M

      

ML

1

TH! STRESS ANALYSIS OF
HELICAL ROUND-WIRE SPRiNGS

Thesis {at he Degree of M. S.
MICHIGAN STATE COLLEGE
Chi-Chum Chang
3943

 

This is to certify that the .
thesis entitled

1310 Stress W18 of Helical
Bound-wire Springs

{,1

presented by

Chi-amen Chang

has been accepted towards fulfillment
of the requirements for

___Lfln_degme in___-o_no_

Major professor

 

.. Mil: A

.~.n J. U

_-.l __1

THE STRESS ANALYSIS OF

HELICAL ROUND¥WIRE SPRINGS

By

CHI -CHUAN gym

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIEECE

Department of Mechanical Engineering

1948

TH ESES

 

 

Contents

Introduction
Approximate Theory
Exact Theory
'Working Stresses
Example

References

53.07.3313

17
22
29

Symbol

fez

List of Symbols

Definition
radius of the spring wire
mean radius of the spring coil
spring index (c==R/a)
load
twisting moment (Mr-PR)
cylindrical coordinates
shearing stress on the plane per-
pendicular to O-axis and along
the direction of f’-axis
endurance limit in shear
yield point stress in torsion
correction factor of exact theory
Wahl's correction factor
stress multiplication factor
stress concentration factor due
to curvature

factor of safety

Units
inch
inch
lb
in-lb

lb/in‘
lb/in1
lb/in‘

INTRODUCTION

Springs of most importance in machine design are helical
round-wire type. They are made in a wide variety of sizes and
used in tremendous quantities. Here we shall discuss the theo-
ries for stress calculation in this type of spring.

For calculating helical springs, the elementary theory as
commonly given in.textbooks on strength of materials or machine
design is based on the assumption that the spring may be con-
sidered essentially as a straight bar under torsion. This assump-
'tion is only approximately true where the spring index is large
and where the helix angle is small. Since the elementary theory
does not take into account the direct shear stress and the differ-
ence in fiber length between.the inside and outside of the coil
'which arises because of the curvature of the spring wire, consider-
able error will be involved if this theory is used for springs
with small or moderate indexes.

In order to take care of the effects both due to the curva-
ture of the spring wire and the direct shear stress,.A. M.‘Wah1
has introduced an approximate theory which is widely used. After
‘Mr.‘Wahl's work is reviewed we shall discuss the exact theory
originally developed by O. Goehner. By the theory of elasticity
we can set up a set of differential equations and try to solve
them under the initial and boundfi—y conditions. Then its result

is compared with that of approximate theory.

Finally the methods of evaluating working stresses are dis-

cussed, and the limitations of the formula derived are specified.

APPROXIMATE THEORY

For a stress analysis, consider a quadrant of a coil as a
free body, as shown in.Fig. l. The load is assumed to act along
the axis of the spring, in which case it produces a torsional
moment MIPR in the body of the wire. This same torsional
moment exists in all sections of the loaded spring.

In elementary theory, the spring is considered as a straight
bar under torsion. It follows that the shearing deformations and
hence the shearing stress will have a linear distribution along a
radius as shown in Fig. 2. The torsional resisting moment is
81/: a sud/2 Equating the applied moment to the resisting
moment, we get

flW¥== FVT==.SVTCZiéz

solving this equation for S, we find

<1) wife?

Where R is the mean radius of coil, a is radius of wire, P is axial
load applied.

As stated before, the stress calculated by Eq. (1) wdll be in
considerable error for springs with small indexes for two reasons:
1. The increase in stress due to the difference in fiber length
between the inside of the coil and the outside produced by wire
curvature is not considered. 2. The effect of direct shear stress

due to the axial load P is neglected.

