) – make axis of plate levels vertical (VV vertical).

4. Focusing: Eyepiece (cross-hairs), objective (focused image).

Permanent Adjustments: Done in workshop to maintain fundamental line conditions (e.g., plate bubble axis perpendicular to vertical axis).

Methods of Traversing with Theodolite:

1. Method of Included Angles: Measure interior angles at each station.

2. Method of Deflection Angles: Measure angle from forward line.

3. Method of Direct Angles: Measure angle from backward line.

Theodolite Types:

• Repeating Theodolite: Measures horizontal angle by repetition (vernier scale moves). Process: Set 0°, sight fore bearing, unclamp, turn telescope to back bearing, clamp, read. Mean of readings = angle.

• Direction Theodolite: Measures each angle independently (single setting). Uses double reading (both verniers). Process: Direct & reverse readings taken for each line.

• Gale's Table: Direction theodolite with divided circle on lower plate; upper plate has verniers.

[!TIP] Exam Focus: Difference between repeating & direction theodolite, and processes of repetition (same angle measured multiple times) and reiteration (all angles from one station) are key.

5.0 LEVELLING

Types: Simple, Differential, Precise, Fly, Reciprocal (for long sights), Barometric, Trigonometric.

Instruments:

• Dumpy Level: Rigid telescope, 3 foot screws, single bubble. Parts: Telescope, eyepiece, objective, diaphragm (cross-hairs), foot screws, leveling head, tripod.

• Tilting Level: Telescope can be tilted slightly via screw.

• Automatic Level: Self-leveling via compensator.

Temporary Adjustments (Dumpy Level):

1. Setting up: Tripod over station, legs spread.

2. Centring: Rough.

3. Leveling: Focus eyepiece, sight rod, use foot screws to center bubble (parallel to two legs, then perpendicular).

4. Focusing: Objective lens for clear rod reading.

Permanent Adjustments: (a) Axis of Bubble Tube ⊥ Vertical Axis, (b) Line of Collimation ⊥ Axis of Bubble Tube.

Booking & RL Calculation:

• Height of Instrument (H.I.) Method:

  H.I. = R.L. of B.M./T.P. + Back Sights (B.S.)

  R.L. of point = H.I. - Fore Sights (F.S.) or Intermediate Sights (I.S.)

  Arithmetic Check: Σ B.S. - Σ F.S. = Last R.L. - First R.L.

• Rise and Fall Method:

  Rise/Fall = Previous R.L. - Current Staff Reading

  Rise if +ve, Fall if -ve.

  R.L. = Previous R.L. - Fall/Rise

  Arithmetic Check: Σ Rise - Σ Fall = Last R.L. - First R.L.

Errors in Levelling:

• Personal: Parallax, improper focusing, staff not vertical, booking errors.

• Instrumental: Bubble not centered, axis not perpendicular, worn screw.

• Natural: Earth's curvature, refraction, temperature, wind.

Curvature & Refraction Correction:

• Curvature Correction (C): Earth's surface falls below horizontal line.

$$C = \frac{d^2}{2R} \quad \text{(in same units as d, R)}$$

Where `d` = distance (km), `R` ≈ 6370 km. For `d` in m, `C (m) ≈ 0.0785 d² (km²)`.

• Refraction Correction (R): Atmospheric refraction lifts image.

  R ≈ 0.11 d² (m) for d in km.

• Combined Correction (C - R): ≈ 0.067 d² (m) for d in km. Effect: Makes staff reading too high (RL appears lower).

[!TIP] Exam Focus: RL calculation problems (both methods) are very common. Remember: In H.I. method, H.I. is constant between two setups. In Rise/Fall, compute difference between consecutive readings.

Longitudinal & Cross-Sectioning:

• Longitudinal Section (L-section): Profile along centreline. Shows ground profile & formation level. Used for earthwork, drainage design.

• Cross-Section (X-section): Profile perpendicular to centreline at intervals. Shows side slopes, width. Used for area/volume calculation, drainage structures.

