RP-S6-4

N um ber of A ctive Coils in Helical SpringsBy R. F. VOGT,1 MILWAUKEE, WIS.

D u e t o t h e t o r s i o n a l d i s p l a c e m e n t b e t w e e n c r o s s - s e c t i o n s o f t h e h e l i c a l s p r i n g b a r o r w i r e i n t h e s o - c a l l e d d e a d o r i n a c t i v e c o i l s o n e a c h e n d o f t h e s p r i n g , t h e t o t a l d e f l e c t i o n o f t h e s p r i n g i s g r e a t e r t h a n t h a t w h i c h c o r r e s p o n d s t o t h e d e f l e c t i o n o f t h e f r e e c o i l s . T h e e x a c t a m o u n t o f t h e s p r i n g d e f l e c t i o n c o r r e s p o n d i n g t o t h e i n f l u e n c e o f t h e “ i n a c t i v e ” c o i l s c a n b e c a l c u l a t e d f o r k n o w n l o a d i n g c o n d i t i o n s o f t h e s p r i n g a n d c a n b e c l o s e l y e s t i m a t e d f o r a l l s p r i n g s s u b j e c t e d t o c o n v e n t i o n a l s p r i n g - p r a c t i s e l o a d s . T h e d e f l e c t i o n s o f t h e e n d c o i l s a r e c a l c u l a t e d i n t e r m s o f t h e d e f l e c t i o n o f a c t i v e c o i l s . A t e s t t o d e t e r m i n e t h e a c t u a l n u m b e r o f a c t i v e c o i l s i s s u g g e s t e d a n d e x a m p l e s a r e g i v e n .

THE predetermination of the correct number of active coils in helical springs is, in many applications, very important. C e n tr ifu g a l sp rin g - loaded regulators, controlling the speed of prime movers, spring-loaded indicators, and many other apparatus require helical springs of which the correct number of active coils is essential.

If, in the use of the conventional helical- spring deflection equation

/ = deflection r = mean radius of coil d = diameter of wire

P = loadG = modulus of elasticity in torsion n = number of active coils

an error is made in determining the correct value of n, the factor G is usually adjusted to offset the original error. The factors /, r, d, and P are always fixed in value and can easily be measured and checked.

In such cases it is erroneously assumed that G is a variable amount for different helical springs, the variation ranging from below 10,000,000 to 12,000,000.

The fact, however, is that the modulus of elasticity in torsion G is proportional to the modulus of elasticity in tension E and is characteristic for each material and constant within the elastic range:

1 Assistant Chief Consulting Engineer, Allis-Chalmers Mfg. Co. Mem. A.S.M.E. Robert F. Vogt was born in Geneva, Switzerland, and had his prim ary and secondary schooling a t Romanshorn, C anton School a t St. Gall, and Swiss Polytechnicum a t Zurich, Switzerland. His professional career began in the United States in 1903. He has been connected with the Allis-Chalmers Mfg. Co. as m echanical engineer since 1907.

Contributed by the Special Research Committee on Mechanical Springs and presented a t the Mechanical Springs Session of the Annual Meeting, New York, N. Y., Dec. 5 to 9, 1932, 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 l E n g i n e e r s .

N o t e : Statem ents and opinions advanced in papers are to b e understood as individual expressions of their authors, and no t those of the Society.

where ju is Poisson’s ratio. According to Htttte, 26th edition, for spring steel

E — 30,000,000 lb per sq in.m = 0.275m — 1/n — 3.63G = 0.392E = 11,700,000 lb per sq in.

Any discrepancy between these given values and experimental values is due to an error in counting the number of active coils in the helical spring under consideration.

I t is the purpose of this paper to show how the correct number of active coils can be determined both by analysis and experiment.

The number of active coils as used in the deflection formula for helical springs does not always equal the total number of coils or the number of free coils. Particularly, in most commercial helical compression springs we find that the number of active coils must be more than the number of free coils, if we assume G = 11,700,000 ~ 12,000,000 as correct and applicable to helical springs.

Tests of regulator tension springs, of which each end coil was held a t two diametrically opposite points, checked closely with G = 2/5 • E when the number of active coils was counted from the middle of the two supporting points on one end of the spring to the corresponding point on the other end, i.e., when the number of active coils was taken as the number of free coils plus V2 coil.

R o d U n d e r T w i s t

Let us consider, as shown in Fig. 1, a rod of the length L twisted by applying a force P perpendicular to a rigid arm of length r, which is perpendicular to the rod.The deflection of the point of application of the force P in the direction of P is expressed by the formula:

in which co is the angle of twist in radians.

H e l i c a l S p r i n g W i t h R i g i d A r m s a t C o i l E n d s

If the rod referred to for Equation [1 ] is coiled into the shape of a helical spring on which the rigid arms extend from the ends of the coil to the center line of the coil and are perpendicular to the coil center line, the force P acting on the arms at the center line of the coil in the direction thereof produces a deflection / of

a = pitch angle n — number of coils

467

468 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS

In this case all coils are active, i.e., subjected to the twist of the full torque moment P ■ r and no coils or part of coils are inactive. The number of coils n is also the number of active coils.