-3-

In.Fig. 3b, if the radial section bb' and aa' rotate through

a small angle with respect to each other and about the bar axis,
the inside (and shorter) fiber a'b' will be subject to a much
higher shearing stress because of its short length than the out-
side fiber ab which is longer. Moreover, the stress on the inside
fiber a'b' is increased because the shear stress due to the direct
axial load P is added to that due to the torque moment PR at this
point. In the outside fiber ab this stress is subtracted from
that due to the torque moment. The result is that the stresses
on the inside of the coil reach values around 2.5 times those on
the outside for springs of index 3, as may be shown both by test
and theory.
Considering these two effects, wahl's approximate solution(1)
of the problem of determining stress in springs of small index,
which is sufficiently accurate for practical use can be derived
as follows:

.A small helix angle is assumed since this assumption.is valid
for nearly all practical springs. Considering an element of an
axially-loaded spring with mean radius of curvature R cut by two
neighboring radial cross sections aa' and bb' as shown in Fig. 4a,
the forces acting on this element are resolved into a twisting
moment Ms W”! acting in a radial plane and a direct axial shear-

ing force P. The stresses set up by this twisting moment are

-4-

first considered and later are superimposed on the stresses due
to the direct shear load.

The shear stress 3 acting over small cross section area d4
in.Fig. 4b may be resolved into two components: iQw parallel
to the axis of the spring and‘.53p perpendicular to the spring
axis.

If it is assumed that the two neighboring cross sections aa'
and bb' rotate relative to each other and about an axis ee' per-
pendicular to their surfaces and passing through their centers 0,
the distribution of the axial components of the stress (‘5;2)
along a transverse diameter perpendicular to the spring axis will
be somewhat as shown by the shaded area of Fig. 5. Such a distri-
bution of stress due only to a moment would not be possible since
the area to the right of the center 0 is greater than that to the
left; hence an external force would be needed to secure equilibrium.
If, however, rotation occurs about some point 0', Fig. 6, which is
displaced toward the axis of the spring, instead of about point 0,
a distribution of stress is obtained which is possible under the
action of a pure moment f4: . From.conditions of symmetry the
transverse stress components lir'will be in statical equilibrium
when rotation occurs about any point on the axis aa', Fig. 4b.

Now we try to find the position of 0'. Under the assumption
of rotation about 0', the stress 8 acting on any element d4 with

coordinates x and y may be found. When the sections aa' and bb'

.-5-

have rotated through a small angle dp with respect to each other,
the relative movement of the ends of filament dd' corresponding
to dA will be dpa‘m)’: and, since the length of dd' is (R-r-xk/a

the shearing stress 8 acting on this element will be

(2) 5, M

m-r-nds
The axial component $03 of the stress will be

3: x64!
(3) 5“ 3 W ' m-roxuo

Under the assumptions made, this distribution of stress is
identical with that in a curved bar and the distribution of the
stresses Sea is hyperbolic in form, Fig. 6. From curved bar

theory (2) the distance 3* may be expressed as:

 

.. .21 .1... 4‘
(4) 0' " om (”fi‘J'T 4R

The term 4743’ is neglected in the denominator since in
practical springs 4/3 seldom greater than 1/3 and hence 4741‘

is smll compared to unity. Putting r in (3),

(5) S” = A

m- :1: -x)da

Further, it is assumed that the ordinary formula for angle
of twist of circular bars will apply with sufficient accuracy for

the calculation of d3/49”). Thus

 

<6) 4% = 5,232

-5-

Where Mt'P/I Putting this in (5),

ZX(,R)R
Ira" (3- go!)

 

(7) 50:

From this equation it is clear that the maximum value of 5'”
will occur when Ava-‘2‘ i.e. at point a'. Putting this value
in (7) and let the spring index R/a = c, the stress at a' in
Fig. 6 becomes
(8) 54’ #2} ( 45:; )

As to the effect of direct shear stress due to axial load P,
it appears reasonable to take for this stress that given by the
theory of elasticity at the outer edges of the naztral surface of
a cantilever of circular cross section loaded by a force P. This
theory (4) gives a value of stress equal to lama/7"“):

= $4244?“ 3' $15) Adding this stress to that due to
the twist moment PR from equation (8) the maximum stress 5m

at a' will be expressed

2P1? -1 .616“
(9) 5W'737 fi-4’C)

This may be written in the form

(10) 50m)! =' {rg—k

Where K known as Wahl's correction factor is

4"” 06/!
(11) K 9 —_4¢'-4 +——c

EXACT THEORY

The approximate theory developed in the preceding section
for calculating stress in helical springs of small or moderate
index is sufficiently accurate for most practical purposes.
Where greater accuracy is desired, the exact method of calcula-
tion (5) developed from the theory of elasticity may be used.

