ON THE STRENGTH OF THIN SHELLS UNDER FINITE DEFORMATION

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000100190015-5
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 29
• Document Creation Date: December 22, 2016
• Document Release Date: March 12, 2012
• Sequence Number: 15
• Publication Date: February 1, 1952
• Content Type: REPORT

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STAT
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STAT
ON TH1 'S 'i NQTH OF THIN_ H1LL UN DJp 1'INITE W~FO!ATION, 1, FtJNDA 1TAL Th1 OTY
S UHAJA Jiro,
1aeheior oi' l incerin# and
AJoainte Member of the Arc 'n.
otc; Th i'o11o%Jir is report hard 3 Apr11 19L3 t the joint neetitg
of the on yoke, / 'hipbui1din AeooeiationJ 'arc the c en ; yo1tai Han'Shin
hurtbu Shipbui1din ic ociation1 Ocnka-Kobe Cit17; and pt1iched in the
Jaandear~aQ jotal Posen -yoai Koen epo~t, Qi' the hipbui1dinj Assoc..
1atio 17, Vo1ui a 72 (Jun e 1 fl 3), p 'e 1O1s1i: t. The ab3tra t in the cri in 1
waa in M1i2h, which ie copiad here.
Tho njithor it a Univorstty ha11 student at Kyushu imps riai Univeraityj
ACT
MT TACT
T'he oneral theory oi' t1?iin nho11a wao ?orl~r1y invecti ;atod by A. E. U. Love,.
~d huh been applied to n problems of etibi1itsy of tht he118 and their
r;cults of cu au1ation compared with ~xperim me by varios a investi atoro.
ener ai1y it way etc r th t bone re it1ar dis r~eps~ cie i exi ted betty wn reo Its
ded ccd ?rom Lov/' ? theory end experimental cvidencoa.
Th. von Kermaf pointed out that dove t a thcor t baeed on inrinitely sma11
defor;nation of oheUc ie not realised in many ca?ea; actual1:, deformntiona of
ohe11a by cxterna1 or end preco it ern ~? called "Dttr th ch1agn ~ote i Tho.
provtuional paneoe` tr n 1ation died by the author for th1e word i tee u
which liter 1 meaeio nr~obending" according to the ccparato Orinoco ch rcctere j
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whereas II rehsoh1 L lI mf~ans "punt thrcu h" ,7, and ccnseq'lently the stability
of thin shei2.3 must be dicc1lssed takin rota account ,finite dt 'o matioh based
)n no ?jne r r 'ereftiai eqi at Mfl o' equiiibrium, Duty his th o r has not
yet been coinpleted~
The author firet rove;;t:t ates th, genera , theory of eias 4city takin into
account L nite distil(icAment, then he obtains the general theory o4 then shells
for he c t@ cal' iar a def ranationa, which ?orr the eheraLi ation of Love's
thin?5heil tncacry; and 1cat1i, its cp cie1 se, he chows the f~mdwnental
e dation of t~Durchschla " by the ; ethhc rf der: vat1Mn enuatn rci'erred
to coano she11 shapc8,
:i: r ifTinDU TION
The enflral thoory~A'dcfor,natic)ns in shells ~tterali', "thin eurvcd plates
wan fret discussed by A. 1i. ;'1. Love 1) on the basis of the assumption that
the43e det'Qrmaticns are infinitely s~na11, ar~d /star finny other investigators
d o it ca det one in application to the probl m of the bucklint oV ahelle
earn. ,
in d fig to make cdnpariaona with eicperiment?1 results; the general conclusion,
hcwevor, was that he bccklir v ,yea of . siie1i which poaaeoaoa initial curvaa
i.iro is sCVCN~1 timed larger hen the oxperimontal va1ue. This waa a plained
t
by Thy von 1aranan as dui to '~D~~chalaa,~ ote; 'o'ho Japanese exprhaf~icn,
tenkut!u employed b,, the author here to translate ~ach3 1 LlTbreak
through7 miens 1ite~aily n'ubendin&'? It is intero?tin to note that the
fig anew word ~akutau, ,for "bucklin~n muans literaliy ~~oe1 apse-bendin~~
Itat is the earl problem of buckling wau` a theory cmp1oyit~g the ao umption
af int'initely c;n11 deformations, and hence it8 buckling vtluee pare souht
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w the j envalUe3 cf' a linear differential equation, cencernind the phenomenon
o;t d?fcrtuattQfs in sho11s 'posy sain initial e rvature, if one takes into Gon~
deration finite def'onmation in shell', it is spparent that there exists
eceilibrial state wivleh can rnalntain equilibrium by an external pure lower
than the rtbnvewsenti('ned bueklin value; bafore the external pressure reaches
the buck1 n value, the shell leaves the state cf' infinitly ssoall deformation
asumed by Love and thin dicontinuously ex;eriencea finite deformations. The
i8tt4er is r ften re riled w the bucklin load in experf.ments This reprosents
th so-daUed nenomenon of burch8ch1 and, if fin?'ite defor,nation is not
takers into considorotidn, cannt,t be interp~aUd.
