Plate Theory Aid II

8/6/2019 Plate Theory Aid II

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, *I ,2, i THEORY OF ST A B I L I T Y OF THIN ELASTIC HETEROGENEOUSAT!IISOTROPIC PLATES OF V A R I A B L E R I G I D I T Y

E . G. Shulezhko

r

T r a n s l a t i o n of K t e o r i i u s t o y c h i v o s t i u p ru g ik h t o n ki khanizot ropnykh neodnorodnykh p l i t peremennoy

Mekhanika, V o l . 6 , N o . 2 - 3 ,\ .

zhes tko s ti. Pr i kZadnaya ?la5:er:ntikaipp. 139-150, 1942.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONWASHINGTON, D. C. 20546 NOVEMBER 1967


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THEORY OF STABILITY OF THIN ELASTIC HETEROGENEOUSANISOTROPIC PLATES OF VARIABLE R I G I D I T Y

P. G. Shulezhko

ABSTRACT. Th is a r t i c l e p r e s en t s a n ex p re s si o n f o r t h e t o t a l~ p o t e n t i a l e ne rg y, t h e d i f f e r e n t i a l e q u at i o n of t h e e l a s t i c

su r f a c e - of a p l a t e , an d t h e b oun dary c o n d i t i o n s f o r . a n is o -t r o p i c h e te ro ge ne ou s p l a t e s w i t h v a r i a b l e r i d i g i t y , a ss um in gt h a t t h e mi dd le s u r f a c e of t h e p l a t e i s s im ul t a ne ous ly thes u r f a c e of b o t h e l a s t i c and geometric symmetry. I t i s shownt h a t K i r c h h o f f s b ou nda ry c o n d i t i o n s f o r t h e be nd in g o f p l a t e sc a nnot a lwa ys be e xte nde d to t he buc k l ing o f p l a t e s .

.. -

An ever inc rea s in g number cf s c i e n t i f i c w orks a r e p r e s e n t l y b e i n g d e v o t e d /1t o the q u e s t i o n o f s t u d y i n g t h e s t r a i n and s t ress of bo th homogeneous and h e t e r-ogeneous an is o t ro pi c bodies .

S. G. Mikhl in [l], S . G. Le khn i t sk iy [ Z ] an d H. M. Savin [3] have accom-p l i s h e d s i g n i f i c a n t work i n t h i s d i r e c t i o n . T he se a u t h o r s were t he f i r s t t of o r m ula te and so lv e se v e r a l e x t re m e ly im por ta n t f undam e nta l que s t ion s r e ga r d ingt h e p l a n e t h eo r y o f e l a s t i c i t y o f a n i s o t r o p i c bo d ie s .

Some pa r t i c u l a r c a se s of a n i so t r opy were cons idere d by Saint-Venant [ 4 ] ,Voigt [ 5 ] , Somigliana [ 6 ] , M. Huber [ 7 ] , Y a . I . Sekerzh-Zenkovich [ B ] , L . I.Balabukh [9] , G. G. Rostovtsev [lo], S . V. Serensen [ll], nd o the r s .

Saint-V enant,Vo igt , and Somigliana were c onc e rne d wi t h th e s tudy o f th es t a t e of s t ress of a n a n i s o t r o p i c body ha v ing th e sha pe of a l o n g c y l i n d e r .M. T. Huber examined the l a t e r a l b en d in g of o r t h o t r o p i c p l a t e s .Y a . I. Sekerzh-Zenkovichand L . I . Balabukh s tu die d th e buckl in g of plywood

p l a t e s .G. G. R os tov t se v de vote d h i s a r t i c l e t o t h e q u e s t i o n o f t h e r e du c ed w i d t h

of a n o r t h o t r o p i c p l a t e .S. V. S er en se n, i n h i s c o ur s e o n t h e t h e o ry of e l a s t i c i t y , d e r i v e s a funda-m e n tal e qua t ion o f th e p la n e pr ob le m f o r a n o r th o t r op ic medium and g ive s the so-

l u t i on f o r a case of th e bending of p l ane beams, and a l s o d iscu sse s th e works ofs e v e r a l o t h e r a u t h o r s d ev ot ed t o q u e s t i o n s co n ce r ni n g t h e t h e o r y of e l a s t i c i t yof a n i s o t r o p i c b o d i e s .

