Surveying (CE-303) - Unit 2 Short Notes

UNIT 2: THEODOLITE SURVEYING, LEVELLING, TACHEOMETRY, CURVE SETTING & OTHER TOPICS

I. THEODOLITE SURVEYING

A. Geometry and Components of Transit Theodolite

A transit theodolite allows telescope to rotate 360° vertically (transit). Key components:

• Telescope: For sighting, with cross-hairs and bubble.

• Vertical Circle: Graduated 0-90° for vertical angles.

• Horizontal Circle (Vernier/Plate): Graduated 0-360° for horizontal angles.

• Index Bar (Vernier): Reads fractions of smallest division (least count).

• Levelling Head: With foot screws for rough/fine levelling.

• Tripod Stand: For support.

• Plunger (Reversing): To flip telescope (transit).

• Clamp & Tangent Screws: For fixing and fine movement.

[!TIP] Exam Focus: Sketch must show vertical axis, horizontal axis, line of collimation, axis of plate bubble, axis of altitude bubble. Label all parts clearly.

B. Temporary Adjustments

1. Setting up: Position tripod over station, roughly centre head.

2. Centring: Use optical plummet or plumb bob to centre instrument over station mark.

3. Leveling:

   • Rough level using tripod legs.

   • Fine level using foot screws. Bubble Down method: centre bubble in two positions (90° apart) to ensure axis truly vertical.

4. Focusing:

   • Eyepiece: Focus cross-hairs for observer's eye.

   • Objective: Focus on target (distant object/staff).

Key Terms:

• Transiting: Rotating telescope 180° about vertical axis.

• Centring: Making instrument's vertical axis pass through station mark.

• Bubble Down: Bringing plate bubble to centre in both positions of telescope.

C. Permanent Adjustments (Fundamental Lines)

Goal: Make 4 axes mutually perpendicular:

1. Axis of Plate Bubble ⊥ Vertical Axis

2. Line of Collimation (optical axis) ⊥ Horizontal Axis

3. Horizontal Axis ⊥ Vertical Axis

4. Vertical Axis truly vertical (when bubble centred).

Adjustments done in specific order using known standards (e.g., distant vertical rod, horizontal line).

D. Methods of Traversing with Theodolite

1. Repetition Method (Direction Theodolite):

   • Measure angle multiple times (e.g., 3-4 repetitions) without changing instrument position.

   • Final reading = (Final circle reading - Initial reading) / Number of repetitions.

   • Eliminates index error.

   • Used for high-precision angle measurement.

2. Reiteration Method (Repeating Theodolite):

   • Set telescope to first station, read angle.

   • Swing to next station, read angle. Continue around traverse.

   • Closing error in sum of angles distributed.

   • Used for traverse surveys.

3. Using Gale's Table:

   • Tabular method for reiteration.

   • Columns: Station, Face Left (L), Face Right (R), Mean, Correction, Final Bearing.

   • Face L & R readings taken at each station to eliminate index error.

   • Mean of L & R gives correct angle at station.

E. Errors in Theodolite Observations

• Instrumental: Non-perpendicularity of axes, eccentricity of verniers, graduation errors.

• Personal: Reading errors, centring error, levelling error, focusing error.

• Natural: Temperature variation, wind, refraction.

F. Least Count of Vernier Scale

$$ \text{Least Count (L.C.)} = \frac{\text{Value of one smallest division on main scale}}{\text{Number of divisions on vernier}} $$

For a theodolite, typical L.C. = 20" (20 seconds) or 10".

II. LEVELLING

A. Principles and Methods

• Principle: Line of sight is horizontal. Difference in staff readings = Difference in elevation.

• Differential Levelling (Spirit Levelling): Determine difference in elevation between two points by taking Fore Sight (F.S.) and Back Sight (B.S.).

• Height of Instrument (H.I.) Method:

  • H.I. = R.L. of B.M. + B.S.

  • R.L. of point = H.I. - F.S.

