132 CITY SURVEYING 



in which c is the coefficient, of expansion of the tape, which 

 for steel averages about .0000065, and I is the measured length 

 of the line. The true length is therefore 



If t is less than to, the correction is negative and should be 

 subtracted from I. 



EXAMPLE. A line was measured with a tape that was 

 standard at 62. The temperature was 90. The length, as 

 measured, was 502.34 ft. If the coefficient of expansion of 

 the tape was .0000065, what was the true length of the line? 



SOLUTION. Here, c(t to) = .0000065 X (90-62) = .000182. 

 The correction c(t-to)l is, practically, .000182X502, the 

 decimal .34 being dropped, as the product of it by .000182 is 

 too small to be considered. Therefore, lo = 502. 34 + .000 182 

 X 502 = 502.43 ft. 



Correction for Pull. If the length of the tape is denoted 

 by L, the cross-section by A, and the modulus of elasticity 

 by E, the true length Lo of the tape stretched by a pull P is 

 given by the formula 



If the length of a line as measured with the stretched tape 

 is /, and the true length of the line is la, then 



For such steel as tapes are made of, E may be assumed 

 without great error as 28,000,000 Ib. per sq. in. A not unusual 

 cross-section is about .002 sq. in. A tape 100 ft. long with 

 such a cross-section would be lengthened about .036 ft. for a 

 pull of 20 Ib. above the normal. Hence, a line measured with 

 such a tape under such a pull, and found to be 400 ft. long, 

 would really be 400 + 4 X. 036 = 400. 144 ft. long. 



Correction for Sag. If a tape is held off the ground so that 

 it is supported only at each end, it will sag and hang in a curve. 

 The effect of sag is to shorten the distance between end gradu- 

 ations, the amount depending on the weight and length of 

 the unsupported part of the tape, and on the pull exerted at 

 the ends of the tape. 



