deflections of belleville springs by william m....
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Deflections of Belleville Springs
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Authors Faust, William Morray, 1936-
Publisher The University of Arizona.
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DEFLECTIONS OF BELLEVILLE SPRINGS
by
W i l l ia m M. Faust
A Thesis Submitted to the Faculty o f the
DEPARTMENT; OF CIVIL ENGINEERING
in P a r t ia l F u l f i l lm e n t of the Requirements
For the Degree of
MASTER OF SCIENCE.
In The Graduate College
UN I VERS ITY OF AR IZONA
196}
STATEMENT BY AUTHOR
This thesis has been submitted in p a r t i a l f u l f i l l m e n t of requirements fo r an advanced degree at The U n iv e rs i ty of Arizona and is deposited in The U n ive rs i ty L ib ra ry to be made a v a i la b le to borrowers under rules of the L ib ra ry .
B r ie f quotations from th is thesis are a l low able w i th out special permission, provided tha t accurate acknowledgement of source is made. Requests fo r permission for extended quotations from or reproduction of th is manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in t h e i r judgement the proposed use of the mater ia l is in the in te re s ts of scholarship . In a l l other instances, however, permission must be obtained from the author.
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
J a n , 2 0 , / $ £ /R~. Schmi dt
Associate Professor ofC iv i l Engineering
Date
ACKNOWLEDGEMENTS
The author wishes to express his app rec ia t ion to
Dr, Robert-Schmidt f o r his heip pnd guTdance- in th is
thesis and to his w i fe Janet f o r her pat ience and en
couragement throughdut i ts w r i t i n g , -
TABLE OF CONTENTS
^
LIST OF FIGURES ' --x c \ ■ ./ -'l': . :: i v
L is t OF tABLES '' "'T ' v " ' ; ; ' v
NOMENCLATURE . i ; ■ v i
Chapter ....; ", ■ ; • '
' I INTRODUCTI ON ^ - . ' -Vv : ■ 1
H is t o r y o f B e i T e v i i i e Spr ings 2
C h a r a c t e r i s t i c s o f B e l i e v i l i e Sp r ings 4
Advantages and V e r s a t i l i t y o f B e l l e
v i l l e Sp r ings v; ■ : ; 6
A p p l i c a t i o n s o f B e l l e v i l l e Spr ings 10
l i - DIFFERENTIAL EQUATIONS AND ENERGY INTEGRALS 12
Equi 1 ibriurn Equations I. 12
Expressions fo r S tra ins 13
Other Important Relat ionships 14
The Ritz ,Method 18
Dimension!ess Forms o f Equations 21
I 11 PRESENTATI ON OF SOLUT IONS 24
, Case I ' ' ' 24
Case 11 - |1
Chapter Page
IV DISCUSSION 36
Case I 36
Case I I 40
BIBLIOGRAPHY 53
APPENDIX 54
L I S T OF FIGURES
Fi gure
la , Di sk Spri ng Approximately to Scale
lb Schematic Drawing of Disk Spring Showingthe Sense of Various Q uanti t ies
2 D e f lec t io n Per Un it Thickness vs. Load/ Factor :
3a Spring Mechanism of B e l l e v i l l e SpringsStacked in Series •
3b Spring Mechanism o f B e l l e v i l l e SpringsStacked in P a r a l le l -
k V a r ia t io n s in Methods of Loading B e l l e v i l l e: Springs ■' .'
5 The R a d ia l ly Tapered Disk Spring
6a Free Body Diagram o f a S e m i- In f in i te s im a lElement of the Shell
6b Free Body Diagram of the Middle Port ion ofthe She l1 .
7a D i f f e r e n t $ a 1 El ement o f Mer idian Line on theMiddle Surface Before and A f te r Deformation
7b . D i f f e r e n t i a l Element of a C ircum ferent ia lLine on the Middle Surface.
8 Load vs. D e f le c t io n
9 Load vs. D e f lec t ion
10 A Comparison o f the Actua1 Shear With That Calculated from the Assumption
Page
43
43
44
45
45
46
46
47.
47
48
48
49
50
551
L I S T OF TABLES
Table
1. Comparison of Certa in Values as Given by the F i r s t Approximate Solut ion and an Exact Series Solut ion
Page
52
v
NOMENCLATURE
r s 0 Polar coordinates in planes perpendicular to theax is of symmetry
s Heridiortal cdord inate (see Fig. 11%;) j
z thickness coordinate (see F ig . l b . )
h y. th i ckness of .th#. sheTTr^y: y;y
d Free cone height
a sb . r coordihate of the i hnef" and outer edges„ respect i v e l y
0 i n i t i a l cone angle (see Fig; lb . )
P Total a x ia l edge load
usw Components of displacement of the middle surfacein the meridional d i r e c t io n and the d i re c t io n normal to the middle surface , re s p e c t iv e ly
§ Rotat ion of a meridian o f the middle surface
e r » e Extensional s t ra in s i n r and 0 d i r e c t io n s , respec-• " : t i v e l y ■ y
E Modulus of e l a s t i c i t y in tension and compression
~P PoiSson's r a t i o
D F lexura l r i g i d i t y of the shell
Nrs N. Normal forces per u n i t length o f sections of thes h e l l perpendicular to the r and 0 d i r e c t io n s , res p e c t iv e ly .
Mr , Mt Bending moments per u n i t length of sections of theshe l l perpendicular to the r and 0 d i re c t io n s , res p e c t iv e ly
Q Shear fo rce per u n i t length of a sect ion of theshe l l perpendicular to the r d i r e c t io n
A' \ ' R a t i o y o f a .toyb-y'
A u x i l i a r y notat ion is explained in the t e x t .
v i
CHAPTER I
INTRODUCTION
The B e l l e v i l l e spring, or disk spring,^ is a truncated
shallow conical she l l of uniform thickness. I t may be. simp
le r to th ink of i t as an annular p la te tha t has been dished
s 1i g h t1y in to the shape of a cone. Such a spri ng is shown
in Figure l a f
The Be1 le y 11Te s p rin g is usua l ly loaded only a t i ts
edges by c ircum feren t ia 11y uniform loads, a x ia l in d i r e c t io n
and with a sense such tha t they tend to reduce the cone angle .
In most cases the edges are completely f re e to move. How
ever, c e r ta in a p p l ica t io ns req u ire tha t e i t h e r the outer edge
be res tra in ed from rad ia l expansion, as would be the case i f
the spring were Inserted in a c y l in d e r , or th a t the innef edge
be res tra in ed from rad ia l c o h trac t io n , as would be the case i f
a shaft were inserted through the spri ng. Some a p p l ica t io ns
may require tha t both of these f e s t r i c t io n s be imposed.
l ju Me Wahl, "Design and Select ion of Disk 'Springs,1’; Machine DesignV Vol.. 11 (March, 1939), 32-37«•
1
HISTORY OF BELLEVILLE SPRINGS2
B e l l e v i l l e springs were f i r s t used in th is country
about 1890 as counter^recoi l springs in c e r ta in large guns.
Production d i f f i c u l t i e s were such th a t no p r iv a te company
would undertake to produce the springs, and the U. S. Army,
a f t e r considerable experimentat ion , produced those necessary.
Short ly t h e r e a f t e r , however, these e a r ly B e l l e v i l l e springs
were replaced by h e l ic a l springs, which were found to be
cheaper and eas ie r to f a b r ic a t e .
During World War I the U. S. Army obtained designs f o r
gun carr iages from the French government in which B e l l e v i l l e
springs were used to place i n i t i a l compression on packings
which had to be capable of holding a pressure of more than
100 atmospheres. 1nasmuch as B e l l e v i l i e spfings were id e a l l y
suited fo r th is purpose. I t became necessary to produce them
in g re a t q u a n t i ty . These springs were a lso used a t tha t t ime
as rebound springs on caissons. A f t e r some m odif ica t ion of
production procedure, these springs proved s a t is f a c to r y ,
and a f t e r the war the Army undertook to design more powerful
guns involv ing la rg er reco i l mechanisms. Consequently,
B e l l e v i l l e springs of d i f f e r e n t dimensions and character
is t i c s were needed. S ince;there was no s a t is f a c to r y design
^Do Ao Gurney, "Tests on B e l l e v i l l e Springs by Ordnance Department, Ul S. Army," Trans. A .S ,M ,E . , Vo l . 51 (January- A p r i l , 1929), 13- 15, , .
