UNIT 4: SURVEYING - COMPREHENSIVE SHORT NOTES

1.0 FUNDAMENTALS & PRELIMINARIES

1.4 Meridians & Bearings

• True Meridian: Line joining true North & South poles.

• Magnetic Meridian: Direction shown by a freely suspended magnetic needle.

• Arbitrary Meridian: Any convenient direction (e.g., a building line) used for small surveys.

• Grid Meridian: Parallel lines used in map projections (e.g., UTM).

Bearing Types:

• True Bearing (T.B.): Angle between True Meridian & survey line.

• Magnetic Bearing (M.B.): Angle between Magnetic Meridian & survey line.

• Fore Bearing (F.B.): Bearing of line in direction of survey progress.

• Back Bearing (B.B.): Bearing of line in opposite direction. \boxed{B.B. = F.B. \pm 180^\circ}.

• Whole Circle Bearing (W.C.B.): Bearing measured clockwise from North (0° to 360°).

• Reduced Circle Bearing (R.C.B. or Quadrantal Bearing): Bearing measured from North/South towards East/West (e.g., N 30° E).

WCB to RCB Conversion:

[!TIP] Local Attraction: Deviation of magnetic needle from true magnetic meridian due to local magnetic influences (iron ore, steel structures, etc.). It causes F.B. + B.B. ≠ 180°.

Detection & Correction of Local Attraction:

1. Detection: Check if F.B. + B.B. = 180° for each line. If not, local attraction exists at one or both stations.

2. Correction (Computation of Corrected Bearings):

   • Method 1 (Using Included Angle): Correct included angles first (sum should be (2n-4)*90° for closed traverse). Then start from a line free from error (or with known correct bearing) and propagate corrections.

   • Method 2 (Direct Correction): Find the mean bearing for each line from the observed FB and BB. Corrected Bearing = Mean Bearing. This works if attraction is equal at both ends of a line.

1.5 Chains & Tapes

Sources of Error in Chaining/Taping:

• Personal: Incorrect alignment, inconsistent tension, improper reading, sagging.

• Instrumental: Incorrect length (calibration error), broken/worn links, stretched tape.

• Natural: Temperature variation, uneven ground, wind.

Obstacles in Chaining (Methods with Sketches):

1. Chaining across obstacles (river/pond): Use reciprocal ranging or set out a perpendicular baseline.

2. Chaining past obstacles (building/tree): Use offsets (perpendicular/oblique).

3. Chaining on sloping ground: Use break-off method or hypotenuse allowance.

4. Chaining in undulating terrain: Use taping on the slope with slope correction or step chaining.

Tape Corrections (Numerical Problems):

True Length (L) = Measured Length (l) + ΣCorrections

1. Temperature Correction: C_t = α (T_m - T_s) * l

   • α = coeff. of thermal expansion, T_m = mean temp, T_s = standard temp.

2. Pull/Tension Correction: C_p = (P - P_s) * l / (A * E)

   • P = applied pull, P_s = standard pull, A = cross-sectional area, E = Modulus of Elasticity.

3. Slope Correction: C_s = \frac{h^2}{2l} (for gentle slopes, h = vertical difference).

4. Sag Correction: C_{sag} = \frac{w^2 l^3}{24 P^2}

   • w = weight of tape per unit length.

2.0 LINEAR & ANGULAR MEASUREMENT

2.1 Chain Surveying & Offsets

Offsets:

• Perpendicular Offset: Offset measured at 90° to chain line. Most accurate & preferred.

• Oblique Offset: Offset measured at an angle ≠ 90°.

• Tie Offset: Offset to a prominent point (e.g., tree corner) not on chain line.

• Side Offset: Offset taken along a line parallel to chain line.

Merits & Demerits:

[!TIP] Why Short Offsets are Preferred? Error in locating offset point is proportional to offset length. Short offsets minimize plotting error and are easier to set accurately in the field.

Area Computation from Offsets:

• Trapezoidal Rule: \boxed{Area = d \left[ \frac{O_1 + O_n}{2} + O_2 + O_3 + ... + O_{n-1} \right]}

  • d = common interval, O = offsets.

