RP-S6-4
N u m b e r o f A c t i v e C o ils il s i n H e l i c a l S p r i n g s By R. F. VOGT,1 MILW AUKE E, WIS. D ue
to
the
t io n s o f th e or
in a c t i v e
t o rs io n a l
h e l ic a l s p r in g c o ils
f le c ti o n
o f
the
sponds
to
the
am ou nt
of
i n fl u e n c e
displacem ent
on
b a r o r w i re i n t h e en d
is
s p r in g
the
o f the
greater
d e f le c t io n
the
of
each
s p r in g
betw een
o f
the
f re e
d e f le c t i o n
“ i n a c t iv e ”
c o i ls
s o - c a ll e d d e a d
spring,
th a n
th a t
the
tota l
w hich
c o i ls .
be
de
E m m G
corre
T he
corresponding can
where ju is Poisson’s Poisson’s ratio . for spring steel
c r o s s -s e c
ex act to
c a l c u l a te d
the for
According to Httt te, 26 th edition,
— 30,000,000 lb per sq in. = 0.275 — 1 / n —3.63 = 0.392 E = 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 e s t im a t e d f o r a l l s p r in 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 in the helical spring under consideration. p r a c t i s e l o a d s . T h e d e fle c tio n s o f t h e e n d c o ils a r e c a lc u It is the purpose of this paper to show how the correct number l a t e d i n t e r m s o f t h e d e f l e c t io n o f a c t i v e c o i ls . A test to of active coils can be determined both by analysis and experi 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 ls i s s u g g e s t e d ment. a n d e x a m p l e s a r e g iv e n . The number of active coils as used in the deflection formula HE predetermination of the correct for helical springs does not always equal the total number of numb er of active coils coils in helical coils coils or the numbe r of free coil coils. s. Particula rly, in most commer commer springs is, in many applications, cial cial helical compression compression springs springs we find tha t the n umbe r of active very very impor importa tant nt.. C en trifu ga l sprin gcoils must be more than the number of free coils, if we assume loaded regulators, controlling the speed 11,700,000 ~ 12,000,00 12,000,000 0 as correc t and applicable to helical G = 11,700,000 of prime movers, spring-loaded indica springs. tors, and many other apparatus require Tests of regulator tension springs, of which each end coil was helical springs of which the correct num held at two diametrically opposite points, checked closely with ber of a ctiv e coils is essen tial. G = 2/5 • E when the number of active coils was counted from If, in th e use of the conven tional helicalhelical- the middle of the two supporting points on one end of the spring spring deflection equation to the corresponding point on the other end, i.e., when the num ber of activ ac tivee coils was take ta ke n as th e numb nu mb er of free coils plus V2coil. k n o w n lo a d in g c o n d i ti o n s o f th e
s p r in g
and
can
b e c l o s e ly
T
R
/ = deflec deflecti tion on coil r = mean radiu s of coil wire d = diameter of wire P = load G = modulus of elasticity in torsion coilss n = numbe r of active coil an error is made in determining the correct value of of n, the factor is usually adjusted to offset offset the original original error. The factors G is /, r, d, and P are always fixed in value and can easily be mea sured an d 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. 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:
1Assistant Chief Consulting Engineer, Allis-Chalmers Mfg. Co. Mem. A.S.M .E. Ro bert F. Vogt was born in in Geneva, Switzerland, and had his primary and secondary schooling at Romanshorn, Can ton School at St. Gall, and Swiss Polytechnicum at Zurich, Switzer land. His professi professional onal career began in the United State s in 1903 1903.. He has been connected connected with the Allis-Chalmers Mfg. Co. as mech ani cal engineer since 1907. 1907. Contributed by the Special Research Committee on Mechanical Springs and presented at 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
So
c i et y
o t e
o f
M
e c h a n ic a l
E
n g i n e e r s
U
n d e r
T
w is t
Let us consider, as shown in Fig. 1, a rod of the length L twisted by applying a force P per pen dicular dic ular to a rigid arm of le ngth ng th r, which is perpendicular to th e rod. The deflection of the point of ap plicatio plic atio n of th e force P in the di rection of P is expressed by the formula:
in which co co is the angle of twist in rad ians. H e l i c a l S p r in g W i t h R ig id A r m s
a t
C oil Ends
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 of the coil coil in th e direction thereof produces a deflectio deflection n / of
.
