Measurement of Horizontal Distances

Measurement of Horizontal Distances

Horizontal distance is the distance between two points measured along a horizontal plane. It is fundamental in surveying.

Methods of Measurement

1. Pacing: Using steps to estimate distances.
   • Pace Factor (PF): The average length of a person's pace.
   • Formula: $D = (\text{No. of Paces}) \times (PF)$
2. Taping (Chaining): Using a graduated tape (steel, invar, cloth) to measure directly.
3. Tacheometry: Optical distance measurement (stadia, subtense bar).
4. Electronic Distance Measurement (EDM): Using electromagnetic waves (Total Stations).

Pacing Calculations

To determine your Pace Factor:

1. Walk a known distance ($D$) multiple times.
2. Count the number of paces ($N$).
3. $PF = \frac{D}{N}$.

Taping Corrections

Tapes are standardized under specific conditions (Standard Pull, Standard Temperature, Supported throughout). Measurements taken under different conditions require corrections.

1. Temperature Correction ($C_t$)

Steel expands with heat and contracts with cold.

$C_t = \alpha L (T - T_s)$

Where:

• $\alpha$: Coefficient of thermal expansion (approx. $11.6 \times 10^{-6} / ^\circ C$ for steel).
• $L$: Measured length.
• $T$: Temperature during measurement.
• $T_s$: Standard temperature.

Sign:

• If $T > T_s$, tape is too long (add correction).
• If $T < T_s$, tape is too short (subtract correction).

2. Pull (Tension) Correction ($C_p$)

Elastic stretching of the tape due to tension.

$C_p = \frac{(P - P_s) L}{A E}$

Where:

• $P$: Applied pull.
• $P_s$: Standard pull.
• $L$: Measured length.
• $A$: Cross-sectional area of tape.
• $E$: Modulus of Elasticity ($2.1 \times 10^6 \text{ kg/cm}^2$ or $200 \text{ GPa}$).

Sign:

• If $P > P_s$, tape stretches (add correction).
• If $P < P_s$, tape is slack (subtract correction).

3. Sag Correction ($C_s$)

Effect of gravity on an unsupported tape (catenary curve). Always subtractive.

$C_s = \frac{w^2 L^3}{24 P^2} = \frac{W^2 L}{24 P^2}$

Where:

• $w$: Weight per unit length of tape.
• $W$: Total weight of tape between supports ($W = wL$).
• $L$: Length of unsupported span.
• $P$: Applied pull.

Sign: Always subtract.

4. Slope Correction ($C_{sl}$)

To reduce slope distance ($S$) to horizontal distance ($H$).

$H = \sqrt{S^2 - h^2} \approx S - \frac{h^2}{2S}$

Where:

• $S$: Slope distance.
• $h$: Difference in elevation.

Correction $C_{sl} = S - H \approx \frac{h^2}{2S}$ (Always subtract).

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