Tape Corrections

Table of Contents

In the field of surveying, the precision of measurements is paramount. A measurement over a horizontal distance using tape is affected by a number of factors that dictate the accuracy of the measurement. In order to obtain the best result, a surveyor makes corrections to the measurement. In practice, these corrections are not made for each tape length as the measurement is taken. Surveyors simply make a note of raw measurements obtained from the tape and later apply corrections to the total distance.

These corrections become necessary because the process of taping itself has a variety of possible sources of error, be they systematic, random, environmental, instrumental, or human. The determination and accounting for these errors will greatly enhance the accuracy in which a surveyor makes his measurements.

To ensure accuracy in tape measurements, the following corrections are applied:

Correction for Standard Length
Correction for Alignment
Correction for Slope
Correction for Tension
Correction for Temperature
Correction for Sag
Reduction to Mean Sea Level (M.S.L.)

1. Correction for Standard Length

Before using a tape for field measurements, it is essential to verify its actual length by comparing it with a standard tape of known accuracy. This comparison identifies any discrepancies between the tape’s nominal length (its designated length, such as 30 m or 100 m) and its absolute length (the actual length under specific conditions).

It is rare for a tape’s absolute length to perfectly match its nominal length, and this difference necessitates a correction when calculating the true measured distance. The correction for standard length is applied using the formula:

Ca = (L * C) / l

Where:

Ca represents the correction for absolute length
L is the measured length of a line
C is the correction to be applied to the tape
l is the nominal or designated length of the tape

It’s important to note that the sign of Ca will correspond to the sign of C. This ensures that the correction is applied in the right direction – either adding to or subtracting from the measured length.

When applying this formula, surveyors must ensure consistency in units. Both L and l should be expressed in the same unit of measurement (typically meters). Similarly, Ca and C should share the same unit.

For instance:

If you measure a 100 m line with a 30m tape that’s 0.02 m too short (C = +0.02 m) (the amount needed to reach the nominal length), the correction would be: Ca = (100 * 30.02) / 30 = 100.0667 m
If you measure the same 100 m line with a tape that’s 0.02 m too long (C = -0.02 m)(the amount by which it exceeds the nominal length), the correction would be: Ca = (100 * 29.98) / 30 = 99.9333m

2. Correction for Alignment

In ideal conditions, survey lines are set out in continuous straight lines. However, obstacles may necessitate following a bent line composed of two or more straight segments at angles other than 180°. This situation requires a correction for alignment.

Consider a bent line ACB where:

AC = l1, CB = l2 <BAC = θ1, <ABC = θ2

The true straight-line distance AB is calculated as:

AB = l1 cos θ1 + l2 cos θ2

The correction for alignment is the difference between the sum of the measured segments and the true straight-line distance:

Correction = (l1 + l2) – (l1 cos θ1 + l2 cos θ2)
= l1(1 – cos θ1) + l2(1 – cos θ2)

Case of Non-Intervisible Stations

When stations A and B are not intervisible, the angle ACB (α) can be measured with a theodolite, and the distance AB computed using the cosine formula:

AB = √(AC² + BC² – 2AC·BC·cos α)

It’s important to note that the correction for alignment is always subtracted from the measured length of the line to obtain the true straight-line distance.

3. Correction for Slope

When measuring distances on sloped terrain, the measured slope distance is always greater than the true horizontal distance between two points. The slope correction accounts for this difference and is always subtracted from the measured length.

The basic formula for slope correction is:

Ch = h²/(2L)

Where:
Ch = slope correction
h = difference in elevation between the two points
L = measured slope distance

This formula is derived from the geometric relationship between the slope distance and horizontal distance. It assumes that the angle of slope is relatively small.

For steeper slopes or when more precision is required, alternative formulas can be used:

Using the angle of slope (θ):
Csl = L(1 – cos θ) = L * versine θ [This formula works for θ in both radians and degrees. ]
For angles in degrees:
Csl ≈ 1.5 * L * θ² / 10,000 [ This approximation is specifically for θ measured in degrees.]

4. Correction for Tension

The correction for tension (or pull) in linear measurements is applied when the pull applied to a tape during measurement differs from the standard pull at which the tape was standardized. This correction ensures that the measured length accounts for the elongation or contraction of the tape due to the applied pull.

Understanding Tension Correction:

When the applied pull is greater than the standard pull, the tape elongates. As a result, the measured distance becomes less than the actual distance. In this case, the correction is positive, meaning you add the correction to the measured distance.
When the applied pull is less than the standard pull, the tape contracts. This results in the measured distance being more than the actual distance, and the correction is negative, meaning you subtract the correction from the measured distance.