The theory of shells twin into consdertion finite deformation has been
studied in connectitan with by 1~ariaan 2'3) i~A'v~l~O tt), FIlZtJ1~I
t3ar Uc 1ar shells ,~ ,
' 'rUY 1 b); but, as eva~ Karman himself has stated, the theory of lurchschla
has not yet been perfected. Conccrnint pht nomenon besides nurchachla, often
ofe mirst CCls dt states of finite t eformatlof in order to d!cide the ultimate
~atrmth of shells if evtrcnae1y thin. Since, hnwever, Ur e hati been no litcra-
a : as far as the author knows, ivin the cner l theory of finite doformed
start?38the a~ thor hay attempted here to treat this subject.
In etudyin the finite deformotione of sheile nne must first review the
~
cneral theory of elaetie bodies cnb Jected to finite defoinn tion1 which theory
veal studiee 7,8,9) on this rave already
basic to the diecuesion hero. 3~
appcarcd, but cinco they are thought unsuitable as a i'owidation for any theory
of dofor?ation ?f ~hclls, tine author will fire e?deavcr to abt~.in the ecpua~
bons oi' equilibrium rel tivn to a coordinate ayatem which dcinne ( trine ornw )
-
.3.
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to..eth(r tirith th ~ b dy; next he w U pp3y these equation to he
deiorn>atiol oi' czriia to obtain their fundarcnt1 equtiAn , "j'hen, after crrr,w
in out o~nc tdies in coection witty 1urchchla, the author will derive
the crier 1 squat orn hind i'irraiiy wit stow a nethod for or~tainin definite
oncrete muatio i'or several a cia1 sae Fectantu1ar aid polar coordinate,7,
I1. INIT pT T'(r1':oN (T1: t) : jfl A T'~C
e '
~re 1 take ' ~r~r~a11 rc):e in th hodfir; to ndicto
In l~iu
he poa Uon of
ex a by any ctf vi1ine r c~ordin to syster~
with th i'orrna ( tranaforns ) along with the body, rid by a reetan u1ar coordinate
syfterfl R(`, x , X 3)j which is held fixed in aace~ (Note; The numbers in
0
xl x2, x are not exponeni,e but exprAae mere indiP!ce. indices amurne the values
o1~c we shall ueo this notations) Ta1a the position of E
1, 2, 3 - n what foil
before deforrflation an A, and the x ition ai'trr defr~rMatic~f ae 'P. Taking Rl
d
bt 1 ore tic for~flation, w@ i'ind the re1,tion between ( ! s ) an
to be
c1e which le very c1o~e to ; take he position
Next, take another Marti. a
cif t19 br~fore d Trnmtion ee A, rind the position ft' deforrnatton ae 1Q.