• Importance in Roads/Railways: Earthwork volume estimation, design of gradients, vertical curves, drainage, and construction planning.

6.0 TACHEOMETRY (STADIA METHOD)

Principle: Rapid determination of horizontal distance & elevation difference using a theodolite/tacheometer and a stadia rod, without chaining.

Types:

• Fixed Hair (Stadia): Fixed stadia hairs (usually 3) on diaphragm. Subtense s = staff intercept.

• Movable Hair (Subtense Bar): Movable hairs; distance proportional to staff reading.

Fixed Hair Method:

• Constants: K (multiplying constant, usually 100), C (additive constant, usually 0 or 0.3). D = K s + C.

• Line of Sight Horizontal, Staff Vertical:

  Horizontal Distance: D = K s + C

  Elevation Difference: Δh = D \cdot \frac{\sin 2i}{2} + v - i \cdot D (simplified: Δh = D \tan i + v - i)

  Where i = vertical angle (+ for elevation, - for depression), v = staff intercept correction for curvature/refraction (usually negligible for short sights).

• Line of Sight Inclined, Staff Vertical (Derivation):

  Let i = vertical angle, s = staff intercept, h = RL of instrument axis.

  Horizontal Distance: D = K s \cos^2 i + C \cos i

  Vertical Difference: Δh = \frac{1}{2} K s \sin 2i + C \sin i + v - i \cdot D

  Reduced Level of staff station: R.L. = H.I. ± Δh (+ for point above, - for below).

Characteristics of Tacheometer:

• Anallatic lens (makes C = 0).

• High magnification (≥ 30×).

• Stadia hairs spaced correctly (usually K = 100).

• Bubble sensitive for precise leveling.

Procedure:

1. Set up tacheometer, level, note H.I.

2. Sight staff at point, read vertical angle i and three hairs (top, middle, bottom).

3. Compute s = (top - bottom), D, Δh, then R.L.

4. Compute coordinates if traverse.

[!TIP] Exam Focus: Derivation of distance formula for inclined sight is very frequent. Remember D = K s cos²i + C cos i. For horizontal sight (i=0), reduces to D = K s + C.

7.0 CONTOURING

Contour: Imaginary line joining points of equal elevation.

Characteristics (with sketches):

• Contours never cross (except in vertical cliff/overhang).

• Parallel, equally spaced contours → uniform slope.

• Close contours → steep slope.

• Widely spaced contours → gentle slope/flat.

• Contours form V-shape upstream in valleys, downstream in ridges.

• Closed contours: hill (values increase inward), depression (values decrease inward, hachures).

Methods:

• Direct Method: Plot points of known elevation directly in field.

• Indirect Method: Determine elevation of selected points, then interpolate contours.

  • By Tacheometry: Fast, used for large areas.

  • By Plane Table: Graphical method.

  • By Levelling (Cross-Sections): Common for roads/canals.

Advantages of Indirect: Faster, less expensive, suitable for large areas. Disadvantages: Less accurate, requires interpolation.

Interpolation: Estimating contour positions between known points. Methods: Arithmetic, graphical, estimation.

8.0 AREA COMPUTATION

From Offsets (Chain Line & Irregular Boundary):

• Trapezoidal Rule:

$$Area = d \left[ \frac{y_1 + y_n}{2} + y_2 + y_3 + ... + y_{n-1} \right]$$

Where `d` = interval, `y` = offsets.

• Simpson's Rule (Parabolic Rule):

  Condition: n (number of offsets) must be odd (even number of intervals).

$$Area = \frac{d}{3} \left[ y_1 + y_n + 4(y_2 + y_4 + ...) + 2(y_3 + y_5 + ...) \right]$$

If `n` is even, use Trapezoidal for last segment.

From Coordinates (Closed Traverse - Latitude/Departure Method):

• Latitude (L): L = l \cos \theta (North/South component).

• Departure (D): D = l \sin \theta (East/West component).

• Area: Area = \frac{1}{2} \sum (L_i D_{i+1} - L_{i+1} D_i) (Bowditch/Coordinate method).

