Flexure of a Flat Elastic Spring. 187 



and others, warrants us in asserting that if it be replaced by 

 any other distribution having the same force- and couple- 

 resultants, the form assumed by the band will not be sensibly 

 altered, except within a distance from the ends comparable 

 with the breadth. It appears from (7) and (8) that 



j: 



Ydx=0, (16) 



■b 



so that the forces on either end reduce to a couple 



J>4^{f-Ml)}- • • («) 



x=b 



On substitution from (10) and (12) this becomes, after a little 

 simplification, 



/** , _ 8 (X + /ji)fju 3 2b f a 2 cosh 2mb — cos 2mb\ ,_ 

 )-b $ \ + 2fju ' p L mb sinh 2m6 + sin 2mb f 



The form assumed by the cross section of the band, and 

 the value of the flexural couple, depend on the magnitude of 

 mb, which is comparable with bj s/ {np) . For sufficiently 

 small curvatures, i. e. so long as p is large compared with 

 6 2 /A, mb is small, and we find without difficulty 



•-if ( 19 > 



whilst the formula (18) for the flexural couple reduces to 

 4{3\ + 2n)fih*b 



X + fj, 



(20) 



which is in fact the value given by the ordinary theory for a 

 bar of breadth 2b and depth 2h, the coefficient (3X + 2p)\i\ (X + //,) 

 being Youno's modulus. 



As the curvature increases beyond the limit above indicated, 

 the flexural couple increases in a greater ratio, until in the 

 other extreme, when p is small compared with b^/h, and mb is 

 consequently large, the expression for the couple becomes 



16 (X + M)Ai W 



3 \ + 2p, p' [ l) 



the same as for a plate. 



* This exceeds the value (20), given by the ordinary theory, in the 

 ratio 1/(1— cr 2 ). 