  • After shifting, new H.I. = previous H.I. + (B.S. - F.S.) of turning point.

• Rise and Fall Method:

  • Compare consecutive staff readings.

  • If previous reading > current reading → Rise.

  • If previous reading < current reading → Fall.

  • R.L. of point = R.L. of previous point - Fall (or + Rise).

B. Calculation of Reduced Levels (R.L.)

• With Instrument Shifted (Multiple Set-ups): Use either H.I. or Rise & Fall method, ensuring turning points are used for continuity.

• With Fore and Back Sights: Standard procedure as above.

C. Sources of Error

• Personal: Staff not vertical, reading errors, bubble not centred, focusing error.

• Instrumental: Collimation error (line of sight not horizontal when bubble centred), axis of bubble not perpendicular to vertical axis.

• Natural: Curvature & refraction, wind, temperature.

D. Corrections in Levelling

1. Curvature Correction ($$\displaystyle C_c $$): Earth's curved surface causes staff reading to appear higher.

$$ C_c = \frac{d^2}{2R} $$

where $d$ = distance (km), $R$ = Earth radius (~6370 km). **Always negative**.

2. Refraction Correction ($$\displaystyle C_r $$): Atmospheric refraction bends light ray downward, partially counteracting curvature.

$$ C_r \approx -\frac{1}{7} \times C_c = -\frac{d^2}{14R} $$

**Always positive**.

3. Combined Correction (for long sights):

$$ C_{combined} = C_c + C_r = 0.0673 \, d^2 \ \text{(in meters for } d \text{ in km)} $$

> [!TIP] **Exam Problem:** Often given "distance visible", "height of eye", "height of object". Use Pythagoras with combined correction.

E. Trigonometric Levelling

• Definition: Determining elevation difference using vertical angles and horizontal distances.

• Principle: $$\displaystyle \Delta h = D \tan \theta + i - s $$

  • $\Delta h$ = Elevation difference

  • $D$ = Horizontal distance

  • $\theta$ = Vertical angle (+ for elevation, - for depression)

  • $i$ = Height of instrument above station

  • $s$ = Staff reading (or target height)

• Height of Object: $$\displaystyle H = h_{inst} + \Delta h + \text{staff reading (if used)} $$

III. TACHEOMETRY (STADIA SURVEYING)

A. Principle and Constants

• Principle: Stadia method uses fixed cross-hairs (stadia hairs) in telescope. Intercept on vertical staff gives horizontal distance.

• Anallactic Lens: Special lens making staff intercept independent of focus (i.e., $$\displaystyle K=100 $$, $$\displaystyle C=0 $$). Common in modern tacheometers.

• Stadia Constants: For a tacheometer,

$$ D = K \cdot s + C $$

where $D$ = horizontal distance, $s$ = staff intercept (top - bottom hair), $K$ = stadia constant (~100), $C$ = additive constant (~0 for anallactic, 0.3-0.5 m otherwise).

B. Fixed Hair Method

1. Line of Sight Horizontal, Staff Vertical:

$$ D = K \cdot s + C $$

Elevation of staff station:

$$ \text{R.L.}_\text{staff} = \text{R.L.}_\text{inst} + i - v + C \cdot \frac{v}{K} \ \text{(approx)} $$

where $i$ = H.I., $v$ = staff reading at central hair.

> For anallactic lens ($$\displaystyle C=0 $$): $$\displaystyle \text{R.L.}_\text{staff} = \text{R.L.}_\text{inst} + i - v $$.

2. Line of Sight Inclined, Staff Vertical:

   Horizontal Distance:

$$ D = K \cdot s \cdot \cos^2 \theta + C \cdot \cos \theta $$

Elevation difference:

$$ \Delta h = D \tan \theta + i - v $$

where $\theta$ = vertical angle (+ up, - down).

C. Movable Hair Method (Subtense Method)

• Principle: Measure angle subtended by a fixed-length subtense bar (or staff length $l$).

$$ D = \frac{l}{2 \sin \frac{\theta}{2}} \approx \frac{l}{\theta} \ \text{(for small } \theta \text{ in radians)} $$

• More accurate, independent of focal length.