C o m m e r c ia l S p r i n g s

Commercial springs are not equipped with rigid arms at the ends. Many tension springs have loop ends, as shown in Fig. 2, which add a negligible amount of bending deflection to the tor

sional deflection given in Equation [2]. In such springs all coils between the base of the loops are active coils. T h e d e ta ile d discussion of this type

;fig_ 2 of sp rin g w ill beomitted in this paper.

In helical springs where the full length of the bar is within the cylindrical part of the spring, as is the case of commercial helical compression springs and also of many kinds of expansion springs, the active coils extend beyond the free coils. The free coils include the coils of the spring which are not connected with brackets or yokes through which the spring load is applied and do not make contact with the end coils. The active coils include all coils contributing to the deflection of the spring. The angle of twist of the bar changes from maximum in the free coil to zero in the end coils. The angle of twist within the end coils adds a certain amount to the total deflection of the spring, which can be determined in many cases with complete accuracy and in all other cases with sufficient accuracy to satisfy fully the practical applications.

H e l i c a l S p r i n g s W i t h D i f f e r e n t T y p e s o f L o a d i n g

In order to illustrate the deflection of the end coils in helical springs, various ways of spring loading will be analyzed:

1 The Two-Point L o ad ing .To an open-wound helical spring the load is applied by means of yokes reaching on each end diametrically from one side of the coil to the other (see Pig. 3).The load P is applied at the middle of the yoke by means of a pivot, so that each end of the yoke transmits the same pull P/2 on the spring.

As is shown in d eve lop ing Equation [2], the effect of the pitch angle a is such that it may be correctly assumed that the end coil is in a plane perpendicular to the center line of the coil and that the forces are perpendicular to the plane of the coil.

Referring to Fig. 3, the load P /2 at A has no part on the deformation of the end coil between the points A and B. The load P /2 at B produces a moment of P /2 • 2 • r = P ■ r a t point Awhich is balanced by the same moment P • r acting on the other side of the spring. All cross-sections of the spring bar between A and B ' are under the influence of this moment P • r.

If the end coil between A and B would be absolutely rigid, the cross-section a t A wrould remain in the same position relative to the end coil as it had before loading took place. But the end

coil is as flexible as the free coils located between A and B ' and the bar between A and B will twist in accordance with the respective moment acting on the bar at its cross-sections. This moment is no longer constant and equals P ■ r for all cross-sections from A to B, but decreases gradually from the maximum of P /2 • 2 • r = P ■ r a t A to P /2 -0 = 0 at B. Between A and B the

P • racting moment is M = P /2 • (r — r cos

RESEARCH RP-56-4 469

The general expression for the deflection of helical springs which are not provided with rigid arms or loops at the ends and to which the load is applied in the common manner then takes a convenient form readily available to designers:

dI1 .8 3 9

II1.122D c= 7 .5 3 4 .1 4 0

n'19V s

6 .0 7 5195/sll1/*n - n ' + 1/2 = 6 .5 7 5 12

P 8V t4580 H»/44600f = 3 0 .7 6 0 1 .6 6 0Q

8 n-D* P f-d* 11,800,000 11,850,000

The springs were tested with a testing machine of 5000-lb capacity. The dimensions were carefully taken with micrometers and averaged from many measurements taken of diameters perpendicular to each other along the full length of the springs. The free coils were carefully determined and fixed by inserting spacers at the contact points with respective end coils.

For compression springs the forces are applied in the opposite direction; the deflection is also in the reversed direction, but the calculations and results are otherwise the same. The design and application of a commerical helical compression spring are such that the load condition ranges between the two cases given. The first condition is the more common.

The foregoing calculations are based on full-bar cross-section in the end coil. This assumption applies to most commercial compression springs, for that part of the coil which must be considered in the calculation. The basic design of the spring is shown in Fig. 5.

The ends of such a spring are closed, 3/< of the end coil is tapered from full cross-section to 1/ i thickness a t the end, the pitch changes at contact point B from p to d. Full cross-section of bar is maintained from B to A, suggesting a load division of P/2 at A and P /2 a t B. The variation is usually not far from this load division and comes within the range of the three- point loading of 3 X P/3, in which extreme case the difference would only correspond to 1/ 3 — 1/ i = 1/ 12 coil for each end correction.

We must bear in mind that the resultant of these forces is in the center line of the coil. If it were to fall outside the center line, the spring would bend out sidewise, which, in most compression- spring applications, does not occur to any appreciable extent. Within the range of free coils, i.e., from B to B ' (see Fig. 5), the bar is subjected to a shear force equal to P /2 and a torque equal to P ■ r. The effect of shear upon the deflection is small and can be neglected. In a well-applied compression spring the torque P ■ r is uniformly the same all along the bar, P acting in line of the center line of the coil.

Due to the fact that the length of contact between the end coils increases slightly during increase in deflection, thereby effecting a slight decrease in active coils, the assumption of the2 X P/2 load division is more justified, as the error in allowing for slightly less active coils than would correspond to an actual, possibly different load distribution, is compensated by the tendency for a slight decrease in active coils during compression. It is therefore logical and practically correct to choose the 2 X P/2 load division.

