﻿Wire or Tape including the Effect of Stiffness. 119 



Thus, so far as the surveyor engaged in traverse work 

 is concerned the sag correction on any slope is given by 



and the higher the tension the more nearly is this correct. 

 In any case of doubt the next term within the curved brackets 



( 1(; 2£2 \ 2 

 — =2 ) 5 which should be increased to 

 / w-l- v . 24T ' 



— 41= 2 ) if the slope is anywhere near 45°. 



Effect of the Elastic Extension. 



The effect of the elastic extension of the tape is investigated 



in the papers above quoted by including it in the equations 



of equilibrium. As it is an effect of a smaller order than the 



catenary curve, it is justifiable to assume the tape to have 



taken the catenary form unstretched and then to add the 



effect of the stretch. The amount of stretch in any element 



Tds 

 of length ds is ^-r-, where T is the tension at that point, E is 



Young's modulus, and A is the area of the cross section. As 

 the curve is very nearly the catenary, we have T = T cosh ?;, 



x 



where u= -, w is measured from the vertex of the catenary 



rp G 



and c= — . We also have ds = c cosh u da. 



ID 



, Tds „ i o 7 



. . ^-r = wc l cosir u an ; 

 JdjA 



and integrating, we have 



CTds 9 T 19 , ivc 2 /sinh 2u , u\ 



J EA = WC J C ° sh " du = EA \—T~ + 2 j 



If the chord is horizontal we have for the stretch in the 

 whole tape 



wc 2 /sinh 2u , \ , . . . , / / P 



g-^l g +wj, and w= sinh" 1 



(37) 



l o .a "+ 



2c 2c 



Expanding this we have 



wc 2 1 2i^ ^ u :} \ 



stretc i 1= _^ + „ + - i +- 7 -+...) 



ivcl /, /- \ 



= Ea( 1 + 21? + "-) ; 