If the helix angle is mall, the element of the spring may
be considered under pure torsion, and in cylindrical coordinates
( 790.1 3 Fig. 7a) All shear stress components except 593 and
59f may be assumed zero. Now, let us consider the equilibrium
of the smll elanent abcda'b'c'd' as shown in Fig. 7b.

Shear force on bb'c'c is Sm-[fdfi-JJJ

Shear force on aa'd'd is [5].. %ij[rf+df)laodi)

Shear force on aa'b'b is Sufzi‘glyf - a?)

Shear force on dd'c'c is (5... €194?” $132149... ’39:]

Taking moment about 2 axis, we get

[5. . ngnq-ufmeu (fur) - 5}» (fee. JaJ-f +
(5» . ’ts’ideflfi-‘E—‘i‘ Semi!) - Surfl’iz‘E-r‘slm ska

neglecting the terms involving (dp)2 and simplifying, we have

fl $419404: 4 5,» cameras: +1» g’itsafdade :3 o

Dividing by f‘dfdadi , and putting $1,334,, 314:5” finally

we have

35
(12) “ST“ 2%.: * 7“" saw

Also this equation may be written as

 

(13) 3%(f’50f) f £(f‘501) '0

Besides this equation of equilibrium, there are certain rela-
tions between components of strain which by Hooke's law can be
changed into a set of equations involving stress components. These
are known as compatibility equations(6) in the theory of elasticity.

Under our circumstances these are

4.95.: ,, aIGu_ s- ‘0

 

IT'
(14) 13:5” #31' 3‘5”

7!” av ’0

 

Thus the problem reduces to the solution of Eqs. (12) and (14).

Knowing from equation (15), we satisfy equation (12) by taking

at - R‘.
(15) 50f! 73%; . 505' F a
where ¢ is a stress function introduced and R the mean radius of

the spring coil. Substituting (15) in Eqs. (14), we find

:wes so =0
.2. . -
e (s - e - an e a
from which we conclude that the expression in the parentheses must

be a constant. Denoting this constant by .25 , the equation for

determining the stress function p is

(16) §;%+§§‘C%%f*2bao

Let us consider now the boundary conditions for the func-
tion 4:. From the condition that the lateral surface of the spring
wire is free from.external forces we conclude that at any point
at the boundary (Fig. 8) the total shearing stress must be in the
direction of the tangent to the boundary and its projection.on the

normal N to the boundary must be zero. Hence
.2 id
Syd: ‘5“75- ‘0

where ¢fl5 is an element of the boundary. Substituting from

eqs. (15), we find that

(i7) +7; gag—«j;- + $5?) =0
This shows that p must be constant at the boundary.

It will be found advantageous to introduce new coordinates
as follows (Fig. 8)

(18) x=R-f. i=3;

Referring to Fig. 8, we know that the total twisting moment
acting over the spring cross section will be

(19) -PR ‘ [7(Sef-24-5oe-x) 44rd:

Up to this point our problem is reduced to how to solve
equation (16) and p satisfies both the boundary condition (17)
and initial condition (19). Generally partial differential
equations like eq. (16) may be solved by the separable varible
method. Before doing that let us put

(20) 95 = 51' + 7%!)

into eq. (16), we get

-10-

3’4 314' 3 34' a ..

If 7"“) + 35 30 then we have

(21) fit?) a-b3‘+ ca + a’

31 214’ J 34’
(22) or? “ 75-7-57"?

where c and d are arbitrary constants.
In order to solve eq. (22), let us assume
(23) 4’ = sappy)
then eq. (22) becomes
2PM am - floss o

£3-1L’ -a‘,
“P fP ’ 2 "1

where A’ is constant, we have
(24) 2" + A‘s} =0
(25) P'4%P’—,\’p=o
The solution of eq. (24) is
(35) Z - C. COS/M‘- + 45.»)2

And eq. (25) is a Bessel's equation, the first kind solution is

 

(27) P =- (c'AfPJ; (Hf)
By definition
. a, x ”Affect
(28) ‘33 (01‘P) -"-' ’g’ ('I) zantx ”1(314')!