E x p e r i m e n t a l V e r i f i c a t i o n o f A n a l y s e s

A large number of tests substantiate the mathematical analysis and the general application of the results. To illustrate, the following test records are presented. Errors in the dimensions of bar diameter, coil diameter, and deflection, on account of their large amount, are relatively small. The examples, therefore, are of especially high value as proofs of the analyses.

The following springs were made by the Railway Steel Spring Company for the Allis-Chalmers Manufacturing Company:

Bar diam eter (average), in ___Coil diam eter (average), in ___Free len g th .....................................Free co ils .........................................A ctive co ils .................... . ..............

Total num ber of coils from tip to tip of bar (tap er)..

Load, lb ............................................D eflection ........................................

M od u lu s...........................................

F i g . 5

The theoretical number of active coils n and the modulus of elasticity in torsion G may easily be determined by testing for deflection two compression springs made of the same material of equal bar and coil diameter, but with a different number of free coils, as follows:

Spring 1 Spring 2C oil d ia m eter ...................................... D = DB ar d ia m eter ....................................... d = dN um ber of free c o ils ....................... n ' n"L o a d ........................................................ P ~ PD eflec tio n ............................................. / i = hN um ber of a ctiv e co ils ................... m = n ' + x m = n" + xM odulus of e la stic ity in torsion . G — G

Since both springs are alike except for number of free coils, the modulus of elasticity in torsion is the same for both springs as well as the effect of the end coils under the same load.

We find

and

If there should be any variation between the two springs in di, Di, or P, then /i o r /2 must be corrected to correspond to values for d, D, and P adopted for the foregoing calculation. I t will be found that x approximates the value of */» very closely and that G approximates 11,700,000 for any size and kind of steel spring bar.

Appendix 1T N order to find the total angle of twist of cross-section at A

under the influence of P • r, the half-circle between A and B is divided into differential lengths A (s) = r ■ A

470 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

acting on any cross-section of the arc A B is M = P /2 ■ a = P /2 ■ r ■ (1 — cos < a

T N calculating deflections of a portion of a circular ring out of its plane by forces perpendicular to the plane of the ring, the

known solution of Saint-Venant can be used.2 If an incomplete circular ring is fixed at A and loaded by force P at B (Fig. 6),

F i g . 7

then according to Saint-Venant’s solution the deflection at any point C, defined by an angle , is given by the following equation:

2 The Saint-V enant solution can be found in Love’s “ M athem aticalTheory of E lasticity ,” pp. 456-457, or in "S trength of M aterials,” vol. 2, p. 469, by S. Timoshenko. This m anner of solution was suggested by R. L. Peek.

in which C is the torsional rigidity and E l the flexural rigidity. This equation is satisfactory for any value of 0 between 0 and a.

T w o - P o i n t L o a d i n g

If there are two forces P /2 acting at A and B, the deflection at B is readily obtained from the direct application of Equation

d * TT[a], substituting in it P /2 for P, a = 0 = it, I = -----, and C =

j 64(l • 7r

G ■ ——. Then the deflection of point B is:oZ

Point 0 deflects

in which form the equation shows that the deflection of an end coil is equal to the deflection of one-quarter of a free coil.

C a s e W h e n > aIf 0 is larger than a, the necessary

deflection can be obtained by using Saint-Venant’s equation in conjunction with the reciprocity theorem.3 From this theorem it follows that the load P applied at C (Fig. 7) produces at B the same deflection as the deflection at C produced by the load at B.Since Equation [a] gives the deflection at any point C in Fig. 6, we can get at once the deflection at any point B for the loading shown in Fig. 7.

T h e e e - P o i n t L o a d i n g

Take now, as an example, the case of three loads P /3 put at points A, B, and C, 120 deg apart (Fig. 8). Point A is considered as fixed. The deflection of the point B consists of the two parts: (1) Deflection produced by the load P /3 at B and (2) deflection produced at B by the load P /3 at C. The first part is obtained by substituting into Equation [a] P /3 for P and 2x/3 for the angles a and 0.

PR3 5This gives: FsduetoS = 7.50 —- (assuming E = ~G) .

(j t CL Z

The second part is obtained from the same equation by puttingPR3

4?r/3 for a and 2ir/3 for 0. This gives: Fsdue to C = 6.70 —— •Crd

Hence the total deflection of the point B is :

P R 3V b = U . 2 0 -

In calculating the deflection of the point C, we again have two parts: (1) Deflection at C produced by the load at C is obtained

• M ethod proposed by Prof. S. Timoshenko.

RESEARCH RP-56-4 471

by substituting P /3 for P and a = = 4ji-/3 into Equation [a],P R 3

which gives Fcdue to C — 33.10 —— and (2) deflection at C pro-Gd

duced by the load at B. This is equal to deflection at B when the load is a t C and is obtained from Equation [a] by substituting in it P/3 for P and taking a = 4x/3 and = 2ir/3, which gives:

P R 3VCdue to B = y Bdue to C = 6.70 Gd4

Then the total deflection at C is:

COS

472 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

B = spring loadSh = deflection

I = length of spring barr = radius of coil a = pitch angle

In the deflection Equations [2], [3], [5], [a], [6], [7], and [8], it is assumed that the spring bar or rod is thin in comparison with the radius of the curvature, i.e., D /d « c o . The error caused by this assumption, when the equations are applied to commercial helical springs where D /d = 3 or more is very small and for all practical purposes negligible.