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

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

2.2 Compass Surveying

Prismatic Compass vs. Surveyor's Compass:

2.3 Theodolite Surveying

Geometry of Theodolite (Fundamental Lines):

1. Vertical Axis (VV): Axis about which telescope rotates horizontally.

2. Horizontal Axis (HH): Axis about which telescope rotates vertically.

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

4. Axis of Plate Levels (PL): Axis perpendicular to vertical axis.

5. Axis of Altitude Bubble (AB): Axis parallel to horizontal axis.

Temporary Adjustments (Setting Up):

1. Centring: Plumb the instrument over survey station.

2. Leveling: Use foot screws & plate levels to make vertical axis truly vertical.

3. Focusing: Remove parallax by focusing eyepiece (on cross-hairs) and objective (on object).

Permanent Adjustments (Key Ones):

1. Collimation Adjustment: Ensure LC is perpendicular to horizontal axis.

2. Horizontal Axis Adjustment: Ensure HH is perpendicular to vertical axis.

3. Vertical Axis Adjustment: Ensure VV is truly vertical when bubble is centered.

Methods of Traversing by Theodolite:

1. Included Angle Method: Measure interior angle at each station.

2. Deflection Angle Method: Measure angle between forward and backward lines (common in route surveys).

3. Direct Angle Method: Measure angle from forward line to new line.

4. Free Station Method: Set up at each station, sight previous & next station.

Repetition & Reiteration Methods:

• Repetition: Used to measure a horizontal angle at one station with high precision. Angle is added repeatedly (e.g., 4-5 times) to eliminate reading error. Mean Angle = Total Reading / No. of Repetitions.

• Reiteration: Used to measure multiple angles at one station (e.g., all interior angles of a polygon). Telescope is turned to each successive station, reading taken, and returned to first station to check closure.

Gale's Table for Traverse Computation:

A tabular method for closed traverse balancing.

1. Compute Latitude (N-S component) = Length * cos(Bearing).

2. Compute Departure (E-W component) = Length * sin(Bearing).

3. Sum Latitudes & Departures. Discrepancies (ΣL and ΣD) indicate error.

4. Apply Bowditch's Rule (Compass Rule) for balancing:

   Correction to Latitude = - (ΣL / Perimeter) * Length of Line

   Correction to Departure = - (ΣD / Perimeter) * Length of Line

5. Apply corrections to get Balanced Latitudes & Departures.

6. Compute Independent Coordinates (starting from known point).

3.0 LEVELLING

3.2 Levelling Methods & Calculations

Height of Instrument (H.I.) Method:

1. Take reading on Benchmark (B.M.) → H.I. = R.L. of B.M. + Backsight (B.S.).

2. Take Intermediate Sights (I.S.) on all points → R.L. of Point = H.I. - I.S..

3. When instrument is shifted, take Foresight (F.S.) on last point before shifting → New H.I. = R.L. of last point + F.S..

4. Repeat.

Rise & Fall Method:

1. For each set of readings (B.S., I.S., F.S.):

   • Rise = Previous R.L. - Current Staff Reading

   • Fall = Current Staff Reading - Previous R.L.

2. R.L. of next point = R.L. of previous point + Rise - Fall.

3. Check: ΣRise - ΣFall = Last R.L. - First R.L..

3.3 Sources of Error in Levelling

Personal Errors:

1. Parallax: Not eliminated.

2. Inaccurate focusing.

3. Staff not held vertical.

4. Error in reading staff (stadium error).

Instrumental Errors:

1. Axis not perpendicular to vertical axis: Causes error over long sights.

2. Line of collimation not parallel to axis of bubble tube: Causes error if bubble not centered.

3. Telescope not focused properly.

4. Graduations of staff not accurate.

3.4 Corrections in Levelling

Curvature Correction (C_c):

Earth's curved surface causes staff reading to appear higher than true level.

\boxed{C_c = 0.0785 \times D^2} (D in km, C_c in m) Derivation: From geometry of circle, d^2 = 2Rh + h^2 ≈ 2Rh. For two points distance D, h = D^2 / (2R). With R = 6371 km, h (m) = 0.0785 D^2 (km).

Refraction Correction (C_r):

Atmospheric refraction bends light downwards, making staff appear lower. Approx. C_r ≈ 0.067 D^2 (D in km).

Combined Correction (C):

\boxed{C = C_c - C_r = 0.0785D^2 - 0.067D^2 = 0.0115 D^2}.

For ordinary levelling (D < 100 m), this is negligible (< 0.001 m).

[!TIP] Numerical Example: Eye height 7.5 m, lighthouse height 50 m. Distance to see top just above horizon?