■ ;■ ■ ■ ' 3 procedure e x is t in g a t tha t t ime, the Army Ordnance Department
c a r r ie d on extensive research between 1919 and 1929. A l
though th is research shed considerable l ig h t on the subject
of B e l l e v i l l e springs, i t did not y i e ld a s a t i f a c t o r y
design procedure.
Not u n t i l 1936 was th is need f o r a s a t is fa c to r y design
procedure f i l l e d a t Which time J. 0. Almen and A. Laszlo
published a paper which contained an approximate solution^ to
the Be l lev i M e spri ng problem and ou t ! ined a method of design
employing th is s o lu t io n . This s o lu t io n y ie lded resu lts in
good agreement w ith experimental data and was l a t e r made the
basis f o r a design manual^ fo r B e l l e v i l i e springs published
by the S oc ie ty of.Automotive Engineers, The Almen-Laszlo
so lu t ion has been used almost e x c lu s iv e ly f o r the las t 24
years as a hasis f o r the design of B e l l e v i l i e springs, and i t
has given adequate resu l ts fo r most design purposes.■ : ■ ..." ’ • 5
However, G. A. Wempner has recen t ly presented a numer
ica l so lu t io n to the governing d i f f e r e n t i a l equations fo r the
Bel l e v i M e spri ng which ind icates tha t there is considerable
3“The Uni form, Section Disk Spri ngsu Trans. A,S.M, E. aVol . 58 : (May, 1936)f 305-314. , : r , ■; y ?, ; ,
4 p u b l ica t ion SP-63» a v a i la b le from S.A.E . Special Publ ic a t io n s Department, 29 West 39th S t re e t , New York 18, N.Y,
5“Axia lTy Symmetrical Deformations of a Shallow Conical Shell,11 (Doctoral Thesis, U n iv e rs i ty of I l l i n o i s , 1957).
TQom fo r improvement in tbeA im en -La s z Io s o lu t io n . Also
Go A* Wempner^ and R. Schmidt and G. A. WempAerZ have recen t ly
advanced approximate solut ions to th is problem which appear
to be b e t te r than the AlmenrLaszIo s o lu t io n . I t is be l ieved ,
however9 th a t these recent approximate solut ions can be im
proved upon, and i t is the endeavor of th is thes is to do so.
CHARACTER8STICS OF BELLEVILLE SPRINGS
The simplest type of spr ing , a p r ism at ic tension spec
imen, e x h ib i ts a c h a r a c te r is t i c which is ty p ic a l of s tructu ra l
members in the e l a s t i c range: a l in e a r lo a d -d e f le c t io n curve.
Accordingly, the h e l ic a l spr ing , the vo lu te spr ing , the r ing
spring, and most common springs have a s t r a ig h t Vine r e l a t i o n
ship between load and d e f le c t io n . On the o ther hand,
Bel leyi l i e springs usual 1 y have a nonlinear re la t io n s h ip be
tween load and d e f le c t io n , and i t is possible to design
B e l l e v i l l e . s p r in g s w i th many d i f f e r e n t l y shaped lo a d -d e f le c t io n
curves. By varying the fundamental parameters, i t is possible
to obtain p o s i t iv e , zero, and even negative spring rates i n .
given port ions of the lo a d -d e f le c t ion curve.
Most a u th o r i t ie s ^ report th a t the most important
^‘■Axial ly Symmetri cal Deformat ions. . . 11 bp. ci t . .
‘The Non 1 i near Conical S p r i n g Jour . App. Mech. „Vol. 2 6 : (December, 1959), 681-682. , .
8 j „ 0. Almen and A. Laszlo , J . J . Ryan and A. M„ Wahluse th is c r i t e r i o n in t h e i r papers 11sted 1n the Bi b l io g ra p h y .
parameter in varying the shape of the lo a d -d e f le c t ion curve
is the r a t i o p f the f r e e cone he ight , dj, to the she l l
th ickness, h. For values of d/h near 2 .0 , a curve such as
th a t labeied "A" in "Fig . 2 wi l l occur. This curve demon
s t ra tes the “snap-through1* c h a r a c t e r is t i c which is a res u l t
of i n s t a b i1i ty of the cone as the cone angle i s reduced to
the v i c i n i t y o f zero. Consequently, large a d d ! t io n a l d e f le c
t ions occur in thi s regioh accompanied by a decrease in load.
Of course, once the spring has reached a s tab le pos it ion
a f t e r ^snap-through^, increase in d e f le c t io n is accompanied
by increase in load. For values of d/h near 1 .5 , th is
“snap-through61 act ion is reduced to the point tha t a f t e r a
c e r ta in load, d e f le c t io n is increased considerably with ho
appreci able change i n load, as i s shown by curve :,,B“ i n
Fi g .2 . Thi s type of Bel l e v i l i e spr i ng is the so-cal led
“constant- load" spring. For values of d/h less than 1.5,
increasing load is accompanied by increasing d e f le c t io n , as
is shown by curves and “ 0“ in F i g . 2. i t is in te re s t in g
to note tha t Ryan^ has found th a t f o r v a 1ues of d/h between
0i. 11 and 0.17 the Bel lev i l i e spring exh ib i ts a l in e a r load- .
d e f le c t io n curve over a considerable range.
Another c h a r a c t e r is t i c o f in te re s t is the fa c t tha t
compared to h e l i c a l , v p lu te , e l l i p t i c lea f spr ings, and other
^^C haracter is t ics of Dished-PI a te (Bel le v !V ie ) Springs as Measured in Portable Recording Tensiometers, “ Trans.A .S . M.E. Vol. 74 (May, 1952), 431-438. '
comraon springs of comparable s iz e , the Bel l e v i l i e spring is
a high load, small d e f le c t io n spr ing .
Also of in te re s t is the fa c t tha t the f l e x i b i l i t y of
disk springs is a funct ion o f the r a t i o , A, o f the inside
to the outside diameter of the spr ing. For instance, Almen
and L a s z lo ^ report th a t fo r an i n i t i a l l y f l a t Bel l e v i 1le
spring, t h a t is d /h / ' is 'zero,' the maximum f l e x i b i l i t y occurs
fo r a value of A near 0 . 5 .
ADVANTAGES AMD VERSATILITY OF BELLEVILLE SPRINGS
To s t a t e tha t the B e l l e v i l l e spring is superior in any
given respect to another type of spring is somewhat mi s iead-
?ng. C e r ta in ly there are spring appl i c a t io n s v in which other
types of springs are f a r super io r . However, in a great many
instances, the B e l l e v i l l e spring has d e f i n i t e advantages.
By f a r the grea tes t advantage of d isk springs and the
predominating reason why they are used in various app l ic a t io n s
is the wide range of lo a d -d e f le c t io n curve shapes tha t can be
obtained. In a p p l ic a t io n s c a l l in g f o r a nonl inear r e l a t i o n
ship between load and d e f le c t io n ; the B e l l e v i l l e spring is
very useful
In cases where heavy loads and only moderate d e f lec t io n s
are desired , the B e l l e v i l l e spring is again useful due to i t s
^ . . : V . . . . 7compactness in the d i re c t io n of loading. A h e l ic a l spring$
fo r instance, w ith these c h a r a c te r is t ic s would require a
very heavy c o i 1, probably w ith a diameter of about the same
dimension as the f r e e cone height of an equ iva len t conical
spri ng. Si nee usual ly a t le a s t three loops are requi red to
make a he l i cal spr ing, the equi v a le n t B e l l e v i 1 le spring
would be several times more compact in the d i r e c t io n of
loading, ^ -. ■ . , . ' . . . . ■
In cases where heavy loads and r e l a t i v e l y large de
f le c t io n s are required, several B e l l e v i 1le springs stacked
in ser ies may be used. As might be expected, th is procedure
does not decrease the load-icarrying capacity but increases
the a 1lowable d e f le c t io n by the number of springs used.
Although ser ies stacking decreases the compactness of a
spring mechanism consisting of Bel 1e v i l i e spr ings, i t is
probable th a t th is mechanism is more compact than an equ iva len t
spring of another type. Another method of arranging disk
Springs, p a r a l l e l s t a c k i n g , 3 enables one to increase the
load carry ing capac ity of a spring mechanism over that of an
ind iv idua l d isk . In ad d i t io n , such m u l t ip le disk mechanisms
have the advantage over other types of springs in tha t f a i l u r e
/ . '"'^^^^ee'Fig. ’sa.'' ^ : '"' X /
T%lmen and Laszlo, "The Uniform Section Disk Spring," pp. 307-308. H ,: , r y; ;..