D. Procedure and Field Observations

1. Set up tacheometer, level, focus.

2. Take staff readings (top, middle, bottom hair) on point.

3. Note vertical angle (if inclined sight).

4. Compute $s$ (intercept), apply formula.

5. For anallactic lens, calculations simpler.

E. Characteristics of a Tacheometer

• Telescope with stadia hairs.

• Anallactic lens (optional but common).

• High magnification (25x-30x).

• Precise vertical circle (1' or 30").

• Often combined with theodolite (tacheotheodolite).

F. Applications and Circumstances for Use

• Rapid detail surveying (contours, boundaries).

• Rough terrain where chaining difficult.

• Hydrographic surveys (from shore).

• Mine surveys.

• Where speed is more critical than high precision (compared to plain taping).

IV. CURVE SETTING

A. Horizontal Curves

1. Simple Circular Curve

• Elements:

  • $R$ = Radius

  • $\Delta$ = Deflection angle (intersection angle)

  • $T$ = Tangent length $$\displaystyle = R \tan \frac{\Delta}{2} $$

  • $$\displaystyle L_c $$ = Length of curve $$\displaystyle = \frac{\pi R \Delta}{180^\circ} = R \cdot \Delta \text{ (in radians)} $$

  • $LC$ = Long chord $$\displaystyle = 2R \sin \frac{\Delta}{2} $$

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

  • $A$ = Apex distance (versed sine) $$\displaystyle = R \left(1 - \cos \frac{\Delta}{2}\right) = M $$

  • Chainage of T1 = Chainage of PI - T

  • Chainage of T2 = Chainage of T1 + Lc

Setting Out Methods:

• Offsets from Long Chord:

  • Offset $x$ at distance $y$ from midpoint of chord:

$$ x = R - \sqrt{R^2 - y^2} \approx \frac{y^2}{2R} \ \text{(for small } y\text{)} $$

• Offsets from Tangents (Rankine's Method / Theodolite Method):

  • Offset $$\displaystyle O_x $$ at chord length $x$ from tangent point:

$$ O_x = R \left(1 - \cos \frac{\theta}{2}\right) \approx \frac{x^2}{2R} \ \text{(for small } \theta \text{ in radians)} $$

where $$\displaystyle \theta = \frac{x}{R} $$ radians.

*   **Procedure:** Set theodolite at T1, direct to PI (angle $\Delta/2$), then set incremental angles $$\displaystyle \theta_1, \theta_2... $$ to stake points.

2. Compound Curve

• Definition: Two or more simple curves of different radii on same side of common tangent.

• Elements: $$\displaystyle R_1, R_2, \Delta_1, \Delta_2, T_1, T_2, T $$ (common tangent length), $$\displaystyle L_{c1}, L_{c2} $$.

• Calculation: Use geometry of triangles formed by PI, centers, and tangent points.

• Setting Out: Set first curve by Rankine's, then set second curve from intermediate tangent point (PCC).

3. Transition Curve (Spiral Curve)

• Necessity: Provide gradual change in curvature from straight to circular, avoiding sudden centrifugal force. Also for superelevation runoff.

• Requirements:

  1. Curvature increases linearly with length ($$\displaystyle \frac{1}{R} \propto l $$).

  2. Rate of change of centrifugal acceleration constant.

• Types: Euler's spiral, Lemniscate, Parabolic, Cubic Spiral (Lemniscate of Bernoulli) – most common.

• Ideal Transition Curve (Cubic Spiral):

$$ \theta = \frac{l^3}{6RL} \ \text{or} \ \theta \propto l^3 $$

where $\theta$ = deflection angle at length $l$, $L$ = total length of spiral.