Equation [8] is given to show the influence of the pitch angle. Other influences which affect the accuracy, and are not considered in this equation are caused by preventing the free ends of the spring from turning freely about the axis of the spring during compression or expansion and the pure shear deflection. The complications affected by properly considering all these facts are too great, and the change in end results too minute to warrant the application of these highly refined methods of calculation in practical engineering work.

As far as the deflection effect of the end coil is concerned, it is, for all practical purposes, sufficient to consider its torsional deflection only in the manner shown in Appendix 1. This is especially justified when we realize that the mathematically more complicated method employed in Saint-Venant’s solution is also not absolutely accurate because the effects of such items as pure shear deflection, coil pitch angle, spring index D/d, weight, and end conditions of the helical springs are neglected. Another factor, which demonstrates the fallacy of striving for accuracy to the extreme in calculating the deflection of the end coils, is the unavoidable variation in cross-section shape and size, coil diameter, and pitch angle in commercial springs. Equa-

64 • n ■ r> ■ P , , 64 • («' + §) • r3 • Ptions / = — ----------- and / = -----------——---------, respectively,

O r * a * (j • a 4can be regarded as being accurate for all practical purposes.

DiscussionT. M c L e a n J a s p e r .6 The paper by Mr. Vogt on helical

springs is exceedingly interesting. I am wondering if the values of E, 0, and l/m are as constant for spring steel in general as is assumed in the paper. My reasons for asking this go back to some tests made in 1924 which were published in the Transactions of the American Society for Testing Materials of that year and some work presented in the Philosophical Magazine for October, 1923, which indicate that the state of the steel as well as the temperature at which the tests were made influences the values of the so-called elastic constants somewhat.

The only way that this should be determined for spring application is to make several tests on identically shaped springs made of different steels.

I am not familiar with the values of O to be assumed for steel when formed into helical springs and when using different steels,

and therefore the values presented in this paper may be an appropriate average for steel springs to be used at ordinary temperatures only.

W. M. A u s t i n . 6 The writer has had to apply helical compression springs, both large and small, to quite a variety of machinery and has often observed the influence of the end turns. Particularly, he has observed that the average spring designed to be made like the author’s Fig. 6, except having a length relative to diameter several times longer than Fig. 6, will usually buckle badly when fully loaded.

Small springs often have their ends malformed. The spring maker winds enough wire on his mandrel to make two or more springs, and then cuts them apart. He then presses the end of the spring against the flat side of a rapidly turning dry grinding wheel. The heat generated makes the end turn red hot at some point about 3/s to Vs turn from the end of the wire. The wire bends at this red-hot place and the end of the wire moves back against the next turn. He then dips the spring in water in an attempt to restore the temper to the heated part, and finishes the grinding.

The end turn, instead of tapering uniformly in thickness for 3/ 4 of a turn to V i the diameter of the wire at the end, tapers for V i turn to a thickness about l / z diameter of the wire, then increases in thickness for another l/ t turn to 3/ i diameter of the wire, then tapers another x/ 4 turn to V4 diameter of wire at the end. This last taper may lie against the next turn for most of its length.

The writer has often had to show the machine assembler (not a spring maker) how to cut off part of the end turn and regrind so that the spring will not buckle in service. Even if the spring were made according to the drawing as usually made, the center of gravity of the load would not be in the extended axis of the spring. If the spring is not more than three times as long as its diameter, the buckling is usually not very noticeable.

The writer prefers to make the end turn so that the end of the wire does not touch the next turn until the spring is compressed solid, and instead of making the ground end exactly perpendicular to the axis of the spring, to make it a helicord of small pitch relative to the pitch of the spring. If this is done, the end of the wire will take its proper share of the load without bending beyond the plane of the part, 3A of a turn away, where the tapering of the wire began.

Most springs are never completely unloaded in service, many of them never more than 1/ 2 unloaded. In cases like this the minimum load brings the end of the spring into a plane perpendicular to the axis. It is probable that the center of gravity of the load is not in the axis of the spring, at the time of minimum load, but as the load increases the center of gravity of the load approaches nearer and nearer to the axis, and when maximum load is attained the ideal condition exists with the center of gravity of the load is in the axis of the spring.

It is then seen that the flat-ended compression spring and its modifications is at best only a compromise, more or less successful, so to load the spring that at no time during the compressing or releasing of the spring will any part of it be stressed beyond its safe load.

In tension springs provided with hooks bent up out of the end turn and having tHe same diameter as the main body of the spring, the hooks have to stand the same bending moment as the torsional moment in the body of the spring. This means that the tension stress on the inside of the hook is about twice the shearing stress on the inside of the body of the spring because, for

1 D i r e c t o r o f R e s e a r c h , A . O . S m i t h C o r p o r a t i o n , M i l w a u k e e , W i s . 6 E n g i n e e r , W e s t i n g h o u s e E l e c . & M f g . C o . , E a s t P i t t s b u r g h ,M e m . A . S . M . E . P a . M e m . A . S . M . E .

