The analysis of thin-walled pressure vessels by relaxation methods
by David H Drummond
A THESIS Submitted to the graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Civil Engineering
Montana State University
© Copyright by David H Drummond (1949)
Abstract:
The following paper is a further application of the method of suc-cessive corrections. The analysis of
thin-walled cylindrical vessels is shown to be easily made by an application of the Hardy Cross
moment and shear distribution procedures.
The paper is divided into two sections. The first section consists of an analysis of thin-walled
cylindrical shells and flat-plate heeds Which is made by separating the membrane stresses from the
bending stresses. such a separation results in the deflection equation for thin-walled shells being
(Formula not captured by OCR) where r' is the membrane deflection, and the remainder of the
expression is deflection due to bending.
Tbe second auction of the paper shows the method and resulting ex- pression for the fixed-end
moments and shears, the carry-over factor and the distribution factors which are necessary for the
application of successive corrections.
Examples are worked by both methods to demonstrate the practicability of the method introduced. TBB ANALYSIS OF IEm^WALLBD PaESEfOKE TBSgRSLSt
BY
mlAXATIOB METHODS
DATID SE* DRQWOND
A THESIS
S u bm itted to th e O raduate G o m itte e
in
p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts
f o r th e d eg ree o f
M aster o f S c ie n c e ■in C i v i l E n g in eerin g
at
Montana S ta te C ollegeA pproved:
In Charge o$H fejor"fori?
.Bozeman, Montana
December, 1949
' ' 1!'fUIli i!lU!i, i :i'j:
IHhl,-!
,'(iz; 'i;.
IV 3 ,1 ^
D S A cv
-S-
0 ^ ,1
ACKHOWLEDOEMEHT
I w ish to express my g r a titu d e to A sso c ia te P rofessor B. C. DeHart o f
th e C iv il
u g ln ee r in g Department o f Montana S ta te C ollege fo r h is sin c e r e
h e lp and guidance throughout the w r ittin g o f t h is paper.
/ / jj/stJJyytr, *?7/kr7/(i /
92G1B
Tabla o f
AbSil^^Ct? ^
* 4 .«
Xtl.i/3?0(1^10ii-3*052*
**
K*
^
** A». A.
W*** , * AW^w, \ M* * w' * W* i ' ' A* * * ' <4' 4a. * * i oi * ' ^M* * w^, ^#* , * ^, ^* , ^* * * * »* ^
^
SeatXoJj $
A n a ly ala o f s tr e s a e o i n TiJiijHWalled ^ re o e u re V daeele
Xxjt^Oouot I ojx
* » ,1 * ,^ ^ ^ ^ .* * * ^ * ,* » * * ,* *
Ti^QJll"ijJ^tiXiG *”>GjX(y*1\5SO W
r W, M* *M-S*' ** "• W rr» rv»
B e f o rm tio n s Due to 'E e stra s e s t r e e s e s
^
Jr* iH, „ Wr nt v^wuWWt uf wi WWW 1*0
w#,.**-*.*-*
20
Tbe Change .to -S h e ll R adiue Due to lA:ere8?rane S tr e s s e d
3,1
Membrane S tr e s s e s and D eform ations in Y eeeo ls w ith Son-Uniforrft
R reeeu re U ie trib u tx o ji . w i w w — <** — w> — —i
w i w
** »>■**.** — -*-<•« « — *<*
BendiBg S t r e s s e s l a C y lin d r ic a l T h in -U allo d S h e lls ~ ~
~ ~
is
R e s is tin g Moments, m& Bheurs in T h in -C a lle d C y lin d r ic a l S h e ila
- - » IV
The R e f le c tio n o f a T h in -T a lle d C y M ad ritial S h e ll Due to
Bending * » SO
T o ta l S e fle e tlO tt o f T h in -w alled c y l i n d r i c a l S
l
h
e
l
s
The SffO tit o f S h e ll le n g th on th e T o ta l D e fle o tio n - - -
—
S t r e s s e s Due to Bending in F l a t C ir c u la r Heado - —
- —
v
si
- - - - gg
»—
33
R e s is tin g Eomenta and C heers i n F l a t C ir c u la r Heeds
gg
D e fle c tio n o f F l a t C ir c u la r Beads u n d er Uniform, lo a d - - - - - - - -
g6-
IrajJJttlti I
— — km — — —, — —
— — —- — —- — — — — — —
Fxomple 3
—
— -w
— — — — — ——
— 33
^ —— — — — — — — — — 33
seotlon i$
Sueeeasive Corrections Applied to Ttott8^ a lled .Presouro Vessels
In fro duo t i 0u
— ———
The Eethod of Successive CforreeMofts
—j^ - - —————
30
35
FSxadHW Koaeiits in 3» d s
to
O y liM r io a l B&ella Gttbjaetad
rw I're a s n re D la tr lb tttlo a
Fixed-Sad 1CWmttta l a lo n g T b itt^ e lla d C y lla d r lo a l G bella Subjeoted
to Uniform Preaeurte D iatribtttlott ^ * - * * - * - - - - * .
«■ ~ «. * * * 40
Fixed-Sad ISssm te in Qbort Maitt-Wellea C yllttdrlttol G b elle su b jected
to iron-Ottiforra Preeeure D is tr ib u tio n ^ * * - - » - ^ - - - - - - 4 0
FlxedHSttd K"ome%t8 Itt Short Tbltt-Walled C y littd rlo a l G belle Subjected
to Unifozai Preosure D is tr ib u tio n
^
w^
w
44
Fixed-Sad Sb ser in Ioug Tbltt-Walled O y lin d r iea l S h e lia su b jec te d to
IiOYi-Uttifo r a Preaeupa D is tr ib u tio n
- « . - * - - " " ^ - + 44
F ixed -M d GhouT In Isag- T hltt-^ alled C y lin d r ic a l O b olls Subjected to
Uniform P ressure D istr ib u tio n
Fixed-Bad Shear in sh o rt T liin-U elleS O ylittd rleol S h e lls Oubjeeted to
Mon-Dttlform P ressure B la tr ih n tio ii
~
~
45
Fixed-Fad Shear in Short Thia-Vtolled O yliarlriool S h elia Subjected to
Uniform P ressure D is tr ib u tio n
4@
Carry-Over Factors fo r Short ThIn -U a lled C y lin d rici I s h e l l s su b jected
to Uott-Uniiorm Pressuz1O D is tr ib u tio n * ^ « # . - . - . - - ^ - . - « - . - 4^
The I iffe c t o f a U n it ./ingle Change on the Toment a t th e Bad o f a lo n g
T h in -S alled C y lin d r ic a l S h e ll
“ ^«46
The B ffe o t o f a B a it Angle Change on th e Toment a t th e kad o f a Short
Thltt-VaXlad C y lin d r ic a l S h e ll
The X ffea t o f s U nit Tosient Change on Shear a t the M d o f a lon g
Thitt-Walled C y lin d r ic a l Ohol I
The E ffe c t o f a U nit Tosiont Change on Cheer in a sh o r t T h in -b a lled
C y lin d r ic a l S h e ll
Si
fhe S ffetit o f a Bzdt MeplaeeBeni' on th e Sheas * t the Md of a tong
tid C ylindsitiel S h e ll
—»■-««■**«..-. SI
The B ffeoi o f a B hit Bleplstiemeht on the Slteai- in a Short ThinW elled O y H n d sle e l S h e ll
»» «»
- •«■ •*■ «««. -i »_ «.
gg
The S f f e tit o f a W l t change' 1» S h ear a t Oao BftS <*» th e Mozaants a t
' Both Wdg o f a Short ih i& ^ s lle d C y lin d rical S h e ll - * - * * ( * « «
SS
Slxed-EM SbBieafcS and Shears in F la t O lren lar Seadh
84
The E ffe ct o f a Cfnit ,angle •Ohsnge on th e Eosents a t the Sdgei o f a
/:
H s t O ifottlar Head w - # *
50
w.. nr «1
Motneafc P ietrib a fclM F actors fo r T eg e e lg 'w ith lo n g Thin^Walled Qylindriw
c a l S h e lls toid F la t Heads
-
S?
- »!•**»•« -
SB
'Moment B is tr ib a tlo n F actors fo r T eseels w ith Sherfs Thin^Walled
C y lin d rical S heila and F la t Iltiadg
X0
«w Wv **
^ ^
O bH c X u b
Wt 1* w* w* k* ^ W" ,-+w W ^ **
^ -Cr
^ 4» ^ ** ^ ^ ^ **" f"* ^ "*
-w ^ w *4, * *r ^ ^ ^ ^ ^ /* ^
M' 4* 4^ -W #
*C« w *w ^
^
w
^ ^ ^ Mf'**
^ **, «
^ ^ -w* w' #- —i T ^ ** ^ ^
^
^ 00
^ ^
©X
Appendix
Tabic I
Coneteats f o r Fixed-Bad Bbaaate and W ears in Short s h e lls ^ SB
Table TI Gonataatg f a r D istrib u tio n s and Carry-Over in Short S hells - 64
Sumraary o f Forraulii fo r the Relaxation SiothOd
-
I f a t o f Abbrotrlstions
.* J*,. *4 fk W' ^ "k «*, *t" Ar
65
S?
f"
^ *» I* *4 a* ** *6 ^
^ ^
00
i fh e f o X lo tia g p a p e r i s a f u r t h e r 8#g&i&atto% &£ t.$B ^netMS o f
o e e a iw eorre#M hh& *
$he a n a l g i a e f
S ^ i M r i e a X tre s s e ls i s
ehomi to be e a s ily meae by m aR plioetloa o f the Bewgy Orow moment aiwl
s h e a r d i s t r i b u t i t e i p ro c e d u re s e
th e p a p er i s d iv id e d IW o two s o o tio n s*
file f i r s t -seo tio tt tionsis'bs
o f ah e*a& y*le-a# th lm ^ aX led , c y H h d r io a l s im ile m d f la t- o p ia te heed©
Which i s made by sep aretto g th e wmbraae ©.trossee item th e beading s tre s s e s
Suoh a sep ara tio n .result© in the defleetjkoti equation fb r I f h ia ^ a lle i -QheiXs
being
fa
(A% -soe Ba f
s in B%) * e'*®58 (% cos Be t % .@ ia B%) * -W
where r'* la the seabraae Heflsetioa^ a a i the remaiaier o f the expression
is deflectloB # w to beaiiag,®'
9%e geeomi w e tlo * o f the ^ a p e r »hw »
p r e s s lo a f o r th e f Ig M m sai o m m t s # # * # * * 8 * #
*#&*&
resu ltin g. #»"
amrrymover f a o t o f ##* '
th e H le tr lb u tio h <% ot6r».iA W k o fH n g # e am $ y A w tW a p p l i o a t l o a o f ***»
c o s tiv e c o rre c tio n ^ *
RmmglHe w o worked by b o #
o f th e w lfh o i Ia tro iu o o tljf
to Hmoaotrate # o pm otloab iilty"
mmoDO&TioB
Duping the Ba8$
years 3 gpoat aeal o f l&tereet hae bee& abawa
in m raepioal methods o f a n a ly s is , partlealG P ly f o r engineoring ^w h lm #
^hieh do not lead themsolvos readily to rig id matbe#ti@al analyela* %bl8
Ia tO fea t m e a tn m le te d i a tb la oooatry a ia o e t e n t ir e ly by p r o fe s e w Bardy
Crossd) Dho presented th e ptooedures fo r moment d is ty lM tio n ,
Snmerioal
methods of bnalyel# had been weed before the presentation o f th is *@rk but
a n m tre n d „ o f accenting th e physical ra th e r than the m theToatieal as*
p a c ts , was introduced by th is method*
The anm erioal methods o f s u c c e s s iv e do fr a c tio n s has no# been ap p lied
t o such en g in e e rin g f i e l d s a s h yareh lios» e l e c t r i c i t y , vib r& tion a, thermo*
dynamics and stress© a n a ly s is *
The wide u se o f th e method a tte st® to i t s
pTsC t io a b lli ty«
T his t h e s is I e a fa r th e r a p p lic a tio n o f th e r e la x a tio n method*
The
eork Io based on th e n&rdy Oroas methods o f momenta and sh e a r d ie tr lb a tio n
and p e a n lte in a slm p liflG a tlo n o f th e co n v en tio n a l method o f thl#*Q aH ed
pressure v e sse l anclyals*
Tbs g en era l eq u ation fa r tbo d e f le c t io n a t any p o in t i n & th in -w e lle d
c y lin d r ic a l s h e l l I s found to be
v*eBK(A4 cos Ba » Bi sin Ba) *
cos Bs + 9% aln Ba) + r*
Dhere A,* %* 0& and % a r e co n sta n ts r e a u ltln g from th e s o lu t io n o f e
d l f f o f e n t l e l eq u a tio n ,
These co n sta n ts a r e shown to b e dependent o n ly on
th e manner o f su p p ortin g th e ends o f th e s h a ll*
(DlW dy Oroes* Analysis o f Contlnuow Frames by D istrib u tin i Fixed*B&a
!'tOSienta., Transactions A, 3«.