The cnrdinateo to and of aro rc ectively (x~ ~ dx,x2 ~ dx2, x3 + dx~ )
4. ~
} p~.71
{
3 i? th~ ~~~~~llws babeeen ~ and C~ ~~foi?~
der'o1 m Lion ie taken as dr, then di' ie ivon by the ?o11owin equation
(2.2)
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Next,' we subtitute into (2.s) the prior ~btind from (2,1
we obtain
where we take
(Note; Uex'caf't r, the J snarl 'ttcrd i,
j
ck
f, '
ueed to deu3ir~te tho' various
(2,3)
quantities relaU.n to o; d to Urt k 1ottora I era used to do l nate
the various q anttt oo elattn tc J, 'n the came Index appf ars twice in
the same terrwe hallreo to the ueual t-eor ouatiori convention that is,
in every such case we Turn from 1 to 3 rel tive to the repeated indox. )
Ne{t let the e3amtlc hod 8UVf r a dpi rt tlnn, U tiro coordlnat?3 oL
ref tiva to re t ken` ae 3'x ` x4' x the ix re +r?oent the c n
o ~ ~ ~ ~ ~ ~ ~ ~ 0~0ents,
relative to R, f tie dhepl cement vector, hmve ex p da ed 'xj and
A
II as functions of ' ` , and i we ouurn
then the o ooordln~tc~ o ' ~Lter a de 'o ton c n
(2.t)
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Corrsea'uent , thA F. coordiniato3 e '" a 'tor do 'orm tton become
there 'are he d agree d'r betweon
ecpreseed ae ?A11QWS:
I,
I *7
flrt'1Y
7*'"
sax s
mm aM
L
Agin, ii we yet r' i r, ,
we het
* ,,1n2
Turtbor, it we ilt3 e, re1tt v to /other co rdie
vcrt:ec with the body, the ereooion
%H clearly cif eetcbliob that
F
.'.
;e 't tern
4
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Y
(2,)
'ter doi'o iri tion ie
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ire ?the components o' a sr cond-srder cover art
thafore, we find that v
tensor rc s.nd mQrec~ver ?gymmttrio ecordin to (2.).
~,rtt~,ve to ~1
e the` com -vnents o tb disp1uoenent vcctr ref?tiva h as
Next take ~
risnt derty tLve 1-v that t~
s
eov I
sin it
it: u
J~9 a dI p d+ --S-- S'
If f wlw.~nRRi~n~ ~R
w,an?' e:rsrn^ ~ ~ ~ 1
! ~ trm
'vi.~
we tran8i'os'm vi in the right side oi' (2.) tto f' we et
(2.G)
(Note: 'rye c~~~. iz'{avisions11y the tensor ' the distortion tensor. See
1o) The third term on the 'right side oi`' (2.6) is Use terra that
;~c~o~~i~e~.~-
.
exp~ cc3c~s tmow the i' ct that we are oonaiderin ; Finite doi'or~nstior.
f which Mrises dui to di~tdrtioi, of to coordii;te syBtem
The v~~ ~.~tio
s~
c)f the Chr ~tOfiM sy~ubo1 , rynistive to thy; coardiTHt? system, which
t
d~iorras t4tethr with the e1s8tic bL)dy) ine1Udc the disp1 CCrneft end its
'
derivative when o e grin in the td{ouilit rium ecustiorrs; hithE$rto it had been
disrcC 4rded, but in the theory oi' finite dormUiti')f it is feoessary t? take
it too into ooosiderr~tinn. Con3qu nU:y ifs t 1cin ttw Chr? s toti'e1 syrnbo1
' ~ ! we take u to cdnd?order tens oi' /i I we
jttr 08 n
' .$
f~}
M9
i'
Ill
(2.7)
where
d1r :
if _J _t 4AfiI/ ?4
'II
(we d? not tike the sti- relative to
k.
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Tensor
F' V
i auantitf CQfccri~cd with
dtortiQ, but i not the
di tartion it 1f. ttow i we tike ~. the compQwnt o' c~,c~n anon
1:1Pi ttve to ooordLn to P then we pct
x
(the Burn e1at1.ve to, is not taker)
where
P'Rx_.
:J
F ;4
oxpreaee the phyic1 components off'
?hcn t rsru
t,n c i t Qt C1O ;rO~ hiIhor thvf the second can be omitted, our work i~ it a ;ree?
zucnt v~ith pr v1,ou Lheorie . Ii' we take 1 ~ the an lc ?Arined by the
r% "i' ?? curve nd , - curve ht 'ore dci'otimatiof, we het
I
;
F?
