Surveying (CE-303) - Unit 1 Short Notes

1.0 FOUNDATIONS & CLASSIFICATION

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

Objectives:

• To prepare maps/plans for engineering projects.

• To determine areas, volumes, and configurations.

• To set out works according to design.

Importance in Civil Engineering:

• Essential for all construction projects (buildings, roads, dams).

• Provides basis for planning, design, and execution.

• Used in land acquisition, boundary disputes, and resource management.

Primary Classification:

Secondary Classification (Based on Purpose):

• Topographic: Natural & man-made features.

• Cadastral: Property boundaries for ownership.

• City: Urban infrastructure mapping.

• Engineering: For construction projects.

• Route: Alignments for roads, railways, canals.

• Hydrographic: Water bodies (depth, shoreline).

• Astronomical: Celestial observations for control.

• Photogrammetric: From aerial photographs.

[!TIP] Exam Focus: Distinguish Geodetic (curved Earth) vs Plane (flat Earth) clearly. Know applications of each secondary type.

Fundamental Principle: Working from Whole to Part – Establish primary control points first, then fill in details. Minimizes error propagation.

2.0 MERIDIANS, BEARINGS & ANGLES

Types of Meridians:

• True Meridian: Line joining geographic N-S poles.

• Magnetic Meridian: Direction of magnetic needle.

• Arbitrary Meridian: Local reference direction (e.g., building axis).

• Grid Meridian: Used in map projections (e.g., UTM).

Bearings:

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

• Reduced/Quadrantal Bearing (RQB/RB): <90° measured from N/S towards E/W (e.g., N30°E).

Conversion Rules (WCB ↔ RQB):

1. 0°–90°: RQB = WCB (NE quadrant).

2. 90°–180°: RQB = 180° – WCB (SE quadrant).

3. 180°–270°: RQB = WCB – 180° (SW quadrant).

4. 270°–360°: RQB = 360° – WCB (NW quadrant).

[!TIP] Common Pitfall: Forgetting quadrant identification. Always sketch the WCB angle first.

Fore Bearing (FB) & Back Bearing (BB):

• FB: Direction of survey progress.

• BB: Opposite direction.

• Relationship: \boxed{BB = FB \pm 180^\circ} (add 180° if FB < 180°, subtract if FB > 180°).

Local Attraction:

• Cause: Local magnetic influences (iron ores, steel structures).

• Detection: Discrepancy between FB and BB (should differ by exactly 180°).

• Correction Methods:

  1. Included Angle Method: Calculate correct included angles from unaffected lines, then correct bearings.

  2. Correction Method: Compute mean correction for affected stations, apply to all bearings at that station.

Included Angles of a Traverse:

• For closed traverse: Interior angle = (n-2)*180°/n for regular polygon.

• From corrected bearings: For line AB to BC, included angle = (BB of AB) to (FB of BC) measured appropriately.

3.0 CHAIN SURVEYING & OBSTACLES

Principle: Subdivide area into triangles; measure sides only (triangulation).

Field Procedure:

1. Reconnaissance.

2. Station marking.

3. Chaining (main line & offsets).

4. Plotting.

Types of Chains/Tapes:

Tape Corrections:

1. Temperature: $$\displaystyle L = L_0 \left(1 + \alpha (T - T_0)\right) $$

2. Pull (Tension): $$\displaystyle L = L_0 \left(1 + \frac{P - P_0}{AE}\right) $$

3. Sag: $$\displaystyle L_{sag} = \frac{w^2 L^3}{24P^2} $$ (if unsupported)

4. Slope: $$\displaystyle L_{horizontal} = L \cos \theta $$

5. Alignment: Correct for crooked alignment (Pythagoras).

6. Standardization: Correct for known tape length error.

Obstacles:

• Obstacle to Chaining (e.g., river, building):

  • Setting off perpendiculars: Measure perpendicular offsets.

  • Reciprocal ranging: Two surveyors sight each other across obstacle.

  • Optical square/prism: Set perpendicular lines optically.

• Obstacle to Ranging (intermediate points not visible):

  • Reciprocal ranging: Two surveyors move along lines to align.

Errors in Chaining:

• Personal: Parallax in reading, inconsistent pull.

• Instrumental: Tape length incorrect, kinks.

• Natural: Temperature, wind, sag.

Precautions: Use consistent pull (spring balance), level tape, avoid kinks, measure in cool part of day.

4.0 LEVELLING

Purpose: Determine reduced levels (RL) of points relative to datum.

Types:

• Differential: RL of points.

• Profile: Along a line (for routes).

• Precise: High accuracy (first-order).