D = \sqrt{\frac{h_1 + h_2}{0.067}} = \sqrt{\frac{7.5+50}{0.067}} ≈ 29.7 km.

3.5 Longitudinal & Cross-Sectioning

• Longitudinal Sectioning: Levels taken along the centerline of a linear project (road, railway, canal). Shows ground profile along alignment.

• Cross-Sectioning: Levels taken perpendicular to centerline at regular intervals. Shows ground profile across the alignment.

• Importance: Used for earthwork calculation (cutting/filling), design of drainage, and setting out construction. Provides 3D ground model.

3.6 Trigonometric Levelling

Principle: Determine elevation difference using vertical angle and horizontal distance (or slope distance). For Inaccessible Point (Single Setup):

1. Measure vertical angle (α) to point from known R.L. station.

2. Measure horizontal distance (D) or slope distance (S).

3. \boxed{\Delta h = D \tan \alpha} (if D known).

\boxed{\Delta h = S \sin \alpha} (if S known).

4. R.L. of point = R.L. of instrument station + Δh ± H.I. (H.I. = height of instrument above station).

4.0 TACHEOMETRY (STADIA METHOD)

4.2 Stadia Method: Fixed Hair vs. Movable Hair

4.3 Distance & Elevation Formula Derivation

Case 1: Line of Sight Horizontal, Staff Vertical

\boxed{D = K s + C}

• D = horizontal distance.

• s = staff intercept (upper hair reading - lower hair reading).

• K = multiplying constant (≈ 100).

• C = additive constant (≈ 0 for anallatic lens).

• Derivation: From similar triangles, s / f = (K s + C) / f' → D = K s + C.

Case 2: Line of Sight Inclined (Angle α), Staff Vertical

Let α = vertical angle (positive for elevation, negative for depression).

1. Horizontal Distance: \boxed{D = K s \cos^2 \alpha + C \cos \alpha}

2. Elevation Difference (Δh):

   • If angle of elevation: \boxed{\Delta h = D \tan \alpha + H.I. + \text{staff reading at axial hair}}

   • If angle of depression: \boxed{\Delta h = D \tan \alpha - H.I. - \text{staff reading at axial hair}}

   • (Signs depend on convention; typically Δh = D \tan α + \text{reading on central hair} if α positive).

Numerical Problem Structure:

1. Compute s from three staff readings (upper, middle, lower).

2. Apply formula for D.

3. Compute Δh using D and α.

4. R.L. of staff point = R.L. of instrument station + H.I. - (staff reading at axial hair) + Δh (or as per formula).

5. For multiple stations, use known R.L. of B.M. to find HI at each station.

5.0 CURVES (HORIZONTAL & VERTICAL)

5.1 Simple Circular Curve

Elements with Neat Sketch:

Degree of Curve (D):

• Arc Definition (common in India): D = angle subtended at center by 30 m (100 ft) arc.

  \boxed{D = \frac{5729.578}{R}} (approx. D = \frac{1720}{R} for R in m).

• Chord Definition: D = angle subtended at center by 30 m chord.

  \boxed{R = \frac{15}{\sin(D/2)}}.

Setting Out by Theodolite (Rankine's Method):

1. Set up theodolite at PI.

2. Sight along forward tangent.

3. Rotate telescope by deflection angle (δ) for first chord point.

   δ_n = \frac{\text{Chord length}}{2R} \text{ (in radians)} or δ_n = \frac{C_n}{2R} \times \frac{180}{\pi}.

   For equal chords: δ = \frac{C}{2R} (degrees) for each chord.

4. Measure distance along chord from PI to set point on curve.

5. Repeat for subsequent chord points.

5.2 Compound Curves

• Definition: Two or more simple circular curves of different radii joined by a common tangent.

• Necessity: Terrain constraints where single radius curve not feasible.

• Elements: R1, R2, Δ1, Δ2, Common Tangent Length (CT), T1, T2.

• Calculation: From given Δ1, Δ2, R1, R2 or from PI positions. CT found by solving triangle formed by centers and common tangent point.

5.3 Transition Curves (Spirals)

Requirement & Applications:

• Provide gradual transition from straight to circular curve.

• Introduce gradual superelevation.

• Reduce centrifugal force suddenly applied.

• Improve safety & comfort at high speeds.

• Used in highway & railway design.

Types: Clothoid (ideal), Cubic Parabola, Lemniscate.