Statements and opinions advanced in papers are to b e understood as individual expressions of their authors, and not those of the Society. N
o d
:
a = pitch angle coils n — numbe r of coils
467
468
TRANSACTIONS OF T H E AMERICAN SOCIETY OF MECHANICAL ENGIN EERS
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 in active. The numbe r of coils n is also the number of active coils. C
o m m e r c i a l
Spr
i n g s
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 re spective moment acting on the bar at its cross-sections. This moment is no longer co nstant 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 at A to P/2 -0 = 0 at B. Between A and B the P • r acting moment is M = P / 2 • (r — r cos
Commercial springs are not equipped with rigid arms at the ends. Ma ny tension springs have loop ends, as shown in Fig. 2, which add a negligible amount of bending deflection to the tor 2 sional deflection given for the cross-section designated by angle ip. The total angle of in Equ ation [2]. I n twist of cross-section at A , due to the torsional resilience in such springs all coils the end coil from A to B and the moment P • r acting at A , is between th e base of 16 • P ■ r2 a = — - ■■- —, and the center 0 of the coil, which we imagine the loops are active * coils. T he d eta ile d rigidly connected with cross-section at A , moves in reference to its discussion of this type original position O', when the spring is not loaded, an amount of ;fig_ 2 of sprin g will be 16 • P • r* omitted in this paper. f = r a = ————— (see Appendix 1). O' may be regarded d • G In helical springs where the full length of the bar is within the as the center of the end coil in its new position. O may be taken cylindrical part of the spring, as is the case of commercial helical as the original location of the center. The distance between 0 compression springs and also of many kinds of expansion springs, and O' then equals/', which is also the deflection of the end coil the activ e coils exte nd beyond th e free coils. Th e free coils 16 - p . rs include the coils of the spring which are not connected with / ' = — - ——— . This corresponds to the deflection of lU coil d 4 • G brack ets or yokes thr ou gh which th e spring load is app lied and do not make contact with the end coils. The active coils include which is subjected to the moment P • r. Th e same, of course, all coils contributing to the deflection of the spring. The angle of occurs at the end A 'B ', so that the total deflection of the helical spring amounts to twist of the bar changes from maximum in the free coil to zero in the end coils. The angle of tw ist within the end coils adds a certain amount to the total deflection of the spring, which can be determ ined in ma ny cases wi th complete accura cy and in all other cases with sufficient accuracy to satisfy fully the practical applications. CL
H
e l i c a l
S
pr in g s
W
it h
D
if f e r e n t
T
y p e s
o f
L
( r
o a d in 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 Loading. To an open-wound helical spring the load is applied by means of yokes reaching on each end dia metrically 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 developing Equation [2], the effect of the pitch angle a is such that it may be corr ectly assum ed th at th e end coil is in a plane perpendicular to the center line of the coil and that the forces are perpendicular to the plane of t he coil. Referring to Fig. 3, the load P / 2 a t A has no part on the def ormation of the end coil between the points A and B. The load P / 2 a t B produces a moment of P / 2 • 2 • r = P ■ r at point A which is balanced by the same moment P • r acting on the other side of the spring. All cross-sections of the spring bar betw een 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 at A wrould remain in the same positio n rela tive to the end coil as it had before loading took place. Bu t the end
A
n a l y s is
a t
T
h r e e
-P
o in t
L
o a d in g
Proceeding in t he same manner for three-point loading, shown in Fig. 4, where three equal forces P /3 are placed 120 deg apart on the end coil, it is found that the de flection, due to twist in the end coils amounts to
and the total spring deflection is
In these analyses the effect of bending and shear have been omitted in favor of F ig simplicity. The error made thereb y is so small as to be of no practical consequence. C
o n c l u s io n s
o f
A
n a l y s e s
f o r
D
. 4
e s ig n
Most commercial spring-loading conditions will conform to the two-point loading and the equivalent deflection of the two ends of a helical spring in terms of the deflection of a free coil will approximate 0.5.