B», M2<s, pp I a-ISS*'
The #oaveBtlo&a& m&tbed a f
g & w l# # # # #gua%&e& th& aefle#*
tlo& a f t&e # b # li.%* %&# Af i w feBWa&a&x#m#b&Be a t tb@ joiaW ,
Biap#
there e t $e#et fob# aabaowa do&etao&e t& thee* aagreaeloaat. i t 1& aeeeaeefy
to equate Qlto t&e aomeat or dba&i&g# f@pe& &qg&t&&a0 + Tbie aea&e #%t #
Ie aeoaeeaey to ao&ve o t la a e t #&%& e&@B%t@aoog# agttatlobe* a%4 e&aea the
r eetra in t 6& the a&e&l ie aepe&Q##t o& %a etlff& ees of the reetto ia lo #
^etaber I t ia sebeesary to oolvo those Ibur eqaotltme fbr each v e s s e l.
By ap p ly!## a r t i f i c i a l r e e ta a ta te to #@ ea@s o f th e s h e l l to ollm*
ln a te any rotaM os o r i e f i e c t l o # o f % #se p eia tS s th er e la on ly one s e t o f
boundary con d ition # f o r a l l s h e ila *
th e work from th e aoltrblos*
Thfe allm lnatga a major p o rtio n o f
The w t t f l o l a l r e a t e a ln ts a re then resiow d W
auooeealve'eorrootlone,.
I n th e B B alyole @f w in * v e ile d e y & ln a eie a l a&aZla th e membrane
etreooea ore aaperatcd from the bending etreaoea*.
This Ie neoeeaary elnoe
the balancing o f momente end Aheere Ie !n # p e n a # t o f the membrane atreacea,
&KOTIQR %
am&BBB o p ,CPPHLfma i a w i w m i K D m E B sm s v m m g
A t&&8*w«#Wi veaeal la (sew* t& which the **11 thleKaoe* Se
mmll la ee^eeleoa with the Qleoetey of the veeeel.
%&e degree to w&ieh
th la Goaaitloa m e t he s a tis fie d le mch th a t the woumptloo *» IKbe wait
Btreoeea w lthia the m aterial, aeglectjkos tboee jpmdiiced by head lag* ere
w ifoim aaroae the longltadihal oroea^eotioa of the Treaeol - caa be made
w ith l i t t l e e rro r »
A thia^wolled Bhellj free from the reotra ilin g effect* of heada or
etiffeim ra end abhjeoW to an in teraal fla id presmme, aa^ande. wifoaa&y
1% the eaaial ana lOBgitudlaol direotio&s and tb@ atreoaeo Iadaaoa are
uniformly diatr&Wt&a over any eroae-aeatioa*, Tbeae atreaaoe are the mem*
braae atreaaeB ea& era t&@ major etmaao* to be ooBoMered la a pre*mx%»
vaeael a t palate where the#* ere so aieeoat&Rultle** For iaateaee, &
v e s s e l w ith a a e a t r e l oarfaoo la th e form o f 8 epheee *&& w ith m dleooa*
tiw&itle# 080b oa flvatoa o# ye&%f@e@ea jo late weaia have ao atreeaea
o th e r than moobrot* otveeaos#
A eoay bubble I e a p erfa o t exasipl® o f eaoh
a atrea# state#
a. preaauro
voaeel with a thia^welled e y lla d r le e l s h e ll ouch a© i s to
be ooaoidorod hero, Baat have a t le a st one e&a elOGed* Obetber the Oloaape
la fomo# byhy&raalio pressor© and a juaetlo* with some other etraotwro
o r by a bead, & dlaeoatlaulty w ill oatet a t the joint*
Tbla dlecoat&awlty
in the uniformity of the etm eturo w ill produoe ©haaria# aa& benaiag
stresses which era not uniformly distributed over the loagltndiaal oresssec tio n .
For # e agpllgatioa of Giicogeaivo
I t la aeoeoeary to
e o a a ld o r th e atreaao B 1$) th e V eeool a s tioiagl mWbro&o o tro o o e o pfoauceA by
the iatoeaol flo ia MWoew#* wpeeWOao* o& bend&ne etr&aeee pro@wee& by
I
r e s t r a is in g .rasienta and eh earo,
f)%STf%W%Q%9 * THe a& gaitw * o f t&a W i t o tr e ea M tiB g on a lo n g l*
t u a ia e l o ro a o -o e o tio a o f a o ylin & rizzl v e s s e l oabjoeteG to m in t e r e a l
f lu id pres e w e p Ie
S1. si JBJSl
^
t
(a)
s&ere 8% I e th e u n it o tr e s s a e tin g &@#&el to th e lo a g it n a in o l oro sa s o o tio a , r Io th e m & iae o f th e v e s s e l m a t la the w a ll thiekaeoo*
The m g n ita d o o f th e u n it Gtreoa a o t is g oa a tre& everee o ro a o -o eetia a
1# g iv e n by th e eXpreoolon
Sg i®
C
where Ga i s th e w a it e tr e e a a c tin g normal t o th e tm n ev o ree e r o a e -o e e tio a
OS th e v e ss e l*
Foe ooBVenie&oe th e a tr e e se a 0% ao& 8%, w i l l h e r e a fte r be referred to
a s th e e lro u m fe r a a tia l membrane e tr e a e and th e W g l W W l mewbreae a tr e e e
r e e p e o tI v e ly *
9%B0R?%Tiam DQB TD KEBBBARB STRB03&8 ~ TbO membrane at****** found above
e l l l b e ageompeaied by deform ations o f the heads end s h e l l o f the v e s s e l*
In th e hcade th e s t r e s s e s w i l l produce a chooge l a the diam eter a t the
juaotloR w ith, the s h e ll*
Sbwevefa l a a v e s s e l w ith f l a t heads ee *111 be
oonaldefed h e r e , t h i s eba&ge w i l l be n e g lig ib le 1% cwmparieoa w ith th e
-1 1 -
o th er deform ations
a p p recia b le errors#
nd a sy be considered zero w ithout Introducing any
The deform ations o f the s h e ll c o n s is t o f an elon­
g a tio n In th e lo n r ltu d ln o l d ir e c tio n and a civ n e In the s h e l l ra d iu s.
s the deformstlo n In the lo n r ltu d ln o l d ir e c t io n has no bearing on the
work to fo llo w , i t .ray be om itted <nd o n ly the r a d ia l d e fle c tio n found.
■
o o t on
m
- The membrane s t r e s s e s
d i f f e r e n t i a l p a r t ic le o f the c y llm lr lc I s h e ll in the manner shown
In Figure I .
The X -axis and Y -axis rep resen t the lo n g itu d in a l and trans­
v e rse d ir e c tio n s r e s p e c tiv e ly end the X -axis represento the d ir e c tio n
normal to the s h e ll su r fa c e .
y
o is a o n ’ e
a t lo
2
c lr c u r fe r o n tin l d ir e c t io n .
I s o by Young’ o Modulus
Figure I
Sy
K 6y
where #y i s the u n it d lo to r tlo n In the c irc u m fe re n tia l d ir e c tio n produced
by the s t r e s s
3 ,,.
,quoting th ese ex p ressio n s end s o lv in g f o r Qy g iv es
Sie&«^
- ——
uUp
eg ■
Il«Pl
g
y
o r , u sin g th e v a lu e s fo r
s
and G2 given by equations (a) and (b)
by l e t t i n g AC bo th e obengo IA t h e l m g t b o f th e o irc w fw e m e e o f
th e e b e l l @a& # w o ti* g tb e b&ea&e 1% tb e . to & W o f tb * a e t t t r e l atw faoo o f
the sboll by Z^i i t Ie st-Meat that
40 » S vrey »
* %);
Bad
r*
or
v
r* A "Hgi® *■ ft) * % .* * 6 4
* t (I )
Tble e%pre88ioA ia derived aalng the etreeeee defimod by o q w tW e
(a) and (b) Abiob are baaed on the aeam&tlon. th a t the preeewe la m lformly Aiatributed Altbih the veeael,
Eqtmtloa (I) muet therefore be
lim ited to veasels with AAifotoi pressure diatrlbatioa*
m o m a im n tm w x m Aim D sF om ^T iots
ns-TniBQTloli ^ W esels enoh as storage tanks and ota&dplpee which oontala
flu id s o f rol& tlvely high d esaltles have zmmbrme streeeea Ablob not only
depend on the preeeore but also on the m oasr of atxpport of the vessel*.
3b r th is reason i t Ie not feasib le to w rite a geaerd equation for #@
q&OBge la radius in a vessel of th is type* h ereafter la th is paper when
i t is desirable to ooaalder a vassal eoatalala# a ooa^mlform presewre,
the ohange la radius w ill be referred to as f(%) raiber thaa r%
BEfmzm agsozooBG m o y i m m c a L
am w
* M e m * a shows a
-1 3 -
th tn - w a lle d c y l i n d r i c a l
s h e l l s u b je c te d to an ex­
t e r n a l l y a p p lie d bending
moment o f
Q pound in c h es
p e r u n i t e re le n g th a t
one end.
O ince th e re a re
H g u re 2
no d i r e c t lo a d s , t h i s i s a case o f p u re bending o f the s h e l l .
The fo llo w in g a n a ly s is w i l l be confined to & lo n g itu d in a l s t r ip o f
th e s h e l l a u n i t o re le n g th in w idth and x u n ite in le n g th .
T h is may be
done s in c e both the s h e l l and the lo a d in g are sy r e t r i e I about the lo n ­
g i t u d i n a l a x is .
This s t r i p i s shown in l i g u r e 3 .
n a rb itra ry d if f e r ­
e n t ia l len g th dx, bounded in the u n stressed S V t e by planes a and b and
in the s tr e s s e d s t a t e by p la n e s a* ami b f , Is re p re s e n te d by th e shaded
a re a s.
F ig u re 3
-1 4 -
The n o ta tto n used in th e d is c u s s io n Is a s fo llo w s :
u* = disn ln o em en t in th e X d ir e c tio n
v
= d i s p l a c e r e n t in th e Y d ir e c tio n
e x a u n i t s t r a i n in th e X d i r e c t i o n
Cy = u n i t s t r a i n in th e c ir c u m f e r e n tia l d ir e c tio n
e yx= u n i t s t r a i n In th e X d ir e c tio n caused by th e c ir c u m f e r e n tia l s t r e s s
Ctx = u n i t s t r a i n in th e X d i r e c t i o n caused by th e bending s t r e s s e s
U sing t h i s n o ta tio n i t is p o s s ib le to w r ite th e e x p re s s io n f o r th e
u n it s t r a i n in th e c ir c u m f e r e n tia l d ir e c tio n a s
e
y
- rd # - ( r - v ) M . %
rdj5
r
(c)
The d i f f e r e n t i a l s e c tio n dx i s shown a g a in in F ig u re 4 .
The p lan e
c 'c " i s p assed p a r a l l e l to p la n e e 'a " th ro u g h th e I n te r s e c tio n o f b 'b " end
th e n e u tr a l a x i s .
The d is ta n c e between p la n e s c 'c " and a 'a " i s then th e
o r i g i n a l le n g th o f th e f i b e r s .
ssu n ln g a p la n e c r o s s - s e c tio n b e fo re
bending w i l l be a p la n e c r o s s - s e c tio n a f t e r ben d in g the f i n a l le n g th o f
th e f i b e r s i s th e d is ta n c e
betw een p la n e s a ’ a " and b 'b " .
I t is th en e v id e n t t h a t th e
change in le n g th o f a f i b e r i s
th e d is ta n c e between p la n e s
b ’b" and c 'c " measured a t th e
fib e r.
L et y be th e d is ta n c e
from th e n e u t r a l a x is to any
fib e r.
Then
F ig u re 4
-1 5 -
(d)
U sing th e fshyi c o n v en tio n th c t d e fo r o tio n n a re p o s itiv e when th ey
ln c re n a e th e le n g th o f th e p a r t i c l e
ex e eyx * e tx
FYora P o is s o n ’s R atio
eyX wXuey
and u sin g th e v a lu e s given In e q u a tio n s (c ) and (d)
®x = "p + ^
(e)
I t now becomes n e c e s s a ry to use e x p re s s io n s g iv in g th e s t r e s s e s In
t e r r a o f th e s t r - I n s .