- ‘ . F ig . 3b. v . '
... ■ . / • .- . . . : , ' . ■ . 8
of one of the disks wi 11 riot cause complete loss of f l e x i
b i l i t y , and in the case of ser ies stacking such a f a i l u r e
w i l l not increase the load on the remaining d isks. ^ Also,
i t is a simple matter to replace a broken d isk .
Another advantage of m u l t ip le disk mechanisms is the
f a c t th a t i t is possible to vary the f r i c t i o n damping e f f e c t ! ^
In the case o f ser ies s tacking, v i r t u a l l y no in te rsp r in g
f r i c t i o n occurs, and only s l i g h t f r i c t i o n occurs between the
outermost springs and the support ing r ings , as is apparent
from the s l ig h t hysteres is loop which can be obtained on a
io a d -d e f le c t io n diagram fo r such a m edian ism .^ In the case
of p a r a l l e l s tack ing, however, considerable in te rsp r in g
f r i c t i o n causes a large hysteres is lo o p .*7 Hence, by employ
ing e i th e r or both methods of stacking disks in a spring
mechanism, i t is possible to vary f r i c t i o n damping.
Brecht and Wahl report th a t uniform heat treatment in
the case of disk springs is considerably e a s ie r than i t is
in the case of heavy h e l ic a l s p r i n g s . Also, B e l l e v i l l e
springs, wi 11 t o le r a t e l a t e r e ! as wel 1 as a x ia l loading. *9
Wo A. Brecht and A. H. Wahl, "The R a d ia l ly Tapered Disk S p r in g ,% Tr&ns. A.SoMfE. , Vol . 5 2 - (May-August, 1930) ,45 .
o* A.lmeri and A. Laszlo , "Disk Spring F a c i l i t a t e s Compactness,11 Machine Des ion . V o l . 8 (June, 1936), 42.
iG ib id : ; ■
* 7 1 bi d . - . '
ISuThe R a d ia l ly Tapered Disk Spring," p . 4 5 .
19 ,b id . i
' 20Almen and Laszlo a lso point out tha t the chara c te r
i s t i c s of a given B e l l e v i l l e spring may be v a r ie d consider
ably w ithout a l t e r i n g the spring. This v a r i a t io n is brought
about by applying e i th e r or both of the c i r c u m fe r e n t ia l ly
uniform loads on the face of the disk between the inner and
outer edges as is shown in F ig . 4. '■ '
Several au tho r i t ies^^ have pointed out the need to im
prove the e f f i c ie n c y tha t is , the un i fo rm ity of stress
d is t r ib u t io n throughout a disk under load - - a n d the f l e x i
b i l i t y o f the B e l l e v i l l e spr ing. According to Brecht and
W a h l t h e f l e x i b i 1i t y of the disk spring can be improved
by rad ia l 1y taper!ng i ts thickness in a manner shown in
F i g . 5. I t should be noted th a t a disk spring which has any
v a r ia t io n in thickness is no longer c a l le d a B e l l e v i l l e
spring. - r
Although none of the more complicated loading arrang e
ments and disk a l t e r a t io n s mentioned above are considered here
in d e t a i l and on ly the r a d i a l l y tapered disk spring is con
s idered a t length in any of the references g iven, these v a r i
a t ions do point out the v e r s a t i 1i t y of th is type of spr ing.
C e r ta in ly fu r th e r study of these more complicated cases is
warranted.
20t‘Disk Spring F a c i l i t a t e s Compactness," p. 42.21 j . 0. Almen and A. Laszlo,- W. AZ Brecht and A. M. Wahl
and Joseph"Kaye' Wood (see discussion of paper by D. A. Gurney, p. 17 of tha t r e fe re n c e )„
22nY^g yy Tapered Bisk Spring," py45.
APPLICATIONS OF BELLEVILLE SPRINGS
Only a few examples from the wide ra n g e of app l ica t io ns
of disk springs W i l l be c i te d .
The most common a p p l ic a t io n of B e l l e v i l l e springs is o f
the "constant- load" type. In app l ic a t io n s where i t is d e s i r
able not to exceed a c e r ta in value of load and i t is not
possible to control d e f le c t io n tod c lo s e ly s the “constant-
load" spring is very u se fu l . One a p p l ic a t io n of th is type
has a lready been c i te d : the device used on large guns to hold
a s p e c i f ie d pressure on c e r ta in packings. Here i t would
c le a r ly be d i f f i c u l t to control the d e f le c t io n of the spring;
y e t , i f the d e f le c t io n of each spring can be held to a range
of 0«,8h to 2 . 25hs the pressure on the packings can be con
t r o l l e d t o w i th in 5 per c e n t . ^ In another a p p l ic a t io n of
thi s type , “constan t- load11 spri ngs are arranged to take the
load from the bearings of the l i v e t a i l - s t o c k center -of. a
la th e . Thus i t is guaranteed th a t these bearings w i l l not be
overloaded due to expansion of the m ater ia l being machined.
A si mi 1ar appli cat i on of d isk springs has been made to support
commutator bearings in e l e c t r i c motors. In pressing, stamping,
and punching machines, B e l le v i l ie springs a r e v e ry useful
because of t h e i r high load c a p a b i l i t i e s . Here a lso the
“c o n s ta n t -1dad“ character i s t I c i s p f i mportance s i nee 1oads
^3Wah1, “Designing Constant-Load Disk Spri ngs,“ p .59 .
24a 1men and Laszlo, “Disk Spring F a c i1i t a t e s Compactness," p. 42. f ■ ; V v:; ’V '
are appl ied with considerable impact, and i f the d e f le c t io n
can be held to the value prescribed above, the load applied
by the pre^si ng, stampi hg,' or puhchi ng devi ce can be mai n
ta ined a t a constant va lue . In app l ic a t io n s of th is type,
■Ryan^S- points out tha t Bel levi l i e springs have been used on
high speed machines I n ; constant operat ion fo r many months
without excessive f a i l u r e s . I t may there fo re be concluded
tha t Be 1l e v i 1ie springs have good fa t ig u e res I stance.
In a p p l ica t io n s where springs with heavy load capa
b i l i t i e s are needed and only l im i te d space is a v a i la b le ,
B e l l e v i l l e springs are of great u t i l i t y . I t is f o r ju s t such
a reason tha t B e l l e v i 1ie springs were chosen to serve as
c o u n te r - re c o i l springs on large guns. Ryan^ has made a
unique and in te re s t in g a p p l ic a t io n of the compactness and
heavy load c a p a b i l i t i e s of B e l l e v i l l e springs. For values
of d/h between 0.11 and 0.17 he found tha t d isk spri ngs ex
h i b i t a l in e a r r e la t io n s h ip between load and d e f le c t io n .
Usi ng spri ngs wi th th i s s p e c i f i cat ion, he was able- ' to bui 1 d
a portab le tensiometer weighing 6 pounds, having an o v e r -a l l
length Of 9 inches, and capable of measuring suddenly app l ied
loads o f the order o f magnitude of those in .fowlines attached
to ships or barges So successful was th is a p p l?ca t io n , th a t
Hr. Ryan fe e ls tha t the grea tes t use of B e l l e v i l i e springs is
ip instrument a p p l ic a t io n s .
25»Character i s t i cs o f Di shed-PI a te ( B e l l e v i l l e ) Springs as Measured by Portable Recording Tensiometers," p. 438.
2&l b i d . , pp. 431-438.
CHAPTER I I
DIFFERENTIAL EQUATIONS AND ENERGY
' INTEGRALS
A so lu t ion of the di§k spring problem requires
considerat ion of la rg er d e f lec t io n s than are usu a l ly taken
in to account in t!the small d e f le c t io n theory'1. ^ In the-= , 1 , " ^ O
fo l low ing ana lys is the s o -c a l le d “ large d e f le c t io n theory"
wi 11 be used.
EQUILIBRIUM EQUATIONS
Reference is made to F ig . pa in which a typ ic a l element
is shown,which has been cut out from a conical she l l of u n i
form thickness by two meridi onal and two c ircu m fe ren t ia l
sections normal to the middle surface . By taking the sum o f
the pro jec t ions of forces act ing on th is element in the
See S, Timoshenko and S. Woi nowsky-Kri egery Theory of Plates and S h e l ls , (Npw Yorks 1959)$ PP« 533-568.
