﻿98 



Mr. A. E. Young on the Form of a Suspended 



in sag, which results the writer thinks may be of interest 

 and perhaps of utility to those engaged in making surveys 

 with the long steel tape. 



That a steel tape is not perfectly flexible is easily seen by 

 the surveyor. If it were, a small piece held out in the 

 lingers would at once droop like a piece o£ thread, and if 

 a long tape were suspended in several sags or bays over 

 supports, it would drop away from the supports leaving 

 a sharp point or cusp thereat, but on the contrary, even 

 with the thinnest tape there is a distinct crest of curvature 

 concave downwards at each support, with points of contrary 

 flexure at some little distance on each side. The influence 

 of the rigidity is to decrease the sag in either case, but it 

 will easily be seen that the effect must be greater when the 

 ends are bent over supports than it is when the tape is 

 simply suspended at the ends with no bending moments at 

 or points of contrary flexure near them. 



We will first take the case originally solved by the writer 

 of a long thin tape stretched by a large tension so that the 

 sag is quite small, and with its ends supported at the same 

 level so that the chord is horizontal. The tape is then 

 symmetrical about its centre, and it will be convenient to use 

 Cartesian coordinates, the axis of x being horizontal and 

 tangent to the tape at its lowest point and the axis of y 

 vertical through the same point. 



(Fig. 1.) Let I be the length of the whole tape, w its 



fic. 1. 



weight per unit of length, T the horizontal component of 

 the applied tension or, which is the same thing, the actual 

 tension at the lowest point, and M the unknown bending 

 moment at the lowest point. Taking M as positive when it 

 tends to produce curvature convex to the axis of x, the 

 bending moment at any point distant x from the origin is 



M = M t-T> 



WX" 