C o n sid e r th e d i f f e r e n t i a l volume shown In F ig u re 5 ,
w ith s id e s d x , dy and d z .
«s b e fo re , l e t Ux , Sy end S56 be th e u n it s t r e s s e s
and ex , e.. and e z be th e u n i t s t r a i n s ,
where Gz end e z o re th e s t r e s s end
th e s t r In r e s p e c tiv e ly In th e r a d i a l
d ir e c tio n
nd th e o th e r n o ta tio n s
a re na p re v io u s ly u sed .
From th e d e f i n i t i o n o f
r a t i o and o f
o ls s o n ’ s
oung’ s modulus i t Is
Figure 5
p o s s ib le to w rite
' ‘®X ” Gx — u (8 y + Gz )
',O y = G y — U (G X + Gjg)
” S z — U (G x + G y )
*4,15'^
HWWet * 1$$. t&e -easfe o f
yli# stt-ess in tb#
r a d ia l direct;5-08. io BefOj th e re fo re
SO^ m Ss, *■ # y
W
Solving W r & f tvwa exprssOioy, (g) and sutsStll/Uting # e r e s u lt ingcxprecoioA into aquation (#)
^ Sig ^
'5*
■and solvin g for* Sjs #om expression If) and onbstittiting the resulting'
expression i s e q m tlo n Ig 5
C* S y
%Sy)
Iiie preceding two expressions- een now he Solved f o r Sx and Sy a@
The sub stitu tio n c f the value o f &%, # v o a in ogpraaslon (a) @n& the
value of @y given in eg re ssio n (d). r e s a l# in
(I)
I t la of In te re st to not* th at the etreae 8% AiatrlbateA aver the
transverse 3%08$*8e#tio& o f the s h e ll'Ie atm ilar in fo ra to B treae-in a
-17
s t r e lg h t he/;;:: su b jected to pure bendlnr an i th at t ‘ie s t r e s s Gy d i s t r i ­
buted over the lo r lt u d ln e l o ro sa -o o ctlo n and a c tin g tangent to the are
o f the s h e l l I s s im ila r In f o r ; to th a t o f a curved beers su b jected to pure
bending.
ICAl '
r n - cin ce i t
Ie n ecessa ry to know th e v a lu e s o f the r e s tr a in in g momenta and shears a t
th e d is c o n tin u ity i t I s n ecessa ry to fin d e x p ressio n s fo r th ese r e s t r a in t s .
The ex p r essio n s must be fu n c tio n s o f both merr.br^ae s t r e s s e s and bending
str e sse s,
"he e a s ie s t fu n c tio n to fin d which Has t h is property Is th e
?
t o t a l d e f le c t io n .
Let V be the sh ea rin g fo r c e per u n it nrc len g th a c tin g perpendicular
to the s x l s ,
be the moment per u n it e re le n g th In the plane perpendleul r
to the radius end 1<0 be the normal fo r c e per u n it len gth a c tin g perpen­
d icu la r 6
lag o f the s h e l l e e shown In lig u r e 6*
By applying the
u n i t a rc le n g th
R
F ig u re 6
equations o f s t a t i c eq u ilib riu m to th e d if f e r e n t ia l s e c tio n i t i s o o s slb le
to <-et r e la tio n s h ip s between th ese fo r c e s
nd or on t o .
The su! n tIon o f th e semento about the l e f t end o f the s e c tio n i s
-4AjUB**1'
Qlsoe I t Ie poeslble to aegleet poROfa o f the dlffe&e&tlal hlghof
then the f i r s t w ithout IntrpdttOiag epprepiahSe'e rro rs the equation oaa be
w ritte n as
T t,
U)
t h e eum awtlon o f th e fcw oeo a n tin g 2# th e B eA lal A ire o tio n #?&&
b a t th e eoot&pR $e # W L t a r e l e a g t b l a ml&tb 88#
r A0 # 1
Glaoa the u n it ere length ie a m ll la comparison, with the re&laa of
the dhell, A# la email and the difference Wteeea B l a ^ enA
IigibleA
I* &e@*
By mklng aae of the tea foregoing etatemeate la the equation
f o r the ew m tlo n o f the fo rces in the r a d ia l S i w t i W
W
%be q lrc M o fa & en tial fo r o a 3# a&a t h e m m * # &% ay he w r i t t e n #8
funotiona of the elrqum ferentlel end &@ng&Wlaal at****** in the folio#*
lng manner
ByAy
$0 5b
t
V
W
'V4Jt
S #
'I'
V
t
•A'Jvi
Where t i s t&g ebaU.
the vaiaea o f By aad'
&a& y ie &s aba*# la Blguye 4 .
By gala#
give# in- eag?r#aaieae (h) a## (I) onS performlnS
the IadloateG iategre’t loae
F0 j»
(i'5 ■
a t&
a»
#-= B f r ^ a e f ' <S - '
Since the term
Feletad -only to the physical Aimea*
sio # e .» f the s h e ll ona is a sea ew e tit i t s ycsistentie- to heading, i t i s
defined an the
rigid ity" and denoted by D,
M= B
The#
a#
Referring to Blgure 4 and making the assumption th a t d@ i s Gmail i t
Is possible to w rite without introducing appreciable error
ten # fe # e» -d#
$he#
$ “S
and
■ a&
K- » » B ?
0 * 6 e -$ * w *'* -a
(a)
D ifferentiating tbi& eapreenlo# with respeot to x end making the
substitution lndlcatod in expression {j)
>H.i
Again I* la ln ta re etlag to note the a W le r lty of the sapreselone
fo r the re e ie tla g moment aa# sheer In a cy lin d rical sh e ll end the eapree*
s l o a s fo r th e eorrospoyidiiag Q u a n titie s % 6 e t r a lg h t beam*
T m ,,B e M K m rn # u i J m # ^ m s p
#
d i f f o r e h t i a t i n g g ^ u a tio a (S) and
j l
Ax
.
.
'
A
fm m eK praaaion (k) fox'
'
.
* *
.
'
.
Br
o r,, by OAtag th@ V eloe o f ## g if e a && gapree&i&a ( I ) e&8 re a rre n g iH g th e
te rs e
^ 0
This te ilie a tffe r a o tie l eqoatloB for the defiedtioix o f e thinmailed cy lin d rical s h e ll due to the restraining ssment %«
fo r th is agnation i s a s'fo lio ® ..
1st.
•'^‘P « 4B^
Br*
Ir a
th a n
6
a*&
6 a^e*
e&&
B^eax ■}- r B % ^ ?» 8'
Srori t h i s
a ^ * B VSi.
or * B B (
"VT)
B it
•/I" «- ods 45° 'r i s i n 45° w
4 i
The solution
@
$ B(K t I) or ^ B{*& * I)
%@ W l^otiOA At eay. # o la t w y thm bo w lt t o a os
, « Aet s • SB)* f Be5*-® * M " ♦ C8I-B - tol® * Ba!8 - iB )«
%be$e A,
& @a& # #&* arbitrary cow W ita?
V*
. ■■
yaatoyleg th is gives
4 DeTlBG; * d43%(B6&B* + OaT#*:)
and egavertl&g t& the taigoaemetric Roam
V # @B#
+ 0*8%
^ g;
B% + {i&.*klDj 9&R D*
^ q; 022
+ { i s + #Q) A&& Bx
Bo® .by roplaolhg ik e eomblhhtioae o f Ootstahts appearihg abov^ by
new arbitrary om ataata t&@ d efleo tlo a eqwstl&a fop the ease o f pure
bending a##eeP8 a&
v «
oos B# +
ab@re the eo&ata&te
Sih Br) + @TBK(Q^ gos BK * B& Sla BK)
W
and % are Qayebdeat on the bobn&ary goa*
d ltion s o f 'bte sh e lls
* I t has previously
IbGmaB pointed out th at the deflebtlon ,Qdr a %i%Mpetled v e sse l can be
thoW%t o f a# a # e ^ e # tl9 a A&e to IBbe mewhtwa.b'bMMMao ssperiaposed o&
the d efieo tiob oansed by b e str sin isg w # sb ts a&& sheers% Ihe to ta l do*,
fle c tlo b OlT any p@l#$ i a tha sh a ll then jus tb# sum o f Ibbe d eflaotlo#
■»2S"
oaueed by the ^aobraaee BtroseeB end th e defleotlo# oeueea by beading,
Lettlag T% b@
to ta l d efleetloa, the egwatloa may be w ritten aa
V^j w v + r*
la the ease o f ttaifors preaaure* or m
v* * f + #(#)
l a th e ease o f a o s^ m ifo n a p r e ssu r e »
3533 m%8# OF IBR&L IMMTE! PM THB SBTAt Dmi&OTlOH - The sh e ll lesgth
h a s no e f f e c t Ca th e d e f l e c t i o n due »
s,$ateene s t r e s s e s ae can b e seen
b y exorotalng oq,m ticm ( I ) *
Honever9 t h is is not so 1% the oaee of th e d e fle ctio n due to bead*
lag.
sin c e 2$ i s measured along th e lo n g itu d in a l m is of the s h e ll the
d e fle c tio n is a function o f sh e ll length#
Gote th at th e f i r s t t e r n to
th e rig h t o f th e equation sig n in equation (#) in creases a s th e s h e ll
len g th increases,.
I t i s apparent th a t i f th e d e fle c tio n is not to be*
QOite unduly large 1% a sh e ll idhooe length approaches in fin ity the cons te n ts Aji and Bj must be extremely stasli#
Since th o se ^constants do not
depend on the physical diwaeionB of the shell* I t oon be concluded that
the f i r s t term may be neglected in s h e lls of ordinary length without
introducing appreciable error*
Iiie lim itin g value o f BL i s approximately
5*5 f o r an e rro r o f l e s s then I per cent#.
On the o th e r hand, i n a v e ry s h o r t s h e l l a l l o f th e c o n s ta n ts m ust
be s m a ll s in c e th e d e f le c ti o n can re a s o n a b ly he ex p ected to be sm all*
In
such a s h e l l the f i r s t term may account f o r e s i g n i f i c a n t p o r tio n o f th e
*•33»
d e f le c tio n and may n o t be n e g le c te d .
In th e su cc e ed in g d is c u s s io n s lo n g s h e l l s o r s h o r t s h e l l s w i l l be
re fe rre d to .
A lo n g s h e l l w i l l be d e fin e d ae one whose t o t a l d e f le c tio n
la giv en by
v a e""''*(Ci cos Bx +
s in Bx) + r * .................. .... (4)
and a s h o r t s h e l l w i l l be d e fin e d a s one whose d e f le c tio n is given by
v a e *(&! oos Bx +
A-Tr-' A
:• ’',y;K TO
a ln Bx) + e"*"x (C^ cos Be +
s in Bx) + r * . (5)
TUIBG IB 1 1 . . T ClBCT I K TIA DO - The s t r e s s e s due to bend­
in g in H a t c i r c u l a r heads may be found by c o n s id e rin g a f l a t c i r c u l a r
p l o t s w ith e moment o f
ed g e.
’Q pound In ch es p e r u n i t a r c le n g th a p p lie d a t th e
Ince both the lo ad and
th e p l a t e o re sy m m e tric a l ab o u t
th e c e n te r , th e d e f le c tio n o f
e l l p o in ts e q u a lly d i s t a n t from
th e c e n te r i s th e same.
This
a llo w s th e a n a ly s is to be made
on a d ia m e tr ic a l s e c tio n .
l e t th e d e f le c ti o n o f any
p o in t on th e n e u tr a l p la n e be
F ig u re 7
denoted by w and th e r a d i a l d is ta n c e o f t h e p o in t from th e c e n te r be de­
n o ted by r - s shown in F ig u re 7 .
Then assum ing only em ail d e f le c ti o n s ,
th e a lo n e a t any p o in t in t h e n e u tr a l s u r f a c e may be w r itte n vs
0 = —
(n)
. *84*
'
# # le t
toe th e raS iu s o f om^zatare l a the d l a w t r l o a l pla&e aad
3?t toe th e I'a llu s o f eurvature aozmal to 1Qxe Slsmetaltoial plane*
further^
assuming th a t fo r a sm all ohaage 40 i& th e angle 0
Sr Sr arcs. Iesgtii SestorlbeS * r^S0
i t i s possible to w rite
if ”
= ~S I
■
■
ia i
Sintoe the l in e ato, when ro ta te d about th e I i s o oto 9 d e scrib e s a eoae
I t must be th e ra d iu s o f su rra ta re In the plane normal to the d lam etrleal
S eotlons and
--**■*'
ab es
Again ustouMng only sm all defleetitohs
ta a 0 # 0 v
r
d#
45 ^ U*
or
I
I #
* r3 r
I f the d e fle c tW ef thto plate is emdLl in oomparleou to %ie thlok*
neaa t , %e assumption o # too mads th a t the neutral plane of the plate la
th e to e n tro id a l p la n s an d a s in th e @*#e o f e toeam^ th e u n i t s t r a i n s a re
proportional to the diatsnoe from the oepter* Denoting the u n it strain in
the rad ia l direction toy e%, and the u n it s tra in in the dlreqtlon normal to
the radius as
and le ttin g y toe the distance from the center
r*
*89*
BnS
By applying HookGt S I m th e
oaa no« be w itte B as
a&d
(a ).