28see Timoshenko and Woinowsky-Kriegers pp. 396-428 fo r a discussion of large d e f le c t io n s of, p la te s . , .
V ■ . . ■■ .A:.;' 12 , . . ■ ■ ■ .
13
meridional d i r e c t io n , neglect ing magnitudes of higher order ,
and l im i t in g considerat ion only to s h a l l o w ^ s h e l ls , we
obta in
N ^ r ^ - N t = 0 (1)
For the same element, by taking moments of a l l forces
with respect to an axis perpendicular to one of the bound
ing meridional sections and by using the foregoing c r i t e r i a ,
we obtain the second e q u i l ib r iu m equation:
- M r - - - r Q (2)
In F ig . 6b the middle port ion of the she l l has been
cut out by a c i rcu m feren t ia l section that is normal to the
middle surface. The t h i r d e q u i l ib r iu m equation may be ob
ta ined by summing the pro jec t ions of a l l forces in the a x ia l
d ire c t io n which act on th is port ion of the s h e l l . Thus,
+ r N r(^+<$)= - r Q ( 3 )
fo r smal1 values of ^ and
EXPRESSIONS FOR STRAINS
Reference is made to F ig . 7a which shows a d i f f e r e n t i a l
^ T h e i n i t i a l cone angle ($) should be less than 0.10 rad ians .
element of a meridional l in e on the middle surface before
and a f t e r deformation of the s h e l l . By use of the Pythag
orean theorem, we f in d
ds' = ds S f - ( I f - + 1
By expanding the r ig h t side of Eq. ( 4 a ) , neglecting
magnitudes of higher order, r e s t r i c t i n g considerat ion to
shallow s h e l ls , and using the fo l low ing d e f i n i t i o n of
s tra in:= _ ds*- ds e - — d i -
we obtain
e f = dy + ( t + $ )p
s inee ds ^ dr
By means of Eq. (4b) and F ig . 7b we f in d
S t= r
OTHER IMPORTANT RELATIONSHIPS
M u l t ip ly in g both sides of Eq. (5) by r and d i f f e r e n
t i a t i n g with respect to r , we obtain an expression for d^.
S u bst i tu t in g th is expression in Eq. ( 4 c ) , we obta in the
fo l low ing compatabi1i t y equation:
15
From Hooke's law we f in d
, e r = E h ( N - - " V N 1t ) • ( ? a )
N t = 7 ^ ( e t + v e r) , N r= ^ ( e r + v e t) (7b)
Eq. ( 6 ) may now be w r i t te n in terms of Nr ,Nt andp
i f Eqs. (7a) are used to e l im in a te e r and et . E l im inat ing
Nt from the re s u l t in g expression by means of Eq. ( 1 ) , we
obtain the f i r s t governing equation.
djSL _ CL /& (8)
In the work that fo l lows i t is convenient to put Eq. ( 8 ) in
the fo l low ing form:
1 8 * >d r
The expressions fo r bending moments in large d e f le c t io n
theory are id e n t ic a l to those in small d e f le c t io n theory
and are g i v e n ^ by
M r = ’ D ( j r + f P) ’ M t = " D( r + v d r ) ( 9 )
imoshenko and Woi nowsky-Kr i eger, p . 52.
16
By e l im in a t in g Q between Eqs. (2) and ( 3 ) , we obtain
an expression in terms of Mt , Mr , Nr , and jg . Mt and Mr
may be e l im inated from th is expression by means of Eqs. ( 9 ) .
Thus the second governing equation is
This equation may be put in the fo l lowing equ iva len t form:
terms of displacements only. I f th is is the case, the f i r s t
governing equation is obtained by e l im in a t in g Nr and from
E q . ( l ) by means of Eqs .(7b ) . Values fo r e^ and e^ given
by Eqs. (4c) and (5) are s u b s t i tu ted in the re s u l t in g
expression, and we obtain
Rewrit ing th is equation as in the case of Eq. ( 1 0 ) , we
obtain
( 10)
( 10a)
I t may be des irab le to proceed in a given problem in
( 1 1 )
( 11a)
The second governing equation in terms of displacements
obtained from Eq. ( 10a) w ith the a id of Eqs. ( 4 c ) , (5)>
the second of Eqs. (7b ) . Thus,
THE RITZ METHOD3 '
18
A so lu t ion of e i th e r Eqs. ( 8a) and ( 10a) or Eqs. (11a)
and ( 12) which s a t i s f i e s the appropr ia te boundary conditions
w i l l be an exact so lu t ion to the disk spring problem. A l
though the general form of such a so lu t io n is known, i t would
be extremely d i f f i c u l t to f in d a s p e c i f ic so lu t io n which s a t
i s f ie s the appropr ia te boundary condit ions. Therefore , one
must resort to e i th e r numerical or approximate methods of
s o lu t io n . One method of obtain ing such an approximate solu
t ion is the R i t z method.
I f a system is in a s ta te of s tab le e q u i l ib r iu m , i ts
to ta l energy is a minimum. In the case of large de f lec t io ns
of a shallow s h e l l , the to ta l energy. I , consists of three
terms: the s t r a in energy due to bending, V j , the s t ra in
energy due to s t re tch ing of the middle surface, and the
p o te n t ia l energy of the load act ing on the s h e l l , . These
expressions in t h e i r respective order are given as follows:
(14)
(13)
33
32
^Timoshenko and Woinowsky-Krieger, pp. 343-346.
3 2 | b i d . , p. 345.
33 I b i d . , p. 400.
p b 34\4 = "2.TT I (wc|)rdr ( '5 )
and
I = V , + V2 +V 5 (16)
We may assume tha t the r o ta t io n at any point of the
shell can be represented in the form of a ser ies
p = K ,F , ( r ) + K2F2 ( r ) + ^ ( r ) + . . . KnFn( r ) (17)
in which F p F 2 ,F^, . . . Fn are functions chosen so as to
s u i ta b ly represent the d e f le c t io n surface and s a t is f y the
boundary condit ions. S u bs t i tu t io n of Eq. (17) in Eq. (16)
resu l ts in an expression fo r I in terms of the c o e f f ic ie n ts
K |,K 2JK3 , . . . Kn . In order tha t I be a minimum, these
c o e f f ic ie n ts must be chosen such tha t
w 0 , f t r 0 ’ - ■ | k 0= o (18)
These condit ions y ie ld a system of n a lg e b ra ic equations
in K p K 2 ,K^, . . . Kn, and each of these c o e f f ic ie n ts can then
^Timoshenko and Woi nowsky-Kr i eger, p. 345.
be determined. By a wise choice of the functions
F s, . „. , Fn we may obtain an approximate so lu t io n which
very close to the exact s o lu t io n .
PI MENS 1ONLESS FORMS OF EQUATIONS
21
In the work that fo l low s , i t was found convenient to
introduce dimension 1 ess v a r ia b les which are defined as
f o l 1ows:
N - ^ N r , W =
In terms of these new v a r ia b le s , E q . ( l ) is
K I + c t d N _ f \ j - o
doc
By using dimension 1 ess va r ia b le s and e l im in a t in g Q
between E q .(2) and (3) we obtain
j i f e t M ) - M = ^
In terms of di mens i onless v a r ia b le s , equations ( 4 c ) ,
( 5 ) , ( 6 ) , ( 7 a ) , (7 b ) , ( 8 a ) , ( 9 ) , (1 0 a ) , ( I l a ) , and (1 2 ) ,
r e s p e c t iv e ly , may be w r i t te n as fo l lows:
(19a)
(19b)
(19c)
( la )
(2a)
_ddd h" V ZttDoc
= i ( ' ' v X z + ? ) ' ( P + ^ ) f e
22
( 4 d )
e , = S (5a)
e r = e t -v c ^ d | t + ( 6 + 4 ) p ( 6a)
N - v>N l - Ve t " 5 e r =
K l -v N iI - P2- (7c)
N= e r + 9 e^ , N = e ^ + v e r (7d)
( 8b)
M = d & + V | , M = v dp_+ g (9a)
23
_d_doc
In the s p e c i f ic case of the B e l l e v i l l e spr ing, i f
dimension 1 ess v a r ia b les are used, Eqs. ( 1 3 ) , ( 1 4 ) , and (15)
may be w r i t te n as fo l lows:
V = ir D
-p ^ (w h e r e wa is the d e f le c t io n at r=a)
I f i t is des irab le to work in terms of displacements
only, s u b s t i tu t in g Eqs. (4 d ) , ( 5 a ) , and (7b) in Eq.(14 ) we
f in dirE h S
V . - Z ( d a ) ( | + § ) ? + ( % + #
■($S ^ ( 3 ) i n + z v ( 3 ) ( |+ $ > ] c x d a
( 1 2 a )
( 1 3 a )
(14a)
( 15a)
(14b)
CHAPTER I I I
PRESENTATION OF SOLUTIONS
CASE I
The most p r a c t ic a l and common 1y used ap p l ic a t io n s of
B e l l e v i l l e springs are those in which the edges are completely
f re e to move. The boundary condit ions in th is case are
As stated in the foregoing chapter, an exact so lu t ion
to th is problem would be very d i f f i c u l t . Hence, an approx i
mate so lu t ion w i l l be presented using the R i t z method.