• Length Calculation (based on design speed $V$):

$$ L = \frac{V^3}{127(R_f + R)} \ \text{(approx)} $$

or from superelevation $e$ and rate of change $C$:

$$ L = \frac{e \cdot V^2}{127 \cdot C} $$

• Setting Out: By offsets from tangent ($$\displaystyle x = \frac{l^3}{6RL} $$) or by deflection angles ($$\displaystyle \theta = \frac{l^2}{2RL} $$).

4. Key Terms

• Degree of Curve (D): Angle subtended at center by 100 ft chord (in US) or 20 m (in India).

$$ R = \frac{5729.578}{D} \ \text{(for } D \text{ in degrees, chord=100 ft)} \quad \text{or} \quad R = \frac{180 \times 20}{\pi \times D} \ \text{(for } D \text{ in degrees, chord=20 m)} $$

• Vedred Sine (Apex Distance): $$\displaystyle R(1 - \cos \frac{\Delta}{2}) $$.

• Superelevation (e): Raising outer edge of road on curve to counteract centrifugal force.

$$ e = \frac{V^2}{127R} \ \text{(for } V \text{ in km/h, } R \text{ in m)} $$

Provided gradually over transition curve.

• Initial and Final Sub-chords: First and last chord lengths on spiral (less than full chord due to varying curvature).

B. Vertical Curves

• Types:

  • Summit Curve (Convex): Crest of hill. Parabolic shape preferred.

  • Sag Curve (Concave): Valley. Parabolic or cubic spiral.

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

• Design: Length based on sight distance, comfort, appearance. Parabolic equation: $$\displaystyle y = g_1 x + \frac{(g_2 - g_1)}{2L} x^2 $$.

V. PHOTOGRAMMETRY AND AERIAL SURVEY

A. Aerial Photography

• Principle: Taking photographs from aircraft/drone to map ground.

• Objective: Obtain planimetric and topographic data.

• Uses in Civil Engineering:

  • Topographic mapping

  • Route surveys (highways, pipelines)

  • Land use/land cover studies

  • Volume calculation (stockpiles)

  • Disaster assessment

B. Geometry of Vertical Photograph

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

• Flying Height (Altitude above Datum): $$\displaystyle H = h + \text{R.L. of point} $$ where $h$ = height of camera above MSL.

• Ground Coordinates from Photographic Coordinates:

  For vertical photo, with $x,y$ photo coords (mm), scale $$\displaystyle S = f/H $$:

$$ X = x \cdot S, \quad Y = y \cdot S $$

(Assuming no tilt, radial line from principal point).

C. Relief Displacement

• Definition: Apparent shift of image of a point from its true orthographic position due to elevation.

• Causes: Object above/below ground level; photo is central projection.

• Derivation:

  From similar triangles:

$$ \frac{r}{R} = \frac{H}{h} \quad \Rightarrow \quad r = \frac{R \cdot H}{h} $$

where $r$ = photo distance from center, $R$ = ground distance from nadir, $H$ = flying height above ground, $h$ = height of object above ground.

**Displacement** $$\displaystyle d = r - r_0 = \frac{R \cdot h}{H - h} \approx \frac{r_0 \cdot h}{H} $$ (for small $h$).

\boxed{d = \frac{r_0 \cdot h}{H}} where $$\displaystyle r_0 $$ = radial distance on ground.

D. Flight Planning and Photo Coverage

• Longitudinal Overlap (End Lap): 60-65%. Ensures stereoscopic viewing and coverage.

• Side Lap (Side Overlap): 30-40%. Ensures no gaps between flight lines.

• Number of Photographs to Cover Area:

  For rectangular area $$\displaystyle A = L \times B $$ (km²), photo size $l \times b$ (cm), scale $1:S$:

  Ground coverage per photo (with overlap):

$$ \text{Along flight: } \frac{l \cdot S}{100} \times (1 - \text{end lap}) $$

$$ \text{Between lines: } \frac{b \cdot S}{100} \times (1 - \text{side lap}) $$

Number of photos $$\displaystyle N = \frac{L}{\text{coverage along}} \times \frac{B}{\text{coverage across}} $$.