Putting (26) (27) into (25) and (23) (21) into (20) finally we have
(29) ¢ a (own was») mph); (elf) stance m!
It is difficult to determine all the arbitrary constants C), C", A ,

b , c , d of eq. (29) by both boundary and initial conditions in

-11-

general terms. Moreover, by this solution the stress will involve

a power series of f . Where fafl-X

so it is no use in our problem
practically.

We had better try other method.

In new coordinate system xzfl-f’, s-é eq. (16) becomes

a: 3 .av ,
(30) fig+j€4+mw +25 0

Since in general ’/,q may be considered small,

 

I - I J: .11 .........
(31) 1-7,, ”+7" na‘m‘
Then equation (30) becomes
W 34 J x _)_r_“_ ,,,,, z
(32) 7—,. 4- a2. 4- 7(/+-;-+ R, + )5‘14-26 a

This makes it possible to solve equation (32) by means of a
series of successive approximations.

(33)

We assume

¢'¢O*¢I*¢¢foc-......

and determine 4., ¢, , ¢,,

in such a manner as to
satisfy the equations

3’45 0
W 4- 03‘ +16 =0

9": 3" 3 3‘0
( 34) an d! fl 6:

av) am a an guide
W‘W‘Ta: *Rn

 

sum.of eqs. (34) approaches more and more closely eq. (32), and
the series (33) approaches the exact solution for the stress

function S.

In the new coordinate system, (15) change to

(35) 50,0 0-53” (,3 , :53: O-Z)‘ at

using binomial expansions

I X - IX
(36) W2 (l—n—II: [+-— R «LLflT-L %4 ----..

we find as the first approximation:
(:57) (Soft: 2% (Soc). 3 3%?- 1

For the second approximation we find from.eq. (35):

($9,) s[(/4.£.’£ 3%? 4 .39]
(33)
.— ’f o M
65b9h-[W 4-%}jl§3% '*--37-]

For the third approximation:
' ax ‘ M at! i a ,
(5991' [0+ r'f- $4417 4-(1 4.3—);4‘ a—é"

(599)..»[04 it! + %‘)%P +0 +%L)§%. .21")

(39)

If the radius of the spring wire is a, the equation of the
boundary (Fig. 8) is
(40) J‘+£‘-a‘=o

and the solution of the first of eqs. (34), satisfying the boundary

-13-

condition, is

(41) 4’, r -§(x‘+e‘-d)
Substituting the above expression for 42, into the second equation
of eqs. (34)? we find

ext _5;‘L_1%_ =0

The solution of this equation, satisfying the condition that Q
vanishes at the boundary, is

(42) 4),: %—-§-m(x+s -4‘)
Substituting (41) (42) in the third of Eqs. (34) we obtain

w. b
‘ £37: " 3322:- + i'?(*'+i‘-a') =0

The solution of this equation satisfying the boundary condition is
(43) ¢ 2 .. “.05 ssi-ISa‘Xx’uz” q, )
Substituting values of “2.2. p, of (41) (42) (43) in eq. (:59) we

can find the third approximation for the stress components

(5")3 ' -b[£ ‘ £19 fizz-5, (27’? 4 52‘- wa’))

(4") (Ssz)i=-b[x*3¥ 33(3’a’)+,%£ 4443-; “4’7

Substituting the results of (44) for the stress components into
Equation (19) the corresponding torque is

(45) PR 2 9—2-3? . + 35%;)
From this expression we can determine constant b, and substituting
it in eqs. (44), we find the stress components as functions of
the applied torque PR. Along the horizontal diameter of the cross

section of the spring (Fig. 8) 220, Sop-=0 and from the second

-14-

of Eqs. (44), we find

(Set): ' -b(x+37-%-+ +4},

-;I;
M a
Isl a
+

For the inner point i. , ran. and we have

I
(503); r ~bq(/+ g;— 4- 435%

For the outer point a , x 3-4.: and

 

(5a).: ran-{— _4_ . 7'55)

Using (45) for b, the value of these stresses becomes

 

”'1‘:ng 43 c

R 7‘ JP)? 6'44; 2 i

(5.1)2— ”a; 1+;E-fié 8 —'—,"q"'(/ 73‘73‘)
- 3PM is. 7 a"

6"" "W 4 n ”‘77:?"

