216 Mr. Horace Lamb on the 



On the Flexure of a Flat Elastic Spring. By Horace 

 Lamb, M.A., F.R.S. 



{Received April 29th, i8po.) 



It is a well known consequence of the ordinary theory 

 of flexure that when a bar of (say) rectangular section is 

 slightly bent by opposing couples in a plane parallel to one 

 pair of faces, so as to form a circular arc of radius p, the 

 remaining faces take an anticlastic curvature or/p, where 

 <r is " Poisson's ratio " ; and this fact has been made the 

 basis of ingenious experimental methods of determining <r y 

 by Cornu* and Mallockf. It appears to have been first 

 remarked by Thomson and Tait (Natural Philosophy, §717) 

 that this tendency to anticlastic curvature imposes in 

 certain cases a limit to the degree of flexure, beyond which 

 the theory in question will not apply. " For unless the 

 breadth of the bar (or diameter perpendicular to the plane 

 of flexure) be very small in comparison with the mean pro- 

 portional between the radius [p/cr] and the thickness, the 

 distances [in any cross section] from [a line through the 

 axis perpendicular to the plane of flexure] to [one pair of 

 corners] would fall short of the half-thickness, and the dis- 

 tances to [the other pair] would exceed it by differences 

 comparable with its own amount. This would give rise to 

 sensibly less and greater shortenings and stretchings in 

 the filaments towards the corners than those expressed in 

 our formulae, and so vitiate the solution. Unhappily,, 

 mathematicians have not hitherto succeeded in solving,, 

 possibly not even tried to solve, the beautiful problem thus 

 presented by the flexure of a broad very thin band (such as 



* Comptes Rendus, Aug. 2, 1869. 

 t Proc. Roy. Soc, June 19, 1879. 



