222 Flexure of a Flat Elastic Spring. 



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



3 V+a« p> [20) 



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

 of breadth 2b and depth 2I1, the coefficient (3A + 2/u)/u/(A + ju) 

 being Young's modulus. 



As the curvature increases beyond the limit above in- 

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

 in the other extreme, when p is small compared with ffi\h y 

 and nib is consequently large, the expression for the couple 



becomes 



I_6{\ + H) F IM 



3 A + 2(1 p 



the same as for a plate. We have also in this case 



A = v / 2 4- e- mb co%(mb + -) j . v 



/ \ f 



w-jj \ 4 / / 



approximately, so that the value of w given by (10) and 

 (13) is insensible except close to the edges. Near the 

 edge x = b we have, for example, 



w= — — z-e~ m< - b ~ x) cosi m(b — x) + [. (2-1) 



s/ 2 ra 2 p t v 4-j 



At the edge itself the deviation from the cylindrical form 

 is comparable with k, but it rapidly diminishes as we pass 

 inwards, at the same time fluctuating in sign. The latter 

 result may perhaps be unexpected, but a little consideration 

 will show that it is intimately bound up with the supposition 

 we have made, that there is no resultant force on the ends 

 of the band. The amplitudes of the fluctuations diminish, 

 however, so rapidly that it is not likely that this feature of 

 the strain could ever be made the subject of observation. 



This exceeds the value (20), given by the ordinary theory, in the ratio i/(l -<J 2 ). 