  Or Area = \frac{1}{2} \left| \sum (x_i y_{i+1} - x_{i+1} y_i) \right| for coordinates.

[!TIP] Exam Focus: Apply correct rule based on number of offsets. Simpson's requires odd number of ordinates. Coordinate method is foolproof for closed polygon.

9.0 HORIZONTAL CURVES

Elements of Simple Circular Curve (sketch essential):

• IP/PC: Intersection Point of tangents / Point of Curvature (beginning).

• PT: Point of Tangency (end).

• Tangent Length (T): T = R \tan(\Delta/2)

• Length of Curve (Lc): Lc = \frac{\pi R \Delta}{180} (arc definition) or Lc = 2R \sin(\Delta/2) (chord definition, approx).

• Long Chord (LC): LC = 2R \sin(\Delta/2)

• Mid-ordinate (M): M = R (1 - \cos(\Delta/2))

• Apex Distance (A): A = R (\sec(\Delta/2) - 1) or A = T \tan(\Delta/4)

• Deflection Angle (Δ): Angle between tangents.

Degree of Curve (D):

• Arc Definition (common in US): D = central angle subtended by 100 ft arc. R = \frac{5730}{D} (approx).

• Chord Definition (common in India): D = central angle subtended by 100 ft chord. R = \frac{10150}{D} (approx).

  Relationship: R = \frac{100}{2 \sin(D/2)} (chord), R = \frac{100 D \pi}{180 \times 100} (arc, D in degrees).

Types: Simple Circular, Compound (two+ curves same direction), Reverse (curves opposite directions), Transition/Spiral (gradual curvature).

Setting Out Methods:

1. By Offsets from Tangent:

   • Long Chord Offsets: x = \text{distance from PC along chord}, O = R - \sqrt{R^2 - x^2}

   • Normal Offsets: O = R (1 - \cos \theta), θ = deflection angle.

   • Tangential Offsets: O = R \tan \theta - x

2. By Offsets from Chord: O = \frac{x^2}{2R} (approx for small chords).

3. By Theodolite (Rankine's/Deflection Angle Method):

   • Set up at PC.

   • Sight tangent, set 0°.

   • For each chord length C, set deflection angle δ = \frac{C}{2R} (in radians) or δ = \frac{C \cdot D}{200} (D in degrees, C in m).

   • Stretch tape along chord direction.

Compound Curve (given radii R₁, R₂ and intersection angles α₁, α₂):

• Check: α₁ + α₂ = Δ (total deflection).

• Common Tangent Length (T₃): T₃ = \sqrt{T_1^2 + T_2^2 - 2 T_1 T_2 \cos \Delta} or via triangles.

• Chainage of PT: Chainage of PC₁ + T₁ + Lc₁ + T₃ + Lc₂ + T₂.

[!TIP] Exam Focus: Deriving δ = C/(2R) for Rankine's method, and compound curve calculations (finding T₁, T₂, T₃, chainages) are very frequent. Sketch the compound curve layout.

10.0 TRANSITION CURVES (SPIRAL CURVES)

Requirement: Gradual change in curvature from straight (∞ radius) to circular (R). Provides comfort, safety, prevents jerk, allows gradual super-elevation.

Ideal Transition Curve: Clothoid/Euler Spiral.

• Property: Radius is inversely proportional to length from tangent point.

$$R \cdot L = \text{constant} = R \cdot L_s \quad \text{or} \quad \rho = \frac{R}{L_s} \cdot l$$

Where `L_s` = total spiral length, `l` = distance from SP, `ρ` = radius at `l`.

• Deflection Angle (θ): θ = \frac{l^2}{2 R L_s} radians.

• Equation: x = L_s - \frac{L_s^4 - r^4}{6 \sqrt{2} R L_s^3} (approx), y = \frac{r^3}{6 \sqrt{2} R L_s} (Cartesian).

Elements:

• Length of Transition (L_s): Designed based on max speed, super-elevation, rate of change of radial acceleration.

• Shift (s): s = \frac{L_s^2}{24 R} (distance by which circular curve is shifted inward).

