## Tension,Sag and Slope

**Tension:**

If the tension in the tape is greater or less than standard the tape will stretch or become shorter. Tension applied without the aid of a spring balance or tension handle may vary from length to length, resulting in random error. Tensioning equipment containing error would produce a systematic error proportional to the number of tape lengths. The effect of this error is greater on a light tape having a small cross-sectional

area than on a heavy tape.

Consider a 50 m tape with a cross-sectional area of 4mm^{2}, a standard tension of 50N and a value for the modulus of elasticity of E = 210 kN/mm^{2}. Under a pull of 90N the tape would stretch by

this value would be multiplied by the number of tape lengths measured it is very necessary to cater for tension in precision measurement, using calibrated tensioning equipment.

**Sag:**

The correction for sag is equal to the difference in length between the arc and its subtended chord and is always negative. As the sag correction is a function of the weight of the tape, it will be greater for heavy tapes than light ones. Correct tension is also very important.

Consider a 50 m heavy tape of W = 1.7 kg with a standard tension of 80 N. From equation and indicates the large corrections necessary.

If the above tape was supported throughout its length to form three equal spans, the correction per span reduces to 0.003 m. This important result shows that the sag correction could be virtually eliminated by the choice of appropriate support.

The effect of an error in tensioning can be found by differentiating equation with respect to T:

In the above case, if the error in tensioning was 5 N, then the error in the correction for sag would be −0.01 m. This result indicates the importance of calibrating the tensioning equipment. The effect of error in the weight (W) of the tape can be found by differentiating equation (4.4) with respect to W:

and shows that an error of 0.1 kg in W produces an error of 0.011 m in the sag correction.

**Slope:**

Correction for slope is always important.

Consider a 50 m tape measuring on a slope with a difference in height of 5m between the ends. Upon taking the first term of the binomial expansion of equation the correction for slope may be approximated as:

and would constitute a major source of error if ignored. The second-order error resulting from not using the second term h4/8L3 is less than 1 mm. Error in the measurement of the difference in height (h) can be assessed using

Assuming an error of 0.005 m there would be an error of – 0.0005 m (δCh). Thus error in obtaining the difference in height is negligible and as it is proportional to h, would get smaller on less steep slopes.

By differentiating equation with respect to θ, we have

= 50m is required to an accuracy of ±5mm on a slope of 5◦ then

δθ = 0.005 × 206 265/50 sin 5◦ = 237 ≈ 04

This level of accuracy could easily be achieved using an Abney level to measure slope. As the slopes get less steep the accuracy required is further reduced; however, for the much greater distances obtained using EDM, the measurement of vertical angles is much more critical. Indeed, if the accuracy required above is changed to, say, ±1 mm, the angular accuracy required changes to ±47 and the angle measurement would require the use of a theodolite.