• Fly: Establishing benchmarks.

• Reciprocal: Across large obstacles (using two setups).

Levels & Staffs:

• Levels: Dumpy (fixed telescope), Tilting (tilt compensator), Automatic (self-leveling).

• Staffs: Solid (aluminum), Folding (4-part), Telescopic.

Levelling Methods & Calculations:

1. Height of Instrument (H.I.) Method:

   • HI = RL of BM + Backsight (BS)

   • RL of intermediate point = HI – Foresight (FS)

   • When instrument shifts: New HI = RL of turning point + BS at new setup.

2. Rise & Fall Method:

   • Rise/Fall = BS – FS (positive = fall, negative = rise).

   • RL = previous RL – rise/fall.

Booking Format:

Curvature & Refraction Corrections:

• Curvature: Earth's surface curves away: $$\displaystyle C_c = 0.0785 d^2 $$ (d in km, C in m).

• Refraction: Light bends downward: $$\displaystyle C_r = 0.067 d^2 $$.

• Combined Effect: \boxed{C = 0.067 d^2} (approx.).

Errors in Levelling:

• Personal: Staff not vertical, parallax, error in focusing.

• Instrumental: Bubble not centered, line of sight not horizontal.

• Natural: Curvature & refraction, atmospheric refraction, temperature.

Precautions: Level instrument carefully, staff vertical, avoid heat haze, consistent observations.

5.0 THEODOLITE SURVEYING

Components & Geometry:

• Transit Theodolite: Telescope rotates 360° vertically.

• Non-Transit: Limited vertical rotation.

• Main Parts: Telescope (eyepiece, objective, cross-hairs), Vertical & Horizontal Circles (graduated), Verniers (readings), Plate Levels (spirit levels), Tripod.

Temporary Adjustments:

1. Setting Up: Place tripod, roughly level.

2. Centring: Plumb bob over station.

3. Levelling: Adjust plate levels using foot screws.

4. Focusing: Eyepiece (for eye), objective (for object).

Permanent Adjustments (Conceptual):

• Collimation Axis: Telescope axis ⊥ Horizontal axis.

• Horizontal Axis: ⊥ Vertical axis.

• Vertical Axis: Through intersection of axes.

• Index Error: Vertical circle reading when telescope horizontal ≠ 0/180°.

Methods of Traversing:

1. Included Angle: Measure interior angle at station.

2. Deflection Angle: Angle from forward line (left/right).

3. Direct Angle: Angle from previous line.

4. Fast Needle: Magnetic bearings using compass.

Repetition & Reiteration:

• Repetition: Measure same angle multiple times on same side to eliminate error (mean).

• Reiteration: Measure all angles from first line to maintain consistency.

Trigonometric Levelling:

• Principle: Determine RL using vertical angles and distances.

• Formula (horizontal distance D):

  • If staff held vertical: $$\displaystyle h = D \tan \theta + (I - S) $$

  • If line of sight inclined: $$\displaystyle D = \frac{k s + C}{\cos^2 \theta} $$, $$\displaystyle h = D \tan \theta + I - S $$

  • Where $I$ = instrument height, $S$ = staff reading, $\theta$ = vertical angle, $s$ = staff intercept, $k, C$ = tacheometer constants.

6.0 COMPASS SURVEYING

Prismatic Compass vs Surveyor's Compass:

Temporary Adjustments of Prismatic Compass:

1. Centring: Plumb bob over station.

2. Levelling: Bubble centered.

3. Sighting: Direct telescope to object.

4. Reading: Through prism, note WCB.

Magnetic Declination & Dip:

• Declination: Angle between true north and magnetic north (varies spatially/temporally). East (+), West (–).

• Dip: Angle magnetic needle makes with horizontal (varies with latitude).

Local Attraction: Same as in bearings – detect via FB/BB discrepancy, correct by included angles or correction method.

7.0 TACHEOMETRY (STADIA METHOD)

Principle: Rapid determination of horizontal distance and elevation differences from one station using a theodolite/tacheometer and a staff.

Fixed Hair Method (Stadia Method):

• Telescope has two fixed stadia hairs (upper/lower) parallel to horizontal hair.

• Staff Intercept: $$\displaystyle s = \text{top reading} - \text{bottom reading} $$.

Derivation of Distance Formula:

1. Line of Sight Horizontal, Staff Vertical:

   • Similar triangles: $$\displaystyle \frac{s}{D} = \frac{i}{f} $$ → $$\displaystyle D = \frac{f}{i} s $$

   • Where $f$ = focal length, $i$ = stadia interval (distance between stadia hairs).

