736 BELL SYSTEM TECHNICAL JOURNAL 



equal the distribution Syp-Syp resulting from the original forming 

 operation. Therefore the stress distribution in the relaxed condition 

 (radius R\) must be the difference between Syp-Syp and S1-S2. or as 

 indicated in Fig. 7(c). The value of ^2 is of course so determined that 

 the moment as specified by equation (21) is the same for the dotted- 

 line as for the solid-line stress distribution. 



The value of Sr = S2 — Syp is relatively easy to determine for 

 rectangular and round wire on the basis of the straight-line stress-strain 

 characteristic if the bending has been sufficiently severe to have caused 

 plastic flow almost to the neutral axis. The moment given by the 

 actual stress distribution will then differ but little from that obtained 

 by equation (21) with 5 replaced by Syp, a constant. The values of 5 

 corresponding to the S2.-S2. distribution are given by 



5 = 2S2ylh. (22) 



For a rectangular wire 6 is a constant and equating the moments corre- 

 sponding to the two stress distributions gives 



I SYpbydy = I l^hyHy, 



»ft/2 /ift/2 C 



SYpbydy =1 



-ft/2 J-hj2 



from which S^ = (3/2) Syp or 



Sr = S2 — Syp — (l/2)5rp (rectangular wire). (23) 



Similar integrations in the case of a round wire of radius h/l for 

 which b = 2V[(/i/2)2 - /] gives 



Sr = 0.7 Syp (round wire). (24) 



These equations give the residual fiber stresses in the extreme inner 

 and outer fibers under the assumed conditions. In any case where the 

 stress-strain characteristic is known a correct value for the residual 

 stress can be obtained by graphical integration. The values given by 

 equations (23) and (24) will, however, be fair approximations even in 

 cases where the stress-strain characteristic is curved provided it does 

 not exhibit strain hardening to a decided extent. Since a limited 

 amount of plastic flow takes place for any stress above the proportional 

 Hmit, which is generally far below the yield point, the residual stress 

 may be sufficient to cause a small amount of creep. Any additional 

 stresses will then cause permanent deformation of the spring. 



The analysis will be continued on the basis of the idealized straight- 

 line characteristic. Figure 7(d) shows the result of placing the spring 