As a f i r s t step in the a p p l ic a t io n of the R i t z method,
we must make an assumption as to the shape of the d e f le c t io n
surface. I t has been found by F . Dubois^S tha t the stress
d is t r ib u t io n in a shallow truncated conical she l l has the
same character as tha t in a c i r c u l a r p la te w ith a hole a t the
center . This discovery suggests tha t the d e f le c t io n s in the
N = 0 a t d = 1; N=0 a t ql = A
M = 0 a t oi = 1; M=0 a t ° = A
( 20a)
( 20b)
3 5 1bi d . , p. 564
24
25
case of a B e l l e v i l l e spring may be s im i la r to those in an
annular p la te . Guided by th is reasoning, we take
A = K(36= K[cxJLnOi + B ot + -£ -J( 21 )
in which ^>s is the r o ta t io n of a meridian on the middle s ur
face of an annular p la te as given by the small d e f le c t io n
t h e o r y . E g . (21) is a s im p l i f i c a t io n of E q . (17) in which
only the f i r s t term in the ser ies is taken. The constants,
B and F , are given as fol lows:
A*2-I / v t A I_B> - , _ i+ v (21a )
r - - j T v ( j ( 2 1 b )
S ubst i tu t ing the assumed expression fo r , Eq .(2 1 ) ,
in the f i r s t governing equation, Eq. ( 8b ) , we o b ta in , a f t e r
one in te g ra t io n ,
dbj = + Aocirvod + A (X + - g - r ^ ^ L +
+
36 l b i d . , p . 5 9 .
26
in which the a d d i t io n a l constants, A 7 } / \ , and T , are
defined as fol lows:
A = ib l - A 1" l+v .
< _ i t i + ^ v a ^ a < a l i - w l i - a 2-
ZA^i^Ai-A’ l+V
SA^jBtv. A _ 4 + v l l -A7, l + A j
A I r Z - A ^ A r A’ -gyvA. 1 . 2^'/V I6 (_ h a 2 L l -A 7 2. I+V
V = l b ( l - A 2)2.A2(lviAjl(A'2'+ 4 f ) - jBrxA(2A4- A A"
-3 U A 4+ a r 2)tlb Az- 8AA* + ifoAA1 - A f
L-I
( 2 2 a )
( 22b)
( 22c)
( 22d)
( 22e)
T = vTa^ + 1 AXi-A) -Z T A ^ I-a ) -Z rA ( l-A ) ( 2 2 f )
We now w r i te Eq. ( 14a) as fo l lows:
27
(14b)
Since the boundary condit ions require th a t N=0 a t
oc =1 and oi =A, we f in d tha t the f i r s t in tegra l in Eq. (1 4b)
vanishes. Hence, we have
S u bst i tu t in g Eq .(22) in E q .( 14c) , we obta in a f t e r
in te g ra t ion
VL= C,K4 ■+ C ^ K 3 + C342K2 (23)
The expressions fo r the constants, C^, C2 and , are ex
tremely long and are presented in the appendix to th is th e s is .
S u b s t i tu t in g the expression fo r ^ , Eq. ( 2 1 ) , in Eq .( 13a)
and in te g r a t in g , we f in d
X= c 4kz (24)
28
where
C .» « T rD ^ -£ ? + t r t i - v )N T *
H-
- ( l+v ) + Z B A ^ A - C i - A ^ e ? (24a)
W rit ing Eq. (2 1 ) in the form
wti
t I
- y /b ^ d c x = ^ /b K o f^ r ia -+ Bex + ^ J d a
we obtain a f t e r in te g ra t io n
V5 = -Pvsnx- CqK,
where
( 21c)
(25)
% i n A(A% Z.r) + ( l -A^X '- 2 6 ) ( 25a)
S ubst i tu t in g Eqs. ( 2 3 ) , ( 2 4 ) and (25) in E q . (1 6 ) , we
obtain the fo l low ing expression fo r the to ta l energy of the
system:
1= C,K + C 3<$rK*+C4K* + C *K
Since the to ta l energy is a minimum fo r a system in
the s ta te of s tab le e q u i l ib r iu m .
( 26 )
t l = A t lKi + 3 C & K 1 + Z C t ^ K Z C 4K + C s =o ( 26a>31a
29
I f we le t A= 0 .5 and V = 0 . 3 , d iv ide a l l terms byZtrEhbCs
o - ^ jpand introduce the dimensionless parameters
(27)
Eq.(26a) becomes
0.046,919** - 0 . 0 6 9 , 8 7 5 ^K5- + O.OI9,777$4K + 0 .(fo6,b67 'T '1'K=p ( 26b)
Using Eq.( 2 5 ) , we may determine the value of K for
any desired d e f le c t io n a t the inner edge of the s h e l l , wa .
Hence, Eq.(26b) gives us the r e la t io n s h ip between load and
d e f le c t io n . This re la t io n s h ip is represented g ra p h ic a l ly in
Figs. 8 and 9 in which p is p lo t te d against w^. For com
parison, these f igures a lso include lo a d -d e f le c t ion curves
fo r a numerical so lu t ion of the governing equation as d e te r
mined by G. A. Wempner.37
A special case of in te re s t is the annular p la te of
uniform thickness. For th is case $ is zero, and Eq.(26b)
becomes
Assuming that p = 0 .000 ,041 ,325 , "Y = 0 .05 and solving
Eq. (26c) fo r K, we obtain
0.046.919K3 + 0 . 166,66772K = p ( 26c)
K = 0 .066 ,326 ( 26d)
3 ' A x ia l ly Symmetrical Deformations of a Shallow Con i c a 1 Shell
Hence» fo r t h i par t igu 1 ar ya 1 u,e o f 1 pad given above,
using the value of K together w ith Eqs. ( l a ) , ( 9a ) , ( 21) , ( 22)
and the in tegra ted form of Eq„(Z2), we may obta in numerical
values fo r ^ N and N a t any po int of the p la te .
Numerical values fo r these q u a n t i t ie s are given a t various
points of the p la te in Table T. For comparison, numerical
values fo r these same q u a n t i t i e s , as given by a series so lu
t io n determined by Wempher and Schmi d t f A a r e a ls o Included.
3®11 Large Def lec t ions of Annular P la t e s ,1* Trans.A.S .M0 E. , Vol. 80, pp. 449-452.
31
CASE I I
Although not as common as Case I , ap p l ica t io n s of
disk s p r in g s . in which both edges are res tra in ed from
l a te r a l displacements, but are s t i l l f ree to r o ta te , are
a lso of in t e r e s t . In th is case, the boundary condit ions
are
T | = 0 a t o( = 1; T| = 0 a t oc = A (28a)
M = 0 a t oc = 1; M = 0 a t oc = A ( 28b)
Due to the excessive length of the previous s o lu t io n ,
we sha l l no longer use Eq. ( 2 1 ) , but instead w i l l make the
fo l low ing assumption:
I t is not immediately apparent that there is any
merit in such an assumption; however, examination of Table 1
ind icates tha t M is very small compared to R and hence may
be considered n e g l ig ib le . Furthermore Schmidt and Wempner^
found tha t such an assumption gave ex c e l le n t resu l ts in the
case of f re e edges, and there is reason to b e l ie v e that i t
w i l l g ive s a t is fa c to ry resu lts fo r other boundary condit ions
also.
39,lThe Nonlinear Conical Spring."