RESEARCH 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:
For compression springs the forces are applied in the opposite direction; the deflection is also in the reversed direction, bu t 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 th at pa rt of the coil which mu st be con sidered in the calculation. The basic design of the sprin g 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 at the end, the pitc h changes a t conta ct 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 at 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 cor rection. 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 cent er line, the spring would bend out sidewise, which, in most compressionspring applications, does not occur to any appreciable extent. Within the range of free coils, i.e., from B to B ' (see Fig. 5), the ba r is subjected to a she ar force equ al to P /2 an d a torqu e eq ual to P r. The effect of shear upon the deflection is small and can be neglected. In a well-applied compression spring th e tor qu e 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 the 2 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 dist ribu tion, is com pensated by th e ten dency for a slight decrease in active coils during compression. It is therefore logical and p ractic ally corre ct to choose the 2 X P/ 2 load division. E
x p e r im e n t a l
Ve
r i f ic a t io n
o f
An
M od ul us ...........................................
Q
c= n ' + 1/2
=
I 1.839 7 .5 3 19Vs 6.075 6.575 8V t
=3
8 n-D* P
f-d *
469
The springs were tested with a testing machine of 5000-lb capacity. The dimensions were carefully taken with microme ters and averaged from many measurements taken of diameters perpe ndi cular to each othe r alon g th e full len gth of th e springs. The free coils were carefully determined and fixed by inserting spacers at the contact points with respective end coils.
F
ig
. 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:
C o i l d i a m e t e r ...................................... Ba r di am et er ....................................... Number of free coils .......................
L o a d ........................................................ D e f l e c t i o n ............................................. Num ber of acti ve coils ................... Modulus of elasticity in torsion.
Spring 1 D d n' P /i m =
n ' + x G
Spring 2 D d n" ~ P = h m = n" + x — 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
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: d Bar diameter (average), in ___ Coil diameter (average), in ___ D Free length ..................................... Free coils ......................................... n' Act ive co ils .................... . .............. n Total number of coils from tip to tip of bar (taper).. Lo ad, lb ............................................ P Deflection........................................ f
RP-56-4
4580 0 .7 6 0 1 1,8 00 ,0 00
II 1.122
4 .1 4 0 195/s ll 1/* 12
H»/4
4600 1 .6 6 0
1 1,8 50 ,0 00
and If there should be any variation between the two springs in di, Di, or P, then/i or/2must be corrected to correspond to values for d, D, and P adopted for the foregoing calculation. It will be found th at x approximates the value of */» very closely and tha t G approximates 11,700,000 for any size and kind of steel spring bar.
Appendix 1 T N o rder to find the tota l angle of twist of cross-section a t 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 TH E AMERICAN SOCIETY OF MECHANICAL EN GINEERS
acting on any cross-section of the arc A B is M = P /2 a = P/2 r ■ (1 — cos < p ) . (See Fig. 3.) The tw ist angle u between two cross-sections separ ated by th e distance A(s) amo unts to 32 • M A(s) ml , ------- - — —— = 4 m. The total twist angle—th a t is, twist angle tt
a* G
of cross-section at A in reference to cross-section at B —equals the sum of the angles of twist for all sections A(s) located b etween po int A and point B. It is
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
Lo
a d in 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 [a], substituting in it P/2 for P, j(l
•7r
G ——. oZ
If we assume the center 0 of the arc A B rigidly connected to the cross-section at A , it will move with the cross-section A the 16 • amount of r -, which distance corresponds to the di
G
a
=
0 =
d
it,
* TT
I = -----, and C = 64
Then the deflection of point B is:
Point 0 deflects
deflection of the center O of arc A B from its original position under the influence of load P/2 at point B. For the three-point loading of the end coil, we find the de flection of the center of the arc of the end coil, proceeding the same as in the case of the two-point loading, as follows:
in which form the equation shows that the deflection of an end coil is equal to the deflection of one-qua rter of a free coil. Ca
s e
W
h e n
W
h e n
< t>> a
If 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.
Appendix 2 Ca
s e
4>< a
T
T N
calcu lat ing defle ction s of a po rtio n of a cir cular ring ou t 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 a t B (Fig. 6),
h e e e
-P
o i n t
L
o a d in g
Take now, as an example, the case of three loads P/3 put at points A , B, and C, 120 deg ap ar t (Fig. 8). Poin t 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 pa rt is obtained by sub stitutin g into Equatio n [a] P /3 for P and 2x/3 for the angles a and 0. 5 PR3 This gives: Fs du eto S = 7.50 —- (assuming E = ~G). j t CL (
Z
The second part is obtained from the same equation by putting PR3 4?r/3 for a and 2ir/3 for 0. This gives: Fsdue to C = 6.70 ——• Crd
F ig . 7
then according to Saint-Venant’s solution the deflection at any po int C, defined by an angle
, is given by the following equatio n: 2T he Saint-Venant solution can be found in Love’s “M athematical Theory of Elasticity,” pp. 456-457, or in "Strength of Materials,” vol. 2, p. 469, by S. Tim oshenko. This m anner of solution was sug gested by R. L. Peek.
