RP-S6-4
N um ber of Active Coils in Helical Springs
By R. F. VOGT,1 MILWAUKEE, WIS.
D u e to th e to r s io n a l d is p la c e m e n t b e tw e e n c ro s s -s e c ­
tio n s o f t h e h e lic a l s p r in g b a r o r w ire i n t h e s o - c a lle d d e a d
o r in a c tiv e c o ils o n e a c h e n d o f t h e s p r in g , t h e t o t a l d e ­
fle c tio n o f th e s p rin g is g r e a te r t h a n t h a t w h ic h c o r r e ­
s p o n d s to th e d e fle c tio n o f t h e fre e c o ils . T h e e x a c t
a m o u n t o f th e s p rin g d e fle c tio n c o rr e s p o n d in g to th e
in flu e n c e o f th e “ in a c tiv e ” c o ils c a n b e c a lc u la te d fo r
k n o w n lo a d in g c o n d itio n s o f t h e s p r in g a n d c a n b e c lo s e ly
e s tim a te d fo r a ll s p rin g s s u b je c te d to c o n v e n tio n a l s p rin g p r a c tis e lo 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 ­
la te d in te r m s o f th e d e fle c tio n o f a c tiv e c o ils . A te s t to
d e te r m in e t h e a c tu a l n u m b e r o f a c tiv e c o ils is s u g g e s te d
a n d e x a m p le s a r e g iv e n .
where ju is Poisson’s ratio.
for spring steel
E
m
m
G
HE predetermination of the correct
number of active coils in helical
springs is, in many applications,
very important. C e n tr if u g a l s p rin g loaded regulators, controlling the speed
of prime movers, spring-loaded indica­
tors, and many other apparatus require
helical springs of which the correct num­
ber of active coils is essential.
If, in the use of the conventional helicalspring deflection equation
T
—
=
—
=
30,000,000 lb per sq in.
0.275
1/n — 3.63
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 experi­
ment.
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 commer­
cial helical compression springs we find th at 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 num­
ber of active coils was taken as the number of free coils plus
V2 coil.
R
/
r
d
P
G
n
=
=
=
=
=
=
deflection
mean radius of coil
diameter of wire
load
modulus of elasticity in torsion
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 mea­
sured and checked.
In such cases it is erroneously assumed th a t 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 A ssistant Chief Consulting Engineer, Allis-Chalmers Mfg. Co.
Mem. A.S.M .E. R obert F. Vogt was born in Geneva, Switzerland,
and had his prim ary and secondary schooling a t Rom anshorn, C an­
ton School a t St. Gall, and Swiss Polytechnicum a t Zurich, Switzer­
land. His professional career began in the U nited S tates in 1903.
He has been connected w ith the Allis-Chalmers Mfg. Co. as m echani­
cal engineer since 1907.
C ontributed by the Special Research Com m ittee on M echanical
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 ­
According to Htttte, 26th edition,
od
U
nder
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
pendicular to a rigid arm of length
r, which is perpendicular to the rod.
The deflection of the point of ap­
plication of the force P in the di­
rection 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 in g W it h R ig id A rm s
a t
C o il 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
S o c ie t y o f M e c h a n ic a l E n g in e e r s .
N o t e : Statem ents and opinions advanced
can
in papers are to b e
understood as individual expressions of their authors, and n o t those
of the Society.
a = pitch angle
n — number of coils
467
468
TRANSACTIONS OF T H E AM ERICAN 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 p art of coils are in­
active. The number of coils n is also the number of active coils.