4
%
: 1 1 'tY vi \i
(the usual unun~itton convcntio does not apply here to
w
1:hen wt cm nv,1t tt~rrn in ,, above tho ocond cue riec, We t
which i n rrfcric;t with previous theories.
A;ain, wt~ cok suitably condo oris, we dot
the third teii on t otti eido i are newly ddod tern.
nI N LAW lt" PJAST 1..(1'Vr A1) UATX )W i M- T1I iN
Tako S ns the uroc oL' ni clot ed curve in an ola, tic bcd r ai't rthe ornitrtion,
grad take V e i the volume oi' the part stirrounJ d h - S. Ii' we dosi ni to T trio
kinet t_ nor jy poua n ied V, wind +oei ;i nto by tho dei it i ' er d o ~~
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tiOf, we h tv
(Notre; r is t1 e ec rd n to that ' on it h he o id, ) The
dot (' ) deinit'a di''crenUtir3th re pt to thr trn~r t, end we use the
desi.nration
e~.Yr=
If
{ yc 4
;." /? b _ k
If we take Kh Born onent the
re5ultBft 5tr f vector open itin on V end the camponont c " the nacnentwn .
vector, arid if we de i rate by ' V t w virtu 1 work doru by K and M we
then have
whera
If :e tbke k uro coinpotl nth t c 1 o1ocity o
t:,;.
I
?urtlior, for ~r two tirnou to rid tl, we
at, introdbein thn ~trot ten or ' by moo of
employe Oguo~ ' dvorjonoo thoorem, we ~t
Ri
S
(3.2)
(3}11)
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Hamilton's p noipie i
8
t2 o 'or, Vii` we ubatitut ' ' obtained by substitution of (,),j ) intl.
(3,2), 11''d (:3.3) into (3, ), wt' oLit tin
C \ / 4'
/j h r V
+f ~
1
In the above L'ornlula, 1 is any arbitr wy virtual f sj~ ccui ent;
t1? creforf), iC a 'virtual d p1 cor rnt d( 1r) con uonti " ' is taken w hcivo for all c1 ~. that, is no di ti t cn i
pToc~ueoc~ r iattve'to zuc ~ virtu?a1 dtspi conic,ntti. A1ooj s no we haves
In, tin
irs
1y t
y
the 1a t term in (3.o) 'ray CQm egta1. to z o 0, Frcin tine ? of tint (3,6) houid
hold ?'nr arbitrary t~ and \T we cat tl,n{ equation c_f motioia
(qi f
i I
(3.7)
;ach tc Am in the above equation exprotaea the compotiont i cf' the v out u nt-
iti. a r 1ativo to th coorc1irH t ay t rn cift r thti'ers' tior-~
A.130, the virtual work at each tiin? bceonio rina1 -.
t fig, '1 ~ Mrs
S ` ~
}~ t an + i '.mss 1 ( ;) ,
>
fittrthn'rtinrt amoo qx},:
= o, we het
If we do i~rinto by tbcz oiaetic potenti, wo dot (3,)
(319)
10-
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since p ?wwtion off'
we h vc
9?