Hence the total deflection of the point B i s: V b
=
U.
PR3 2 0 -
In calculating the deflection of the point C, we again have two pa rts : (1) Deflection at C produced by the load at C is obtained • Method proposed by Prof. S. Timoshenko.
RESEARCH by s ubsti tut ing P /3 for P and a = = 4ji-/3 into Equation [a], PR3
which gives Fcdue to C — 33.10 —— and (2) deflection at C proGd
duced by the load at B. This is equal to deflection at B when the load is at 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: V Cdue to B
=
y
Bdue to
C
= 6.70
RP-56-4
471
par alle l to this cente r line, i.e., perp endicular to the plane . The component P / 2 sin a is negligible, as it does not contribute directly to the deflection considered and has very little influence on the diameter of the coil. The problem of finding the total spring deflection may, there fore, be illustrated by Fig. 10.
PR3 Gd 4
Then the total deflection at C i s:
It will be seen that the method described can be used for any number of conc entra ted forces. I t can be easily extended also to th e case of distribut ed loads.
Appendix 3 HTHE Saint-Venant solution may be used directly to determine 1 the tot al deflections of helical springs under variou s load condi tions. The most simple case of its appl icatio n for helical springs is a spring with the two-point loading as shown in Fig. 3, and Fig. 9. For this analysis the spring is assumed to consist of two equal
The deflection f a of point H representing the center of the bra cket BC is the mean of the deflection at B and C. The de flection fn of point B is composed of the deflection f u of point B due to the load P/2 at B and fn " due to the load P/2 at C, and the deflection f c of point C is composed of the deflection fc ' of point C due to the load P/2 at C and/c" due to the load P/2 at B. The total deflection
S b "
= fc ," according to the theorem of reciprocity, so that
The tota l deflection of the spring is / = 2 fH
The deflections/s",/s', and/c' can be determined by equa tion
parts, equa lly loaded, which me et a t th e center cross-section A of the spring bar. (See Fig. 9.) Poi nt A is now considered the fixed point of two circular arcs. The center angles of these arcs are equal, a = (»'/2) 2ir + ir = (n ' + 1) x, 2n' being the number of coils be tween B and B '. If ip is the pitch angle of the spring, the actual length of arc AC is L = r • r (n' + 1) COS
and
the
loads on the arc perpendicular to the plane of the arc would be P/2 cos
To this result a correction must be added to compensate for the influence of the pitch angle and the deflection due to pure shear. Saint-Venant’s equation4for the deflection of a helical spring with rigid end levers is:
4Love, “ Mathem atical Theo ry of Elasticity,” p. 422; S. Timo shenko, “Strength of Materials,” part 1, p. 289.
472
TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS B = spring load Sh = deflection I = length of spring bar r = radius of coil a = pitch angle
and therefore the val ues presented in this paper may be an appro priate average for steel springs to be used at ordinary tempera tures only.
If we transform this equation in terms used in Equation [2], we find
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 com mercial helical springs where D /d = 3 or more is very small and for all practical purposes negligible. Equation [8] is given t o show the influence of the pitch angle. Other influences which affect the accuracy, and are not con sidered 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 war rant the application of these highly refined methods of calcula tion 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. Equa64 • n r> P ,, 64 • («' + §) • r3• P tions / = — ---- ------- and / = -----------——---------, respectively, Or *
a *
( j
• a 4
can be regarded as being accurate for all practical purposes.
Discussion T. 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 Transac tions of the American Society for Testing Materials of that year and some work presented in the Philosophical Magazine for Oc tober, 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 applica tion 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, 1
W. M. A u s t i n . 6 The writer has had t o apply helical compres sion springs, both large and small, t o quite a variet y of machinery and has often observed the influence of the end turns. Particu larly, he has observed tha t the average spring designed to be made like the author’s Fig. 6, except having a length relative to diame ter 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 V s 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 in creases in thickness for another l/ t turn to 3/ i diameter of the wire, then tapers another x/ 4 turn to V 4 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 th e 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 t he 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 perpen dicular 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 w ith 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 suc cessful, so to load the spring that at no time during the compress ing 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 t he body of the spring because, for 6E n g i n e e r ,
D i r e c to r o f R e s e a r c h , A . O . S m i th C o r p o r a t i o n , M i lw a u k e e , W i s .