#b@#e 8 ^ a&4 8 ^ ar@ t h e #&&t a tp ea p a # l a # » $ad#o& m s @l#@UB#ep%a#$6&
& ly @ e tla w # e e p e & tW ly «
Th* #B%t#w&*t*eee#@ % the &aa& w *
by reglaolng y la the
©Opstiono a to y e by t / 8*
it;siis%KiK%}.ta)&w%B!R8 ,a m amBAAs m mAT
m enta m y W
i m m * %@ r o a i s t i a e
w # o a e tio a $ o f # * e tr o o a e s h y u a la g th e r e l a t i o a *
sbl%#
I
S
y dy
and
t
2
.y
%be*e % W
%&% swee t h a y e e le ttn g ; m e m a te a#t&a& % t h e e & y c W e r e a ti e l
and r a d i a l oroa$*8@ 0tlons reep ectiv s& y *
3 # b e tlt% tl8 8 e x p re s s io n s (%) a&&
(r) for the atreesee 8%, and 8% and i&tegaating* the momenta are fogad la
terma o f th e a w a t g a ’ee m
■ana
where 5 Is t h e f l e x t t r a l r x g t a i t ? d e fin e d
A nw#@ oowenleaii form fo r t&eae siosentB &&y b$ had by anbatiituning
th e r e t e e s A>r t|ie o u r fa tu r es s iv e a in, e q u a llo to (A) OskT (P ) , ;S w
* (6 )
an d
(f)
BbeerlA g f o r g e s ^ e e e m t a c t o n ly o n th e S i re W e r e n t l e l oroee*
e e o tlo n sAd a re o o % o e w o f % e IoaA * 1 # 1 & tW e l r o n i o r aeoiilon o f * * # #
r d iv id e d by # 6 l o n g # o f t h e c r o a e - a e o t l w o r
(8)
v h e re ? Ie t h e e h e e r la g f o r o e p e r u n i t a r e Z eag th o f t h e # * 4 w f# re @ 4 # a t
the point under ooaaideratlon.^
W tS G T iom QF #W # OlROhMA BSAW
.
x
.IW ) # % e W l e e t W
Of
a f l a t eiraulap head under * uniform fluid preeeuro m y W found by eon*
elder lug a p&*M@bep#& a lle e of the head $6 a%o%o In, Figure 8,
Thera are
:
no eheorlng foreee aetlug 6% the aiaea of @uoh a alioo duo to the eyin*
m try of the ali$* end the Wdlug* md e l l o f the foreee' ehidh net on a
o m ll olemont ore ehoua In the figure tlt& the mtaMouo previously need*
l
-2 7 -
-ipplying th e e q u a tio n s o f s t a t i c
V+fd r
e q u ilib riu m to th e s e c tio n and ta k in g
;
N
*
O' e n ts a b o u t p o in t c , assum ing thr t
-u
th e a n g le
i s sm a ll
>r
y~
<- d r -»• •r---- r
t
—
T
--------—
A
c “ ( Lr ♦ -dr- r Clr)( r + d r)d #
-
&
t= H
r r d0 - Vt d r 60
F ig u re 8
+ (V ♦ 7^- dr) ( r + d r)d 0 d r
+ p [r6 0 d r +
r + dr
d r di
« O
Py n e g le c tin g d i f f e r e n t i a l q u a n t i t i e s o f h ig h e r o rd e r th'm the f i r s t
t h i s e q u a tio n becomes
;-'T d r ♦ r ~ r + 7 r - f-t ■ O
The s u b s t i t u t i o n o f t h e v a lu e s f o r
r and
given by e q u a tio n s (6)
end (7) r i v e s th e d i f f e r e n t i a l e q u a tio n f o r th e head d e f le c tio n as
<=>
The e q u a tio n ra y be so lv ed by n o tin g th e t i t can be w r itt e n tie
d_
dr
B I?1i^S1]
SB
JLE
2
and i n t e r r a t i n g to g e t
.1 i - f r ^L)
r d r' d r7
where
i s a c o n s ta n t o f in te g r a tio n dependent on th e boundary co n d i­
tio n s o f th e pli t e .
m o th e r in te g r a tio n g iv e s
# 0 p iste and i t s IosQtng ease epmae^rtcal about' %e aeafef,
the slope at: the deutet must toe seye,
i*> $ &
SK
S O “ Op
Zfttegsattng a g ain » th e QefleeMop. Is given as
Vf * GTB ^
4
^4 -0 # f » * * » * f * * * * (9)
# e r a Og la a eonata&t o f ln te g n a tio a M ke 0^«
^XAKPiai I -
4 ppeseore t e a s e l e e W n ta o f a th i» * w 6 lle a e y lin Q r io a l s h e l l
80 l&obes 3& a&emeter* 80 taebea long* m& ^ ln c b la t a l l thickness, ena
p
two circu lar f l a t bes&sIg inches In thickness* The in tern al pres&nre is
50 p el.
Watt the m aterial of th e aboil anQ beads ^opng*s moanins Is
3 0 ,000,000 pel and Poieeo&'e ra tio i s 0 , # «
I t Is W ireG to find the
moments ex istin g a t the joints*
Solutions
B% * 1*G4&6 B ' * M158&
BL * 80 X 1»199S s 28*166
Oince BL is larg er than S.& th is Ie a long sh ell end the to ta l Ge*
flectio n , i s given by equation (4) as
v s*
O3
cog
as *
siQ. Bk) +
M e w l t h a t the yadiel 6iatesa?tioR #
v w O %t K #: o»
th e iieads i s n e g lig ib le ,
Ift the A eflectien eguatla.ft thea,?
# ^ rf ,
Tm AoSlm*--
tio n 1$ then
lr#e*Bx:('**%'+
el& Bz) t*! *
B iffersniila tls g th le gives
_
^
«e
qea B% +
a la W
*
a i n BK +
doe 5%)
Beneting the rotation o f the s h e l l at the j o i n t s inhere x - 0 a s 0Bi
ana noting th a t 0^ & ^
The sedonh d eriv atio n o f the d e fle e tid ft equation gives
w *«-
sin Bx *
cos Bx)
$fo\i denoting the moment a t the ead of %e sh e ll where %e O by
denotlA g th e f l e x u r a l r i g i d i t y o f th e s h e l l ah B&, and em ploying e q u a tio n
(B)
n .
% !sD&ta &3M)%)
IW oons i Coring th e M ad th e fo llo w in g aquntiona oan be w r itte n
*30,
idaere % G eaotee t h e CLeaoraL p ig & d ity o f tb a booGe.
lo ttin g a be the mxWmx m@ltie ia th e heed* and
&* the r& W lea
of the heede a t the jo la ta , a&d a#ply&ag axpaeeeLaa (a)
dw Sg
Gr
#1%
16D ^ 4 P
Denoting the moment La the ra d ia l Glreetloa a t the edge o f the he&de
by Rp #a& w ing egnetioa (a)
p g
-T
#% * + Bbj^&Db
# o four equations
%»
+ *U
and Mp eoatala too onkaema eooBte a ts t
Booever, by aasomiag * rig id oon&eatloa W teem the sh e ll and the heede
I t i& poeelhle to flag one ceaataat la te#B& o f the other elnoe # *#(&*
or
-$- !DjJtI ■#* “' f ^ ^ +
th e n
131 “
S ia e e t h e j o i n t m a st be i n e q a llib r ig m
op
and oela& the value o f is found abo#
r ‘
^
8BJB&
Cs + u} 4'
( l -i-
n)J
cmd so Ivi Sg f a t
*8
7
mrnB
% * ............... " 8 D ^ ' T %
^ ^ r
T&$m&at&80 mgmaBts aea #9% be #eu%&$ Kbese ays*
the Bomeat 3#
the abe%l* the re d ia l mao&at In' the bead m& th e
mom&a#
la tbe bead* W e t l t a t W the mla@ Io et ^oaad Aw o, Iato the @#B*e#8io&
to r Mp g iv e s
(8 + %)
% S=
sb^
f e i ' * * ”) - I 2 (s + 'l! (X +
d )]
SGaBa.+ 0^ (1 * ?)
and. f o t th e o iro u W e re a t i a l moment in th e Jiead
[JmS Ll 1 3 a , D „
1[1Q%
# t * ~ %b)
16%
SDgBa + D%(1 + %;
Cl -i- ti j aj
jB e wlv.ee■ o f the ooaataate to he applied to these equations are;
g.
s
. ■g
D*
*'
W a K tbeae value# l a th e aomahti eapr& ssloae above g iv e
,
%L * 188 # .* i b ^ e e / l a e b
% * 189 # ~ looheB/iagh
*" A
gyaa#)*# ?#*##&. ta m6e of #%oel
*&$&* p* 0*2* m a % 6 80% 10^+
ia @ia&8##,
aolB*aa*a
§* t&i#k* %&'lo%B**4 4d0G
I f $b&Wt#z%a& B&aagwBe 1# 000 g*% eM the sh a ll la &#<*
etm iaea agai&$t BMlal tra&alat&&& a t th# #a&& hat igatherw ipa fpee*
find the t#8&8ver*@ ahea# a t the ahd#*
S oteM dtii
n e S its^ w * m ^ ’o^m r ” ms>ms
■
sBt » S s b l a ^ i ^ W k 8
,
B * 0 ^880
#
* 0.0984,
.
B&* &kO&196
SL * 0+880 X 10 * 8,80
S
**
^
.0 # .% ..8 i* .m .
(8 * 0 ,2 3 ) * 0,0099098"
sS his Ih a ah03?t sh a ll and the aefle'-etion la defined hy
T SS e®K(Ag_ GQB The 4- B% SXE Bx) f
tiO'S W + Bj, 3111 Bx) t #
Qlaoa a l l red ia l Sefleotion i s pfevestea a t the- eado o f the sh e ll
the following two eguationa m f he written
0 <*
4' Ol 4-
••
.m , ai&.g& e + @*9L eg* $%,
0 * 9 ^ 008 BL A% +,G^L
+ a*BL gfa BL
fh e BQuation f o r th e moment- existing; i n th e s h e ll is
4 ?'
M*
g m m t % 60» W
+
ooa Bx)e~Bx
81*8# th » $b@ ll 1# fp e e t o yot& t# a t th e 9R&8f tb # m&mgata a t tb&ee
point® m ust %$ z e r o , m i
0 *
0 * BD#e8&(»Ai 81* a , + aoa BK,) + 88B»B*8&(Q
fh e re a r e now f o u r e q a a tlg a e
rS e oob,StE a ta i o r e # #
■
-
8 00# 8 B& * 8 OOeb 8 B&
% -
^
Q1 R
*4
8 #06 B B& * & 60#& & BL
04 A
*
& 4&8 8 SL * 8 OoMb 8 BL
@oe % )
# 0 four- unknown co u o tau te,.
4 .60». 8 . K . i » : 6 W% BL,Ap^
A
*&& #% * ^
a 0*&@08&&00
i - ^"W = O M
.
.at B #o048®s%
*
.# 0*000099088
%@ etUBtion f o r the Sraaararao #Wmr i s
V a B b a P a ^ j^ ^ o o * B% + s in Ba) + % [699 BK * a la BK)?
{t m Bx <4 S iu Ac) ■¥ B^faea 6a + a la Bxjj
>
At the ss,6 # # # K M 0
V a 8BB
^
+ 9% -t C& + 9&
* 8 K 388*38$ %0,08&@8 [*0*O0O2#X> * 0*000890083 ^ O+OO48&806
* 0*000090088]
404^
Xt aho&IW be
th a t
aiajoi1- pe^tidh. ■&$ th e *Ark Iw o lireS ta
t h l e Goiwt&oa i& &a th e A eteB w iaatloa o f th e a&pMBeieaa fo e th e eoaetd& ta
A1, %* %
%* rS e e%$;ya8$ioa$ afopfO oem o-bta in sd by solving a
fourth PtSer aptetwiaent^
-3 5 DECTIOM I I
-U C C
i
COTii '.C TIO l53
iL I l O TO T i 'I L - ..,L L E D
T
L C V M C lM
Ii1TTPOr LCTIOK - In ■Die p re e ee d in g a n a l y t i c a l a n a ly s is th e ends o f th e
s h e l l r o ta te d and tr a n e l u t e d a ira il ta n eo u a ly in going from tiie u n s tre s s e d
to th e s tr e s s e d e q u ilib riu m c o a itlo n s .