If we transform this equation in terms used in Equation [2], we find

RESEARCH RP-56-4 473

Thus, with the additional fact that the wire is often damaged by making the hooks, accounts for the observed fact that tension springs, if they break, always break where the hook connects to the body of the spring. There is a way to reduce the excessive stress in the hooks. I t is to make the end turn a spiral and bend up the hook from the inner end of the spiral, making the mean diameter of the hook about one-half the mean diameter of the main body of the spring.

The author’s tests on the two large springs would be much more valuable if the springs had been loaded to near their maximum safe loads instead of limiting the stress as calculated by the old formula to 34,400 for the small spring and 14,100 for the large one. In any event, I believe they should have both been loaded so as to produce the same stress.

I t is quite generally known that Hooke’s law gives only the first term of a rapidly converging series, so that Young’s modulus E and the shearing modulus G both have higher values when determined by stress-strain measurements using low stresses than

they do when using stresses near but below the elastic limit.16 • r3 ■ P

The author’s deflection r • w = ----------- due to the torsion ind4 ■ G

the end turn is the deflection due to torsion of the point B, Fig. 3, and not the deflection of the pivot hole in the bar connecting points A and B. This would make the deflection of the

8 • r3 ■ Ppivot hold only — —— • In the author’s analysis no account

d r • (jt

is taken of the deflection at B due to the bending of the end turn by the load P/2 a t B. This would produce a deflection about as large as that due to torsion.

In order to get experimental data on the deflection of the end turn, the writer had a piece of Vi-in. pretempered spring steel wire bent into 3/ t of a turn of 2n /ie mean diameter, as shown in Fig. 11.

I t was loaded at B with a 70-lb weight and the deflection at B was 0.29 in. If we let G = 11,400,000, the deflection per turn of the main body of the spring is 0.488, when loaded to 140 lb. The deflection at B is then seen to be 0.595 of the deflection of one turn, and the deflection at the center of the bar connecting A and B would be 29.7 per cent of the deflection of one turn, and for both ends the deflection due to the end turns is 59V2 per cent of one turn.

J. P. M a h a n e y . 7 At the beginning of his paper the author shows that the value of G should be nearer 12,000,000 than 10,000,000. This is true provided Poisson’s ratio is taken as 0.30 to 0.335 rather than 0.365. In some instances attempts have

7 Assistant Professor, Industrial Engineering, Virginia Polytechnic Institute, Blacksburg, Va. Jun. A.S.M.E.

been made to prove G equal to the lower value by substituting test data in the conventional spring formulas, but, since it is generally admitted that these formulas are approximate for “closely coiled” springs, such computations are not adequate proof.

The author states that the bar in the free coils is subjected to a shear force of P/2. This is incorrect. The total load on the spring is P; consequently, the single bar must transmit this total load from one end of the coil to the other and the shear in the bar will be P instead of P/2.

Since present conventional spring formulas can be proved inaccurate, it does not follow tha t illogical corrections are acceptable. Adding one-half a coil to the number of free coils admits that a portion of the seated end coils deflect, which is beyond comprehension. I t is true tha t torsional deformation extends beyond the free into a portion of the seated coils, for if this were not true, the first free coil a t each end would not contribute its full share of deflection. To infer that there is axial deflection derived from the seated coils is a process of creating one error to compensate for another.

As the paper shows for balanced loading the load P on a compression spring may be resolved into two components of P /2 each acting 180 deg apart. If the spring in Fig. 3 is loaded in compression, P /2 at B will produce torsional stress at A, and P /2 at A increases this stress to the final value within the spring. The stress in the seated coil must build up to the proper value at A in order that the active end coils may be completely effective in contributing deflection. The stress within the seated coil A-B produces deflection indirectly but its contribution to the total should not be counted twice. The author’s mathematical deduction clearly shows that the torque available in A-B is sufficient to produce deflection equivalent to one-quarter of an active coil provided it were free to move, which is of course impossible.

R. L. P e e k , J r .8 H o w accurately the solutions given for two- and three-point loading apply to helical springs compressed between parallel plane surfaces requires further analysis. Following the treatment given in Love’s “Mathematical Theory of Elasticity,” pp. 456-457, I have evaluated the force required

to keep the extreme end of the inactive turn in contact with the point A (Fig. 3), a condition that must be satisfied under compression of this sort. I find this force trivial in comparison with the reaction at A under two-point loading, and this consideration therefore does not affect the validity of applying the result for two-point loading to compression between parallel plane surfaces. On the other hand, in such compression the change in pitch angle of the active coils w'ill cause their axis to be no longer normal to the parallel plane surfaces applying load and their deformation will not be that corresponding to a purely axial thrust. Whether this effect will appreciably change the result, I have not ascertained.

A. M. W a h l .9 The exact solution of the additional deflection produced by the end turns of a helical compression spring is undoubtedly a very complicated problem, since it depends on the exact shape of the end turns and on the distribution of load thereon. The author has simplified the problem by assuming the end turns to have the full bar cross-section throughout their length. In addition he assumes various distributions of load on the end turns, finally choosing that which seems to agree best with test results.