M e m . A .S . M . E .
Pa.
W e s ti n g h o u s e
M e m . A . S .M .E .
Elec.
&
M fg. Co., E ast
P i t ts b u r g h ,
RESEARCH
RP-56-4
473
round wire, the section mod ulus for tor sion is ird3/16 and f or 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
been made to prov e 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 tota l load on the Thus, with the additional fact that the wire is often dam spring is P; consequently, the single bar must trans mit this aged by making the hooks, accounts for the observed fact that total load from one end of the coil to the other and the shear in tension springs, if they break, always break where the hook con the bar will be P instead of P/2 . nects to the body of the spring. There is a way to reduce the Since present conventional spring formulas can be proved excessive stress in the hooks. It is to make the end turn a spiral inaccurate, it does not follow that illogical corrections are ac and bend up the hook from the inner end of the spiral, making ceptable. Adding one-half a coil to the num ber of free coils the mean diameter of the hook about one-half the mean diameter admits that a portion of the seated end coils deflect, which is of the main body of the spring. bey ond comprehension. I t is true th a t torsiona l defo rmation The au thor ’s tests on the two large springs would be much more extends beyond the free into a portion of the seated coils, for valuable if the springs had been loaded to near their maximum if this were not true, the first free coil at each end would not safe loads instead of limiting the stress as ca lculated by the old for contribute its full share of deflection. To infer tha t there is mula to 34,400 for the small spring and 14,100 for the large one. axial deflection derived from the se ated coils is a process of c reat In any event, I believe they should have b oth been loaded so as to ing one error to compensate for another. produce t he same stress. As the paper shows for balanced loading the load P on a com It is quite generally known that Hooke’s law gives only the pression sprin g ma y be resolv ed into two com ponents of P /2 first term of a rapidly converging series, so that Young’s modulus each acting 180 deg apart. If the spring in Fig. 3 is loaded in E and th e shearing modulus G both have higher values when de compression, P/2 at B will produce torsional stress at A, and termined by stress-strain measurements using low stresses than P / 2 a t A increases this stress to the final value within th e 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 tota l should not be counted twice. The auth or’s mathem atical 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 im possible. R. L. P e e k , J r .8 H o w accurately the solutions given for twothey do when using stresses near but below the elastic limit. and three-point loading apply to helical springs compressed be 16 • r3 P The author’s deflection r • w = ----------- due to the torsion in tween parallel plane surfaces requires furth er analysis. Follow d4 ■G ing the treatment given in Love’s “Mathematical Theory of the end turn is the deflection due to torsion of the poin t B, Elasticity,” pp. 456-457, I have evaluated the force required Fig. 3, and no t the deflection of the pivot hole in the bar connect to keep the extreme end of the inactive turn in contact with the ing points A and B. This would make the deflection of the point A (Fig. 3), a condition that must be satisfied under com 8 •r3■P pivot hold only — —— • In th e au th or ’s analysis no accou nt pression of this sor t. I find thi s force tri vial in comparison d r (jt with the reaction at A under two-point loading, and this con is taken of the deflection at B due to the bending of the end turn sideration therefore does not affect the validity of applying the by the load P /2 at B. This would produce a deflection about result for two-point loading to compression between parallel as large as that due to torsion. plane surfaces. On the othe r han d, in such compression the In order to get experimental data on the deflection of the end change in pitch angle of the active coils w'ill cause their axis to turn, the writer had a piece of Vi-in. pretempered spring steel be no longer normal to the paralle l pla ne surfaces app lyin g load wire bent into 3/ t of a turn of 2n/ie mean diameter, as shown in and their deformation will not be that corresponding to a purely Fig. 11. axial thru st. Whe ther this effect will appreciably change the It was loaded at B with a 70-lb weight and the deflection at B result, I have not ascertained. was 0.29 in. If we let G = 11,400,000, th e deflection per tur n of the main body of the spring is 0.488, when loaded to 140 lb. A. M. W a h l .9 The exact solution of the additional deflection The deflection at B is then seen to be 0.595 of the deflection of produc ed by th e end tu rn s of a helical compression spring is one turn, and the deflection at the center of the bar connecting undoubtedly a very complicated problem, since it depends on A and B would be 29.7 per cent of the deflection of one turn, the exact shape of the end turns and on the distribution of load and for both ends the deflection due to the end turns is 59V2 thereon. The auth or has simplified the problem by assuming the per cen t of one turn. end turns to have the full bar cross-section throughout their •
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 tru e provided Poisson’s ratio is tak en as 0.30 to 0.335 rath er than 0.365. In some instances attem pts have
length. In add ition he assumes various distributions of load on the en d turns, finally choosing tha t which seems to agree best with test results. Since, in most practical cases, the deflection due to the end
8 Bell Telephone Laboratorie s, N ew York, N. Y. 7 Assistant Professor, Industrial Engineering, Virginia Polytechnic 9Westinghouse Research Laboratories, East Pittsburgh, Pa. Institute, Blacksburg, Va. Jun. A.S.M.E. Assoc-Mem. A.S.M.E.