E. Stereoscopy

• Principle: Viewing two overlapping photographs (stereopair) with each eye seeing a different photo → perception of 3D depth.

• Used for contouring, feature identification.

VI. HYDROGRAPHIC SURVEY

A. Sounding

• Definition: Measuring depth of water from surface to bed.

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

• Methods:

  • Echo Sounder (Sonar): Most common (acoustic).

  • Lead Line: Traditional weighted rope.

  • Pole: In shallow water.

B. Sounding Equipment

1. Sounding Cable: With weight (lead) and marking (armed lead).

   DiagramSEARCH: "sounding lead line diagram"

2. Nautical Sextant: Measures angle between object (e.g., light house) and horizon. Used for horizontal distance from known height.

$$ \text{Distance} = \frac{\text{Height}}{\tan \theta} $$

DiagramSEARCH: "nautical sextant principle diagram"

3. Tide Gauge: Records water level variation (tide) relative to a datum. Types: staff, float, pressure.

4. Surface Float Method: Float released upstream, time taken between two stations gives current velocity.

5. Pitot Tube Method: Measures current velocity by differential pressure.

   DiagramSEARCH: "pitot tube flow measurement diagram"

C. Marine Survey (Brief Procedure)

1. Planning: Define area, determine sounding lines (grid), set control on shore.

2. Tide Observation: Record tide at gauge station.

3. Sounding: Run lines, record depth + position (by GPS/radio).

4. Reduction: Reduce soundings to chart datum (lowest low water) using tide corrections.

5. Plotting: Plot depth contours, generate bathymetric chart.

VII. TRAVERSING AND COMPUTATION

A. Bearings and Angles

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

• Reduced Bearing (RB): Acute angle (0°-90°) from N/S, with quadrant (NE, SE, SW, NW).

• Conversion:

  • WCB to RB:

    • 0°-90°: RB = WCB (NE)

    • 90°-180°: RB = 180° - WCB (SE)

    • 180°-270°: RB = WCB - 180° (SW)

    • 270°-360°: RB = 360° - WCB (NW)

  • RB to WCB: Add quadrant angle (e.g., S30°E → WCB = 180° - 30° = 150°).

• Fore Bearing (FB): Direction of travel on line.

• Back Bearing (BB): Opposite direction. BB = FB ± 180°.

B. Local Attraction

• Definition: Magnetic attraction from nearby iron/ore, causing compass needle to deviate from true magnetic meridian.

• Detection:

  • Difference between FB and BB ≠ 180°.

  • At a point, bearing of same line from two stations differ significantly.

• Correction:

  1. Find correct bearing of a line unaffected by LA (using known true bearing or by closing error distribution).

  2. Apply correction to all bearings at that station.

C. Latitude and Departure

• Latitude (L): Projection of line on N-S direction. $$\displaystyle L = l \cos \theta $$ (positive for N, negative for S).

• Departure (D): Projection on E-W direction. $$\displaystyle D = l \sin \theta $$ (positive for E, negative for W).

• For closed traverse: $$\displaystyle \sum L = 0 $$, $$\displaystyle \sum D = 0 $$ (if no error).

D. Closing Error and its Adjustment

• Closing Error (misclosure): $$\displaystyle \Delta L = \sum L $$, $$\displaystyle \Delta D = \sum D $$.

• Magnitude: $$\displaystyle e = \sqrt{(\Delta L)^2 + (\Delta D)^2} $$.

• Direction: $$\displaystyle \theta = \tan^{-1} \left( \frac{\Delta D}{\Delta L} \right) $$.

• Adjustment: Distribute error proportionally to line lengths (Bowditch's Rule / Transit Rule).

  • Bowditch (Compass): Correction to latitude/departure of a line:

$$ \delta L = -\frac{L \cdot e}{L_{total}} \cdot \frac{\Delta L}{e}, \quad \delta D = -\frac{D \cdot e}{L_{total}} \cdot \frac{\Delta D}{e} $$

*   **Transit (Theodolite):** Corrections proportional to latitudes/departures.