Solving E q .(29) fo r p , we obtain
( 3 = - C o T
S u bst i tu t in g Eq. (3 0 ) in E q . ( l l b ) and in te g ra t in g
the re s u l t in g equation, with due regard to the boundary
condit ions, we obtain
( i -3v)C^ 1 8 v (i-v )
where
- A1' 2” _ I -F' " "PaT1 ’ F^ _ | -A "
a2( i - a - w ) f _ a20 - a - , v)^ ------------------------------------- 4 ' I - A 1
Subst i tu t in g Eqs. (3 0 ) and (31) in Eq . (14b) and
in te g ra t in g , we f in d
V, - K.C4 +■
where
K i - i -A1 )[_l Av + llv1- - 18V1 + 1 4 - / 1 +
- [ l - 2v + V t - l 2v3] i72' + [_3- <4v + tqvl - k 2v’ ] | z j
^ - - W- v l f e - v ) ^ ♦ 'fcv‘ - K V i
- [ l - Z v + ^ - T v ^ F T F , + [ 3 - i 2 v + t 4 v l - 7 v 1,] ^ j
32
(30)
(31)
( 31a)
(32)
( 32a)
\3v4^ +
(32b)
33
K * = + l l v I “ 'z ^ + 5 v4 ] F' +
-2 [ l - Sv -H v ' - fo v 3 + 4 ^ ] ^ + Z [ 3 - I 3 y + W - I4 v 3+4.y4] % j (32c)
Subst i tu t ing the expression for E q . (3 0 ) , in the
energy in t e g r a l , Eq. (1 3a ) , we obtain
V, = K4C2 (33)
where
„ _-rrEH$(A"'lv- 0-----------7 ^ (33a)
In te g ra t in g Eq. ( 3 0 ) , as in the case of Eq. (2 5 ) ,
we obtain
V3 = K5C = -Pwa , (34)
where
K5- ^ ( A ' - V- 0 (34a)
The to ta l energy of the system is , th e re fo re , the
sum of Eqs. ( 3 2 ) , ( 3 3 ) , and (34 ) :
I = K,C*+ K ^ C 1 + K ^ C 1 + + KbC (35)
and
^ = 4.K,C3 + 3 K ^ C ,- + 2.Ka$ tC + + K B= O (35a)
34
I f we le t A = 0 .5 and x> = 1/3 and d iv id e a l l terms
by - ^ h 1? ^ , Eq. (35a) becomes
0 .5 0 b ,Z 5 C a - l.fo4l,75ci>C* + O . S Z I J S i T o 0 . 0 9 4 , 4 9 3 7 ^ = p (35b)
In the case of an annular p la te s , Eq.(35b) reduces to
0 .506.25C3 + 0 .094 ,493 z7 ' e C = p (35c)
Sett ing C = 0.1 and T = 0 .0 5 , we f in d
P = 0 .000 ,184 ,960 Eb2 (35d)
Since i t was not possible to f in d another so lu t ion to
the disk spring problem which s a t i s f i e s the boundary condi
t io n s , Eqs. ( 2 8 ) , we must resort to another method of
checking the resu l ts given by Eqs. (35b) and (3 5 c ) . Although
the assumption, Eq. ( 3 0 ) , is an approximate s o lu t io n to the
problem a t hand ( th a t is , the disk spring loaded by uniform
a x ia l edge loads ) , i t w i l l be the exact so lu t io n of the disk
spring problem with something other than uniform a x ia l edge
loads. Hence, by computing some funct ion of the load, such
as the shear, Q, by means of the assumption, Eq. ( 3 0 ) , and
comparing th is assumed shear w ith the actual shear, we may
obtain a check.
We shal l consider f i r s t the case of an annular p la te
fo r which A = 0 .5 and V = 1 /3 . By s u b s t i tu t in g Eqs. (3 0 ) and
(31) in Eqs.(4d) and(5a) and s u b s t i tu t in g the re s u l t in g expres
sions fo r e^ and e^ in the f i r s t of Eqs. ( 7d ) , we obtain
35
N - ^ (36)
Then by s u b s t i tu t in g Eqs. (3 0 ) and (36) in Eq.( lOb) and
m u lt ip ly in g the re s u l t in g expression by 0/ we
obtain the fo l low ing expression fo r the shear:
Q " g t f o t * + ( 3 7 )
By comparing the shear Qf as given by Eq. (3 7 ) with
the actual shear Q given by
^ _E_^ Tjrbot (38)
we may obtain a check of Eq. (3 4c ) . Such a comparison is
represented g ra p h ic a l ly in F ig . 10.
A s im i la r comparison was made in the case of Eq. (35b)
y ie ld in g doubtful r e s u l ts . For a va lue of of 0 .08 radians,
i t was found tha t when the load was applied in the usual
sense, the assumed shear deviated from the t ru e shear by as
much as several hundred per cent. However, when the load was
taken in the d i re c t io n opposite to that shown in F ig . 4a, the
assumed shear d i f f e r e d from the actual shear by no more than
the dev ia t ion between these values as found in the case of
the annular p la te . Nevertheless, since the d i r e c t io n of the
load is as shown in Fig . 4a in most p ra c t ic a l a p p l ic a t io n s ,
i t was decided not to include these resu lts here.
CHAPTER IV
DISCUSSION '
CASE I : ■ -Since we have Vi ml ted our considerat ion to small values
of $ s i t seems reasonable to assume tha t the conical d isk
spring w i l l act in a manner s im i la r to the annular p la te , and
i t has been v e r i f i e d by F. D ubo is^ tha t the stress d i s t r i
bution is s im i la r in these two cases. Therefore , i t seems
qu ite log ica l to assume as in E q . (21) th a t , in the range of
small d e f le c t io n s , the r o ta t io n a t any point in the conical
disk w i l l be of the same character as tha t given by the exact
l in e a r so lu t io n fo r an annular p la t e . We might expect, t h e r e
fo re , th a t Eq:. (21) would give e x c e l le n t res u l ts i n the ranges
of small d e f le c t io n s . Since Figs. 8 and 9 i ndi cate th a t there
is l i t t l e dev ia t ion between the numerical and approximate
solu t ions in th is range, we may conclude th a t such a p red ic
t ion is q u i te accurate .
As we might expect, th is assumption is hot as accurate
fo r excessive ly large d e f le c t io n s , and Figs, 8 and 9 show
dev ia t ion between the approximate and numerical solutions in
^Timbshenko and Woinowsky-krieger, p. 564.
V n
th is range. This dev ia t ion is not too important, howevers,
since in almost a l l ap p l ic a t io n s o f disk springs, d e f le c t io n
canriot Continue a f t e r the spring is f l a t t e n e d . In o t h e r :
words, fo r most p ra c t ic a l a p p l ic a t io n s , is less than(§| „
In th is range. Figs. 8 a n d 9 in d ic a te that the assumption
stated in Eq .(21) gives good r e s u l ts .
For the boundary condit ions s tated in Eqs„ (20a) and
( 2 0 b ) , o ther approximate solu t ions have been o f fe re d th a t
have shortcomings not present here. Almen and Laszlo
have o f fe re d an approximate s o lu t io n in. which they assume
that ^ is constant and that the meridional s t r a i n is n e g l i -
g i b i e , these assumptions do nOt s a t is f y the boundary condi
t io n s , Eqs,(2 0 ) , and there is reason to b e l ie v e tha t they
w i l l not g ive good resu lts fo r smal1 values of A. In order
to sub s tan t ia te the la s t conclusion, consider the case in
which A=0 and cjj ==0. We have thus reduced the prob 1dm to th a t
of a s o l id p la te w ith a concentrated load a t the center .
For t h is case, ^ is zero a t the center and increases to a
maximum value a t the outer edge. 11 seems 1 i k e ly , th e r e fo r e ,
t h a t ^ w il l a lso vary considerably in cases where ^ ' i s not
zero and A is small . G, A. Wempner has found another
approximate so lu t io n In which he assumes only th a t is
‘'The Uriiform Section Disk spr ing j11 pp. 305-314.