Calculation of further approximations shows that the final
expression for the greatest shearing stress can be put in the
form”)

'4 a z
“LV'rffi-(T’

%
l + i (213"
A.

(46) (Sal’s-v?“ 2P” '

par .
I
6 l 4%?

I
Substituting spring index ‘8 this becomes

.R
(25— + ‘ 4- —'—-)
-I 47- 6 I
(47) (SaLa'r—rfin' 3 ,’ ‘
This may be written in the form

48) (Salt—I" 3755—6

Where the correction factor G is

-15-

AAAAAA

 

 

Ht

;

 

 

 

 

C + I ’ I
(49) G a C‘I 4E I6Ca

3 I
[+75'-CT.,—

 

By the expressions (11) and (49) we can calculate the cor-
rection factors of the two theories and plot curves against spring
index C. From both the Table I and the curves in Fig. 12, we can
conclude that:
(l) The value of K is larger than the corresponding value
of G. So approximate theory is on the safe side.

(2) K and G are so close that at C as small as 2 their dif-
ference is only 2%.

(3) The approximate theory is simpler and accurate enough

for the practical use.

WORKING STRESSES

For the purposes of working stress evaluation, spring applica-
tions may be grouped into two fundamental classes as follows:

(1) Statically-loaded helical spring; it may be defined as
one subject to a constant load or to a load repeated but a rela-
tively few'times during the life of the spring. These include
safety-valve springs and springs in mechanisms which operate only
occasionally.

(2) Fatigue or repeated loading helical spring; for example,
automotive valve springs are usually subject to a load (or stress)
which varies from.a minimum value to a maximum. Now we shall treat
these two kinds of springs separately.

In the design of springs subject to static loading it is sug-
gested(8) that for the usual spring material which has some duc-
tility, stress concentration effects such as those due to curva-
ture may be neglected, since the localized peak stresses could be
relieved by plastic flow as a consequence of the material ductility.

To calculate the stress in the spring under static-load condi-
tion by neglecting stress concentration due to wire curvature, the
procedure is as follows: Assuming a helical spring of mean coil
radius R, and wire radius a, under static axial‘“ load P, the tor-
sion moment at any radial settion of the spring will be equal to PR
while the direct shear will be equal to P. In neglecting curvature

effect of wire the shear stress S, due to moment PR alone will be

-17-

that given by the usual formula. Thus

29!?
S’ ' W43

On this stress must be superimposed the shear stress S. due to
the direct shear load P, which for our purposes may be considered
uniformly distributed because of neglecting stress concentration

and is

 

5- 2mwg.-m_._5
’ m‘“ Ira} 2R " 1m! 4

So the resultant shear stress will be the sum of S, , 5';

£35

5: 54-518 ”—2—, ([4- 0__5)

This equation may be written

- 2P3

Where K3 3 ’4' '4; Known as a shear-stress multiplication
factor.

Because the Wahl factor K takes care of both the effect of
direct shear and that of wire curvature, we can write Ksk'c‘K;
where Kc accounts mainly for the effects of curvature and Ks ac-
counts for the direct shear as mentioned above. Then eq. (50)

may be written

(51) 5= ‘%%'r"flc
and (52) k. = {7 = (j-f—j,’ L—i”) (/+— 2’"

Fatigue or repeated loading springs are subjected to a con-

tinuous cyclic stress between the minimum stress 3 min and the

-18-

maximum value Smax as shown in Fig. 9. This is equivalent to a
static or mean stress 3m equal to half the sum of maximum and
minimum.stresses on which is superimposed a variable stress.S} .
The variable stress is equal to half the difference between Smax
and Smin, the proper algebraic sign being considered.

