Transactions A.S.M.E.

C alculation of the E lastic C u rv e of aH elical C om pression S pringB y H. C. KEYSOR,1 CHICAGO, ILL.E q u a tio n s for th e e la s tic curve o f a h e lic a l c o m p r essio nsp rin g are develop ed , w ith o u t m a k in g th e u s u a l a s s u m p ­tio n o f a x ia l lo a d in g . T h e lo a d is fo u n d t o b e e ccen tric ,in gen era l, b e in g a x ia l o n ly u n d er c e r ta in c o n d itio n s.E la stic curves are g iv en for b o th a x ia l a n d e c c e n tr ic lo a d ­in g . T h e se curves are sin u so id a l, sh o w in g t h a t d e flec tio na lo n g th e bar is n o t lin ea r , a s is c u sto m a r ily a ssu m e d .A reversed d e flec tio n is fo u n d to o ccu r in th e c lo sed en d ,back o f th e tip c o n ta c t p o in t, w h ic h in c r e a se s t h e d e ­flec tio n o f th e sp rin g a s a w h o le, a n d sh o w s t h a t it is n o tcorrect t o co n sid er t h e e n tire c lo sed e n d a s in a c tiv e . A naxial load is th e o r e tic a lly p o ssib le u n d er c er ta in c o n d i­tio n s, b u t p r a c tic a lly a x ia l lo a d in g is d ifficu lt to o b ta in a n dc a n n o t be c o u n te d o n w ith a n y c e r ta in ty . T h e effect o fload e c c e n tr ic ity o n str e ss is c o n sid ered , a n d a n a p p ro x i­m a te fo rm u la for e s tim a tin g t h is effect is g iv en . R eferen ce is m a d e to la b o rato ry te s t s by a n o th e r in v e stig a to r 2w h ich are in good a g r ee m en t w ith th e th e o r y here p resen te d .A N ANALYSIS of helical-spring deflection has been made/-% by the author which yields three results:1 The load is, in general, eccentric with consequentincrease in stress.2 The correct deduction for inactive end turns is shown tobe approximately 1.2, based on the usual practice of taking thetotal turns as solid height divided by bar diameter.3 The deflection at any point along the bar is obtained, thuspermitting the elastic curve to be drawn.The first two items have been discussed in a previous paper3by the author. The third item forms the subject of the presentpaper.Fig. 1 shows the lower portion of a helical compression springwith ends closed and squared. Assume that the spring is compressedbetween parallel plane surfaces, as this is the usualcondition of loading. Let P be the load. Any point on thebar, as point a', must therefore sustain the direct load P, atorque T, and a vertical bending moment M , the line of actionof P being at some point Q. It is apparent that Q must be on aradial line through a, the longitudinal center point of the bar,for otherwise the forces and reactions on the top and bottomhalves of the spring would not be symmetrical and the necessarycondition of static equilibrium would not obtain. We do notrestrict the position of point Q on the radial line through a; thatis, we consider the load to be eccentric by some variable amounte, leaving the magnitude of e to be determined by the conditionsof the problem.The load P can be replaced by two component loads, P i ata and P 2 diametrically opposite a. If P i + P 2 = P and thecenter of pressure of Pi and P j is at point Q the torque, moment,and direct load at the general point a' will not be affected. Thereactions which balance P are P3 at the tip contact point b,and (P — P 3), distributed in some manner in the vicinity ofpoint 0 where the bar makes contact with the loading plane.There is also a torque reaction at point 0. The angular distancewbetween b and 0 is roughly 180 deg for most springs; the determinationof v will be explained later.The location of the contact point b is subject to variation dueto the manner of forming the tapered end, and due also to theload, for it is evident that, with increasing load, b will move awayfrom the tip so that the angular distance between contact pointswill diminish. Consideration must be given to this fact ininterpreting the following analytical results in their relation toactual springs.From the foregoing discussion, it is evident that we require thedeflection equations for a helix fixed at the origin and sustaining1 Mechanical Engineer, American Steel Foundries.’ “Helix Warping in Helical Compression Springs,” by D. H.Pletta and F. J. Maher. Published in this issue, page 327.‘ “Analysis of Deflection and Stress in Helical CompressionSpring,” by H. C. Keysor, presented at the Annual Meeting of T h eA m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s , December, 1934;not published.Contributed by the Special Research Committee on MechanicalSprings and presented at the Annual Meeting, Philadelphia, Pa.,December 4 - 8 , 1939, of T h e A m e r i c a n S o c i e t y o f M e c h a n i c a lE n g i n e e r s .N o t e : Statements and opinions advanced in papers are to b eunderstood as individual expressions of their authors, and not thoseof the Society.