474
TRANSACTIONS OF T H E 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 ap proximation, 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 prickpunch marks on diametrically opposite points of the coil in the body of a helical spring and is described in a previous publica tion.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” modu lus 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: Sp rin g N o ...........................................
1
2
3
G X 1 0 - « l b p er s q i n ..................
11.45
11.46
11.50
Three springs having indexes of about 6.5 from another manu facturer gave the following values: S p r in g N o ........................................... Q X 1 0 " M b p e r s q in ..................
A
B
C
11.19
11.12
11.30
These values are all definitely higher than the value of 10 or 10.5 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 te sts b y the writer,10wherein it was found that t he ordinary deflection fo rmula for helical roundwire springs was correct within 3 per cent for springs having indexes varying from 2.7 to 9.5. In other words, a fourfold in crease 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 calculation s 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. Gohne r,11 who used more exact methods of calculation involving the theory of elasticity. The effect of curvature may be checked up experimentally by
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 mea sured in the body of the spring between prick-punch marks, so that the “effective” value of G could be found by use of the ordi nary 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 num ber 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
it is found that n = 12, whence the added coils become 12 — l lV 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 valu e 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 mea sured. 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 neces sary 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/2 G — 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 ob tained 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-indicator springs which are used for high-temperature steams and *• A. M. Wahl, “F ur th er R esearch on Helical Springs of Rou nd gases as well as for cold air have been accepted as accurate for and Square Wire,” Trana. A.S.M.E., 1930, paper APM-52-18, practical purposes without using any correction factors for the p. 217. 11 O. Gohner, “Die Berechnung zylindrischer Schraubenfedern,” various temperatures to which they are exposed. The author, however, mainly considered springs used in at Z.V.D .I., March 12, 1932.
RESEARCH 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. How16 • r3 P ever, he errs in his conclusion th at the deflection r ■o> = ------- - —
RP-56-4
475
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 th e com par ativ ely soft copper clamp to prevent twisting of the wire and consequently the turning of the wire cross-section at A. This torsional displacement of dl G cross-section A of course increases the actual deflection due to is the deflection of point B (see Fig. 3). This deflection is derived the twist in half coil BA which stamps a test made according from the product r a which (as is clearly explained in the paper to Mr. Austin’s arrangement, shown in Fig. 11, as unreliable. and in Appendix 1) is nothing else than the deflection of the The author has made a number of experiments according to original end-coil center 0, which, as well as that of the pivot Fig. 12 in which he found the deflections to check very closely hole between points A and B, is at a distance r from the center with the Saint-Venant results. of the bar cross-section subjected to torsion. J. P. Mahaney mentions that the pure shearing force in the Mr. Austin’s claim that the bending effect of load P/2 on free coils due to the load P m ust be equal to P which is quite cor the deflection of point 0 would be as large as th a t of torsion only rect. However, according to the explanation given by the au is unfounded as may be seen from the Saint-Venant solution thor in answer to A. M. Wahl’s discussion, the shearing force P (see Appendixes 2 and 3) which includes the bending effect of is divided into halves. One-half balances an excess of the sum of load P/2 at B and shows that the deflection of point O is even the torsional shearing-force components parallel to the axis somewhat less than that given by the author for torsion only. of the spring and acting in a direction opposite to P, while the In Mr. Austin’s experiment shown in Fig. 11 the deflection other half adds a pure shear deflection to the torsional deflection of point B is claimed to have been 0.29 in. for a load of 70 lb 8 n D3 P at B; bu t according to Sain t-Ve nant’s solution this deflection as given by the