In calculating the static or mean stress Sm, the consensus
of opinion at present is that stress concentration effects due
to wdre curvature may be neglected for ductile materials. This
is consistent with neglecting stress concentration effects where
static loads only are involved. In figuring the variable com-
ponent Sr , however, stress concentration may not be neglected,

Some evidence in support of this method lies in certain
fatigue tests(9) on notched bars under combined static and vari-
ble stress. As shown in.Fig. 10, the mean or static stress
represented by the dot and dash line is not diminished by the
stress-concentration effect, while the variable stress repre-
sented by the vertical distance between either the full lines or
the dashed lines is diminished in a more or less constant ratio
.by the stress concentration effect of the notch. Moreover, the
fact that all spring stresses under fatigue load calculated by
means of the wahl factor K are too high is shown by the results
of a series of carefully made fatigue tests on small helical

springs of different indexes carried out by Zimmerli(lo).

If Wahl's formula with correction factor K is used in
figuring Smax and 8min the static component of stress Sm, when
figured by neglecting stress concentration effects due to

curvature, then becomes

;’ SM 50”! +5”?!
(53) 5.. :- -— . 3“

And the variable components of stress 5v is

In accordance with the previous discussion and the Seder-
ber’ methodwthat the relation between the limiting value of
the static and variable stress components at failure follows a
linear law, we can evaluate working stress in hilical springs
under variable loading as follows:

Referring to Fig. 11, the dashed line shows a typical experi-
mental curve of failure for materials under a combination of
static and variable stress. The ordinates represent values of
variable stress which will just cause failure when superimposed
in the static stresses shown by the abscissas.

We represent the shear endurance limit in a zero to maximum
stress range by 3.: . Then for this case (0 to maximum stress)
the mean stress 52.. and the variable component 5v are both equal
to Smax/Z. . If we locate a point B on the curve such that the

mean component 5... a s‘"‘/;_ and the variable component Sy: 5"”72

-20-

then the point B represents the condition where failure will

occur for a repeated stress from.0 to Smax. With the point E3
located, we can approximate the experimental curve as in Soder-
bery's diagram by the straight line 3A, where A is located by
the yield-point stress of the material in torsion 5; . Again,
dividing S,’ and 5"'/,by the factor of safety (N) defines the line

CD, such that any point C; on.this line represents a safe comp
bination of mean and variable stresses. From.the similar triangles

QGD and EBA , we get

Safih“:sfififi 1‘ 5%!
5,1- 55/: Sit/2

from which
N ch‘y S'-S,’.,
2i in... $2.
or (56) N - K. +5455 I)

The eq. (55) or eq. (56) may be used in design or to check for the
factor of safety N if we keep in mind the following limitations:
(1) In deriving'Wahl's formula we assume that the helix angle
of the spring is small, maximum stress is below the elastic limit,
and effects of eccentricity of loading due to end turns are neg-
lected.
(2) Data concerning the endurance limit and yield point stress

of spring materials are not plentiful, and those data available

-2 1...

from various experimenters are not in good agreement. Because
they are influenced by many factors such as surface condition,
shot blasting, overstressing, corrosion, and wire size, etc.,
‘we should choose them carefully, otherwise a larger factor of
safety may be used.

(3) For higher temperatures, the effects of creep or relaxa-
tion must be considered, but not many data are available for
springs under such conditions(11).

(4) Because some materials are not fully sensitive to stress
concentration the full stress-concentration effect corresponding
to the curvature correction factor K does not always occur even
for fatigue loading. In other words, when such materials are
tested by means of specimens having notches, holes, or fillets,
the fatigue strength reduction produced by the presence of such
"stress raisers" is not as great as that to be expected based on
theoretical stress-concentration factors. This so called “notch
sensitivity" of material is still unknown to metallurgists.
Until we can get more such informations about materials, we may
introduce other factors to rationalize our design procedure
farther.

Example: The force on an automobile valve spring of 27/32
in. inside diameter is 52 lb. when the valve is closed; and
140 lb. when the valve is open. Determine the diameter of wire

if the factor of safety is 1.2.

-22-

Solution: Choose valve-spring wire of $’=.98,000 “Va"

and S". 3 70/000 [5/0” . The mean load is F», 3 Edit/£1 :36"

and the variable component is then Fr ' wile?“ =44'6.
From eq. (56) we have
98,040 arséflx z_x__ufi
1.2 a "a: + [at —-—£?;L‘K(z ago-l)

or
(57) 8/600 = 43%;.“ 4 ég—kXI-e

Assume 61’ 0-094", {be}; Ream ”1'32; 3 0,034 40,423 =45/6 °
c 8 "/4 = “"1094 95-5

From Fig. 12 0: (=55, 5.7/19, k=/.£8L The right side of
Eq. (57) becomes

z .516 O
€M§m§4j x/ ‘5 + 333;? “‘3’!“ ”4/4“” 44200 ski/640%

hence diameter of wire : 0.188"

-23-

 

 

 

 

-24-

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(61.)