Since, in most practical cases, the deflection due to the end

8 Bell Telephone Laboratories, New York, N. Y.9 Westinghouse Research Laboratories, E ast Pittsburgh, Pa.

Assoc-Mem. A.S.M.E.

round wire, the section modulus for torsion is ird3/16 and for bending is «23/32, so if S t be the maximum tension stress in the hooks and S s be the maximum shearing stress in the coils, then

474 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

turns is but a relatively small part of the total deflection of the spring, a considerable error in estimating the effect of the end turns could be made without introducing much relative error in the total deflection of the spring. For this reason a rough approximation, such as the author has introduced, might be of value in practical work, provided it has been confirmed by a number of accurate tests.

The question of the effect of the end coils is closely bound up with that of the modulus of rigidity of the material. For years some spring manufacturers have used modulus values of 10.5 X 10® or 10 X 106 lb per sq in., as the author points out. It is well known that these values do not agree with modulus values obtained by means of torsion tests on ordinary spring steels. It has been the writer’s opinion that these modulus values have been used largely to compensate for inaccuracy in estimating the effect of the end turns and possibly for errors in the spring dimensions.

To illustrate this point, some tests made on different springs at the Westinghouse Research Laboratories will be mentioned. The method used was to measure deflections between prick- punch marks on diametrically opposite points of the coil in the body of a helical spring and is described in a previous publication.10 The coil diameter and wire diameter were carefully measured at several points on each coil and the results averaged. By measuring deflections in the body of the spring, the effect of the end turns was eliminated. The values of “effective” modulus 0 could then be found from the known formula

Three springs from one manufacturer, having indexes of about ten, when tested in this manner, yielded the following values for the modulus:

Spring N o ........................................... 1 2 3G X 10 -« lb per sq in .................. 1 1 .4 5 1 1 .4 6 1 1 .5 0

Three springs having indexes of about 6.5 from another manufacturer gave the following values:

Spring N o ........................................... A B CQ X 10"M b per sq in .................. 1 1 .1 9 1 1 .1 2 1 1 .3 0

These values are all definitely higher than the value of 10 or10.5 X 106 as assumed by some spring manufacturers.

It should be noted that this method of determining the modulus assumes that the effect of the spring curvature is small, i.e., that the spring acts like a straight bar subjected to a torsion moment Pr. This of course becomes more nearly true for springs of large index. As far as spring deflections are concerned, this assumption is born out by previous tests by the writer,10 wherein it was found that the ordinary deflection formula for helical round- wire springs was correct within 3 per cent for springs having indexes varying from 2.7 to 9.5. In other words, a fourfold increase in curvature of a spring having a given wire diameter did not seem to have an appreciable effect on the modulus. The same thing is known to be true of curved bars in bending; i.e., in general, a curved bar in bending may be computed within a few per cent accuracy as far as deflections are concerned by using the fundamental methods applied to straight bars, although this is not true when stress calculations are made. The effect of curvature on deflection was also found to be small in the case of helical springs of circular wire by O. Gohner,11 who used more exact methods of calculation involving the theory of elasticity. The effect of curvature may be checked up experimentally by

*• A. M. Wahl, “Further Research on Helical Springs of Round and Square Wire,” Trana. A.S.M.E., 1930, paper APM-52-18, p. 217.

11 O. Gohner, “Die Berechnung zylindrischer Schraubenfedern,” Z.V.D.I., March 12, 1932.

it is found that n = 12, whence the added coils become 12 — llV i = Vs- But suppose G = 11.6 X 10® instead of 11.85 X 10* (a variation not at all unreasonable). Then we would find n = 11.65, from which the added coils would be found to be 11.65 — 11.5 = 0.15, a value which differs greatly from ‘/a as found by the author. This example shows the necessity for an accurate knowledge of the “effective” value of G. This could be determined, as mentioned previously, by measurements between prick-punch marks in the body of the spring, after which the average dimensions of the spring would be accurately measured. In this connection, the writer has found it to be extremely difficult to obtain accurately the average wire diameter of a spring, without cutting it up after the test, since, due to coiling, the wire section becomes slightly oval.

The method of determining the number of active coils, as proposed by the author, consisting of using two springs similar in every respect except in number of turns, would no doubt give an approximation which would be useful in practical work. For purposes of checking the theory, however, it would be necessary to find the average dimensions of each spring accurately. This would involve more labor than would the testing of one spring, as suggested above. Furthermore, there is a possibility that the modulus would vary some between the two springs, and this again would involve an additional error. For these reasons it is the writer’s opinion that tests on one spring would be preferable in order to confirm the theory.

The value of Poisson’s ratio 1/m = 0.363 reported in the paper seems rather high for steel. Using G = 11.7 X 10a, E = 30 X 106, this would give 1/m = E/2G — 1 = 0.283. Taking 1/m — 0.3 (a value commonly used for steel) and E = 30 X 10®, this would give G = 11.53 X 106, which is not far from the values obtained in the writer’s tests mentioned above.

A u t h o r ' s C l o s u r e

Answering Mr. McLean Jasper’s discussion in regard to the constancy of the modulus of elasticity E and Poisson’s ratio m for spring steel at various temperatures, we may, according to Hiitte, for all practical purposes assume E and m and therefore G constant at temperatures between 0 F and 400 F.