The formula for tension correction is:
Cp = [(P – P0) / (AE)] * L
Where: Cp = Correction for tension
P = Applied tension during measurement (in Newtons)
P0 = Standard tension (in Newtons) A = Cross-sectional area of the tape (in mm²)
E = Young’s Modulus of Elasticity of the tape material
L = Measured length (in meters)

Typical E values:
- For steel: 2.1 × 10⁵ N/mm²
- For invar: 1.5 × 10⁵ N/mm²

If applied tension (P) is greater than standard tension (P0), the correction is positive.

If applied tension is less than standard tension, the correction is negative.

5. Correction for Temperature

The correction for temperature is applied to account for the expansion or contraction of a measuring tape due to temperature variations during measurements. The length of the tape changes when the temperature during measurement differs from the temperature at which the tape was standardized. This correction ensures accurate distance measurements, considering thermal expansion or contraction.

Understanding Temperature Correction:

If the temperature during measurement is higher than the standard temperature, the tape expands, and the measured length is less than the actual length. In this case, the correction is positive.
If the temperature during measurement is lower than the standard temperature, the tape contracts, and the measured length is greater than the actual length. In this case, the correction is negative.

The formula for temperature correction is:

Ct = α(Tm – T0)L

Where:

Ct = Temperature correction

α = Coefficient of thermal expansion of the tape material

Tm = Mean temperature during measurement

T0 = Temperature at which the tape was standardized

L = Measured length (in meters)

Typical α values:

For steel: 0.0000035 per °C
For invar: 0.000000122 per °C

If Tm > T0, the correction is positive (tape has expanded).
If Tm < T0, the correction is negative (tape has contracted).

Invar tapes, with their very low coefficient of thermal expansion, are preferred for high-precision measurements where temperature variations are a concern.

6. Correction for Sag

The sag correction is applied when a tape is suspended between two supports and forms a catenary due to its weight. Since the tape sags under its own weight, the measured distance along the curve of the tape is greater than the horizontal distance between the supports. The sag correction is therefore always negative, as it compensates for this overestimation by subtracting the correction from the measured length.

The formula for sag correction is:

Cs = -L/24 * (W/P)²

Where:
Cs = Sag correction
L = Horizontal distance between supports
W = Total weight of the tape
P = Applied tension

Understanding Sag Correction:

Sag correction is negative because the apparent (curved) length of the tape is longer than the actual horizontal distance.
It is important to apply this correction, especially for longer spans or tapes with significant weight, to ensure that the measured distance is accurate.

This correction accounts for the difference between the tape’s suspended curved length and the true horizontal length, ensuring more accurate measurements.

In practice:

Minimize sag by applying proper tension and using shorter spans when possible.
For high-precision work, consider using invar tapes which are lighter and less prone to sag.

Normal Tension

Normal tension is the tension (or pull) applied to a tape supported in air over two ends, which balances both the correction due to pull and the correction due to sag. This means that when the tape is under normal tension, no correction for either pull or sag is required, as they cancel each other out.

When a tape is stretched due to tension, its length increases, and the correction for pull is positive. On the other hand, sag, due to the weight of the tape, decreases the tape’s effective length, resulting in a negative sag correction. Normal tension is achieved when these two opposing corrections are equal in magnitude, neutralizing the need for further adjustment.

Derivation of Normal Tension

We start by expressing the corrections for pull and sag separately:

Correction for pull

When the tension applied during measurement is different from the standard tension $_{0}$ , the correction for pull is given by:

Correction for sag :

The sag correction, due to the tape sagging under its own weight, is given by:

C2 = -L·W² / (24P²)

Equating and solving for P:

(P – P0)L / AE = L·W² / (24P²)
24P²(P – P0) = AE·W²
24P³ – 24P²P0 = AE·W²

This equation can be solved by trial and error, or approximated as:

P = 0.204 W √(AE)

7. Reduction to M.S.L

When measuring distances at elevations above sea level, a correction is necessary to reduce the measured length to its equivalent at mean sea level. This correction accounts for the Earth’s curvature.

The formula for the correction is:

C_msl ≈ L * h / R

Where:

C_msl = Correction for reduction to MSL

L = Measured length on the ground

h = Elevation above mean sea level

R = Radius of the Earth (approximately 6,371 km)

Understanding Reduction to M.S.L

This correction is always subtractive, as distances measured at higher elevations are longer than their MSL equivalents.
The correction increases with both the measured length and the elevation above sea level.
For most practical surveying work at moderate elevations, this correction is very small and often negligible.
It becomes more significant for very long distances or measurements at high elevations.

In practice:

This correction is typically applied in geodetic surveys or when working over large areas with significant elevation differences.
For local surveys at relatively low elevations, the correction may be omitted unless high precision is required.