^ 2 "A x ia l ly symmetrical Deformations o f a Shallow Conical S h e l l . ' 1 .
constant , Here again the boundary condit ions are not
completely s a t i s f i e d and i t is be l ie ve d tha t res u l ts w i l l
be poor f o r sma11 values Of A . A t h i r d approximate solu-
t ion was di scovered by R. Schmi dt and G, A, Wempner i n
which i t was assumed tha t M=09 as i n Case I I above. A1-
though th is assumption s a t i s f i e s the boundary condit ions,
Eqs.(20a) and (2 0 b ) , there ' s reason to belt eve tha t i t wi l l
not be v a l i d f o r Small values o f A. I f we consider the case
where A=0, We reduce the Problem to tha t of a sol i d cone w ith
a concentrated load a t the apex. For such a problem, we f in d
tha t the assumption tha t M=0 gives an in f i n i t e value fo r (S -
under the load. . This statement fo l lows from Eq. (3 0 ) in which
Poisson1s r a t i o v is a p o s i t iv e constant and C is not zero
when A is zero.. I t appears reasonable to assume, th e re fo re ,
tha t Eq. (30) wi 11 resul t in very large values of (3 fo r
values of A near zero, and we may conclude th a t the assumption
M=0» is not va 11 d fo r sma 11 ho les ,
On the other hand, the funct ion assumed fo r ^ given
by Eq. (21) i s cont i nuous when A=0 and Oc =0 . Also i t has
been shown^ th a t , fo r Small d e f le c t io n s , Eq.(3 0 ) reduces to
the exact .solut ion fo r a p la te w ith a conQentrated load a t th e
center when A=0. We may th e re fo re conclude th a t Eq .(21) is
v a l i d fo r a 11 values of the parameter.A. Furthermore, F ig s .8
^' IThe. Non 1 inear Conical Spring," pp. 681-682.
^Timoshenko and Woinowsky-Krieger, p. 60.
. - ' ; / 39 and 9 and Table 1 in d ica te t h a t , fo r comparable values of
parameters, E q . (21) gives resu lts a t least as accurate as
any of the approximate so lu t ions mentioned above.
Apparently , the major shortcprnings of the so lu t ion
presented in Case I is the length of the f i n a l expression,
. E q , (2 6 a ) . However, s u b s t i tu t io n of numerical values fo r .A
and v in th is expression reduces i t to one which is short
and contains a l l o f the other parameters as v a r ia b le s .
Therefore , although i t would be a laborious process, i t is
poss i ble tp p resen^ C and
g ra p h ic a l ly as functions of A. We would then be able to
evaluate Eq. (26a) for any vplue of: A, fand i t wouId be a
simple matter to obtain values fo r w, , M, M, N and N
fo r any combination o f parameters, AlsP, the so lu t ion
given In Case I could be programmed f o r a d i g i t a l computer,
and the v a r ia b le s , w, p ,H, M, N, and N, could be be evaluated
very ra p id ly fo r a disk spring of any dimensions.
Si nee the assumpt ion, Eq . ( 2 1 ) , was obtai ned f rom the
exact l in e a r s o lu t io n fo r an annular p la te , we would expect
the results obtained ip Case I to be excel lent in the case
of a p la t e . Table 1 substant ia tes such a b e l i e f . With the
exception of M, none Of the values given by the approximate
so lu t io n d i f f e r s frpm the exact values by more than 2%, and
many of the approximate values agree with the exact values to
three s i gni f i cant f igtires. Even 1 n th e case of the meri di onal
bending moment, H, the absolute d i f fe re n c e between values
; : . ' ■ - ■ v : ;x r ■ ■ : : 40
given by the approximate and exact so lu t io ns is o f about the
siame magnitude as the absolute d i f fe re n c e between the ap
prox i mate and exact values of H. Furthermore, compared to
M, the magnitude of M i s very small and would th e re fo re
never be c r i t i c a l In the analys is of d isk springs. We may
th e re fo re conclude th a t the large percentage e rro rs 1n values
of M are not as serious as they seem to be.
Since the small d e f le c t io n theory does not d i f f e r e n t i a t e
between the boundary cOnditibns s ta ted in Eqs. (20a) and (20b)
and those stated in Eqs.(28a) and (28b ) , Eq.(21 ) could have
been employed In Case I I a lso. However, i t was considered
d es irab le to f in d a somewhat shorter so lu t ion than that pb-
ta ined in Case 1, and the assumption that the bendi ng moment
M be zero was used instead. Such an assumption is based on
the fa c t th a t M i s very sma11 compared to M in a grea t many
cases, and a lso i t was found to g ive excel lent res u l ts in
c e r ta in cases fo r d isk springs w ith f re e edges.
In the case of the annular p la te . Fig. 10 seems to
indicate that the assumption, M=0, gives good resu1t s , Also,
since deflections are re la t iv e ly insensit ive to small vafia^
tions in the loading function, i t may we11 be that fhe de
f lec t ions given by E q. (35a) are even.closer to the exact
deflections than the assumed shear in Fig. 10 is to the exact
sheaf. We should keep in mind, however, that the same argu
ments used in the discussion of Eq. (30) in the f i rs t section
of Chapter I V hold hrere a lso , so th a t the assumption, Eq (30 ) ,
should not be expected to give good resu l ts f o r sma11 values
; « f a :
, Since such promising re s u l ts were obtained in the
case o f the annular p la t e , we may conclude tha t the assump
t io n , E q . (3 0 ) , w i l l g ive good resu l ts in the case of a cone
a lso , provided tha t is very smal l . However, i t was d is
covered tha t f o r values of $ even as large as 0 .0 5 , Eq.(35a)
gives very large discrepancies when shears are compared as
in F ig . 10. We may th e re fo re conclude t h a t , fo r r e l a t i v e l y
large values of ^ , the boundary condit ions, Eqs.(28a) and
( 2 8 b ) , are so r e s t r i c t i v e tha t considerable bending of the
meridian l ines is necessary before apprec iab le d e f le c t io n
may take p lace. Therefore , H may not be neglected in such
cases.
I t is in te re s t in g to note tha t i f the d i r e c t io n of
loading is taken opposite to the usual d i r e c t io n of loading,
resu lts comparable to those shown in Fig. 10 may be obtained.
Such res u l ts should probably be expected, s ince when the load
is taken in the d i r e c t io n opposite to that'.shown in Fig . 4a,
there is a tendency fo r the meridian l ines to be extended
ra ther than compressed. Since i t is compressive forces which
cause the bending of the meridian l ines mentioned in the
previous paragraph, i t seems reasonable to assume tha t ex
tending the meridian l ines w i l l reduce t h e i r tendency to bend
Therefore* i t seems l i k e l y tha t M w i l l be a t lea s t as small
in the case of a conical disk .with the load reversed as i t
is in the case of an annular p la te . Although such reasoning
leads us to b e l ie v e tha t Eq . (30) may give good resu l ts in
the case of a conical disk w ith the load reversed* such a
case is o f T i t t l e s ig n i f ic a n c e in p r a c t ic a l a p p l ica t io ns
of B e l l e v i l l e springs.
DISK SPRING APPROXIMATELY TO SCALEFIG. la
MERIDIAN OF DEFORMED MIDDLE SURFACE
MERIDIAN OF ^ UNDEFORMED MIDDLE SURFACE
FIG. lb SCHEMATIC DRAWING 1 OF DISK SPRING SHOWING THE SENSE OF VARIOUS QUANTITIES
LOAD
FA
CTO
R30
A(d/h = 2.0) B (d/h = 15)
/ — C(d/h = 0)2 0 -
i . o -
x 7
301.0 2.00
DEFLECTION PER UNIT THICKNESS
FIG 2 (SEE RYAN, P 432)
SUPPORTING RING
FIG. 3a SPRING MECHANISM OF BELLEVILLE SPRINGS STACKED IN SERIES
SUPPORTING RING
FIG. 3b SPRING MECHANISM OF B E L LE V ILL E SPRINGS STACKEDIN PAR ALLEL
46
(b) (d)
FIG. 4 VARIATIONS IN METHODS OF LOADING BELLEVILLE SPRINGS
FIG. 5 THE RADIALLY TAPERED DISK SPRING
Q + dQ Mr + dMj
FIG. 6 q FREE BODY DIAGRAM OF A SEMI-INFINITESIMAL ELEMENT OF THE SHELL. ALL QUANTITIES ARE SHOWN IN THEIR POSITIVE SENSE.
MERIDIAN OF DEFORMED MIDDLE SURFACE
^M E R ID IA N OF UNDEFORMED MIDDLE SURFACE
r
FIG. 6b FREE BODY DIAGRAM OF THE MIDDLE PORTION OF THESHELL. A LL QUANTITIES ARE SHOWN IN THEIR POSITIVE SENSE.
n o t e : s t a r r e d q u a n t it ie s r e f e r t o
THE DEFORMED SHELL
FIG. 7o DIFFERENTIAL ELEMENT OF A MERIDIAN LINE ON THE MIDDLE SURFACE BEFORE AND AFTER DEFORMATION OF THE SHELL. ALL QUANTITIES ARE SHOWN IN THEIR POSITIVE SENSE.