   • Multiplying Constant: $$\displaystyle k = f/i $$

   • Additive Constant: $C$ (due to lens position).

   • \boxed{D = k s + C}

2. Line of Sight Inclined, Staff Vertical:

   • Horizontal distance: $$\displaystyle D = (k s + C) \cos^2 \theta $$

   • Elevation difference: $$\displaystyle \Delta h = D \tan \theta + I - S $$ (if staff intercept used) or $$\displaystyle \Delta h = D \tan \theta + I - \text{mean staff reading} $$.

   • Where $\theta$ = vertical angle, $I$ = instrument height, $S$ = staff reading at central hair.

Movable Hair Method (Tangential Method):

• Stadia hairs are movable; measure vertical angles to top & bottom hairs.

• Distance: $$\displaystyle D = \frac{1}{2} ( \cot \theta_1 - \cot \theta_2 ) \cdot i $$ (if $i$ = stadia interval).

• Less common; used when staff intercept is difficult.

Tacheometer Constants:

• $k$ (multiplying) typically 100.

• $C$ (additive) usually 0 for anallatic lens.

• Anallatic lens: eliminates $C$ by making effective focal point at stadia plane.

Stadia Readings & Computations:

Given: vertical angle $\theta$, staff readings $a$ (top), $b$ (middle), $c$ (bottom), constants $k, C$, instrument height $I$, RL of BM.

• Staff intercept $$\displaystyle s = a - c $$.

• Horizontal distance $$\displaystyle D = (k s + C) \cos^2 \theta $$.

• Elevation of staff station: $$\displaystyle RL_{staff} = RL_{HI} + D \tan \theta \pm (I - b) $$ (sign + if angle of elevation, – if depression).

• RL of instrument station: $$\displaystyle RL_{HI} = RL_{staff} - [D \tan \theta + (I - b)] $$.

[!TIP] Exam Focus: Derivation of $$\displaystyle D = k s + C $$ and inclined sight formula are frequently asked. Remember sign conventions for elevation.

8.0 CONTOURING

Definition: Imaginary lines joining points of equal elevation.

Characteristics:

• Contour Interval (CI): Vertical distance between consecutive contours.

• Horizontal Equivalent: Horizontal distance between contours (varies with slope).

Methods of Contouring:

1. Direct Method:

   • Survey points on each contour directly.

   • Radial: From a central point (hills).

   • Cross-ridge & draw: Along ridges and valleys.

2. Indirect Method (Spot Levels):

   • Survey points on grid, spot levels taken.

   • Contours interpolated later.

   • Procedure: Establish grid, take spot levels at intersections, compute RLs, draw contours by interpolation.

   • Area/Volume: Trapezoidal & Simpson's rules for irregular boundaries.

Uses & Interpretation:

• Map preparation, slope determination, route selection, earthwork estimation.

• Steep slopes: close contours; gentle: wide apart.

• Valleys: V-shaped contours pointing upstream.

• Ridges: V-shaped pointing downstream.

• Cliffs: overlapping contours.

9.0 CURVE SETTING (HORIZONTAL CURVES)

Circular Curves:

• Elements:

  • IP (Intersection Point): Where tangents meet.

  • TP (Tangent Point): Where curve meets tangent.

  • Δ (Deflection Angle): Angle between tangents.

  • T (Tangent Length): $$\displaystyle T = R \tan \frac{\Delta}{2} $$

  • Lc (Curve Length): $$\displaystyle L_c = \frac{\pi R \Delta}{180} $$

  • LC (Long Chord): $$\displaystyle LC = 2 R \sin \frac{\Delta}{2} $$

  • M (Mid-ordinate): $$\displaystyle M = R (1 - \cos \frac{\Delta}{2}) $$

  • A (Apex Distance): $$\displaystyle A = R (\sec \frac{\Delta}{2} - 1) $$

  • E (External Distance): $$\displaystyle E = R (\sec \frac{\Delta}{2} - \tan \frac{\Delta}{2}) $$

Degree of Curve (D):

• Angle subtended by 20m/30m chord at center.

• For 20m chord: $$\displaystyle R = \frac{5729.578}{D} $$ (approx.)

• For 30m chord: $$\displaystyle R = \frac{1720}{D} $$

Setting Out Methods:

1. Offsets from Tangents:

   • Long chord: $$\displaystyle O_x = R - \sqrt{R^2 - x^2} $$

   • Small chord: $$\displaystyle O_x = \frac{x^2}{2R} $$ (approx.)

2. Offsets from Chord Produced: $$\displaystyle O_x = \frac{x^2}{2R} - \frac{x^3}{24R^2} + ... $$

3. Rankine's Method (Theodolite):

   • Divide curve into $n$ equal chords.