C
o m m e r c ia l
S p r in 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 s p rin g w ill be
omitted 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 tw ist 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 ic a l
S p r in g s W
it h
D
if f e r e n t
T
ypes of
L
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 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 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 <p) = ----- (1 — cos ip)
2
for the cross-section designated by angle ip. The total angle of
twist of cross-section at A , due to the torsional resilience in
the end coil from A to B and the moment P • r acting at A, is
16 • P ■r2
a = — - ■■- —, and the center 0 of the coil, which we imagine
CL
* (r
rigidly connected with cross-section a t A , moves in reference to its
original position O', when the spring is not loaded, an amount of
16 • P • r*
f = r ■a = ————— (see Appendix 1). O' may be regarded
d •G
as the center of the end coil in its new position. O may be taken
as the original location of the center. The distance between 0
and O' then eq u als/', which is also the deflection of the end coil
16 - p . r s
/ ' = — - ——— . This corresponds to the deflection of lU coil
d4 • G
which is subjected to the moment P • r. The same, of course,
occurs at the end A 'B ', so th a t the total deflection of the helical
spring amounts to
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 L o a d in g .
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 th a t each end of the
yoke transm its the same pull P /2
on the spring.
As is shown in d e v e lo p in g
Equation [2], the effect of the
pitch angle a is such th a t 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 def­
ormation 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 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
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. B ut the end
A n a l y s is
at
T
h r e e - P o in t
L
o a d in g
Proceeding in the 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 th at 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 thereby 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 lyses
for
D
4
e sig 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, but the
calculations and results are otherwise the same. The design and
application of a commerical helical compression spring are such
th at 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 con­
sidered 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 at B. The variation is usually not far from
this load division and comes within the range of the threepoint 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 th a t 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 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
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 th a t 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 distribution, is compensated by the ten­
dency 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 im e n t a l
V e r if ic a t io n
of
M o d u lu s...........................................
d
D
n'
n
c=
- n ' + 1/2
Q
I
1 .8 3 9
7 .5 3
19V s
6 .0 7 5
6 .5 7 5
8Vt
P
f
=
8
n-D* P
f-d*
=3
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
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.
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, b ut with a different number of
free coils, as follows:
S p rin g 1
C o il d ia m e te r ......................................
B ar d ia m e te r .......................................
N um b er of free c o ils .......................
L o a d ........................................................
D e fle c tio n .............................................
N u m b er of a c tiv e c o ils ...................
M od u lu s of e la stic ity in to r sio n .
m =
D
d
n'
P
/i
n'
G
Sprin g 2
D
d
n"
~
P
=
h
x
m = n"
—
G
=
=
+
+
x
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 na lyses
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:
B ar diam eter (a verage), in ___
C oil diam eter (average), in ___
Free le n g th .....................................
F ree c o ils .........................................
A ctiv e c o ils ..................... ..............
T o ta l num ber of coils from
tip to tip of bar (ta p e r )..
L oad, lb ............................................
D eflec tio n ........................................
RP-56-4
4580
0 .7 6 0
11,800,000
II
1.122
4 .1 4 0
195/ s
ll1/*
12
H»/4
4600
1 .6 6 0
11,850,0 0 0
and
If there should be any variation between the two springs in
di, Di, or P, th e n /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 th at x approximates the value of */» very closely and that
G approximates 11,700,000 for any size and kind of steel spring
bar.
Appendix 1
T 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 <p. The moment
470
TRANSACTIONS OF TH E 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 < p). (See Fig. 3.) The twist angle u between
two cross-sections separated by the distance A(s) amounts 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 between
point 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 in t
L o 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
a
=
0 =
d * TT
it,
• 7r
I = -----, and C =
64
G ■——. Then the deflection of point B is:
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
deflection of the center O of arc A B from its original position
under the influence of load P /2 a t 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:
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.
Case W
hen
<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 th at 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
C a se W
hen
4> < a
T
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),
h e e e - P o in 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 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
5
This gives: F sduetoS = 7.50 —- (assuming E = ~G) .
(j t CL
Z
The second part is obtained from the same equation by putting
P R3
4?r/3 for a and 2ir/3 for 0. This gives: Fsdue to C = 6.70 ——•
Crd
F ig . 7
Hence the total deflection of the point B is :
PR3
then according to Saint-Venant’s solution the deflection at any
point C, defined by an angle <f>,is given by the following equation:
Vb = U.2 0 -
2 The S aint-V enant solution can be found in L ove’s “ M athem atical
T heory of E lasticity ,” pp. 456-457, or in "S tren g th of M aterials,”
vol. 2, p. 469, by S. Tim oshenko. T his m anner of solution was sug­
gested by R. L. Peek.