/
FIG. 7b DIFFERENTIAL ELEMENT OF A CIRCUMFERENTIAL LINE ON THE MIDDLE SURFACE BEFORE AND AFTER DEFORMATION
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
A =0.5 0 = 0 1 7 = 0 .05----------------- APPROXIMATE SOLUTION------------------NUMERICAL SOLUTION
/_____i______ i______ i-----------i-----------1----------- 1---------- :-----------1-----------1-----------1----
0.02 0 0 4 0 0 6 0.08 0.10
DEFLECTION, wab
FIG. 8 LOAD vs. DEFLECTION
24
22
20
18
16
14
12
10
8
6
4
2
0
A =0.5 $=0.1 7 = 0.03----------------- APPROXIMATE SOLUTION----------------- NUMERICAL SOLUTION
/
/001 0.02 0.03 0.04 0.05
DEFLECTION, wab
0.06 0.07 0.08 0.09 010
FIG. 9 LOAD vs. DEFLECTION
SH
EA
R,
60 \
50
40
& 30
20 A = 0.5 $ = 0 y - 0.05
---------------TRUE SHEAR, Q
---------------ASSUMED SHEAR, Q*
0.5 0 6 0.7 0 8 09 1.0
DIMENSIONLESS RADIUS, <x
FIG. 10 A COMPARISON OF THE ACTUAL SHEAR WITH THAT CALCULATED FROM THE ASSUMPTION
* Denotes Series Solution , Denotes Approximate Solut ion
: ; 1 0 .5 0 .6 0 .7 0 .8 0.9 L 1.0
* * 0 .1 1 4 *0 .1 08 +0.104 +0.101 +0.0982 *0 .0956*0 .113 *0 .1 08 +0.104 +0.100 +0.0976 +0.0948
* 0 *0.00505 +0.0105 *0.0121 i +0.00862 0M ' - 0 *0.00868 +0.0100 +0.0080 * +0.00441 0
: ' ■' .. * ' | *0 .207 *0 .1 65 +0.138 +0.118 l +0.102 v +0.0870M - *0 .2 06 *0 .166 +0 .138 ; +0.117 ..i +0.100 . +0.0863
' * NxlO3 - 0 *0 .2 4 9 ! +0.282 +0.221 +0.119 0 , ' \
0 *0 .2 4 8 +0.281 +0.221 +0,119 : 0 ■ |
• 'klNxlO^ - * 2 .1 8 *0 .913 +0.099 - 0.473 i -0.901 ■ - 1.23 :
* 2 .1 7 ; +0.912 +0.0976 - 0.475 r - 0.899 -1 .22
TABLE K COMPARISON OF CERTAIN VALUES AS GIVEN BY THE FIRST APPROXIMATE SOLUTION AND AN EXACT SERIES SOLUTION BY GV A, WEMPNER AND R. SCHMIDT.^5 VALUES PERTAIN TO AN ANNULAR PLATE FOR WHICH A * 0 . 5 , . ^ = . . 0 3 , Y ® 0 .0 5 , p-« 0 .000 ,041 ,325 .
^5"Large Def lect ions of Annular P la t e s ,11 Trans. A .S.M.E. , Vol. 80, pp.449-452.
BIBLIOGRAPHY
1. A1men3 J. 0 e and A. Laszlo, "The Uniform Section DiskSpring ," Trans. A .S .M .E . » --Vol, 58, pp. 305-314;.Hay, 1936.
2. A 1men, J. 0„ and A. Laszlo, "Disk Spring F a c i l i t a t e sCompactness, 11 Machine Design, Vol. 8 {June, 1936),40-42 . .
3. Brecht, W. A. and A. M. Wahl, "The R a d ia l ly Tapered DiskSpr ing ," Trans. A .S .M .E . , Vol . 52, No. 15, pp. 45-55; May-August, 1930, ..
4. Gurney, D. A . , "Tests on B e l l e v i l l e Springs by OrdnanceDepartment, U. S. Army," Trans. A .S .M .E . , Vol. 51,No. 10, pp. 13-18; January -A pr i1, 1929.
5. Ryan, J, J . , "C h a ra c te r is t ic s of D ished-P late (B e l le -v i l l e ) Springs as Measured in Portable Recording Tens 1ometers," Trans. A .S .M .E . , Vol. 74, pp. 431-438; May, 1952. ,
6. Schmidt, R . , and G. A, Mempner, "The Nonlinear ConicalS p r in g ," Journa1 of AppTied Meehanics. Vol . 26, pp. 681r682, December, 1959. ~
1L Timoshenko, S . , and S. Woi nowsky-Kri eger. Theory of , Plates and S h e l ls . New York: McGraw-Hi11, 1959.
8. Wahl, A. M., "Design and Se lect ion of Disk Springs,"Machine Design, Vol. 11 (March, 1939), 32 -37 .
9. Wahl, A. M . , "Designing Constant-Load Disk Springs,"Machine Design, Vol. 13 (October, 1941), 59-60. ,
10. Wempner, G. A. and R. Schmidt, "Large Def lec t ions ofAnnular P l a t e s ,11 Journal of Applied Mechanics, Vol. 25, Trans. A .S .M .E . , V o l . 80, pp. 449-452; 1958.
11. Wempner, G. A . , " A x ia l ly Symmetrical Deformations of aShallow Conical S h e l l . " Doctoral th e s is . U n iv e rs i ty of I l l i n o i s , Urbana, 111. , 1957.
' ' 53 .
54
APPENDIX
The expressions fo r the constants, C,, C2 and C3, in Eq. (23) are given as fo llows:
C, = -TTEUk1[(iwA)4- |1 .»aA)S+ 9 * W" WM
+ * b (iv^ “ i b + 3fa(»)] +
4 le ‘ l i l t " ) ] * % [(Iv-A)3- l (L .A ) l + | ( U k ) - 1 4. - ^ ( - y ] +
+ e " g ( ^ + -4^" f ( ^ A')1+ i (V A ')-3.+
• *4 (i» )] + L A + i - H i ) ] + ^ [ ( W A f - %(JLA) + ^ +
- H i ) ] * ■£ 4- - g ( l , ) ] 4- A TA 4 - 1 - 1 ( 1
+ ^ [ 4 . A - l . l (1 .)] + ^ [(in +
+ &Y a? Qiv-A-{+1 (i)] + [a6- ll + rX^v'A "i 4 Hi4)]+-»• y [a 4_ i ] + r - - ^ [ - ^ A - ' | + Hif + A V [ a- . ] + ^ [(M *4 -
-juA+i-Hi)] ■* [-LA-i + H i ) ] + H t ^ A),l + t[(1v ]+
+ ( [ f - ' ] + ^ [ ( ^ A ) 1] -4 Z 1 y [ ^ a ] - ^ [ ( J L A ) \ J U A + H H 5]
- ^ [ i - A + 1 - l A 4] + ^ [ a1- ' ] ^
4
55
+ i s i ^ A) + s + ^ - § (? ) ]+
+ i t - a«a + i _ M i 1) ] + 4 r [ ^ A)l_ l ^ A) + w _ ^ ( i sj 4
+ AT A4 [.£*»&- 7 + Mi1)] + 4 'x [a3"1] +
/ A t [ a ? - ' ] + ^ [ ( K A f - |C £^A )+ | - I f r ) ] - 4- X ^ [ • ^ AX 4 3(-A')]+
+ r l A[iMA-1 + i ] + '^•[C.Cv'A)1] + [ X a --^ 4 iC i i j ] + X " t-A 5" ']+
+ 7 .r^ [A -l] + Z Z T ^ a] + ^ [U v ^ A f- 2 (£m A) + 2 - ’z( i j ] 4" ' ^ X a - n £|4.
+ r 2T A [ e v , A - i + i ] + z tm ,[a - i] - x X a + i - a ] - § X A ^ - 4 At] +
Cj = -trEhb1 [ C ^ A ) 2- jC ^ A )+ i - i X ) ] +
+ x ' X A x 4 4 ( i 4 ] ^ ^ X A- i 4 M i 4 ] + x ' K A- | + i ] +
+J^JaS i] + r r [ A 2- i ] + z t t [ a - i] + r 2 [jU a] + I L J ^ - 1 ] +