These r o ta tio n s and tr a n s la tio n o
were unknown and depended on the p h y s ic a l p r o p e r tie s o f the s h e l l and i t s
r e s t r a i n i n g members.
Pecauae o f t h i s i t was n e c e ssa ry to s o lv e th e
e q u a tio n s f o r the d e f le c tio n s o f th e s h e l l s im u lta n e o u s ly w ith th o se o f
i t s r e s t r a i n i n g members*
s i p l l f l e d a n a ly s is can be made by a n p ly in g a r t i f i c i a l r e s t r a i n t s
s t th e ends o f th e members to p re v e n t r o t a t i o n and t r a n s l a t i o n .
The de­
f l e c t i o n o f eoch o f th e me fibers i s th en known a t a s u f f i c i e n t number o f
p o in ts to allow the mo ie n te and s h e a rs in any
p e n d e n tly o f th e o th e r members.
ember to be found in d e ­
The a r t i f i c i a l r e s t r a i n t s can then be
removed by on a p p lic a tio n o f th e method o f su c c e ss iv e c o r r e c t io n s .
T"
11 T:on Th
UCC
I '■ OCR-; CTiOLO - F ig u re 9 (a ) Is o diagram o f th e
lo n g itu d in a l c r o s s - s e c ti o n o f th e n e u tr a l s u rfa c e o f an u n s tr e s s e d p re s ­
s u re v e s s e l.
The p o in ts a and b r e p r e s e n t th e j o i n t s betw een the s h e l l
F ig u re 9
W
%be
dbes# $&& #@8» va&eg& eo& j& eW t o a a W w
&al f l a i l &e@w#wf8i e#a a r s a f i c l a l
ozwF
*&& # w w *#%#&
(Of t m w A A tW (Rf 3b* #>%&%&* % eee y e a t W e t * care
yWMM* WW (A Sk $ W # i l a W S # a 8 #%e # t w r #>&&# fK%4& TRtIWb IRhSk W
m l& W aeetr&tAte*
8t%@* t i ^ y s a t m t a t a are &9p&t8a #
# e j # W # 6 ao* to m y &&*
m ab e# ia*wS e&a#@ tb e M a t m a t - %* &a
#ML1 lbs* M a W W W t m w
«w at W
W
a l l o f AW ms&Wra *% W
iw tW w w
jW%A* IRio ama&t Ag
M# m a w t o f o h w » #h&0b w # w A w w oWL aosmme Ka a o l W
sfk*b&@,4*%M& & w *bt o&& tke f
t&@
i &m#
%f IBtB y o o t m W a g m a w t Ga* x w a N A A w w e o f 4W 3& W e oaf # o
veeedL && # a # m s 0 ( b ) , "Mb** # W
#&&1 b# e ta M a a lk y m W osm aa*
W a
w W a a w e %&&& omw# (Bat* 3 * S # t o %o%e# tm tS l o ^ A l t W w w W t t W o are
a g a ia '@ *W wA*
%& smwmte W e t W
dbamg#* SKGKi .W M K m W m& W
w W W * aa# ojpfioaW *% M ga*
-m m W e a t W
# % w WMMwAeg W%1 # e y w e o f aqw lSSue oa&wat o f aamest o W g e &a ea@h mm*
beep 9&1& be pawpo^lGaml M th e BtSfAteae o f SM
W W W I A w o w Se *&& o f W
# e t o ill
Ilwa eim o f SM
rnmaww %&lt W e q w l W W
w ow t of
SQKtSMMBGaL 2 8 8 W 3m8KK*f<H&t
ZMbBdHa w
a m t e a l B&ocae #
a f *a& 3&&8& W * e w W A a f W W o A w w
IKbe V a w A M W o w W t
W
tM
o f th e
jW W #
W w a a a w w o W L W e o w A y a%W W a a& m w W # # m t e a t W
rataW k
#*&st* at dbeRge l a t&a w A w w l A w a * a # l% a # a t W w a aal& te i f W ay
era a o t t o tm a a ta ta * eoa si A w g a M Wa m a e m o l w o w
BdaaeeBt je i& ta i f th ey ora a a t to m te ta *
w # l# e d a t W e
(EbIb) e W i t w a i e eBosa sat
-? 7 -
F ir u r e 1 0 (a) w hich hao th e r e s t r a i n i n g moment removed from th e l e f t hand end.
A lthough th e rem oval o f Uie r e s t r a i n i n g moiaent a t one end In c re se e s
th e r e s t r a i n t s a t the o t h e r end and a chen e in th e s h e a r a t th e r o ta te d
end, th e changes e r e Elways l e s s th en th e amount o f r e s t r a i n t removed
end th e v e s s e l in F ig u re 10(a) i s n e a r e r to th e c o n d itio n o f u n re s tra in e d
e q u ilib riu m than th e v e s s e l in F ig u re 9 ( b ) •
V+aV
V+AV
F ig u re 10
F ig u re 10(b) shown t h e v e s s e l and i t s r e s t r a i n t s a f t e r rem oval o f
r e s t r a i n i n g .am ents a t b o th th e l e f t and r i g h t e n d s.
The change In th e
r e s t r a i n t s due to th e se rem ovals i s d en o ted byaV a n d tf'.
The r e io v a l
o f th e r e s t r a i n t s I s accom plished by a llo w in g the J o in t to r o t a t e o r
t r a n s l a t e to i t s e q u ilib riu m p o s itio n by a p p ly in g fo rc e s o r moments eq u al
in m agnitude to th e a r t i f i c i a l r e s t r a i n i n g fo rc e s o r momenta, b u t o p p o s ite
in d i r e c t i o n , a t th e p o in t o f r e s t r a i n t .
The rem oval o f a r e s t r a i n i n g f o r c e a t a j o i n t cau ses a change in
th e d e f l e c t i o n o f th e n e u tr a l p la n e
s does th e rem oval o f a r e s tr a in in g
moment end c r e a te s a change in th e r e s t r a i n i n g moment a t t h a t j o i n t
nd
s change in the r o ta tio n and d e f le c tio n r e s t r a i n t s a t th e a d ja c e n t j o i n t s .
Kggavawp a#
the a&e@ of @ #a#&t %mov&& <b& metaMWo # W a#@
cye&tw# @*s *&QBye &B8&XG# #e% #@
KomGyediSBa # $ vaeo#! *@8*
tkm w to f#B#&gG& tk& 608dl%%@& o f m#o@tm&8c@
Sm %*
a&g* of #986 pwwwA voeee&a W »oo%#B&8*# wwAmW aftor rawpvla#
meb o f t&o
a# a jw*&* W #e* %I1& be
.
%W p&oeoaa o f #&ao#8im @#p#aK&Qwt&9*e && G&apla to wawy #&e& %&tl to Am MMteeW i# the #wa*a@&&B& go##* # e *eok w W ao oa*
tlr e ly
aa %0BB a# oa&y o&e #oetr&&B$ &G wawW a t a e w o#&
tb e o tb w W M e eoBB&R fi%@& l# poe&t&oia*
3& tbg Buao«#d&B& WK%%bei&&08%MNk& 8&#a o f tw %waMltWW%
ba&a &Rnoroa Gaoept In t&e ow e of a&o*@#&8&R& paBowam &W*ibu%&&a*
$h%» OBBbIe the
to he oameW *w* Qemt tb# j^aplmA W h w
tbea the m tham tlool atW&ol&t*
o f o lo&a
o g H a W e a l o b e l i m b j e e W to o ao&»B&i$#M8 #n&&*
a w e AlotM W tAem oom h e f e w
b y ae& vla# th e
A a f lo o tie a a g p a tlo a *
%&o SQGGtWa &ik
v m o"^&{c% o o e ss: + % #&a W
w ince W e # f l W ^ A
t fW
*#a8& # o a # TF * & a t % # O sdtth W
oaaa&aoBaa to he a t ooe #& of the dbe&&
O^i # « f ( 0 )
o rig in
ghepe
W th # (Bkcawgw;
Z A ttiA g 0
#%&&%& 6%#%& t& e « m W R $ @t#e@a@a a t # « 0 *
SGbo VAtBtlOA A t t&* GRA O f #L0 B&G&l
{Ka
G#
e#
0 * #. 5*y*S%&(<&^ ooe BK * D& @%* W
sdbeee
*
(3# it# -Gtio f l r a t (BaMkMwtasnB
e&& %* *
# 9 B%) + f* ( * )
t h e 6Wt%mw 6bf3&ot&oa*
B W tt f* B * * # W O M A itW w I#** O Ot %#* Gt &B&
G JG a*
###9 f* W
^ % ) * SC*
So tdbe iGe%&&\RaiRky@ o f idk(>;%KBBkKBaaws #afW&3kp& Q t # * * 0 *
O o W ttttW g # 0 v o W o f
% e t b w t o#*8 t&oA m& M W t g f t #
g iv e #
«s * f { 0 }
W t b O t %*&VOt%tR O f t&o W lB O tlO O O Q jA tW #V@9
01» IBs * % 000 w
+ !T*(x}
By o w a W o o (@& tk a f W & » W m w o o t o t 3* # # m y SWbaat t o w l t w
mz *
o a d tb@ f
0#
4 *#»(9 ) * * f * * . + w
, oo%oot ait # *9 & as*
m i * m A r ^ %(o)
000
m * # * ) @&&m,] + w (& ) * + * taa)
eliem
8&&
the eows#
o f the membrane de#&eq*
ts * a xit as <; O 8B& at * & ^ % » e i* M A y v
* *&e (&%»#«*& e m m w $* 3az%3 W ^ m W l e a
(#lib#fW G & d&&3&8 008 b e o b M a a # d&fciet&y
%h@ W A m
(9 ) a a # ( 10K
a # # iw & l@ a
tiO R .a* :&Gde t h e ipswp&is&db*
m *A m # W W #
M w o # a # r i w t l % G o f a w w t w t e##
m eo * og& atiaa* i[9 ) a&& (%D) bx&ooae
. * ( 1#
39BZ M mxA"* *
Igp
IGbe QS##@88im w A )r 3) AM y* W t foom#*#&a@ # # -
A m * * t&im e # m % W ow t W
a&a&le c m b e A W
By a p w W w a W & e * #
A W w m A m n m ta 1» SkGoai
e h e ll# *
t b a t w#& && f i a d l a # $b@
A w w , a** et%& W *&#&
A e N W t A R m ^ P w w W 1# m d b m m W W m m
A g t h e 0 # m a l $BB#*
A t A A w A t t e^W & m t f o r at *&e*& 8B&3& A #&re& %y a g m tA A (8 )
88
-9 * i*8SK(iy& W
Eae * % a l a Bad * s**89S(<%& w
e#& a*%#@ A t Bdkope e t w
o f IKbo 6#e&(#t*o& e w a tle &
% * D^ *&% W
* fM
p o A t &a t h e e W t A *#ie& to t h e d l f f # m t W ,
# * B e * * !^ * * Re *
*
a*) + %(#& At * am3%g
m + »1% Dar) ^
Tbo bw m w y w a i# 0 9 * W # e
s a l q%8 t&o a # & e *
ifaOidGWa. %* 0;
^
' -
* M a W l "#" f * M
o w m t w M * db@r$ ima*
f e e * l@a@ T m a d l* S W &@* v * O * b e a * * * @ ;
<&!&iBbea
# 0 #&&n **& *
Oeias Mwaee
OoMWoaa 1% SM Sm egg&s&gae w l # m above I S la Boeeib&e to waste toe
IOiaoialmgfW og&at&oae*
A& * g& j# * f ( e )
Ag@S& iso# m + %*8& Sflb % 4» %o~B& #06 % o IttaTSB*
K a* # *(&)
BOL *- e&o BR,) * %e*&(m9» % o # » A )
* % & ^ ( 4 0 9 ZKL 4» M a B&) * D ^ ^ o e o 3& * a l a m ,) a* *
Fmm SbdMS*;
i t i&
#%&* M l o f me ooaetaata *%* a%,,
#& *&& % *&&& &av* * w m m # w W w t # * # 8 & to s h e W w o o f a G oW *
w oe o# o f M l of t&a W # @bl#
to # e l e f t o f -Kbj* oqB&l
Mess* TBhws amwetara f w *a<& o f SBwe o m a w to
Swa She vsltme o f
<&& aotemsamtG # e p o o # o f Mho Wmo &> # # SueiMGo f %o <w&wal # # B
#U& the tegs# So W * 4 # # o f IKbtt o w l OlsBhe NjpW&m the oeo m o W to
o f Sb* <W9&wt W*& W lvel &>** MmetlRg m& Wommatep aebmaSBeat
by xo «#& # o m w o W Wwm&MOte o f A&* 8%* (%
&aa&
ft r*9#Q #9ely
*%? *# &» ## m&
0
4- i
#
#-1
- S
"■f- X
+ 4^#$# ^
* d * # " @p#
* &*%, 0j,lK ^
4“
4k %■
%#
4-
3%
4r
■» #1% BfcJ'
^ »%{Qg8$ gg, # e T ^ l a o e Bk 4 $ T ^ (o o # %
4- # B BfcJ
' . ^ lA A BW
* #6)
0
e l |B
* &
* SVU
4
Bk
^ SsiSil
S
+
Bk
* aW m )
*
4 &
* I
# 3,
» %
% #3
* #8&
+ a*%&8a a Bfc 4
*
-
i
RtR %
* e * ^ (Q 9 9 B &
* a l a BfcJ
O
4%
m.