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
by substituting P /3 for P and a = <t> = 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:
VCdue to B = y Bdue to C = 6.70
RP-56-4
471
parallel to this center line, i.e., perpendicular 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
Gd4
Then the total deflection at C is:
I t will be seen that the method described can be used for any
number of concentrated forces. I t can be easily extended also
to the case of distributed loads.
Appendix 3
HTHE Saint-Venant solution may be used directly to determine
1 the total deflections of helical springs under various load condi­
tions. The most simple case of its application 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
parts, equally loaded, which meet at the center cross-section
A of the spring bar. (See Fig. 9.) Point 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)
a n d th e
COS
<p
loads on the arc perpendicular to the plane of the arc would be
P /2 cos <p. But as shown by developing Equation [2], the prod, r • x (»' + 1) P
uct ------------------ - — • cos eliminates cos <p.
cos ip
2
Therefore, it may be assumed th at cos = 1 without interfering
with end results. This means th a t it may be assumed that arc
ABC is r ■it • {n' + 1) in length, located in a plane perpendicular
to the spring center line and loaded by the forces P /2 which are
The deflection f a of point H representing the center of the
bracket 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 fc of point C is composed of the deflection
fc ' of point C due to the load P /2 a t C a n d /c " due to the load
P /2 at B.
The total deflection
Sb " = fc ," according to the theorem of reciprocity, so that
The total deflection of the spring is / = 2fH
The d eflectio n s/s",/s', a n d /c ' can be determined by equa­
tion
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 equation4 for the deflection of a helical spring
with rigid end levers is :
4 Love, “ M athem atical T heory of E lasticity,” p. 422; S. Timo­
shenko, “ Strength of M aterials,” p a rt 1, p. 289.
472
TRANSACTIONS OF TH E AM ERICAN SOCIETY OF MECHANICAL ENGINEERS
B
Sh
I
r
a
=
=
=
=
=
spring load
deflection
length of spring bar
radius of coil
pitch angle
and therefore the values 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 to 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,
W. M. A u s t i n . 6 The writer has had to apply helical compres­
sion springs, both large and small, to quite a variety of machinery
and has often observed the influence of the end turns. Particu­
larly, he has observed that 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 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 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 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 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 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 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 the body of the spring because, for
1 D ir e c to 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 tio n , M ilw a u k e e , W is .
M e m . A .S .M .E .
Pa.
6 E n g in e e r , W e s t in g h o u s e E le c . & M fg . C o ., E a s t P i t t s b u r g h ,
M e m . A .S . M .E .
RESEARCH
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
Thus, with the additional fact th a t the wire is often dam­
aged by making the hooks, accounts for the observed fact that
tension springs, if they break, always break where the hook con­
nects 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 for­
mula 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 th a t 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 de­
termined 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 in
d4 ■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 connect­
ing points A and B. This would make the deflection of the
8 • r 3 ■P
pivot 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 at 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.
It 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 attem pts have
RP-56-4
473
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 th a t these formulas are approximate for
“closely coiled” springs, such computations are not adequate
proof.
The author states th a t 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 transm it 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 th a t illogical corrections are ac­
ceptable. Adding one-half a coil to the number of free coils
admits th a t a portion of the seated end coils deflect, which is
beyond comprehension. I t is true th a t torsional deformation
extends beyond the free into a portion of the seated coils, for
if this were not true, the first free coil at each end would not
contribute its full share of deflection. To infer th at there is
axial deflection derived from the seated coils is a process of creat­
ing one error to compensate for another.