Examples of springs applied at high temperatures are springs in steam indicators and on valves for internal-combustion engines and steam engines. As far as the author knows, the steam-indi- cator springs which are used for high-temperature steams and gases as well as for cold air have been accepted as accurate for practical purposes without using any correction factors for the various temperatures to which they are exposed.

The author, however, mainly considered springs used in at-

using the following method suggested by R. E. Peterson, of the Westinghouse Company. A heat-treated round bar of spring material is first tested in torsion, thus determining the technical value of the modulus G. This bar would then be wound into a spring, and heat treated, after which deflections would be measured in the body of the spring between prick-punch marks, so that the “effective” value of G could be found by use of the ordinary spring-deflection formula. The two values of G thus found should be nearly the same if the effect of curvature is small.

The writer would like to suggest that in determining the number of coils to add to the free coils to find the active coils, it is necessary to know the “effective” value of G accurately; in other words, a small error in G would produce a big error in the number of added turns. For example, in the case of the author’s spring II, if G is assumed 11.85 X 10®, then from

RESEARCH RP-56-4 475

mospheric temperatures where accuracy in deflection values are essential.

Mr. Austin calls attention to irregularities in the shape of commercial compression springs especially in small sizes. How-

16 • r3 ■ Pever, he errs in his conclusion that the deflection r ■ o> = ------- -—dl ■ G

is the deflection of point B (see Fig. 3). This deflection is derived from the product r ■ a which (as is clearly explained in the paper and in Appendix 1) is nothing else than the deflection of the original end-coil center 0, which, as well as tha t of the pivot hole between points A and B, is a t a distance r from the center of the bar cross-section subjected to torsion.

Mr. Austin’s claim that the bending effect of load P /2 on the deflection of point 0 would be as large as th a t of torsion only is unfounded as may be seen from the Saint-Venant solution (see Appendixes 2 and 3) which includes the bending effect of load P /2 at B and shows that the deflection of point O is even somewhat less than that given by the author for torsion only.

In Mr. Austin’s experiment shown in Fig. 11 the deflection of point B is claimed to have been 0.29 in. for a load of 70 lb at B; but according to Saint-Venant’s solution this deflection should have been 0.231 in. for G = 11.4 X 106 or 0.225 in. for G = 11.7 X 10«.

Mr. Austin would have found more accurate and reliable results had he arranged his experiment according to Fig. 12. This arrangement consists of a helically and closely coiled spring- steel wire of one and a fraction of a turn. The coil diameter is about 20 or more times the diameter of the wire which latter should be about V< in. The wire and coil diameter and the deflection should be large enough to make unavoidable errors negligible in reference to the deflection. In Fig. 12 B A B ’ is

exactly one full coil and BA = A B ' and each is one-half coil. In order to eliminate errors due to initial tension or deflection, the deflection ei — e2 for the load Qi — Q2 is determined. The

1 --- 62deflection of a point B in reference to point A is / = ---------

2for the load Qi — Q2 at B. In order that the stress in the wire is within the elastic limit of the spring steel Qi must be less than

15,000 lb where d and D are given in inches. The

sum of the differential deflections in the two half coils BA and A B ' is alike and opposite in direction. For this reason the wire cross-section at A does not turn and therefore does not cause a change in the true deflection of B in reference to point A.

In Mr. Austin’s test, however, the cross-section at A, Fig. 11,

will turn and thereby increase the deflection of B, an amount corresponding to the torsional twist in the wire within the copper clamp near point This clamped portion of the wire cannot be held securely enough by the comparatively soft copper clamp to prevent twisting of the wire and consequently the turning of the wire cross-section at A. This torsional displacement of cross-section A of course increases the actual deflection due to the twist in half coil BA which stamps a test made according to Mr. Austin’s arrangement, shown in Fig. 11, as unreliable.

The author has made a number of experiments according to Fig. 12 in which he found the deflections to check very closely with the Saint-Venant results.

J. P. Mahaney mentions that the pure shearing force in the free coils due to the load P must be equal to P which is quite correct. However, according to the explanation given by the author in answer to A. M. Wahl’s discussion, the shearing force P is divided into halves. One-half balances an excess of the sum of the torsional shearing-force components parallel to the axis of the spring and acting in a direction opposite to P, while the other half adds a pure shear deflection to the torsional deflection

8 ■ n ■ D 3 ■ Pas given by the conventional deflection equation / = ------——-----

a4 ■ GMr. Mahaney, after admitting that torsional deformation ex

tends beyond the free coils into a portion of the seated coils, elaborates considerably on his conception that since the end coils in a compression spring are not free to move they cannot contribute to the deflection of the spring. The deformation of the end coil, Mr. Mahaney claims, makes it possible for the first free coil to contribute its full share of deflection. If this statement were true, the conventional spring-deflection Equation [2] as developed in the paper under the heading “Helical Spring With Rigid Arms at Coil Ends” would be faulty, as the first free coils in this case do not have the benefit of torsional deformation in end coils and therefore wrould not contribute their full share of deflection. Obviously, such a contention is against sound reasoning as the development of the deflection Equation [2] includes the contribution of the full share of deflection of all coils.