* o ^ e a # Bk
# $ # Bk)
#
'
"i' %
* % '
8 **
B
+ f (fc)
0
+ &
* $ # a t)
4 S
e
4« %-
* &
4 8* ^ ( G 0& 8 X,
w RtR W
O
* I
&3A
*#w
*
B
a #
* * & * # * s&
iifc
# w# %)
O
+
4 %
4 &
4»
a
ao» Bk
Bk
W
# R ia Bk)
* A
ir Si
»
+ @ ^ e % % 8&
*.
* 2(0)
B
4 fW
< m 8fc
* 3,
^ I
* # * * # % 2&
+ ^ ( m a Bk *
- * M n Bk)
Rk
+ - 6 $% Bk)
*
Bfc
4 R ia m
s&a
o f Skea*
a m #%#& so &e
# A S 'MSli S1Il8 f SBId *» &
m * ffO H o iB #B& * #08 KW + %
*
8&*& ER, #0* %& * 8 069&8& #&B W
'»
8$ tA & )
f* (& l
* n G T ^ ^ ***& BK, #<& % )
% * f{ 0 X#*%B&w @2* a # ,, *@ 08 8B&) * ^ p k % * 8&&8BL * **#%&%
* #{&)(&
+
a l a z& * a Giah B L e e a B L * 8 #&%& a& 'ai* a&)
a 4 & G i*B & * 8 at#** B L a o a W
c * #{0 ) ( 8 * a&% eg& * a># aa& * e%8&)
*
% &&&%,)
@&8& W ao@ B L + ;@e*a& B& a # * RWt- + '& W < * 3 ak # k # . *&* W
A * f{ O H o aa %&, + *i& BB& *
2 # , * % * @GB&)
+ f ( i , H 8 (float m m e iBi, + 1%9 & # K @&& %&'+ as # - *&% a q
*
o in b % w e %& * a #B&
The oooeW tta a m oov eemlwatW a s
%&)
a
% a b/R*
** e/& #o&
Bti * S/fflei
9#ot% 8# IKba jMsK;&*»8*w& BomeSo #
a* 4
DW esGa at * # m d a? a I aa
B W ^K A tlealy W t W W @ q w # W (Sf)
W
BKRBKNbt Gd; a&y p e W
t a igbe aW &
8
S *
8B & V * (* A& ei& Ra * % (me W
I- *aB%fSaF*%8s((%L a l e Ba ^ % - w e W
%> M #
+ 1)
a #
fW
*
* D M O k 4 (W
% % « 8#BP9^&{* A& Ota BI, + % 998 B&)
+ 3DafA**&(0& ei& BI, * &1 *oe BU + . O M U
* 0 # |% g 0 ) *
* % 1 ? ^ ^ + D f"W
* * (IG )
v h e re
sx e « S ~ S i ,S S n
f8 *
S m V oloeo v f Mua Z fw sel& aaa obGVe *3# # v m t o %a&&e I o f IWbo
sgxpKM&aabg*
a p p lla a S la B asp & W W k m fp e o e w e M eW W S & oa PWwooa # o o % i# e W # a a # w
t h e A m W u O Qgaaawte,
%y W
Sb* v a r i a b l e A W i » % # W W l a SM W # W L
#*- m O t #
ma * $
mia&se? rm:.u%jin3
o y lla W C v l g b e l l I *
+
SftwadL m #as*& o& W o w w
%) *+ * * + * « + , # * * + - » ( W
* % # a b e o r @s « # , # i & t 1» a w a - ^ W W
b? e%#a#Loa (# ) a@
V a I) TgSge
and
tb*
1*3 Sbe
408&itld%&
%# Aaoe na t W e wed for flag*
&oa*9ae, the aKswoaatoBa 3&r&&@flaad*8na aaeaee my
be obW%@& %qr w w W vnl&oe o f the oaKotaste a* feaad i# the d&ei*
instdboAai? t&* aPisKRafagsaS ysooiGMt b**mww»$<Ka»#
81& #
23%# eSa 3») + Q%(a&B % * @i& #x)7 *
eksap W #U a$ # tb@ ead %a* O Wmme
/■
m s # # * m w f [& # o ) + j%Lia&L]+ &
, , . » . ##)
#a& the aWhewap a t tba *m& 3%** & Ww%#
% % ** *
Idkwape
« 0 ) M* B& +
BOl + *3& w ] 4 D
a&a #*(&) daaete tbe # i s 6 W 'ite.tW e e f the WB##aa dm*
f laetlcm a t the mde %%O aW %** & ##^ M voay*
jSI%3SS±4S%3)J[%:Sa31_32%_lj%%<lJ3?GKRb
ZiksxwKwdswi a w r
a*i \ W g #&&+
WKlloa eylW%^@e% xsW& mdw imtAxma ymwmte lo Amd by W w sttw W
r* for $ ( # % %» e&we eomtlene* W b givm Axp the A WHmd eMer
p m « <0 #
r$ ^ Rms*,? * , . * , # * » . * . , * »
(3»)
t w r ^ #& # w , #»&!#*
Apimk ^ ek I m y M Imi Kgr «eiw@ t w m m tm te @# @*wi $m the derlvetw#
of
m w & e%^rew$oa 1% %e abort eyMhdriW. sheit*
—4 6 -
Equation (S) gives
V= SDB^e®8"£*• A^(cos Bz + sin Bz) + (cos Bz - sin BxjJ
f SBB^e ^^(eos' Bz - sin Bk) 4- B3_(eos Bk•$- sin Bk)J + D f"?(k)
L e ttin g , EES^j and
be th e fix e d -e n d shears- a t X= = O and x = L
r e s p e c tiv e ly
EES0; = S B #
A l + B i + Cl -t- D1J -J- D f " '( 0 )
= SDB45 [% f ( 0 ) - J1 £1121 + J 6 f ( L) + J5 H A l J 4- D ,f"M0)
4 r F 1 + J « f ! « + Jb
b
(18)
-,Si BL [ - A1 (cos BL 4- s i n BL) + IS1 (eos BL - s i n BL)]
ESSli K 2BB^eBL
4- 2DB3e"BL[ci(.co s BL - s i n BEL) + D1I c o s BL t s i n -BL)] + D f ’” (L)
= SDZp [ - Jg f ( 0 ) ^
f(L ) *
B f" '( U .( 1 9 )
where
r
S s in k SBL - S s in , SBL
j S " co,bh SBL -J- ■cos SBL -"B
4S ”
4- Sinh BL cos BL -J 4 cosh BL s i n BL
cosh SHL + Cos SBL - 4'
The v a lu e s o f Jg and Jg may be fo u n d i n T able I o f th e appendix,
BIXED^EBD SHEAR g ; SHORT CYLIWDRIGAL TEKN-WALIED SHELLS SUBJECTED TO BNIEORM BBS38BBE DIgTBlSDTIOB - I n th e c ase o f un ifo rm p r e s s u r e d i s t r i b u t i o n
where th e c o n s ta n t membrane d isp la c e m e n t i s used in p la c e o f the. g e n e ra l
d is p la c e m e n t5 e q u a tio n s ( 18) and (19) become
EES = SDB ^ 5 +
,
r l * * 8 * • * .i t * ’ *
.
# # m O B f m^^TW&BD OYLIMD^A;.
* %
thib-w alled
»
OGBJEGTBD W
f $e t wp A w * « w *
B bell caa be !BotmA by taelag t&@ equatloa for the
ABAeeIAOA o f dtiob a a h e ll eBA ^ a o ta g a wait QKmeat a t oae aaA* Tbo
momeat a t IWbe other aa& le thea tbe
oerry-over fa cto r or the
OMOimt o f moment produced, at ®m esd by a m it ohaage ia the momeat at
the other ea6.«
Siaoe the cmly load on the s h e ll i s a u n it heaSiag moineat at one end
the msKibrettO terms are non,-Untisteet am#
v ss
cos Bs f % a la Be ) 4 ©"^{O^ cos Bst u
s ia Br)
Using th e boundary conditions t h a t v # O a t & a 0 ,
I a t % # (3#
v = 0 a t K s- 1» im& 0 » 0 a t r * L3 th e follow ing equations can be w i t t e a
a o
4'
+ % 39&
Q@@ %
* % 8B#
@09 28*
4-
A l@ ^ 0 0 8 Bh + % A w a m . # C # ^ { C C B BL
t iela B&)
* .» 8&* m )
* s i h Bh)
4
+ il.cfSS' 9 ia EL
+
S» I
s in EL # O1
#
# 0
* s in Bh)
As before l e t the denominator o f the constants be m end the
Btors o f
Ci and % he Bi b, e end A* respeotivoly.,. then
s i«
m l - s in k sm ,
a = H a sin% i>
b « <**%&;& + g * a %%& _ %
c as 18 ela&81,
SSL
d « *•! s in BH, 4 e1
m e d o a e to a tB a e e :
(% * o /m e o a % * a /m ,
TM
GGbatitwtloa of tW e v a lu e a i a # tb&mmaa# eqaetioa
B * BDSPe%&(*Ai d in % + % &oa BL) + R O ^ e ^ O ^ e l a BL # B& gae BK.)
e e a u l t e l a t b e oa%)WM%r ta e iio r b ein g
. . . . . . (m,
Tbe values Of 0 aye tabulated la Tablo II of tbo apnendlK,
^
oap, A m i # ,&;%#& .mmo-a; ow m s K coam Ag s m z m o ? ,& t o m in m *
W\M>M .0TlIf@BI0 A t OIIIII, »» S ln e e th e n o ta tio n s o f a l l "mestisens a t th e j o i n t
are take aeate, the ^ le trib u tlo a fee tore m y be eeloulated from th le angle
ohaage* f o f t h i s re a s o n i t i s a e o e s s e r y .t o h a v e th e r e l a t i o n s h i p between
Idbe ro tatio n and th e mmmt a t -the # 3 of the shell*
This may tie aoae Tbgr
assuming bonudary oondltlons o f if a O a t & # 0& # * I a t %= o$ and sol*
v in g fo r th e moment created*
Slncte
V»
cos Bse •?'
sin 1%)
and
$ P * S e ^ OjCcos Bx « sin Ux) $ BjCcos Bx * sin Bsc)
the followiag equations m y be written
O e 0% o r O j * 0
on#
I a * I) D j I)? Djj = - 1/B
\
%^ V
#hs
{"-B-) ) # 4 %RB w
? f
( 88)
iBaotio? f& r ^ e m o m e b to &t any jo i& t may % e# b e h a #
iby o a lo b la tlM g %be e f f e e t e # & s a t t a a g l e dbeoga # a th e aam eat t n th e
j o la lb g m&m&eya a&a #letr& b% tl% g tb&$ aom eat Mhbaleaoa i n th e r e l a t i o n
a
•?%.
.p
%
4
+ *1
where £ Ss the siomeht # $ t r i b u t i o a fa c to r f e r the member iM leatefi by
th e eab sc rip t.»■
%&B&g&%Nm a*&B8B#oa&fm#LL * # e b o m W y Ooaait&oae f a r t h i s oaaemay
W e a t tip a s , # t % * o, r « 0$ at, x & hi t ■« Of a t % # 0» pf * I ; smct a t
% # I , 0 Kt o«
IM s allow s th e f e l l W la^ eguatioas to be w ritte n <>,
* 0% * O
A& + % * d% + Bi s t g
Al*& 903 BL + BieBL *1% Q& + G%.a*9& # # m + G%eTBL #&% g&, # O
AYeBt(ooa Bh 4 Bt&B&fQpa B& * G i s f ^ c a a B i + B t e ^ o o e Bh e e
* e ia EU
* # ia Bh)
" * e&a EL)
^ a la B&)
As before th e ooaeteata may ha eraln&tad by neing G e taraiasate^
L ettin g in be th e ieaoasiaetor and a , b , e m& G1be W a w e ra to ra o f
®ls ex wd. % *eB#eatiT#Xy,
IU # 6 c o # ,
«f*' 0 <3# SBZf * 4
a„ *„ - L EBj a f e
S
^ J U fM
EHbMWJk * a/bt* % *
■
% a a/%
^ a a W a f e * tbe ex*
fo^ tit$ reau ltisg mosimt a t the jo iiit is f o a # as
M * 8B8^*B*(4A% gdaBar+ % (xxalBd + B D B ^ M % e la Ba - D% cos Bx)
03?