As the paper shows for balanced loading the load P on a com­
pression 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 th a t 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 th a t 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 twoand three-point loading apply to helical springs compressed be­
tween parallel plane surfaces requires further analysis. Follow­
ing the treatm ent 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 th at must be satisfied under com­
pression of this sort. I find this force trivial in comparison
with the reaction at A under two-point loading, and this con­
sideration 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 th a t 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 th a t 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.
7
Assistant Professor, Industrial Engineering, Virginia Polytechnic
9 W estinghouse Research L aboratories, E ast Pittsburgh, Pa.
Institute, Blacksburg, Va. Jun. A.S.M .E.
Assoc-Mem. A.S.M .E.
474
TRANSACTIONS OF TH 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:
Spring N o ...........................................
G X 1 0 -« lb per sq in ..................
1
1 1 .4 5
2
1 1 .4 6
3
1 1 .5 0
Three springs having indexes of about 6.5 from another manu­
facturer gave the following values:
Spring N o ...........................................
Q X 10" M b per sq in ..................
A
1 1 .1 9
B
1 1 .1 2
C
1 1 .3 0
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 tests by the writer,10 wherein
it was found that the ordinary deflection formula 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 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
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 — 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 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/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 ob­
tained in the writer’s tests mentioned above.
A u th o r' s C losure
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, “Further Research on Helical Springs of Round
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 atZ.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> = ------- -—
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 th a t of the pivot
hole between points A and B, is at 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 re­
sults had he arranged his experiment according to Fig. 12. This
arrangement consists of a helically and closely coiled springsteel 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 de­
flection 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 --- 62
deflection of a point B in reference to point A is / = --------2
for 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,
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 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 th a t the pure shearing force in the
free coils due to the load P must be equal to P which is quite cor­
rect. However, according to the explanation given by the au­
thor 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 ■P
as given by the conventional deflection equation / = ----- -——----a 4 ■G
Mr. Mahaney, after admitting th at torsional deformation ex­
tends beyond the free coils into a portion of the seated coils, elabo­
rates considerably on his conception th at 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 de­
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 th at the end coils are held so th at 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., th at 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
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, re­
spectively, (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 (/) un­
certainty 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 de­
flection of Vi coil due to maximum torque, is so small in compari­
son to other discrepancies th at its disregard is fully justified.
This is very apparent when we realize th a t a 5 per cent error in
determining the end-coil deflection results in an error of less than
476
TRANSACTIONS OF TH E 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 =
G =
G /E =
E —
G =
m =
2 ,1 0 0 ,0 0 0 kg per sq cm or
8 2 2,000 kg per sq cm
0 .392
v
30 ,0 0 0 ,0 0 0 lb per sq in .
11,700,000 lb per sq in .
3 .6 3 , 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 (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 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
th a t of 0.15 free coils instead of 0.5 as demonstrated in this
paper. Mr. Wahl’s analysis is, on this point, incorrect and de­
ceiving.
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, say
11.5 X 10°, then the deflection/ would have been 1.715 in. in­
stead 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 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 th a t 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. The deflection
t
P ■L
of direct or pure shear for the spring is /" = yL = - L = ———,
G
Fi ■G
where L = 2Rirn,
d2
* 7T
Fa = ------- (for circular cross-sections from
4- 1. 2
Hiitte), P = spring load, or /" .= ———----------- . About half
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 th at 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 th at 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
t
R
D
d
n
P
G
/
L
/"
=
=
=
=
=
=
=
=
=
=
=
shear angle in radians
shearing stress
mean radius of coil
mean diameter of coil
diameter of spring wire or bar
number of active coils
spring load
torsional modulus of elasticity
total spring deflection
effective length of wire or bar
deflection due to pure shear.
Applying the extended deflection
accurate spring tests will result in
for G.