conventional deflection equation / = ----- - —— ----should have been 0.231 in. for G = 11.4 X 106or 0.225 in. for G = a4■G 11.7 X 10«. Mr. Mahaney, after admitting that torsional deformation ex Mr. Austin would have found more accurate and reliable re tends beyond th e free coils into a portio n of the seated coils, elabo sults had he arrange d his experim ent according to Fig. 12. This rates considerably on his conception that since the end coils in arrangement consists of a helically and closely coiled springa compression spring are not free to move they cannot contribute steel wire of one and a fraction of a turn . The coil diam eter is to the deflection of the spring. The deformation of the end coil, about 20 or more times the diameter of the wire which latter Mr. Mahaney claims, makes it possible for the first free coil to should be abo ut V< in. The wire and coil diam eter and th e de contribute its full share of deflection. If this statement were flection should be large enough to make unavoidable errors true, the conventional spring-deflection Equat ion [2] as de negligible in reference to the deflection. In Fig. 12 B A B ’ is veloped 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 reason ing 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 ac cordance 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, w^hen we consider the error range due to (a) using exactly one full coil and BA = A B ' and each is one-half coil. the conventional spring-deflection equation instead of the In order to eliminate errors due to initial tension or deflection, Saint-Venant equation given in Appendix 3, (6) unavoidable the deflection ei — e2for the load Qi — Q2 is determ ined. The variations in spring-bar and coil diameters of commercial springs 1 --- 62 which appear in the equation in the fourth and third power, re deflection of a point B in reference to point A is / = --------2 spectively, (c) change in pitch angle and coil diameter during for the load Qi — Q2at B. In order th at the stress in the wire is compression, (d) uncertainty as to spring end loading conditions, within the elastic limit of the spring steel Qi must be less than (e) neglecting the influence of the spring index D/d , and (/) un certainty as to the actual value of the modulus of elasticity E or 15,000 lb where d and D are given in inches. The G, respectively, the variation of the actual deflection of the end coil from the one calculated, and given as being equal to the de 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 flection of Vi coil due to maximum torque, is so small in compari cross-section at A does not turn and therefore does not cause a son to other discrepancies that its disregard is fully justified. This is very apparent when we realize that a 5 per cent error in 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, determining the end-coil deflection results in an error of less than
476
TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGIN EERS
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 G = 8 2 2 , 0 0 0 k g pe r s q c m v G / E = 0.392
E — 30,000,000 lb per sq in. G = 11,700,000 lb per sq in. m = 3 . 6 3 , f r o m 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 (re gardless of small or large number of active coils), as given in the author’s paper, and were therefore accepted by the author as being depen dable. H iit te is considered one of the out sta nding 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 pape r. Mr. Wa hl’s analy sis is, on this poi nt, incorr ect and de ceiving. In the author’s example, G is determined from actua l values of n ' = 11.5, d ± 1.122, D = 4.14,/ = 1.66, and P — 4600 and (in conformity with the th eory developed in the paper) n = n ' + '/ 2. If, in the example, the value of G had been different, say 11.5 X 10°, then th e de fle ct io n/ would have been 1.715 in. in stead of 1.66 in. as it actually showed in the test, and n = 12 and no t 11.65. The num ber 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 dimen sions 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-deflection equation for helical round-wire springs is correct within 3 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. Th e deflection t
P
L
of direct or pure shear for the spring is /" = yL = - L = ———, G
where L = 2Rirn,
d2
Fi G
* 7T
Fa = ------- (for circular cross-sections from
4-1.2
----------- . About half Hiitte), P = spring load, or / " .= ——— G • 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 straigh t 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 deflectio n is considered for both. The total deflection of the helical spring under load P is:
or
or
= shear angle in radians = shearing stress R = mean radius of coil D = mean diameter of coil d = diameter of spring wire or bar n = number of active coils P = spring load G = torsional modulus of elasticity / = tota l spring deflection L = effective length of wire or bar /" = deflection due to pure shear. 7 t