The fact that the end coils are held so that they can move only axially and parallel to their plane does not prevent the bar of the end coils from twisting due to the torque applied. Thus the axial deflection of the spring is increased proportionally to this twist and corresponds to one-fourth of an additional free coil per spring end beyond the deflection of a spring with rigid arms at free coil ends.

The author fully agrees with R. L. Peek, Jr., that in cases of compressing helical springs between parallel plane surfaces, the deformation of the spring as a whole and in particular of the end coil, will be different from the deformation as calculated in accordance with assumptions made in the analyses in the paper. This difference will vary with the different shapes of the spring ends as furnished in commercial helical compression springs.

However, ŵ hen we consider the error range due to (a) using the conventional spring-deflection equation instead of the Saint-Venant equation given in Appendix 3, (6) unavoidable variations in spring-bar and coil diameters of commercial springs which appear in the equation in the fourth and third power, respectively, (c) change in pitch angle and coil diameter during compression, (d) uncertainty as to spring end loading conditions, (e) neglecting the influence of the spring index D/d, and (/) uncertainty as to the actual value of the modulus of elasticity E or G, respectively, the variation of the actual deflection of the end coil from the one calculated, and given as being equal to the deflection of Vi coil due to maximum torque, is so small in comparison to other discrepancies that its disregard is fully justified. This is very apparent when we realize that a 5 per cent error in determining the end-coil deflection results in an error of less than

476 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

0.5 per cent in reference to the total deflection of a spring with five active coils.

An objection-free determination of the actual deflection of the end coil of a commercial helical compression spring with an accuracy within such a small error range would be very difficult.

Referring to A. M. Wahl’s discussion, the values of E, G, and m were taken from the latest edition of Hiitte, 1931, first volume, p. 689, where the following data for spring steels is given:

E = 2 ,100,000 kg per sq cm or E — 30 ,000,000 lb per sq in.G = 822,000 kg per sq cm G = 11,700,000 lb per sq in .G / E = 0 .392 v m = 3 .63 , from G = E / 2(1 + 1/m)

These data have always corresponded with spring tests made under the consideration of the proper number of active coils (regardless of small or large number of active coils), as given in the author’s paper, and were therefore accepted by the author as being dependable. Hiitte is considered one of the outstanding sources of reliable engineering information.

Mr. Wahl questions the accuracy of determining the value of G by testing two springs as suggested by the author, and in his example assumes G = 11.5 X 10® instead of 11.85 X 106, in which case Mr. Wahl calculates the effect of the end coils to be that of 0.15 free coils instead of 0.5 as demonstrated in this paper. Mr. Wahl’s analysis is, on this point, incorrect and deceiving.

In the author’s example, G is determined from actual values of n ' = 11.5, d ± 1.122, D = 4 .14,/ = 1.66, and P — 4600 and (in conformity with the theory developed in the paper) n = n ' + ' / 2.

If, in the example, the value of G had been different, say11.5 X 10°, then the deflection/ would have been 1.715 in. instead of 1.66 in. as it actually showed in the test, and n = 12 and not 11.65. The number of effective coils is fixed by the spring design and does not depend on the value of G.

The value of G cannot vary much for commercial spring steel. The skeptical engineer, however, can determine its value and concurrently the actual effect of the end coils, with satisfactory accuracy, by the method of testing two springs of equal dimensions but with greatly differing numbers of coils, as suggested by the author in the last part of his paper.

Mr. Wahl, in referring to the influence of the spring index on spring deflection, mentions that the conventional spring-de- flection equation for helical round-wire springs is correct within3 per cent for springs having indexes varying from 2.7 to 9.5.

The author determines the effect of the spring index on the spring deflection definitely by adding the direct shear deflection to the torsional deflection of the helical spring. The deflection

t P ■ Lof direct or pure shear for the spring is /" = yL = - L = ———,

G Fi ■ Gd2 * 7T

where L = 2Rirn, Fa = ------- (for circular cross-sections from4-1.2

Hiitte), P = spring load, or /" .= ———----------- . About halfG • o2

of this deflection is already included in the conventional spring equation, as in helical springs under load P about one-half the shear load P is balanced by the total sum of torsional shearing-

stress components parallel to the spring axis, as explained by Dr.-Ing. A. Rover in Z.V.D.I., Nov. 20, 1913, p. 1907.

By using the conventional spring-deflection equation, it is assumed that a curved bar has the same torsional deflection as a straight bar of the same length. The stress distribution in the cross-sections of the straight and curved bars is, however, slightly different and causes a small difference in deflection, amounting to one-half the deflection due to pure shear. The deflection of the curved bar is less than that of the straight bar, when the pure shear deflection is considered for both.

The total deflection of the helical spring under load P is:

or

or

7 = shear angle in radianst = shearing stressR = mean radius of coil

D = mean diameter of coild = diameter of spring wire or barn = number of active coilsP = spring loadG = torsional modulus of elasticity/ = total spring deflectionL = effective length of wire or bar

/" = deflection due to pure shear.

Applying the extended deflection equation in conjunction with accurate spring tests will result in finding more uniform values for G.

Plotting the shear deflection, in per cent of torsional spring deflection, against the spring index D/d we find the curve given in Pig. 13.

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