Plotting the shear deflection, in
deflection, against the spring index
in Pig. 13.
equation in conjunction with
finding more uniform values
per cent of torsional spring
D /d we find the curve given
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Idaho
Moscow................ University of Idaho
Illinois
Chicago................ John Crerar Library
Western Society of Engineers
Library, Armour Institute of Technology
Public Library of Chicago
Moline.................. Public Library
U rbana................. University of Illinois
Indiana
Evansville............ Public Library
Fort W ayne......... Public Library
Indianapolis.........Public Library and Indiana State Library
Notre Dame.........Library, University of Notre Dame
Terre H aute.........Rose Polytechnic Institute
West Lafayette.. .Library, Purdue University
Iowa
Ames..................... Iowa State College
Des Moines..........Public Library
Iowa C ity.............State University of Iowa
Kansas
Kansas C ity .........Public Library, Huron Park
Lawrence..............Library, University of Kansas
M anhattan...........Kansas State Agricultural College
W ichita.................Wichita City Library
Kentucky
Lexington.............University of Kentucky
Louisville..............Speed Scientific School
University of Louisville
Louisiana
Baton Rouge....... Louisiana State University
New Orleans........Howard Memorial Library
Louisiana Engineering Society
Public Library
Maine
Orono....................University of Maine
Maryland
Annapolis..............United States Naval Academy
Baltimore............. Johns Hopkins University
Engineers Club of Baltimore
Public Library
Massachusetts
Boston.................. Northeastern University
Boston Public Library
Cambridge........... Harvard University (Engineering Library)
Massachusetts Institute of Technology
Fall R iver............ Public Library
Lowell................... Lowell Textile Institute
Lynn..................... Free Public Library
New Bedford....... Free Public Library
Springfield............Springfield City Library
Tufts College....... Tufts College
Worcester............. Worcester Polytechnic Institute
Free Public Library
Michigan
Ann A rbor............University of Michigan
D etroit..................Public Library
Cass Technical High School
Highland Park Public Library
University of Detroit
E ast Lansing....... Michigan State College
F lint...................... Public Library
Grand R apids. . . . Public Library
Houghton............. Michigan College of Mining & Technology
Jackson.................Public Library
Minnesota
D uluth.................. Public Library
Minneapolis......... University of Minnesota
Minneapolis Public Library (Engineering
and Circulating Libraries)
St. P au l................ James Jerome Hill Reference Library
Mississippi
State College........Mississippi State College
Missouri
Columbia..............University of Missouri
Kansas C ity.........Public Library
Rolla..................... Missouri School of Mines and Metallurgy
St. Louis............... Engineers Club of St. Louis
Public Library
Washington University
Mercantile Library
Montana
Bozeman...............Montana State College
TRANSACTIONS OP TH E AMERICAN SOCIETY OP MECHANICAL ENGINEERS
Nebraska
Lincoln..................University of Nebraska
Omaha.................. Public Library
Nevada
Reno..................... University of Nevada Library
New Hampshire
D urham ................University of New Hampshire
New Jersey
Bayonne............... Free Public Library
Camden................ Free Public Library
Elizabeth..............Free Public Library
Hoboken...............Stevens Institute of Technology
Jersey C ity...........Free Public Library
Newark.................Newark College of Engineering
Free Public Library
New Brunswick... Rutgers University
Paterson............... Free Public Library
Princeton..............Princeton University
Trenton................ Free Public Library
New York
Albany.................. New York State Library
Brooklyn.............. Polytechnic Institute
P ra tt Institute
Brooklyn Public Library
Buffalo.................. The Grosvenor Library
Engineering Society of Buffalo
Buffalo Public Library
Ithaca................... Cornell University
Jamaica, L. I ....... Queens Borough Public Library
New Y ork............ Engineering Societies Library
Public Library
College of the City of New York
Cooper Union
Columbia University
New York Museum of Science and Industry