• Super-elevation (e): e = \frac{V^2}{g R} (theoretical), provided partly on spiral, partly on circular curve.

Setting Out:

1. Compute L_s from design criteria.

2. Compute θ at intervals (e.g., every 20 m) using θ = l²/(2RL_s).

3. Set theodolite at SP, set tangent as 0°, turn by θ for each interval.

4. Offsets from tangent: y = \frac{l^3}{6 R L_s}.

Design Considerations:

• Max Speed V: V^4 = 127 (R + s) (e + f) (f = side friction factor).

• Centrifugal Ratio (e + f) ≤ 0.07 - 0.10.

• Rate of Change of Radial Acceleration: \frac{V^3}{R L_s} ≤ 0.3 - 0.6 m/s³.

11.0 VERTICAL CURVES

Types:

• Summit Curve (Convex): Crest of hill. g₁ > g₂ (both +ve or +ve/-ve). Equation: y = \frac{g_1}{2L} x^2 + (g_1 - g_2) \frac{x}{2} (parabolic).

• Sag Curve (Concave): Valley. g₁ < g₂ (both -ve or -ve/+ve). Equation similar.

Applications: Provide smooth transition between different gradients, ensure sight distance, comfort, drainage.

Length (L): Based on sight distance (SSD/ISD) and algebraic difference A = |g₁ - g₂|.

For summit (SSD): L = \frac{A \cdot SSD^2}{2(\sqrt{h_1} + \sqrt{h_2})^2} (h = driver eye height, object height).

For sag (SSD): L = \frac{A \cdot SSD^2}{2(h_1 + h_2)}.

12.0 EARTHWORK COMPUTATION

Cross-Sectional Area:

• Level Ground: A = b d + s d² (b = formation width, d = depth, s = slope ratio²).

• Sloping Ground: A = \frac{h_1 + h_2}{2} \cdot b + \frac{(h_1^2 + h_2^2)}{2} \cdot s (h₁, h₂ = side heights).

Volume (between X-sections at interval L):

1. Trapezoidal Rule (Mean Area Method):

$$V = \frac{L}{2} (A_1 + A_2) \quad \text{or} \quad V = L \cdot \frac{A_1 + A_2}{2}$$

For multiple sections: `V = L [ (A₁ + Aₙ)/2 + A₂ + A₃ + ... + Aₙ₋₁ ]`

2. Prismoidal Rule (More Accurate):

$$V = \frac{L}{6} (A_1 + 4A_m + A_2)$$

Where `A_m` = area of mid-section. For `n` sections (odd number): `V = \frac{L}{3} [ (A₁ + Aₙ) + 4(A₂ + A₄ + ...) + 2(A₃ + A₅ + ...) ]`

Mass Haul Diagram: Plot of cumulative volume (cut +ve, fill -ve) vs. chainage. Used to balance earthwork, locate borrow pits/dumps. Balance Point where net volume = 0.

13.0 PHOTOGRAMMETRY & AERIAL PHOTOGRAPHY

Photogrammetry: Science of obtaining reliable information about objects through measurement of images.

Aerial Photography Uses: Topographic mapping, route surveys (roads, railways), land use/forestry, disaster assessment, mining.

Terminology:

• Focal Length (f): Distance from lens to photographic plane.

• Flying Height (H): Altitude of camera above datum.

• Altitude (h): Height above ground (if ground level ≠ datum).

• Scale of Photograph (S): S = f / (H - h) for vertical photo. S = f / H if h negligible.

Overlaps:

• Longitudinal (End Lap): 60% (typical). Ensures stereoscopic viewing.

• Side Lap: 30% (typical). Ensures complete coverage, avoids gaps.

Number of Photographs (N):

For a rectangular area L × B with photo size l × b:

$$N = \frac{L}{l (1 - p)} \times \frac{B}{b (1 - q)}$$

Where p = end lap fraction, q = side lap fraction.