E. Computation of Coordinates

• Independent Coordinates: Assign (E, N) to starting point. For each line:

$$ E_2 = E_1 + D, \quad N_2 = N_1 + L $$

• Missing Length/Bearing: Use $$\displaystyle \sum L=0 $$, $$\displaystyle \sum D=0 $$ to solve.

F. Area Calculation (Coordinate Method)

For polygon with vertices $$\displaystyle (E_i, N_i) $$:

$$ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (E_i N_{i+1} - E_{i+1} N_i) \right| \quad (N_{n+1}=N_1, E_{n+1}=E_1) $$

Or using latitudes/departures:

$$ \text{Area} = \frac{1}{2} \left| \sum (L_i \cdot D_{i+1} - L_{i+1} \cdot D_i) \right| $$

VIII. CONTOURING

A. Methods of Contouring

1. Direct Method: Directly locate points of equal elevation.

   • Radial: From a central point (hilltop).

   • Linear: Along parallel lines (plane table/chain).

2. Indirect Method: First survey entire area by tacheometry/plane table/Aerial photo, then interpolate contours.

   • Advantages: Faster, covers large area, useful for difficult terrain.

   • Disadvantages: Less accurate, requires interpolation.

B. Characteristics of Contours (with sketches)

• Steep Slope: Contours close together.

• Gentle Slope: Contours far apart.

• Flat: Contours widely spaced or absent.

• Hill: Closed circles with higher values inside.

• Depression: Closed circles with hachures (short lines) on lower side.

• Valley: V-shaped, apex upstream.

• Ridge: V-shaped, apex downstream.

• Cliff: Contours coincide or very close.

• Summit/Saddle: Specific patterns.

C. Interpolation of Contours

• Linear Interpolation: Between two known points, contour at proportional distance.

$$ x = \frac{h}{h_1} \cdot d $$

where $h$ = contour interval, $$\displaystyle h_1 $$ = difference in elevation between points, $d$ = distance between points.

• Curves: Use graphical method (template) or mathematical (parabolic).

D. Uses and Applications

• Determine elevation at any point.

• Plan routes (roads, canals) with optimum gradient.

• Estimate earthwork volumes.

• Determine watershed/ drainage.

• Site selection for construction.

IX. EARTHWORK CALCULATION

A. Volume of Earthwork

• For Embankments/Cuttings: Volume from cross-sections.

• Methods:

  1. Trapezoidal Rule (Mean Area Method):

$$ V = \frac{A_1 + A_2}{2} \cdot d $$

    where $$\displaystyle A_1, A_2 $$ = areas of two successive cross-sections, $d$ = distance between them.

2.  **Prismoidal Rule (More Accurate):**

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

    where $$\displaystyle A_m $$ = area of mid-section (at $d/2$).

    For $n$ sections (even number of intervals):

$$ V = \frac{d}{3} \left[ A_1 + A_n + 4(A_2 + A_4 + ...) + 2(A_3 + A_5 + ...) \right] $$

B. Longitudinal and Cross-Sectioning

• Longitudinal Section (Profile): Along centerline, shows ground profile and proposed formation level. Used for earthwork mass haul diagram.

• Cross-Section: Perpendicular to centerline at intervals. Shows side slopes, widths. Used to compute area of cut/fill.

• Importance: Accurate volume calculation, balancing cut/fill, planning equipment, cost estimation.

X. CHAIN SURVEYING AND OFFSETING

A. Types of Chains

1. Gunter's Chain: 66 ft, 100 links (0.66 ft each). Old.

2. Engineer's Chain: 100 ft, 100 links (1 ft each).

3. Metric Chain: 20 m or 30 m, 100/150 links (0.2 m/0.2 m).

4. Steel Band: 20-30 m, 1 mm wide, graduated in meters. More accurate.

5. Tape: Steel, Invar, Nylon. 5-50 m.

B. Obstacles in Chaining

• Types:

  • Perpendicular: River, building.