3/W
B ' # -d # o 4, fs o « «■■
»' » * 4 * C )
Tbs &&#WW#@a feetora &p& f#%& w Bo# lao# dbeila*
m g mFEOT cap A
(8ANGB #
WAjaam m immiGAl, S^rnLL + Ia f W W
mm # # »
tbe ow oge t& t&e ehcarlag forces
@8%eea by bala& aw .tbs %&&&&%& mmmte i t i s aeoaae&ry to kso* tbs
dre&atiooe&lp W a g e# a W t # W « t o&e&g# aa& the csepespoaaiBg a&aaga
ia aB@ar* S&le aay be obW&ea by plaoiog a W t mo&mt o& tbe e&& o#
the v@ea@% ea& ea&vis# #g& 'tke,&eaattt%# 8b$*e* %b@ W m W y eo&a&t&a#*
,&&&
the»j> a t * « 0> t « 0# ®8& a t %# Of
M
fe #
V * **BK(Ol 608 BX + Di * la BK)
,a ir ,
an#
m
a t # S* f ^
COa W
%, * "*
asd th e siiear 1$ ijhesiU* 65 S
v. ■§• > « f -e.
■■* ( S4.)
SSEB mPSGT Olf A Tm* W B ## OiSASKaB (%
BI SWBSaf* B A fSSOIKC
c?L iaD ^^A l, WBU, * 8 &&@B th e e#a&tG*&e have a l r e a # bee# ev alu ated : f o r
a u a it a#m#at; oheage ta abort ehalla I t la eaey to 3%aa the relatloaebl#
beteaati the ahear m $
someat*
%# oohatahte ee found -lb the solatioh
fo r the oarry^ovey fa c to r ahen eabattthted lafo the gqyatloh for the shear
g iv e
Tj^ #
'
- *
e l * 8% ^ e i h b a B b
*
* * » * » W )
The T a la e e o f & .a re given, I h ^ h l e 11 o f th e appeh&ix*
W A im ;,^ m a m z o ^ a m , ! . #
t o r e heW ee*
# » abea# a i e t r i b u t w # # . .
e t a j o l o t i f l a * e # # a w * y to M o* b o * m ob. th e a h e #
w i l l O hm )# * l t h a %%*&$ M ep W M W h t h lb o e Al& m W A *» *111 d ie p la o e , th e
home ametmf*.
and ((<* O
T b le oaeo b a a th e h o u rd a ry o o M ltla n e o f v ^ I a t % # p#
. T bW
"
Qj. A I
'
'' '
8B&
Vg ~
TBs
oe
*- * 4 -i *' f>
% .m s
# » » *
# .(-S05
.A
* 2a th e c*e6 o f s w a it dle#leG@m8&t l a sh o r t s h g lle
the bouadary om41ti-oae are Ir # I a t -sc m 0* ? # @ &t z e &, gf * O a t
ji w O3f aaS # s* 9 & t % # Z * %om these; th e
egw atloae may he
w itte a
"t" Ojf
3,
A% + % * - 0l, + D & * O
%##, @9* % *
3%s B& +
A s e ^ t a o e m , + % e9& (oae % *
* a l a BW
+ a ia B L )
see * f
"
ala. W * 9
% * D ,6**&(ooe BL » 9
+ 8 l* B W
- i d # RU
Osiiig the Ssme BOt&tioa as before,
in .«s 8 ooah SSL + B ees SSL * 4
a &
603 g t^ f ^
1%. BBL ** S
' b * *e^B & + t #8 2# L + A laS B L
& * g#BL + a&*, #%&'* 098 # & * %
@8BL *
^ 9@8 BBL
Sieae VAlaaa for th e -oowtaatA # m iseil # 1 # the e # # i q a for the
s h e a r g iW
T
ooa 02? * s i s
t B3J eios Bx ^ g l# Bx)]
4 3l)Bve""^ jc^i ^oB Bx *• ala Sx) ^ D3Jqoa Es -> aia Sx)]
#e a t * * O
V * 2D # T-A3 + #% + 4% + % ,]
th e a
'
%
=
»
« b\
* •-**■- W)
$be vela## ## the qOaateat Kg, # e @&v#a in Table II of #@ agpeaA&a*
m iT o m a m .m m u m j # , o m m s o? # m w m m
AT a e s #
a m m A a m .m # 3 @ a L s * m m # L @ m - » = » « vwn-on*
f&qtg# in # e qaee of a tiait traaelatiOA i s ftofy aeoesaaMly tho same aa:
tlie
fo:j? a tm it ro ta tio n i t 1# amooaaar^ to have a means of
fin d in g th e moment qhaage cauged by a olmnga in sheas?^ -Foxi th is case the
bcuadany o o M itio ss ays ^
O a t x ^ 0, 0 * O a t % # I*, ? # I a t % a O9
and. f » o a t x # &* This $lv es th e fo ito n tn g foun aguet ions«A3 f B3 » 03 f D3 «- 0
-A3 + B3 * O3 + D3 %-gjggSA3S^L G@8 Bh + B3O ^ s l a Bh + €%**%& c o s BL * D3 S ^ e in BL * O
A g -g B & fa o s %
^ s i n B h ) + B3C ^ W o s B L * @1% D& )
^ O3C+^ i c c a Bls * s i n BL) + S3O ^ J cos BL *• s l h BI.) « O
U sing detewiBahta and the n o ta tio n a s b e fo re
m =S 4 sin b # L + 4 s i n BBL
I
,»2BL * Cos 8B& * a la BBL * B)
i) '«±
6
I
(e ta SBi + cos a s t •*
{ 6 ^ + C08 8Z& + s in SM, # 8)
*
ont
4 * " g g # (»
+ s i n BBL » eos 8BL)
These v a ln s s f o r th e c o n e te n ts yhesi s ti h s t i t u t e d i n t o th e e q u a tio n
fo r th e Kioment a t th e @id x - 0 and th e end & * I r e o p e e titr e ly give
t
^
008 BBL
co sh ,,BBL
? * * -* * » * o 4 <28)
^ @3(si% 8B& + e i n h % T ^ W
and
< 9 # * s -W t ar [SG)
Bie Talnoe o f % and % ere tab u la te d in Table I I of th e appendix*
F n n w m i ) w:%vTa.
m rw im m w -.T om oui^m m tm s * % a f iz e d ^ m a
EidsieW' in a f l a t p late tsrhen need && the head of & cylin d rical pressure
v e sse l can be found in a iaamer sim ilar to that need for the fixedness
moments In the s h e lle The head frm a fixod-end shearing force however,
which can be found muoh more simply then that of the s h e ll as i t is the
tfiesbrone s tr e s s transmitted to the sh e ll end given by
oheorlng area ' ' ;8hvr
S a
* ♦■ » o- », 4 » « <$t . (30)
I f th e t h l Ohnees o f the head h is- replaced by th e s h e ll thickness t#
th is w ill be recognised a s th e lo n g itu d in a l m e # m m s tr e s s in the shell=
The d e fle c tio n is given by the equation '
w *s
-*. Jil$2 ❖ eg
■
*§ii£w
but
Ke ziSro # e a y i& e m&tmqmw
By I ^ ttA s g t h e radittA o f th e v e s s e l
be*at
O ■*'
or
Thg S e d e e -tio a e q ^ a tlo * i s th ah
” ” 1.1% ‘ j Z T i F •■• %
end since the Reflection, ia' aero Tahga r = e
8# " "
of
;Aj
The f i n a l R e f le c tio n e je c tio n can no# bo w l t t e ^ a o .
-
e fn
■end
The ^ q m tiO n e & r th e f ix e a -s n d toamente i n the p l a t e are then given
t . (313
4' 'tf' t' iff $ *i *• & Pr 6* * >. -d * ( 0&)
w here 3? 16 th e araftius o f t h e
€lSg0l»/iS jBBAD <* fh e Sla W lb u t3,(?a f a e to r a jfta? e f3 4 h S ^ la a l p raastip e W eseels
w ith f l a t h eaS e Aqh b e # t a l $ e & by p la c in g a u n i t a&g3,e ohewgo on then
j o i n t betwo&a th # o b e l i and th e
@#d to lv lh g I b r t h e . r e a u l t i o g ohapgo
In th e ioomehte i'si th e p la te s and the sh e ll*
Blace Ih the ease o f te a s e ls
w ith f l a t Ikeaae* th e t m a s l a t i o a o f th e j o i n t i s a e g i i g i b l o t th e r e i s no
s h e e r tilo'bribut.M fi f a c t o r ,
O o ssito rlag th e p la te wo have
A
****»$
and
Bar a r b it r a r ily p la e lh g a # i t r o ta tio n a t the ea$e o f the p la te ea#
s a in * th # same s o t a t l o w a s b e fo re
end
th e re fo re :
S e i I T B t I i ' " (Mh
m fi
* 8^
'
dgelga&t&ng t&@ f&e%wdL r ig id ity o f t&e p late aa % aaa w t W
t&et t&e pre&BBBe i& &$#*, t&@ raau&tiag aamaata are* where r Ie the
radius o f the feeaeli
Mg
SS
5»
S!
■*
d
«
i
*■
■*
»
f
BfoesBBKp D iam im m w m ow m m% maoma mtm ia m
*,
S
*
f
(
58}
o tw m ig m
SSBLia ABKD BLAf Im D S "» Tbememg&t d l s W h u t l e n f a o t o r f i e
4
fa * y n g - '.
where M* daaotea the moment dwmeea by a m l t angle (Change* Boy & long
sh e ll by etuatioh (SiB)
Bi* s* 8DB
and fo r a f l a t p late head by equation ■($&}
•
tey ta Mt 43 ■$>h X lw A lSl
then denoting the d istrib u tio n faotor fo r tb e e W l by
and the d ls tri*
button faotor for momenta 1% the heed Jba the radial said taogeatial dtraolions by
and
raapaotlvoly
"d * * * * % * t
■ajid
&P *
I- 4- tx)
£yf, A-
^
# * .<, # e, t * »
. , (SG)
Tb@aa m%uaK$#%*A #<m, IuvayteA g ty # a aim pley eoW Ao& <$"
(3 4 a )^
'9
l a tW fo re g o in g e x p re s e lo n s
, (08%)
W l % demote th e f l e x u r a l r i g i d i t y
,o f th e shell, said head yaspeotiveXsv
BDBBW? DimiiBUTiQB m o # B 8 ?OR
IMP
w im ,moBT,
SBADS * Tbe S f B a o tia f a w a it e a g le O h aaae-Ia She wmemt
a t the end of A short thia^walled h ^ ia a r ie e l u W l W givea by eqaetwa
W ) as
#' «
^
'
'
and fo # t h e f l a t head Dy egam tlee ( # ) e e
The d is tr ib n tio a f a c to rs e re M m
. . 4 (00)
*-59 ^
8#
^
^
J p . . : . ____ ,
*
8% ^
A gaia # a
+
f' k *. t « » * a a P » ' (3?)
f u)
s im p le r
& ( ! + *)
4 $' <1 <f 4
I fa
. % ^ iJ "Jvi
■"%
%
%
f i l + v)
li > # Ir -*• f .p. 4, (36a)
6 » ,* & * * .*'» * » » f #' (&v%>
^ * ^ o l w sxomplo l_ b y meano o f so ae o e a lv o o o r f o o tlo a s y
SOlatloa)
%e fi%e&4ea& momeAW fo r tM oholl by equation (Il) i&
y m * # % & ; * a % 6 ,&o8 % l*34&& % ovoooY * e . e &&*#/*&,
%b@ flX 0 A*ea@ momeats in %o bead# a#o by equation# (31) and (38)
*, 8 %&
grgKLgs,
a
^ M in ^ /in *
The A ia tr ib u tie n o f t&e maiaont W b& laaoe a t th e j o i n # b y e q u a tio n
(04a) I*
S & fl * u )
"S? ^ a +
*a
n* IQbMAOd
je ( I 4 0 '*8S)
=* % +
* * '# ?
