Flexure of a Flat Elastic Spring. 



219 



The functions which occur in these equations have now 

 to be expressed in terms of the deformation of the middle 

 surface of the band. If a rectangular plate of thickness 2I1 

 undergo extensions <n, o- 2 parallel to the edges, the corres- 

 ponding tensions are, per unit length, 



p = 4(\ + fi)ft 



K+ 2/J, 



(<7i+ 0-0-2) • 2/1 



A + 2/X ' 



(4) 



when A, n are the elastic constants of Lame, equivalent to. 

 m — n and n, respectively, in the notation of Thomson and 

 Tait. If, further, the plate experience curvatures i/pi and 

 1 /p a parallel to its edges, the corresponding flexual couples. 

 are 



G _ 4(\ + /zW i + er\ 2& 

 X+2fX \ P i pj' 3 



H = 4(\ + fl)fl / 1 + ff \ 2^_ s 

 \ + 2/x \p 2 pJ' 3 



Now let w denote the deviation at any point of the 

 middle surface from the cylindrical surface (of radius p} 

 drawn through the medial line, in the strained condition ; 

 and let <ro be the extension of this medial line. We have, 

 evidently, in the present case, 



a x = (To + iv I p 



pi = p + ZV 



i/(0 2 = -d*7vjdx 2 . 

 The equations (1) and (4) show that 



<r 2 = — atrx , 



and substituting from (4), (5), and (6), in (2), we get 

 ~~dx\dx* p + w) p V* + ~ 9 )~ 



(6) 



