﻿Wire or Tape including the Effect of Stiffness. 99 



Since the curvature is supposed to be small, we may put 



d 2 u M 



~| = ^j , where E is Young's Modulus and I is the moment 



of inertia of the cross section, which gives 



d?y %_Mo w^ m 



^ 2 *"E1 -EI~2EI [L) 



The well-known solution of this differential equation is 



a • u /% , j> x.- /% M o ™J wEI ,~ 

 #=Asinhy/|jj&' + Bc<)shy/ jvj-^— ^r + i^r + -rjrr, (2) 



where A and B are constants to be determined by the end 

 conditions. Since v = and -'- = when .r = 0, we have 

 B= T u F^ and A = 0, so that 



y 



/M ioEI\/ /T -\ wx 2 . 



The value of M will differ according as the ends are " free" 

 or constrained so that the tangent to the tape is horizontal 

 thereat. In the former case there will be no bending 



72 7 



moment at the ends or -^|=0 when a'= ^ ; and in the 

 latter the slope will be zero at the ends or -f- = when 



1 die 



Taking first the case of free ends, we have 



d?y _ T_o /Mo wEI\ /To Z w _ n 



da 8 "" EI I T " T 2 / V EI 2 + T 



and M =^fri * T 



} L cosb \/Ei , 2-' 

 Inserting this value in equation (3) We have 



mEI^ a/e^- 1 ) . «* 



x o 1 / lo ' " ± o 