   • Deflection angle for each chord: $$\displaystyle \delta = \frac{\Delta}{2n} $$

   • Set out by measuring $\delta$ from tangent at each TP.

Compound Curves:

• Two or more simple curves with different radii, common tangent.

• Elements: $$\displaystyle R_1, R_2, \Delta_1, \Delta_2 $$, common tangent length $$\displaystyle T_1 = T_2 $$.

Transition Curves:

• Necessity: Gradual change in curvature for comfort/safety (railways/highways).

• Ideal Transition Curve Properties:

  • $L \propto R$ (length proportional to radius).

  • $L \propto \delta$ (length proportional to deflection angle).

  • Curvature varies uniformly from 0 to $1/R$.

• Types: Railway (Cubic parabola $$\displaystyle y = \frac{x^3}{6RL} $$), Highway (Clothoid/Euler's spiral).

• Elements:

  • Shift (s): $$\displaystyle s = \frac{L^2}{24R} $$

  • Super-elevation (e): $$\displaystyle e = \frac{V^2}{gR} $$ (for comfort).

  • Length (L): Based on comfort criteria (rate of change of centrifugal acceleration).

Vertical Curves:

• Types: Summit (convex), Sag (concave).

• Shape: Parabolic ($$\displaystyle y = g_1 x + \frac{(g_2 - g_1)}{2L} x^2 $$).

• Length: Based on sight distance or super-elevation requirements.

10.0 AREA & VOLUME COMPUTATION

Area from Field Notes (Offsets from Chain Line):

• Mid-ordinate Rule: $$\displaystyle A = d \left( O_1 + O_2 + ... + O_n \right) $$ (ordinates at midpoints).

• Trapezoidal Rule: $$\displaystyle A = d \left[ \frac{O_1 + O_n}{2} + \sum_{i=2}^{n-1} O_i \right] $$

• Simpson's 1/3 Rule (n even): $$\displaystyle A = \frac{d}{3} \left[ O_1 + O_n + 4(O_2 + O_4 + ...) + 2(O_3 + O_5 + ...) \right] $$

Earthwork Volume (Cross-sections at intervals):

• Trapezoidal Rule: $$\displaystyle V = \frac{d}{2} \left( A_1 + A_n + 2 \sum_{i=2}^{n-1} A_i \right) $$

• Prismoidal Rule (more accurate): $$\displaystyle V = \frac{d}{3} \left( A_1 + A_n + 4 \sum_{\text{odd}} A_i + 2 \sum_{\text{even}} A_i \right) $$

  • For railway embankment: $$\displaystyle A = (b + z h) h $$ (side slope z:1).

[!TIP] Exam Focus: Simpson's rule requires even number of intervals. Prismoidal is more accurate but needs odd number of sections (or use prismoidal correction).

11.0 HYDROGRAPHIC SURVEYING

Sounding: Measurement of depth of water bodies (bathymetry).

Sounding Equipment:

1. Sounding Rod/Lead Line: For shallow waters (lead weight at end).

2. Sounding Cable:

   • Construction: Steel wire with insulation, marked at intervals.

   • Weight: Sinker at bottom.

   • Echo Sounder: Ultrasonic pulse sent, echo received → depth = (velocity × time)/2.

3. Echo Sounder: Electronic, continuous recording.

Methods of Sounding:

• Boat/Launch: Most common.

• Shore-based: For narrow rivers.

• Helicopter: Rapid coverage.

Position Fixing for Soundings:

• Land-based Triangulation/Intersection: From shore stations.

• Radio Acoustic Ranging: Sounding boat releases explosive, time of sound heard at shore stations.

• Satellite Positioning (GPS/DGPS): Modern standard.

Tidal Observations:

• Need: Reduce soundings to a common datum (chart datum, usually lowest low water).

• Tide Gauges: Float type, pressure type (automatic).

• Reduction: Sounding at time $t$ minus tide height at $t$.

Nautical Sextant:

• Principle: Measures vertical angle between celestial object and horizon.

• Use: From boat, measure angle to fixed shore object of known height → distance = $h / \tan \theta$.

• Procedure: Sight object, adjust micrometer, read angle.

12.0 PHOTOGRAMMETRY & AERIAL PHOTOGRAPHY

Introduction: Science of making measurements from photographs.

Aerial Photography:

• Cameras: Frame (single image), Strip (continuous), Panoramic (wide angle).

• Flight Planning:

  • Scale: $$\displaystyle S = \frac{f}{H} $$ (f = focal length, H = flying height above datum).

  • Overlap: Longitudinal 60–65%, Side 30–40% (for stereoscopy).