  • Oblique: Pond, thicket.

  • Chaining obstructed, ranging possible: Hill, valley.

  • Both chaining & ranging obstructed: Dense forest.

• Methods:

  • Perpendicular: Set perpendiculars by triangulation or optical square.

  • Reciprocal (Oblique): Measure $AB$, $BC$, $CD$, $DA$; compute $AC$ by $\cos$ rule.

  • Ranging only: Use triangulation or subtense bar.

  DiagramCANVAS: "Sketch showing reciprocal method for oblique obstacle: points A,B,C,D with measurements AB, BC, CD, DA and diagonal AC to be computed"

C. Sources of Error in Chaining and Precautions

• Cumulative Error: Incorrect chain length → calibrate frequently.

• Temperature/Pull/Sag: Apply corrections if high accuracy needed.

• Not fully stretched: Use sufficient pull.

• Not straight: Use intermediate points.

• Wrong reading: Read from correct end.

• Kinks/bends: Avoid.

D. Offsetting from Survey Lines

• Perpendicular Offsets: At right angles to chain line. Most common.

• Oblique Offsets: At an angle (used when perpendicular difficult).

• Tie Offsets: From a point on boundary to chain line (to fix irregular boundaries).

• Why Short Offsets Preferred? Errors in offset measurement and direction have less effect on area calculation. Long offsets amplify angular errors.

E. Area Calculation from Offsets

Given offsets $$\displaystyle o_1, o_2, ... $$ at interval $d$:

1. Trapezoidal Rule:

$$ \text{Area} = d \left[ \frac{o_1 + o_n}{2} + o_2 + o_3 + ... + o_{n-1} \right] $$

2. Simpson's Rule (1/3rd Rule): (n must be even)

$$ \text{Area} = \frac{d}{3} \left[ o_1 + o_n + 4(o_2 + o_4 + ...) + 2(o_3 + o_5 + ...) \right] $$

XI. TAPE SURVEYING AND CORRECTIONS

A. Types of Tapes

• Steel Tape: 5-50 m, 1/16" or 1 mm wide. Common.

• Invar Tape: Low thermal expansion, for high-precision work.

• Nylon/Fiberglass: Non-metallic, for general use, less prone to kinks.

B. Corrections to Tape Length

Standard tape length $$\displaystyle L_0 $$ at standard temperature $$\displaystyle T_0 $$, pull $$\displaystyle P_0 $$.

Measured length $$\displaystyle L_m $$ at $T$, $P$, on slope $h$, unsagged.

1. Temperature Correction:

$$ \Delta L_t = \alpha \cdot (T - T_0) \cdot L_m $$

where $\alpha$ = coeff. of thermal expansion (steel: $$\displaystyle 11 \times 10^{-6}/^\circ C $$).

2. Pull (Tension) Correction:

$$ \Delta L_p = \frac{(P - P_0) \cdot L_m}{A \cdot E} $$

where $A$ = cross-sectional area, $E$ = Young's modulus.

3. Sag Correction (Unsupported Lengths):

$$ \Delta L_s = \frac{w^2 L^3}{24 P^2} $$

where $w$ = weight per unit length, $L$ = unsupported length, $P$ = applied pull.

4. Slope/Gradient Correction:

$$ \Delta L_g = \frac{h^2}{2L} \approx \frac{h^2}{2 \cdot \text{horizontal distance}} $$

where $h$ = vertical difference.

5. Standardization Correction: If tape not exactly $$\displaystyle L_0 $$:

$$ \Delta L_s = \left( \frac{L_0}{L} - 1 \right) \cdot L_m $$

6. Combined Correction: Apply sequentially or algebraically to get true length $$\displaystyle L = L_m + \sum \Delta L $$.

XII. COMPASS SURVEYING

A. Types of Compasses

B. Temporary Adjustments of Prismatic Compass

1. Centring: Plumb bob over station.

2. Leveling: Spirit bubble centred.

3. Sighting: Align object vane with target.

4. Focusing: Focus eye-piece for observer.

C. Bearing Observations and Local Attraction

• Take Fore Bearing (FB) and Back Bearing (BB).