comp 'in (:3a9). ncl (RaO) w~ t
a~pt;f
is an invL:rint, w must btin "(; in the L'om Q~
mSR~
: 4 .~ t'
, Furth rrnc r),
I o also is a
?unoUcn Qf 3 the ratio with tie cenaity / b+ i'ore dcfor~nation is
i'i erg by
tease
(3,10)
(3.11)
Consequently, now tskin > and ? as constants (Note; Tki re are no ref ticns
Gaon the iNjice )' ' / , and setting
B--
we het
3
(3.12)
(3.13)
In Finite dittortinns, wlun we cur cisr at{d I, I2, l3 in comparison with
unity 1, we het
d1"
l1
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iN () 2,) 1,2 4
(3.2)
 how we tike
wh:,cth divas us hooke I a Law, I ' we take s Y Aunt I s mQdu1us and
?o - ones r tic (Nora; ire c obtain ?r3
on
and
)
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i
the f oUowin re L tions among > ,/2
and Il, 2
a
y)
Iv, DEFfW~1JtTI, AN ~,I', THIN SU r1J4 s~ A1i1j ilFW 1~)c~UATION1 ci NAMiIC$
c1ow, assurnc constant thioknecs and ccnaidcr very thin shops; eaf a
curved surface (shell) whr tl ickm~s is divided Qvc ~ly into two pantu a
,Iceratral surf'ace', Use
e the indiccc for thlr vcctar n d tcnsdr corn-
pQncnt r l ;tive to a contra uric; we adopt the cnvar tidn t. tat thcir value
are taken a 1 aid 2. Furtliermoro, before dcf'r1 ntatt.on) w' assume b /for thand
th t FiF is 0o c1Te)sen that tt~e surf;'ace, whero constant coincide with
the central surfac~esa The pezp ndicular lino From aj the position of any
particle before deformation, in the shoi1 to b) the centre Surface i~ei'ore
deformation is asf ur ld to intersect the pernondicuii ' to ttic cer,trri surface
o. 5o of tc5r th fo ruat is . Wo of fix the ort symbol
fl /A
fries r 1atfn % to tlw ce'=tr l 91trfIf'?, alit dittinCni$h'?'rom the cneral point
guLntitiee inside the at~ll. If we de~i~nteby z the I istanco measured on
the norm, b foro deforniation, to the central gurTce, thon the distortion
tensor of a + arbitrary point within the she11 k'eeomos u f'oilow
ors
mtrtra
L ;
A
. 12 .
II to he 1TViOI gucnt?
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i
Va
rbitf'ary law Qf' e asti~ity doppndin
wf )t
r_4
r_--t? ,- -1-' --t
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toptiQf o the centrt1 Suri'ace~ Also, it we designate by 1
the componnrnt relatiuc to the coordinate sy t 'm, ai't ' dcrmation ' the re-
i tiv..diap)acenient vector cQr ~ pandir to a point ?tx o cn the centrai surf cc,
wo ex1)res the 'r it iva distctiQR with rospcot to the c ritz J. pL by
r--T_
rYaa
urid the quantity r1 tin the change in to curvature of the antral plane is
#i
f
Again, , delign tin by y /iand Z he s odnd i'undamentc1 quantity of the
cer~tr' aura"ao bei'ore nd iter t~ ~rration, r n peotivo1y, w la vo
u thcranore, deli ni~tin by T and "y)thc radius o curvatu bei'ore
I \ 4, j J i (p p.1r s'P
1 and eftor, reopeotivo j deformation relative to the direction detern~i u d by the
ratios oi' P a fi f" of the central suruc?, and by H and ' h the evorao
c1 rvbt? 4 rea and by K and 'K the tot1 curveture8, reap4otiVO1y, W~ het the
:praeeian a ..
( the uaua1 8u w11ntion rciative to C' d+~e~ not hold)
From tie ~lasumptions rearc~in the relative distortion with reepoct
to true central ouriace, n?e1y that the aha11 is very thin Inc th+t even ui'ter
dei4trtn1tion the normal to the central suribce io maintained, we c n oet
--
,1L and also whon dime tirdin ; torm3 in W of the nccond do reo,
1s13a
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ws can sxpriess 1'fOriZ L tiv&~ iunctiOf ti ` rod ativ~a A ~ tea
#)
( , , ""1R and in k~c ?OUOWin approxim tc Form;
Conscquent1y, ii w use
fi
(f t9,1D
C,2'+ 2:;
"L )
,) at the boundary condition
z ' h (thy thickness oi' the sho11 is 2 h) in the ut,cve ?'ornu1a and both sur-
f'cee, w Cet
Corsoquontly' we have
Sii ilir1ya
r~w+ww+
C
2
~t k
I ~ c1_ ta c
1_ 'i
t3)(
~4
d
h O)
C
(b.2)
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Aatn, satin
,.,
~--,
PI+ T ^
d
a
i-I
1 * aM
_
and rnultiplyiru (3.7) by k and zh, we into rata over the sh U's thi:cknss
intexv 1; i Jf u c (t,2) and (L.3)
txcomcs
hR ~qu&4on8 o~ motion o?' the the s w13.
since the hell i v thin, w can as usual set P33 0 0, zrtl~srmaxe, i
w t (L.) w ~
f f
~. rt
saz:?as
tth r~ wo take
1
0