• Number of Photographs:

  \[ N = \frac{\text{Area}}{\text{Photo area} \times (1 - \text{overlap})} \times \frac{1}{\text{side overlap factor}} \]

  More precisely: $$\displaystyle N = \frac{L}{l(1 - q)} \times \frac{W}{w(1 - r)} $$ (L,W area dimensions; l,w photo dimensions; q=long. overlap, r=side overlap).

Photograph Coordinates & Ground Coordinates:

• For vertical photo, no tilt: $$\displaystyle x = f \frac{X}{H} $$, $$\displaystyle y = f \frac{Y}{H} $$ (X,Y ground coordinates).

• Where $x,y$ are photo coordinates from principal point.

Relief Displacement:

• Definition: Shift of image of elevated object from its true position on photo.

• Cause: Object above datum (e.g., tree, building).

• Derivation: Similar triangles: $$\displaystyle \frac{d}{r} = \frac{h}{H} $$ → \boxed{d = \frac{r h}{H}}

  • $d$ = displacement, $r$ = radial distance from principal point to object image, $h$ = object height above datum, $H$ = flying height above datum.

• Applications: Measure heights, correct for displacement in mapping.

Uses in Civil Engineering:

• Route surveys (highways, pipelines).

• Land use/cover mapping.

• Disaster management (floods, landslides).

• Volume calculation (stockpiles).

13.0 PLANE TABLING

Principle: Simultaneous plotting of field observations on a drawing board mounted on a tripod.

Advantages: No measurement errors, immediate map, checks errors. Limitations: Weather-sensitive, less precise than total station. Suitable Conditions: Small areas, detailed mapping, magnetic areas.

Instruments & Accessories:

• Plane table (drawing board with tripod).

• Alidade (sighting device with parallel edges).

• Spirit level.

• Ranging rods.

• Drawing sheet, pencils.

Temporary Adjustments:

1. Centring: Plumb bob over station.

2. Levelling: Bubble centered.

3. Orientation: Alidade parallel to survey line (using back sight).

Methods:

1. Radiation: From one station, sight all points, plot rays, measure distances.

2. Intersection: From two stations, sight common point, plot intersection.

3. Traversing: Along traverse lines, plot each station by intersection.

4. Resection:

   • Two-Point Problem:

     • From unknown station P, sight two known points A, B.

     • Draw rays from A, B on plan, intersect at P.

     • Procedure: Rough orientation, sight A, draw ray; sight B, draw ray; adjust orientation until rays intersect at correct P.

   • Three-Point Problem:

     • Graphical method (Cassini's, etc.) or instrumental (using tacheometry).

14.0 FIELD PROBLEMS & CALCULATIONS (INTEGRATIVE)

Computation of Ground Distance from Vertical Photograph:

• Given photo coordinates $(x,y)$, focal length $f$, flying height $H$.

• Ground coordinates: $$\displaystyle X = \frac{H}{f} x $$, $$\displaystyle Y = \frac{H}{f} y $$.

• Distance $$\displaystyle PQ = \sqrt{(X_2-X_1)^2 + (Y_2-Y_1)^2} $$.

RL from Trigonometric Levelling:

• With multiple staff readings and vertical angles:

  • For each observation: $$\displaystyle h_i = D_i \tan \theta_i + I - S_i $$ (or using intercept formula).

  • RL of instrument station = RL of BM – mean $$\displaystyle h_i $$.

Solving Traverse Problems:

• Closed Traverse Adjustment (Bowditch's Rule):

  • Error in latitude $\Sigma L$ and departure $\Sigma D$.

  • Correction for each line: $$\displaystyle \delta L = -\frac{L_i}{\Sigma L} \Sigma L $$, $$\displaystyle \delta D = -\frac{D_i}{\Sigma D} \Sigma D $$.

  • Corrected coordinates: $$\displaystyle L' = L + \delta L $$, $$\displaystyle D' = D + \delta D $$.

• Missing Length/Bearing: Use sum of latitudes=0, departures=0 for closed traverse.

Correction of Bearings due to Change in Magnetic Declination:

• Old bearing (magnetic) at time of survey.

• Present magnetic bearing = old magnetic bearing + (present declination – old declination).

• If converting to true: add/subtract declination accordingly.

Area Calculation from Offsets:

• Use Trapezoidal or Simpson's rule as in Section 10.

• For irregular boundaries along chain line at fixed intervals.

[!TIP] Exam Strategy: For traverse problems, always check $\Sigma L \approx 0$, $\Sigma D \approx 0$ first. Bowditch's distributes error proportionally to side lengths.