Flexure of a Flat Elastic Spring. 221 



The distribution of applied force and couple over the 

 ends (supposed straight) necessary to produce the strain in 

 question is given by 



y = _^_ rr i - ff-yi.n 

 X + 2/JL v ' 



xT^^ ; V 0+ p> (I4) 



G -IT4r*V _, sO- <I5) 



This distribution is somewhat artificial, but the theory 

 of " local perturbations " developed by St. Venant, Bous- 

 sinesq, 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 



/p<& = 0, (l6) 



-b 

 so that the forces on either end reduce to a couple 



(*Gdx = *<^A a l-Jpl l (*7) 



J_ b 3 \ + 2fi (p \dxj ) 



x=b 



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

 little simplification, 



Gdx=-\ — £!£&. — 1 . ,—=, — . ■} (18) 



•J_ b 3 \+2fj, p mb %vah.2tnb + simmb ) 



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\/(ho). For sufficiently 



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



b' 2 //i, nib is small, and we find without difficulty 



<tx 2 , , 



w= (19) 



2 



P 



* Since i=x/2(x + /j.). 



