Corrections in linear measurement

When a tape or chain is used in the field, the measured length differs from the true horizontal length because of the tape's condition and the field conditions. The following corrections are applied. Let $L$ be the measured length.

1. Correction for incorrect length of tape (standardisation)

$$C_a = \frac{L\,(l' - l)}{l}$$

where $l'$ is the actual tape length and $l$ the nominal length. It is positive if the tape is too long, negative if too short.

2. Correction for temperature

$$C_t = \alpha\,(T_m - T_0)\,L$$

where $\alpha$ is the coefficient of linear expansion, $T_m$ the mean field temperature and $T_0$ the standard temperature. Positive if $T_m > T_0$.

3. Correction for pull (tension)

$$C_p = \frac{(P - P_0)\,L}{A\,E}$$

$P$ = applied pull, $P_0$ = standard pull, $A$ = cross-sectional area, $E$ = Young's modulus. Positive if $P > P_0$.

4. Correction for sag

$$C_s = -\frac{L\,w^{2}\,l_s^{2}}{24\,P^{2}} = -\frac{n\,W^{2}\,l_s}{24\,P^{2}}$$

where $w$ = weight per metre, $l_s$ = length of each span between supports, $W = w\,l_s$. It is always negative when the tape is standardised on the flat and used suspended.

5. Correction for slope

$$C_h = -\frac{h^{2}}{2L} \quad\text{(approx.)} \qquad\text{or}\qquad C_h = -L(1-\cos\theta)$$

where $h$ is the height difference. Always negative.

6. Correction for alignment (tape not in line): always negative.

7. Reduction to mean sea level

$$C_{msl} = -\frac{L\,h}{R}$$

where $h$ is the elevation above MSL and $R$ is the radius of the earth (6370 km). Negative.

True length $= L + C_a + C_t + C_p + C_s + C_h + C_{msl}$.