• Check: FB ± 180° = BB (if no LA).

• If difference ≠ 180°, local attraction present. Correct using known bearing or closing error method.

XIII. PLANE TABLE SURVEYING

A. Principle and Applications

• Principle: Simultaneous observation, measurement, and plotting. Plane table represents field map directly.

• Applications: Small areas, fillings, open cuts, topographical details, cadastral maps.

B. Two-Point Problem

• Objective: Plot position of plane table station using two known points.

• Procedure:

  1. Set up at unknown station $P$.

  2. Sight known point $A$, draw ray $PA$.

  3. Sight known point $B$, draw ray $PB$.

  4. Position of $P$ is intersection of rays from $A$ and $B$ (but from which orientation?).

  5. Solution:

     • From a third known point $C$, sight $P$ and draw ray $CP$.

     • Move tracing paper until rays from $A$ and $B$ pass through their plotted positions, and ray $CP$ passes through $C$.

     • Intersection of $PA$ and $PB$ on tracing paper gives $P$.

     • Prick down $P$, orient table using $CP$.

     • Check with $A$ and $B$.

  DiagramCANVAS: "Two-point problem sketch: known points A, B, C plotted; plane table at P; rays PA, PB, CP; showing orientation and intersection"

XIV. MERIDIANS, BEARINGS AND DECLINATION

A. Types of Meridians

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

2. Magnetic Meridian: Direction of magnetic needle (varies with time/location).

3. Grid Meridian: Parallel lines on map projection (e.g., UTM).

4. Arbitrary Meridian: Any chosen reference direction (e.g., from a building).

B. Magnetic Declination

• Definition: Angle between True Meridian and Magnetic Meridian.

  • East Declination: Magnetic N east of True N → Magnetic bearing > True bearing.

  • West Declination: Magnetic N west of True N → Magnetic bearing < True bearing.

• Variation: Changes with time and location.

• Conversion:

$$ \text{True Bearing} = \text{Magnetic Bearing} + \text{Declination} $$

(Add for East, subtract for West).

• Problem: Given old bearing with old declination, find new magnetic bearing for new declination:

  1. Find true bearing from old data.

  2. Apply new declination to get new magnetic bearing.

XV. MISCELLANEOUS SURVEYS

A. Rainfall Measurement

• Necessity: Water resource planning, hydrology, design of structures.

• Location of Rain Gauge Station:

  • Avoid wind effects (open area, away from trees/buildings).

  • Level ground.

  • Representative of region.

• Measurement using Rain Gauge:

  • Symon's Rain Gauge: Funnel + measuring cylinder. Depth = Volume / Area of funnel.

  • Recording Type: Float, weighing, tipping bucket.

  DiagramSEARCH: "symon's rain gauge diagram"

B. Classification of Surveying

• Primary:

  • Geodetic: Large areas (>250 km²), considers Earth's curvature, high precision.

  • Plane (Plain): Small areas (<250 km²), assumes flat Earth, ignores curvature.

• Secondary:

  • Topographic: Natural & man-made features.

  • Cadastral: Property boundaries.

  • Engineering: For construction projects.

  • Route: Highways, railways, canals.

  • Mining: Underground.

  • Archaeological: Historical sites.

  • Military: Defense.

C. Fundamental Principle of Surveying

"Work from whole to part."

First establish primary control (high accuracy, widely spaced), then secondary/tertiary control, finally details. Minimizes error accumulation.

D. Importance of Surveying in Civil Engineering

• Planning & Design: Base maps, alignment, earthwork.

• Construction: Setting out, alignment, as-built.

• Land Development: Subdivision, boundary.

• Infrastructure: Roads, bridges, dams, pipelines.

• Legal: Property disputes, land records.

• Environmental: Monitoring, impact assessment.

END OF UNIT 2 NOTES