Relief Displacement (d):

Apparent shift of elevated point from its true orthographic position. Derivation (from similar triangles):

d = \frac{r h}{H} or d = \frac{r h}{f} \cdot S (where r = radial distance from principal point, h = height of object above datum). Key: d ∝ r (max at photo edge), d ∝ h.

Ground Coordinates from Photo Coordinates:

For vertical photo with principal point origin:

X = (x - x₀) \cdot (H/f), Y = (y - y₀) \cdot (H/f)

Where (x₀, y₀) = principal point coordinates, (x, y) = photo coordinates.

14.0 HYDROGRAPHIC SURVEYING & SOUNDING

Sounding: Measuring depth of water at selected points.

Equipment:

• Lead Line: Weighted line with marked depths. Simple, for shallow water.

• Sounding Cable: Wire rope with electrical contact (depth recorder).

• Echo Sounder: Ultrasonic pulse to bottom, measures time → depth. Fast, continuous.

Methods of Sounding:

1. By Boat: Most common, systematic grid.

2. By Raft: For very shallow, calm water.

3. By Shore: For near-shore, small areas.

Positioning of Soundings:

• Triangulation: From shore stations.

• Traversing: From boat using radio/radar.

• Radio/Satellite (GPS): Modern standard.

Nautical Sextant:

• Principle: Measures angle between celestial body and horizon (or two objects).

• Working: Two mirrors (index, horizon). Align images, read angle on arc. Used for positioning (latitude/longitude) or horizontal angles between shore objects.

Tide Gauge: Records water level variations.

• Types: Float type (still well), pressure type, acoustic.

• Purpose: Reduce soundings to a common datum (chart datum).

Current Measurement:

• Surface Float Method: Float released, time between two stations → surface current.

• Pitot Tube Method: Measures dynamic pressure → current velocity at depth.

15.0 MISCELLANEOUS TOPICS (From Specific Past Questions)

Trigonometric Levelling (RL Calculation):

Given instrument station A, staff station B, vertical angle α, staff readings s₁ (top), s₂ (middle), s₃ (bottom), H.I. = h.

1. Horizontal Distance: D = K (s₁ - s₃) cos²α + C cosα

2. Vertical Difference: Δh = D \tan α + v - i \cdot D (or simpler: Δh = h + D \tan α - \text{staff reading at point of zero inclination})

3. R.L. of B: R.L._B = H.I._A + Δh

Tape Correction (Temperature & Pull):

Given: L_meas = measured length, L₀ = standard length at T₀, P₀.

Correction per tape length: δ = L₀ [α(T - T₀) + (P - P₀)/(A E)]

True length: L_true = L_meas + n δ (if n full lengths) or L_true = L_meas + (L_meas/L₀) δ.

Compound Curve Elements (given R₁, R₂, α₁, α₂):

1. Deflection angles: Δ₁ = 180 - α₁, Δ₂ = 180 - α₂? No! For compound, angles at intersection points. Typically given: ∠(Tangent1, Common Tangent) = θ₁, ∠(Common Tangent, Tangent2) = θ₂. Then Δ₁ = θ₁, Δ₂ = θ₂.

2. Tangent lengths: T₁ = R₁ \tan(Δ₁/2), T₂ = R₂ \tan(Δ₂/2).

3. Curve lengths: Lc₁ = π R₁ Δ₁ / 180, Lc₂ = π R₂ Δ₂ / 180.

4. Common tangent T₃ from triangle formed by T₁, T₂, and T₃ with angle (180 - Δ₁ - Δ₂).

Apex Distance (A) to Curve Elements:

Given apex distance A and degree of curve D (or radius R):

• A = R (\sec(D/2) - 1) → solve for R or D.

• Then T = R \tan(D/2), Lc = π R D / 180, LC = 2R \sin(D/2).

• Chainage of TP: If chainage of IP = C_IP, then Chainage of PC = C_IP - T, Chainage of PT = C_IP + T.

Chainage of Tangent Points (for simple curve):

Chainage of PC = Chainage of IP - Tangent Length

Chainage of PT = Chainage of IP + Tangent Length

END OF UNIT 5 NOTES