(b)

 

 

Ari: of JP”:

 

 

 

-25—

Arés’ of 5%?!

 

{59.9

 

/7\

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Tins;

 

 

flak)

 

 

 

“ID

a;

4;
5M 1552::

 

 

 

 

 

 

 

-25-

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0 1
Cu + 3(- + lbc‘

3 l
HT???

«4 ,

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.L.
léc'

.JL.
“VH6

“3

 

12
16
20
24
28
32
36
4O
44
48
52
56
6O
64

.125
.083
.063
.050
.042
.036
.031
.028
.025
.023
.021
.019
.018
.017
.016

64
144
256
400
576
784

1024
1269
1800
1936
2304
2704
3136
3600
4096

.016
.007
.004
.003
.002
.001

.062
.023
.012
.008
.005
.004
.003
O 002
.002
.002
.001
.001
.001

2.016
1.554
1.373
1.292
1.237
1.198
1.170
1. 150
1.133
1.120
1.109
1.100
1. 093
1.088
1.082

2.057
1.580
1.403
1.310
1.252
1.213
1.183
1.161
1.144
1.130
1.119
1.109
1.101
1. 094
1.088

1.250
1.167
1. 125
1.100
1.083
1.071
1.062
1.055
1.050
1.045
1.041
1.038
1.035
1.033
1.031

1.645
1.353
1.248
1.191
1.157
1.132
1.113
1.100
1.090
1. 080
1.073
1.068
1.063
1.058
1.054

 

Correction factors

helical round-wire springs

Table I

-27-

of

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. n . - - -l
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. . n -
u . -
_ _ u _
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_ . u . .
. . u _
. . . _ ..
w _ _ I.
_
e . .m - _,.
_ . ._
_ _
—
u

   

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l

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I

I

I

I
l
I
n
I
I

 

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A 1 . .

i

v

           

 

 

23 7-":
22 "_"-'

‘3 _-_...“ ..

15 . —..—. .—

I6

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.—-—-— _._ .
I .

lO

 

        

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.
.
A.
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—28-

(1)

(2)

(3)
(4)
(5)

(6)
(7)

(8)

'(9)

REFERENCES

A. M.‘Wahl, "Stresses in Heavy Closely Coiled Helical
Springs" Transactions A.S.M.E. 1929 paper A.P.M.,
51-170

Timoshenko, Strength of Yaterials, Van Nostrand, Part 2,
2nd Edition, P. 65, Also Timoshenko, loc. cit.
P. 74.

A. M. Wahl, Mechanical Springs, Penton, Chapter IV.
Timoshenko, Theory of Elasticity, MoGrawaHill, P. 290.
O. Goehner, "Schubspannungsverteilung im Guerschnitt
einer Schraubenfeder" Ins-Arch. Vol. I, 1930,
P. 619; also "Die Bevechnung Zylindrische Schraw-
benfedern" JeDeIo Mar. 12, 1932, P0 2690
Timoshenko, Theory of Elasticity, McCrawbHill, P. 357.

Timoshenko, Theory of Elasticity, McGrawaHill, P. 360,
Footnote (1)

C. R. Soderbeny, "working Stresses", Transactions
A.S.M.E. 1933, APM 55-16.

Federstaehle-Hondremont and Bennek, Stahl und Eisen
Vol. 52, P.660; Also Proceedings A.S.T.M. 1937,
Vol. 37, Part I, P. 162.

(10) Transactions A.S.M.E. Jan. 1938, P. 43.

(11) F. P. Zimmerli, ”Effect of Temperature on Coiled Steel

Springs Under Various loadings", Transaction.A.S.M.E.
Hey 1941, P. 363.

-29..

 

It. 1...! V i w 1

USE 031‘!

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