New York University Library
Potsdam ............... Clarkson College of Technology
Rochester............. Rochester Engineering Society
Schenectady.........Union College
Syracuse............... Syracuse University
Public Library
T roy...................... Rensselaer Polytechnic Institute
Utica..................... Public Library
Yonkers................ Public Library
North Carolina
Chapel Hill...........University of North Carolina (Engineering
Library)
Raleigh................. North Carolina State College
North Dakota
Fargo.................... N orth Dakota State Agricultural College
G rand Forks........ University of N orth Dakota
Ohio
A da........................Ohio Northern University
Akron....................Public Library
University of Akron
C anton..................Public Library
Cincinnati............ University of Cincinnati
Public Library
Engineers Club of Cincinnati
Cleveland............. Public Library
Case School of Applied Science
Cleveland Engineering Society
Columbus............. State of Ohio Library
Public Library
Ohio State University
D ayton................. Engineers Club of Dayton
Toledo...................Public Library
University of Toledo
Youngstown......... Public Library
Oklahoma
N orm an............... Oklahoma University
Oklahoma C ity .. .Public Library
Stillwater..............Oklahoma Agricultural and Mechanical
College
T ulsa.....................Public Library
Oregon
Corvallis............... Oregon State Agricultural College
Portland............... Portland Library Association
Pennsylvania
Allentown.............Free Library
Bethlehem............Lehigh University
E aston.................. Public Library
Lafayette College
E rie....................... Public Library
Lewisburg............ Bucknell University
Philadelphia.........Engineers Club
Drexel Institute
University of Pennsylvania
Franklin Institute
Pittsburgh............ University of Pittsburgh
Engineers' Society of Western Pennsylvania
Carnegie Institute of Technology
Carnegie Library (Schenley Park)
Carnegie Free Library of Allegheny
Reading................Public Library
Soranton...............Publio Library
State College........Pennsylvania State College
Swarthmore......... Swarthmore College
Villanova..............Villanova College
Wilkes-Barre........Public Library
Rhode Island
Kingston.......... .Rhode Island State College
Providence........... Brown University
Providence Engineering Society
Public Library
South Carolina
Clemson College. .Library, Clemson College
Tennessee
Kingsport....... ... .Public Library
Knoxville..............University of Tennessee
Memphis.............. Goodwin Institute
Nashville.............. Vanderbilt University
Texas
Austin................... University of Texas
College S tatio n .. .Agricultural & Mechanical College of Texas
Dallas................... Public Library
El Paso.................Public Library
Forth W orth........Carnegie Public Library
Houston................Rice Institute
Public Library
Lubbock............... Texas Technological College (School of
Engineering)
San Antonio.........Carnegie Library
Utah
Salt Lake C ity .. .University of Utah
Public Library
Vermont
Burlington............University of Vermont
Virginia
Blacksburg...........Virginia Polytechnic Institute
Charlottesville.. . . University of Virginia
Norfolk.................Public Library
Richmond............ Virginia State Library
Washington
Pullman................State College of Washington
Seattle.................. Public Library
Engineers Club
University of Washington
Spokane................ Public Library
Tacoma................ Public Library
West Virginia
Morgantown........ West Virginia University
Wisconsin
Madison............... Library, University of Wisconsin
Milwaukee............Public Library
Board of Industrial Education, Vocational
School Library
M arquette University
Wyoming
Laram ie................ Wyoming University
Depositories for A.S.M .E. Transactions
Outside the United States
Argentine
Buenos Aires........Biblioteca de la Sociedad Cientifica
Australia
Adelaide............... Public Library of Adelaide
Melbourne............Publio Library of Victoria
P erth.....................University of Western Australia Library
Sydney..................Public Library, N. S. W., Sydney
Brazil
Rio de Janeiro— Bibliotheca da Escola Polytechnica
Bibliotheca Nacional
Sao Paulo............. Bibliotheca da Esoola Polytechnica
Canada
Montreal.............. McGill University
