Correction for pull or tension in a tape is given by
Understanding Tape Correction for Pull or Tension in Surveying
When measurements are taken using a tape or chain in surveying, the tension or pull applied to the tape can affect its length. Tapes are standardized at a specific tension (standard pull, \(P_0\)) and temperature. If the actual pull (applied pull, \(P\)) applied during measurement is different from the standard pull, the tape will either stretch (if \(P > P_0\)) or contract (if \(P < P_0\)). This change in length needs to be accounted for to obtain the accurate distance. The adjustment made for this difference in pull is called the correction for pull or tension.
Derivation of the Pull Correction Formula
The change in length of a material under tension follows Hooke's Law, which states that stress is proportional to strain within the elastic limit. For a tape of length \(L\), cross-sectional area \(A\), and made of a material with Young's Modulus \(E\):
- Stress (\(\sigma\)) is the force per unit area: \(\sigma = \frac{\text{Force}}{A}\)
- Strain (\(\epsilon\)) is the change in length per original length: \(\epsilon = \frac{\Delta L}{L}\)
- Young's Modulus (\(E\)) is the ratio of stress to strain: \(E = \frac{\sigma}{\epsilon}\)
Substituting the expressions for stress and strain into the Young's Modulus formula:
\(E = \frac{\frac{\text{Force}}{A}}{\frac{\Delta L}{L}} = \frac{\text{Force} \times L}{A \times \Delta L}\)
Rearranging the formula to solve for the change in length (\(\Delta L\)) due to a specific force:
\(\Delta L = \frac{\text{Force} \times L}{A \times E}\)
The force causing the change in length in the case of pull correction is the difference between the applied pull (\(P\)) and the standard pull (\(P_0\)). This difference is \((P - P_0)\). If \(P > P_0\), the force \((P - P_0)\) is positive, causing elongation. If \(P < P_0\), the force \((P - P_0)\) is negative, causing contraction.
The correction for pull, denoted as \(C_p\), is this change in length \(\Delta L\). Therefore, the formula for the correction for pull is:
\(C_p = \frac{(P - P_0)L}{AE}\)
Where:
- \(C_p\) is the correction for pull or tension.
- \(P\) is the applied pull during measurement.
- \(P_0\) is the standard pull at which the tape was calibrated.
- \(L\) is the measured length of the tape or tape segment.
- \(A\) is the cross-sectional area of the tape.
- \(E\) is the Young's Modulus of the tape material.
Analyzing the Given Options
Let's compare the derived formula \(C_p = \frac{(P-P_0)L}{AE}\) with the provided options:
- \(C_p=\frac{(P-P_0)L}{AE}\): This matches our derived formula exactly.
- \(C_p=\frac{(P-P_0)}{LAE}\): This formula incorrectly places \(L\) in the denominator.
- \(C_p=\frac{(P-P_0)AE}{L}\): This formula incorrectly places \(A\) and \(E\) in the numerator and \(L\) in the denominator.
- \(C_p=\frac{L}{AE(P-P_0)}\): This formula has the difference in pull in the denominator and is also incorrect.
Based on the derivation from basic principles of material behavior under tension, the correct formula for the correction for pull or tension in a tape is \(C_p=\frac{(P-P_0)L}{AE}\).
| Symbol | Description | Units (typically) |
|---|---|---|
| \(C_p\) | Correction for Pull | Same as Length (e.g., meters) |
| \(P\) | Applied Pull | Force (e.g., Newtons, kgf) |
| \(P_0\) | Standard Pull | Force (e.g., Newtons, kgf) |
| \(L\) | Measured Length | Length (e.g., meters) |
| \(A\) | Cross-sectional Area | Area (e.g., mm², m²) |
| \(E\) | Young's Modulus | Pressure/Stress (e.g., N/m², Pa, GPa) |
Revision Table: Key Surveying Tape Corrections
| Correction Type | Formula (Common Form) | Purpose |
|---|---|---|
| Pull/Tension (\(C_p\)) | \(\frac{(P-P_0)L}{AE}\) | Accounts for length change due to difference between applied pull \(P\) and standard pull \(P_0\). |
| Temperature (\(C_t\)) | \(\alpha (T - T_0) L\) | Accounts for length change due to difference between field temperature \(T\) and standard temperature \(T_0\). \(\alpha\) is the coefficient of thermal expansion. |
| Sag (\(C_s\)) | \(\frac{w^2 L^3}{24 P^2}\) or \(\frac{W^2 L}{24 P^2}\) | Accounts for the tape hanging in a curve (sagging) when supported only at ends. \(w\) is weight per unit length, \(W\) is total weight, \(P\) is applied pull. Always negative correction. |
| Slope (\(C_{slope}\)) | \(\frac{h^2}{2L}\) or \(L(1 - \cos \theta)\) | Converts a slope measurement \(L\) to its equivalent horizontal distance. \(h\) is height difference, \(\theta\) is slope angle. Always negative correction. |
Additional Information on Surveying Measurements
Accurate distance measurement is fundamental in surveying. Steel tapes and chains are elastic and sensitive to changes in temperature and tension. Therefore, applying corrections is crucial for precise work, especially over longer distances or in varied environmental conditions. Besides the pull correction discussed, surveyors also commonly apply corrections for temperature, sag, and slope to reduce errors and determine the true horizontal distance between points.
Understanding the factors that influence tape length (material properties like Young's Modulus \(E\), physical characteristics like area \(A\) and length \(L\), and external forces like tension \(P\) and temperature \(T\)) is essential for conducting accurate surveys and applying the appropriate corrections.
Important Questions from Linear Measurement
The engineer's chain is typically:
Cross-staff is used for:
The scale of a drawing is given as 1 : 20.
What is the representative fraction?
An Engineer measured the distance between two locations on a plan having a scale of 1 cm = 50 m as 600 m. Later, however, he found that he used a wrong scale of 1 cm = 30 m to measure the distance. The true distance between the locations is:
The length of the chain is equal to _____.