Ideal Transition Curve (Clothoid) Derivation:

• Condition: Radius should be inversely proportional to length from tangent point.

  \boxed{L \propto \frac{1}{R}} or R \cdot L = \text{constant}.

• Equation: L = \frac{R_0 \cdot L_s}{R} where R_0 = radius at spiral end, L_s = total spiral length.

• Deflection Angle (δ): δ = \frac{L^2}{2 R L_s} (in radians) for clothoid.

Design Problem (Given Speed V, Superelevation e, Rate of change of radial acceleration ψ):

1. Radius (R) from superelevation: R = \frac{V^2}{127(e + f)} (V in km/h, f = side friction factor).

2. Spiral Length (L_s): L_s = \frac{V^3}{R \cdot ψ} (ψ in m/s³, V in m/s) or L_s = \frac{0.0216 V^3}{R ψ} (V in km/h, ψ in cm/s³).

3. Shift (s): s = \frac{L_s^2}{24 R} (approx.).

4. Tangent Length (T_s): T_s = \frac{L_s}{2} + \sqrt{(R+s)^2 - R^2}.

5.4 Vertical Curves

Types:

• Summit Curve (Convex): Crest of hill. g1 > g2 (both positive or g1 +ve, g2 -ve).

• Sag Curve (Concave): Valley. g1 < g2 (both negative or g1 -ve, g2 +ve).

Setting Out: Use tangent offsets. y = \frac{e}{2L} x^2 (parabolic), where e = g1 - g2, L = curve length, x = distance from start.

Earthwork Calculation (with Cross-Sections):

1. Take cross-sections at regular intervals (e.g., 50 m).

2. Compute area of cutting/filling for each section (using trapezoidal rule on cross-section profile).

3. Compute volume between sections:

   • Trapezoidal Rule (Mean Area Method): V = \frac{A_1 + A_2}{2} \times d

   • Prismoidal Rule (More Accurate): \boxed{V = \frac{d}{6} (A_1 + 4A_m + A_2)}

     where A_m = area at mid-section.

6.0 TOPOGRAPHIC SURVEYING & MAPPING

6.1 Contouring

Characteristics of Contours (with Sketches):

• Spacing: Close = steep slope, wide = gentle slope, parallel = uniform slope.

• Shape: V-shape across ridge (point upstream), U-shape across valley (point downstream), closed circles = hill (values increase inward) or depression (values decrease inward, hachures).

• Crossing: Contours never cross (except in overhanging cliffs).

• Concentric: Indicate hill or pond.

Indirect Method of Contouring:

1. Radial Method: From a central point (peak/station), take radial lines & measure offsets to contour points.

2. Square/Grid Method: Divide area into squares, take spot levels at corners & center, interpolate contours.

3. Triangular/Trapezoidal Method: Establish triangulation network, spot levels at vertices, interpolate within triangles.

Advantages & Disadvantages:

Area Computation from Contour Maps:

• Trapezoidal Rule: Area = d [ \frac{N_1 + N_n}{2} + N_2 + ... + N_{n-1} ] (N = number of contours between two parallels).

• Prismoidal Rule: Area = \frac{d}{3} [ N_1 + N_n + 4(N_2+N_4+...) + 2(N_3+N_5+...) ].

6.2 Plane Table Surveying

Principle: Simultaneous plotting and surveying. Map is drawn in the field directly from field observations.

Two-Point Problem (Procedure with Sketch):

1. Set up plane table at unknown point P, orient roughly.

2. Sight point A & draw ray pa.

3. Shift to known point A, orient by back-sighting ap.

4. Sight point B & draw ray ab intersecting pa at p'.

5. Draw auxiliary line aa' parallel to ab.

6. Shift to B, orient by back-sighting ab.

7. Sight A & draw ray ba intersecting aa' at a''.

8. Line a'a'' is true position of AB. Draw perpendicular bisector to get true position of P at intersection with pa.

9. Final orientation achieved by sighting A & B from P.

7.0 ADVANCED SURVEYING & PHOTOGRAMMETRY

7.1 Aerial Photography & Photogrammetry

Definition: Science of making measurements from photographs.

Uses in Civil Engineering:

• Topographic mapping.

• Route surveys (roads, railways).

• Land use/land cover studies.

• Volume computation (quarries, stockpiles).

• Disaster assessment.

Scale of Vertical Photograph:

\boxed{Scale = \frac{f}{H-h}}

• f = focal length of camera.

• H = flying height above datum.