Engineering Institute of Canada
Toronto................ University of Toronto, Library
Chile
Santiago............... Universidad de Chile, Facultad de Ciencias
Fisicas y Matematicas (Engrg. School)
Cuba
H avana.................Cuban Society of Engineers
Czechoslovakia
Prague.................. Masaryko va Akademie Prace
Society of Czechoslovak Engineers
Danzig Free City............. Bibliothek der Technischen Hochschule
Denmark
Copenhagen......... The Royal Technical College
England
Birmingham.........Birmingham Public Libraries
B ristol.................. University of Bristol
Cambridge........... University of Cambridge
Leeds.................... University of Leeds
Liverpool..............Public Library of Liverpool
•
Liverpool Engineering Society
London................. City & Guild Engineering College
Institution of Automobile Engineers
Institution of Mechanical Engineers
Institution of Civil Engineers
Institution of Electrical Engineers
The Junior Institution of Engineers
The Royal Aeronautical Society
M anchester..........Manchester Public Libraries (Reference
Library)
Oxford.................. University of Oxford
Newcastle-uponT yne................. 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 et Manufactures de
Paris
Soci6t6 des Ingfinieurs Civils de France
Germany
Berlin....................Verein deutseher Ingenieure
Bibliothek der Technischen Hochschule
Breslau................. Bibliothek der Technischen Hochschule
Cologne (Koln).. ,Universita,ts- und Stadtbibliothek
Dresden................Bibliothek der Technischen Hochschule
Dusseldorf............Bucherei des Vereines deutsoher Eisenhuttenleute
Frankfort............. Technische Zentralbibliothek
Germany {continued)
H am burg.............. Bibliothek der Technischen Staatslehranstalten
H anover................Bibliothek der Technischen Hochsohule
K arlsruhe............. Bibliothek der Technischen Hochsohule
Leipsic.................. Stadtbibliothek
M unich.................Bibliothek der Technischen Hochschule
Bibliothek des Deutschen Museums
S tu ttg art.............. Bibliothek der Technischen Hochschule
Hawaii
Honolulu...............University of Hawaii Library
Holland
Amsterdam...........Koninklijke Akademie von Wetenschappen
D elft......................Bibliotheek der Technische Hoogesohool
The Hague........... Koninklijk Instituut van Ingenieurs
R otterdam ............Nationaal Technisch Scheepvaartkundig
Institut
India
Bangalore............. Mysore Engineers Association
C alcutta................Bengal Engineering College
Poona....................Poona College of Engineering
Rangoon............... University of Rangoon
Ireland
Belfast.................. Queen’s University of Belfast
Italy
M ilan.................... Biblioteca della R. Scuola d’Ingegneria
Comitato Autonomo per l’Esame della
Invenzioni
Naples...................Biblioteca della R. Scuola d’Ingegneria
Rome.................... Biblioteca della R. Scuola d’Ingegneria
Consiglio Nazionale delle Ricerche presso il
Ministero della Educazione Nazionale
T urin.....................Biblioteca della R. Scuola d ’Ingegneria
Japan
Kobe..................... Kobe Technical College
Tokyo................... Imperial University Library
The Society of Mechanical Engineers
Yokohama............Library of Yokohama
Mexico
Mexico C ity......... Asociacion de Ingenieros y Arquiteotos de
Mexico
Library of the Escuela de Ingenieros
Mecanicos y Electricistas
Norway
Oslo....................... Den Polytekniske Forening
Poland
W arsaw.................Bibljoteka Publicazna
Porto Rico
Mayaguez.............University of Porto Rico
Portugal
Lisbon...................Institute Superior Technico
Roumania
Bucharest............. Scoala Polytechnica din Bucharest
Scotland
Glasgow................Royal Technical College
Mitchell Library
South Africa
Cape Town.......... University of Cape Town
Johannesburg.......South African Institute of Engineers
Sweden
Stockholm............ Kungl. Tekniska Hogskolan
Svenska Teknologforeninger
Gothenburg..........Chalmers Tekniska Institut
TRANSACTIONS OF T H E AMERICAN SOCIETY OF MECHANICAL ENGINEERS
Switzerland
Zurich................... Eidgenossische Technische Hochschule
Turkey
Istanbul................ Robert College
U.S.S.R.
K harkov............... Supreme Economic Council of Ukraine
Leningrad.............Leningrad Polytechnic Institute
Moscow................ Supreme Council of National Economy
Tomsk..................Tomsk Polytechnic Institute