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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Rio de Janeiro— Bibliotheca da Escola Polytechnica Bibliotheca Nacional Sao Paulo ............. Bibliotheca da Esoola Polytechnica Canada
Haw aii
Honolulu...............University of Hawaii Library Holland
Amsterdam ........... Koninklijke Akademie von Wetenschappen Delft......................Bibliotheek der Technische Hoogesohool The Hague ........... Koninklijk Instituu t van Ingenieurs Rotterdam ............Nationaal Technisch Scheepvaartkundig Institut
Montreal .............. McGill University Engineering Institute of Canada Toronto ................ University of Toronto, Library Chile
Santiago............... Universidad de Chile, Fa cultad de Ciencias Fisicas y Matematicas (Engrg. School)
Ind ia
Bangalore ............. Mysore Eng ineers Association Calcutta................Bengal Engineering College Poona ....................Poona College of E ngineering Rangoon............... University of Rangoon
Cuba
Havana.................Cuban Society of Engineers Czechoslovakia
Prague .................. Masaryko va Akademie Prace Society of Czechoslovak Engineers Danzig Free Ci ty ............. Bibliothek der Technischen Hochschule
Irela nd
Belfast .................. Queen’s University of Belfast Ita ly
Milan .................... Biblioteca della R. Scuola d’Ingegneria Comitato Autonomo per l’Esame della Invenzioni Na ple s................... Bibliotec a della R. Scuola d’Ingegne ria Rome .................... Biblioteca della R. Scuola d’Ingegneria Consiglio Nazionale delle Ricerche presso il Ministero della Educazione N azionale Turin.....................Biblioteca della R. Scuola d’Ingegneria
Denmark
Copenhagen ......... The Royal Technical College England
Birmingham ......... Birmingham Public Libraries Bristol. ................ University of Bristol Cambridge........... University of Cambridge Leeds.................... University of Leeds Liverpool.............. Public Library of Liverpool • Liverpool Engineering Society London ................. City & Guild Engineering College Institu tion of Automobile Engineers Institution of Mechanical Engineers Institu tion of Civil Engineers Institu tion of Electrical Engineers The Junior Institution of Engineers The Royal Aeronautical Society Manchester .......... Manchester Public Libraries (Reference Library) Oxford .................. University of Oxford Newcastle-uponTyne ................. The North East Coast Institution of Engineers and Shipbuilders Sheffield ................ Sheffield Public Libraries .
WaZes Cardiff .................. Cardiff Public Library France
Lyons ....................University of Lyons Paris..................... IScole Nationale des Arts et Metiers Ecole Nationale Supfirieure de L’Aeronautique ficole Centrale des Arts e t M anufactures de Paris Soci6t6 des Ingfinieurs Civils de France Germany Berlin....................Verein deutseh er Ingenieure Bibliothek der Technischen Hochschule Breslau ................. Bibliothek der Technischen Hochschule Cologne (Koln).. ,Universita,ts- und Stadtbibliothe k Dresden ................ Bibliothek der Technischen Hochschule Dusseldorf ............ Bucherei des Vereines deutsoher Eisenhuttenleute Frankfort ............. Technische Zentralbibliothek
Ja pa n
Kobe ..................... Kobe Technical College Tokyo ................... Imperial University Library The Society of Mechanical Engineers Yokohama ............Library of Yokohama Mexico
Mexico City ......... Asociacion de Ingenieros y Arquiteotos de Mexico Library of the Escuela de Ingenieros Mecanicos y Electricistas Norw ay
Oslo....................... Den Polytekniske Forening Poland
Warsaw .................Bibljoteka Publicazna Porto Rico
Mayaguez.............University of Porto Rico Portugal
Lisbon ................... Institute Superior Technico Rou mani a
Bucharest ............. Scoala Polytechnica din Bucharest Scotland
Glasgow ................ Royal Technical College Mitchell Library South Africa
Cape Town .......... University of Cape Town Johannesburg .......South African In stitu te of Engineers Sweden Stockholm............ Kungl. Tekniska Hogskolan
Svenska Teknologforeninger Gothenburg .......... Chalmers Tekniska Institu t
TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS Switzerland Zurich ................... Eidgenossische Technische Hochschule Turkey Istanbul ................ Rob ert College
U.S.S.R. Kh arko v ............... Supreme Economic Council of Ukraine Leningrad ............. Leningrad Polytechnic Institute Moscow ................ Supreme Council of N ational Economy Tom sk ...... ........... Tomsk Polytechnic Institute .