• h = average ground elevation above datum.

Overlaps:

• Longitudinal Overlap (60%): Overlap between successive photos along flight line. Ensures stereoscopic vision.

• Side Lap (30%): Overlap between adjacent flight lines. Ensures complete area coverage.

Number of Photographs Required:

\boxed{N = \frac{\text{Area to be covered}}{\text{Ground coverage per photo}} \times \frac{1}{(1 - \text{Long. Overlap}) \times (1 - \text{Side Overlap})}}

Where Ground coverage per photo = (Photo size) / Scale.

Relief Displacement (d):

• Definition: Radial displacement of image of a point from its true orthographic position due to its elevation above/below datum.

• Derivation: From similar triangles, d = \frac{r h}{H}.

  \boxed{d = r \cdot \frac{h}{H-h} ≈ r \cdot \frac{h}{H}} (for small h/H).

  • r = radial distance of image from principal point.

  • h = height of object above datum.

  • H = flying height above datum.

Ground Coordinates from Photograph Coordinates:

Using perspective geometry and collinearity condition:

x = -f \frac{X}{Z}, y = -f \frac{Y}{Z}

where (X,Y,Z) are ground coordinates (with Z = H-h), (x,y) are photo coordinates, f = focal length. Requires resection or spatial intersection.

7.2 Hydrographic Surveying

Sounding: Measurement of depth of water from surface to bed.

• Purpose: Charting for navigation, dredging, underwater construction, pipeline/ cable routing.

Sounding Equipment:

• Sounding Rod: Wood/metal pole for shallow water.

• Sounding Lead: Weighted line for moderate depth.

• Echo Sounder: Electronic, uses sound pulses (most common).

• Sounding Cable (with sketch): Wire rope with weight & recorder for deep water. Cable marked at intervals, counter measures depth.

Methods of Sounding (with Sketches):

1. Surface Float: Float on surface, time taken to travel known distance gives current velocity.

2. Subsurface Float: Float at desired depth, connected to surface marker for depth measurement.

3. Pitot Tube: Measures velocity of current. Tube facing upstream, pressure difference gives velocity.

Nautical Sextant (Principle & Sketch):

• Principle: Measures angle between two objects (e.g., celestial body & horizon) by reflecting one ray through a half-silvered mirror.

• Use: Celestial navigation to determine position (latitude & longitude) by measuring altitude of sun/stars.

• Sketch: Show telescope, index mirror, horizon glass, arc scale.

Tide Gauge:

• Purpose: Record water level variations (tides) relative to a fixed datum.

• Types: Float type, pressure type, acoustic type.

• Use: Reduce sounding to a common datum (chart datum).

8.0 AREA & VOLUME COMPUTATION

8.1 Area from Field Notes (Chain & Offset)

Trapezoidal Rule: \boxed{A = d \left[ \frac{O_1 + O_n}{2} + \sum_{i=2}^{n-1} O_i \right]} Simpson's 1/3 Rule: \boxed{A = \frac{d}{3} \left[ O_1 + O_n + 4 \sum_{\text{odd } i} O_i + 2 \sum_{\text{even } i} O_i \right]}

8.2 Area from Coordinates (Latitude & Departure Method)

For a closed traverse:

\boxed{Area = \frac{1}{2} \left| \sum (E_i N_{i+1} - E_{i+1} N_i) \right|}

Or using balanced latitudes & departures: Area = \sum (L \times D) for each line? No, standard is coordinate summation.

8.3 Earthwork Volume

Trapezoidal Rule (Mean Area Method):

\boxed{V = \frac{d}{2} (A_1 + A_2 + 2A_3 + 2A_4 + ... + 2A_{n-1} + A_n)}

Where d = interval between sections, A = cross-sectional area.

Prismoidal Rule (More Accurate):

\boxed{V = \frac{d}{3} (A_1 + A_n + 4(A_2 + A_4 + ...) + 2(A_3 + A_5 + ...))} Condition: Number of sections must be odd (even number of intervals).

Mass Haul Diagram (Basic Concept):

• Graph of cumulative volume (ordinate) vs. distance along centerline (abscissa).

• Purpose: Visualize earthwork distribution, plan hauling & balancing.

• Key Points: ∑V = 0 line (balance line), free haul distance, overhaul distance.

[!TIP] Common Pitfall: Applying Simpson's rule with even number of intervals. Always check: (n-1) must be even → n odd.