'
,, .
Ginoo th ey # e r e o n ly W> mWoo## a t t b a j o i n t t h e rm g in & e r o f th e
ABment u n b a la n c e # u # t b e A ia tr iW te d i&o Gba h ead o r
-6 0 -
f r « f t » I - 0 .4 0 6 « 0 .5 9 4
The m m ent d i s t r i b u t i o n i s now c a r r ie d o u t a s fo llo w s , assum ing
clo ck w ise r e s is tin g : moments a re p o s i t i v e .
Mp
0 .5 9 4
Mt
0.594
- 148
+ 375
- 217
+ 94
- 217
- 158
+ 158
- 183
Ms
L
4 -
'O’ve
x ample 8 by s u c c e s s iv e c o r r e c t io n s .
S o lu tio n :
The fix e d -e n d moment To t th e s h e l l i s filven by e q u a tio n (14) as
FEi ■
♦ J 3 )r* « 2 x 333,335 X 0 .0 7 8 4 )( I . 03 - 0.082)
(0.0039978) = 199 - i n / i n .
The fix e d -e n d s h e a r in th e s h e l l i s g iv en by e q u a tio n (20) oe
FBl; a 2DB3 ( J 5 + J 6 ) r 1 = 2
x
333,333 x 0.08105 (-2 .0 0 9 - 0.148)
x 0.0039973 = l3 3 # /in E/ i n .
Ttie s h e a r c o r r e c tio n f a c t o r i s g iv en by e q u a tio n (25) ea
V« = BK1 ■ 0.280 x 0.9897 = 0.277
-Cl*'Ihe moment c a r r y - o v e r I b giv en by e q u a tio n (S i) mad T ab le I l me
c * 0.154»
The d is tr ib u tio n Is then
0 .1 5 4
T 7 -M
T T = -Q .277
+
37
V r r o .2 7 7
I
+ 199
+ 31
- 250
+
5
5
- 199
+ 199
- 35
+__35
+
I
I
69
0
0
+ 133
-
M
63
+ 133
- 54
-
10
+
69
CO' CTl.; ICI, - The r e la x a tio n method e l in in tea most o f the ted io u s work
Involved In th in -w e lle d pressure v e s s e l a n a ly s is by introducing an a r t i ­
f i c i a l s e t o f boundary c o n d itio n s end then removing the a r t i f i c i a l con­
d itio n s by the
crdy Cross methods o f nomcnt
nd shear d istr ib u tio n *
The a r t i f i c i a l boundary c o n d itio n s ore introduced so th a t th e fix ed -en d
moments end sh ears may be e a s il y computed using e ith e r formulas or t a b le s .
Tables o f th e n ecessa ry con sta n ts rre provided as a fu rth er aid to
com putation.
This method is in d ica ted a s being v a lu a b le s in c e the
constant©
a n lf os sSeH len g th
m
<5 s ttf f n e e c snd w ill apply to a l l
veoeels with tliin*aiallerl c y lin d ric a l B h ellsw
fh ie paper dociC no t cover a l l o f the c a lcu latio n s which are necessary
fo r tho analyele of a l l types of thiu^w allea pressure v e s s e ls ,
The ves­
s e ls tre a te d e re th e sim p lest to analyse tout they demonstrate the pro-*
CeSures anti method o f a tta c h without confusing th e read er to an. unnecessary
degree» On th e o th e r bond, the payer of the Hiethod la not shorn to the
b e s t advantage os the savings in time end lab o r ore p ro p o rtio n al to the
complexity o f tho problem, being treated-,
F u rth er a p p lica tio n s m n he made to v e sse ls with e l i p t i c a l s h e lls
and s h e lls w ith non-uniform, diam eter in the, lo n g itu d in a l cross-section*
as- w ell as to v e sse ls with hem ispherical sad e li p t lo a l heads * ISes t h i s
has boon done the f i e l d Of pressure v e sse l a n aly sis w ill have been, sim­
p l if i e d end th e tru e value o f the a p p lic a tio n of re la x a tio n methods top ressu re v e sse ls w ill hm e been rea lised ,.
&
-essTA3&B I
OWGmKB FOB PixmMMB WOMBR% AMD ORBARB IQiBKORT 98m&*
BL
I
I
. ,fa.....
..F# ,
,.f*
. fs'
fa
1*0
1.8
1,4
1*0
1.8
+ 8.104
8,88&
1.781
1,480
1.846
T 8.0BQ
1,099
1*490
1,988
1*315
2.939
1,986
1.419
1*023
0,943
t 0.986
*800
,690
.090
*499
* 6.740
4,356
3*211
2.684
2.315
+ 5,745
3.170
1,840
1,080
0*618
8,0
8.8
8.4
8.6
8.8
+ 1,184
1.06S
1*081
1,018
1,009
1*108
1,084
1*049
1,08?
1.014
0.538
*0?S
*881
,158
*088
4' O»40Q
*329
*260
*297
.186
- 8,158
2.090
8,051
8*014
8+009
+ 0,310
*111
* 0*019
,101
*148
8,0
8.8
3.4
8,0
8.8
%
3t* X»000
1,000
1.001
1*001
1*008
> 1*00?
1.000
1*001
1.001
1*001
0*088
. ’i* 0*010
*034
*048
*985
t 0*110
,07?
.048
- *080
.008
* 8.008
8,007
8.007
8*009
8.008
* 0 ,1 #
*179
,165
,14?
,186
4,0
4.8
^*4;
4*8
4,8
+ 1.008
1,001
1*001
1,001
l.Odl
1*001
1.001
1.001
1*001
1,000
f 0,056
,058
*047
,040
*033
* 0.004
*011
.015
*018
.019
* 2,004
8.003
8,008
2.001
2.001
- 0,105
*088
*068
404:4s
*030
8*0
5,8
5,4
5.0
+ 1,000
1.000
1,000
1,000
4 1.000
1,000
1.000
1.000
4* 0.026
,019
*014
*009
r-o-N—
q
* 0 ,017
,015
,013
,010
~ 8,000
8.000
8,000
2*000
. * 0.018
*009
.008
+ 0+008
W.nIm.W.HHKWMW.
m m
c m & tm m
u
3pob v i t m x m r s i Q W
%
1.0
1. 8 '
1 ,4
1.6
1. 8 -
*
I .538
1.914
1+170
1,080
1*086
8 ,9
8*2
8 *4 '
8.6
8 ,8
M, 0f 0@?
.995
,989
.986
.990
8,0
3 +3 '
3 ,4
3 +6 '
3 .8 -
«
4.0
4. 8 '
4+4
4 ,6
4, 8 '
-
5 .0
9,8
9 .4
9*6
*
0.994
.999
+999
1,000
1,001
1.001
1,001
1,000
1,000
1,000
1,000
1,000
1*000
1,000
'
'
'
'
1+076
I +039
1.019
1.009
1.004
+
0+527
.816
+900
,902
*500
'
"
'
'
4- 1 , 0Q4
1.004
1.004
1.003
1+003
**
*
*
'
'
'
1,008
1+002
1,001
1.001
1.000
f 1,000
1*000
1,000
1.000
0,498
+498
.499
+499
+499
0,800
,800
*800
.500
,800
*
0*500
+800
*500
.600
-.
. .
A 0,461
.918
, 530 .
.949
.898
'
'
'
'
.................%
8*390
8.139
1,606
1.918
1,189
+
'
'
'
'
m sm m
*$*
*
'
0.249
,181
*184
,079
*041
0,014
0.005
*019
.084
+089
0*488
+498
*459
,430
.895
+
'
f 0+358
+303
.858
.802
. ,164
0*118
, 09 ?
,047
.085
+008
+
'
'
'
'
'
'
'
'
'
*
0.028
+086
.083
*080
.018
m 0,004
* 0&1 '
+OlG .'
.010
.019
^
0,015
*010
.009
,005
»
'
'
'
'
0
..
'h 0,436
.458
■ *440
,590
.321
+
'
%4
m m s
-
0.017
+015
+013
.010
'
*■***
r
m c 'a R ? op m m & i i poa Tns a m m a m m m o a
G aaem l Form alii ApB&gi&a to IM a -B a ileG O y lin a r ia a l S h e lls and F la t Beada
D*
I e ( i - SB?"
*■»&
s - n)
P o m a a il # p ly t& & 1w» lo&g T b l n - W l a d C # W # i o a l S h e lle O aaer DaKorm
Pj?esoure D ia trlb M lan
FSB « 8DB&<
'
.
'
#88 * 4BB#r*
Si* «, SKBB
9 If
'
#& e 4 9 #
F o r a o l i i A pplying t » S h o rt T h in /w a lle a D y lln d rlo e l G k e lla D M er Dnifomw
P r e s n w e D ia tr lb n tlo n
#38*
-t J g lr+
IREEf * SDEPtls + Z g lr t
I*' = # B ?&
V ^ a TB %&
Tg * 4DE^%
M& * "5^
*91 * #&
'
'.
F o m n lh im plying to F ia t O irouler Heads IJnder H aifom ,Pressure
W tlo a
0%
e !
m i^ s u
/f'
# 0 16 m
Wy * # t = p.l& j& JA l
3 bm u 3. i l
ApplyiBB to D lo ty ib u tio a o f B&me&t# i a Ta& sele %lt& lo n g Tbi&*
W alled O y lW r I o e l O h o lle ead B la t Beade
BBeB + J & f l 4 a )
(I + s)
9 % B *" &
( i + %)
F am ulli Applyia# to D W tlbutioa of Moment 1% Veaeele With TblM M l#*
Cylindeieal Ghelle and F la t Beade
i i s i r o r A BmmnATiom
0
«■ Hoiaeat carry-over fa c to r fo r sh o rt gfcello*
D
n F le n a ra l RlgM ltiy
Dg
*
Dil
&H e x to a l BiglSitiy o f t h e heed
1
si Toimgt S B W elu#
Rlgialtiy of the# e l l
Ay* % *»« .i^ W lti ^tifalhS
f
« H orsai fdroe
FSM
«•■F ised^eiiS somenti
FBML a flxedMoad Bomeat 1& a diametrldel plo&a of a fla t head
###$ * 3M%e&*W mmmt la a Blame &e#peBa&oul@r to a diameter of a
f l a t head
"FBS
* 3l%@&Mena gheef
fg
» 1SbmeWti tllstiribwtiiba fa c to r f o r the s h e ll
f_
* Boyteati aistiribatiioa faetior in a dlametirloel plaac o f a fla t bead
« Bomeat dletirtbatios factor la a plan* perpeodioalar to e diameter
o f a f l a t bead
It
» Soad tirlokwesa
I,
» $hell leagtih
I
Np
.
S=MOmeati
* Moment la a diametrloal plane o f a fla t bead
*&. ■ » Hosioati in a p lan e perpendio alas? tie a dlametW? o f a f l a t head
M*
a Wbmeat re s u ltin g from a u n it angle ohan&e
#*
e Moment at a joint resulting from a nn&t eheer change at the joint
"^SSw
t
'%
^ WWM&t a t & J o W r e a u ltiK #
a tm $ t @Wat AhaAgA a t #* M jA W b
'joint
p
a lfsl t preeenne
r
sa Testiel MSlns
r*
» !.Teiabrane R eflection of $ sh e ll
s ' %str e ss
t
=i S h ell tlx Iolifiess
u
« Folason9a Ratio
T
*= S h e l l R e f le c tio n
V
5= S hear
%
« Ohenge in shear Rue to
Vg
« Change In shear Rtie to a unit Sisplaoeaent
w ■ss HoaS d e f l e c t i o n
&
u n it moment ehnage
BeeAy oeoaa * aag ly ei # o f caaM ew i # Fwmm by M ^ te lb e tln s
T=Mimts,! S w a se A tio s e A* 9 a
Jht &#&,. Pp1 I - ^ s e *
1 « B* QriBtofi Bomeflcal Mbtboda o f Aaalyele 16 BbgiaaarlBg , MagMllloa/
Be* York* 1949»
8* Slmoe&oakG* T&eory of Blatea and Bbella * BagAaeerlBg Gocletlea Moao^
S ra p h s ,
Book Company?- M m Y o rk , 1940»,
8 . stmoebeako* Sbeory of Blaetiolty,. BRGiaeariag oooletlea RbBOgrspha,
KefJrax^-Hill BOok OompBay# lew York, 1994»
t
‘ 1I i ' ! t
MONTANA STATE UNIVERSITY LIBRARIES
001 3594 4
3 1762
i*£lS78"
'
'
_______________
‘
___92_63jL„_
""
m ire v e s s e ls Dv^____========-—=
issUEo TO
^===Z======F=^ / f*
Z _ .' JlY
>
92 6 / 3