Methods of measuring with a tape have been dealt with although it must be said that training in the methods is best undertaken in the field The quality of the end results ,however can only be appreciated by an understanding of the errors involved .
of all the methods of measuring taping is probably the least automated and therefore most susceptible to personal and natural errors , The majority of errors affecting taping are systematic not random and their effect will therefore increase with the number of bays measured the errors arise due to defects in the equipment used :
natural errors due to weather conditions and human errors resulting in tape - reading errors , etc . They will now be dealt with individually .
STANDARDIZATION
Taping cannot more accurate than the accuracy to which the tape is standardized it should therefore be routing practice to have one tape standardized by the appropriate authority .
This is done on payment of a small fee : the tape is returned with a certificate of standardization quoting the 'true ' length of the tapeand standard conditions of temperature and tension .
This tape is then kept purely as a standard with which to compare working tapes .
Alternatively a base line may be established on site and its length obtained by repeated measurements using , say , an invar tape hired purely for that prupose , The calibration base should be then checked at regular intervals to confirm its stability .
TEMPERATURE
When measuring with a steel tape , neglecting temperature effects could be than main source of error .For example , in winter condition in the UK , with temperatures at 0∘C , a 50 m tape , standardized at 20∘C would contract by
11.2 × 10 ⁻⁶ × 50 × 20 = 11.2 mm per 50 m
Thus even for ordinary precision measurement , the temperature effect cannot be ignored ,
Even if the tape temperature is measured there may be an index error in the thermometer used m part of the tape may be in shade and part in the sun , or the thermometer may record ground or air temperature which
may not be the same as the tape temperature , Although the use of an invar tape would resolve the problem , this is rarely , if ever , a solution applied on site , This is due to the high cost of such tapes and their fragility the effect of an error in temperature measurement can be assessed by differentiating equation , i,e
੪ Ct = KL ੪ ( ⧊ t)
if L = 50 m and the error in temperature is + 2 ∘C then ႙ C t = 1.1 mm , However if this error remained constant the total error in the measured line would be proportional to the number of tape lengths . Every effort should therefore be made to obtain an accurate value sor tape temperature using calibrated thermometers .
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 4 mm² , a standard tension of 50 N and value for the modululs of elasticity of E = 210 kN /mm² . Under a pull of 90 N the tape would strtch by
50 000 × 40
Ct = ________________ = 2.4 mm
4 × 210 × 10³
As 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 . Form equation (4,4).
1.7 × 9.81 ) ² × 50
Cs = ( _________________ = 0.090 m
24 × 80 ²
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 suspport .
The effect of an error in tensioning can be found by differentiating equation (4.4) with respect to T :
੪ Cs = - W² L ੪ T /12 T ³
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 :
੪ Cs = WL ੪ W/ 12T²
and shows that an error of + 0.1 kg in W produce an error + 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 5 m between the ends .
Upon taking the first term of the binomial expansion of equation (4.4) the correction for slope may be approximated as :
Ch = -h²/2L = - 25/100 = - 0.250 m
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
႘ Ch = - h ႘ h / L
Assuming an error of + 0.005 m there would be an error of - 0.0005 m (႘ Ch ) . Thus error in obtaining the difference in height is negligible and it is proportionan to h , would get smaller on less steep slopes by differentiating equation (4.4) with respect to ⊖ , we have
৪ C⊖ = L sin ⊖ ੪ ⊖
so ੪⊖ " = ৪ C ⦵ × 206 265/ L sin ⦵
IF L = 50 m is required to an accuracy of -+ 5 mm on a slope of 5 ُ then
৪⊖ " 0.005 × 206 265 /50 sin 5ُ = 237 " ≈ 04 َ
This level of accuracy could easily be achieved using an abney level to measutr slope . As the slopes get less steep the accuracy required is further reduced : however , for the much greater distance obtained using EDM , the measurement of vertical angles is much more critical . indeed .
if the accuracy required above is changed to . say ,-+ mm . the angular accuracy required changes to -+ 47 ًً and the angle measurement would require the use of a theodolite .
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