- -. . _ h i s a r t i c l e- ons ide r s_ the q u es t io n _- o f ghe s t a b i l i t y of- a n i s o t r o p i c -.- - __*Numbers i n t h e m ar gi n i n d i c a t e p a g i n a t i o n i n t h e f o r e i g n t e x t .

Commas shou ld be in t er p re t ed as d e ci m al p o i n t s i n a l l m a t e r i a l t h a t. B . :ha s be e n re p roduce d d i r e c t l y f rom the o r i g i na l f o r e ig n docume nt .

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h et er og en eo us t h i n p l a t e s o f v a r i a b l e r i g i d i t y w i t h a m id dl e s u r f a c e t h a t i s s i -m ul tane ously t he p la n e o f e l a s t i c and geometric symmetry.

The a r t i c l e g i v e s t h e d e r i v a t i o n o f a Brian-Timoshenko eq ua ti on , an exp res-s i o n f o r t h e t o t a l p o t e n t i a l e ne rg y, and a l s o a d i f f e r e n t i a l eq ua ti on f o r t h ee l a s t i c s u r f a c e of a p l a t e and boundary con di t i on s.As f a r as w e know, n ei th er the Brian-Timoshenko equ ati on, th e ex pr es si on of

t o t a l p o t e n t i a l e ne rg y, n o r t h e d i f f e r e n t i a l e q u a t i on of t h e e l a s t i c s u r f a c e ofan a n i s o t r o p i c p l a t e o f v a r i a b l e r i g i d i t y a r e e nc ou nt er ed i n t h e l i t e r a t u r e , n o tt o spe a k o f the boundar y c ond i t ions , wh ic h ha ve no t be e n de r ive d f o r t he case ofbuckl ing even fo r an i so t r o pi c homogeneous p l a t e of c o n s t a n t r i g i d i t y .dary con di t ion s der ived by Kirchhof f E121 f o r t h e p a r t i c u l a r c a s e of l o a di n g o f/ 1a p l a t e t h a t i s under c ond i t ion s of l a t e r a l be nd ing ha ve alwa ys be en a u tom at i -c a l l y e x te n de d t o t h e case of buckling.c ond i t ions c a nno t a lwa ys be a pp l i e d t o th e case of buckl ing .

The boun-

.As w e w i l l . see below, th es e boundary

1. Basic P r e m i s e s and HypothesesL e t us imagine a p l a t e whose l a t e r a l s u r f a c e i s formed by a c y l i n d e r f ( z , y ) == C, w h i l e i t s upper and lower sur faces a re r e ? r e s e n t e d b y t h e e q u a t io n

.

Zk= - )k h, ( 5 , y ) (k =I 2) .

L e t . u s f u r t h e r assume t h a t t h e p l a t e i s t h i n and h es s u r f a c e s of e l a s t i cand geometric symmetry.

We s h a l l t a k e t h e mi dd le s u r fa c e of t h e p l a t e as t h e p l a n e nj.The co nd it io n of g eome tr ic symmetry has t h e foll owi ng form:

The condi t ion of e l a s t i c symmetry i n th e ca se under cons ide ra t ion w i l l b e1131:

b --a,, = a*, =a16=a,&=a3 ,=a, =a =ab6=0,- i -

(3 )

where a (z,y) a r e t h e c o e ff e ci e n ts of e l a s t i c i t y of t h e g i ve n p l a t e m ate r i a l .L e t us assume th at th e known hypotheses [13, 1 4 , 1 5 1 t h a t a r e w e d when de-

r iv in g th e e qua t ions o f e qu i l ib r ium o f th in i s o t r o p i c homogeneous p la t e s o f con-s t a n t r i g i d i t y c an b e ex te nd ed a l so t o t h i n a n i s o t r o p i c h e t er og en eo us p l a t e s ofv a r i a b l e r i g i d i t y , t h e mi dd le s u r f ac e of w h ic h i s s i m u lt a n eo u s l y t h e s u r f a c e ofe l a s t i c and geometr ic symmetry (under th e co ndi t io n t h a t th e upper and lowers u r f a c e s of t h e p l a t e a r e smooth sur faces) .

i k

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2 . Brian-Timoshenko EquationL e t us c ons ide r a p l a t e t h a t i s i n a p l a n e s t a t e of s t r e s s u n de r t h e a c t i o nof f o rc e s p a r a l l e l t o i t s p la ne .If t h e f o r c e s a c t i n g on t h e p l a t e a r e e s s e n t i a l l y c om pr es si ve , t h e p l a t emay be i n one of two s t a t e s of s t r e s s :

f l e c t e d , b u t i s bei ng deformed i n i t s p l a n e ; and de f l e c te d , when, i n a dd i t io n t ode f or m a tion i n i t s p la ne , i t i s a l s o be ing be n t .plane, when i t ha s no t ye t be e n de-

The work of t h e e x t e r n a l f o r c e s i n t h e f i r s t s t a t e , which i s e q ua l t o t h ework of t he in te rn a l s t r e s ses , c a n be r e p r e se n te d i n t h e f o l lowing f o rm [15, 16,171: - -.

k'here A1 i s t h e work of t h e e x t e r n a l f o r c e s i n t h e f i r s t s t a t e o f t h e p l a t e ; uand v a re displacements of po in ts of t h e mi dd le s u r f a c e of t h e p l a t e i n t h e d i -r e c t ion of the x an d y axes ; T and IT a re t h e f o r c e s a p p l ie d t o a u n i t l e n g t hand ex t en d in g t h e p l a t e i n t h e d i r e c t i o n o f t h e x and y axes; and S i s a tangen-t i a l force , which i s a l s o a p p l ie d t o a u n i t l e ng t h an d l i e s i n t h e p l an e of t h ep l a t e .

1 2

The'work o f t he i n t e r na l f o r c e s i n the se cond s t a t e of t h e p l a t e w i l l con-/1s i s t of t w o p a r t s 115, 161: t h e work of t h e l o n g i t u d i n a l f o r c e s w i t h p l a t e d i s -t o r t i o n t a k en i n t o a c c o un t

and the work of p l a t e bending

where M1, M 2 and H a r e t h e e l a s t i c moments a p p l i e d t o a u n i t l e n g t h of t h e p l a t e ,and which have the f o l lowing f o rm f o r a n a n i so t r o p ic he te r oge ne ous p l a t e o f var i -able r i g i d i t y [18]:

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. I

J-h.

where a (x,y) a r e t h e c o e f f i c i e n ts of e l a s t i c i t y of t h e g i v e n m a t e r i a l , andr e p r e s en t e d i n terms of t h e c o e f f i c i e n t s g i v e n ab ov e by means of t h e formula [18ik

The c o e f f i c i e n t s ci (z,y), i n t u r n , c an b e r e p re s e n t e d i n terms of theikf o l l o w i n g " t e c h n i c al " c o e f f i c i e n t s , w hich a re more f a m i l i a r t o u s :

where

E an d E a re Young 's m odu l i f o r e x te ns ion ( c om pr e ss ion ) i n the d i r e c t io n of th ex and y a xe s ; v1 and v2 are Poisson's r a t i o s w hi ch c h a r a c t e r i z e t r a n s v e r s e com-1 2p r e s s i o n of t h e m a t e r i a l upon exten s ion a long th e x and y a xe s ; G i s t h e s h e a rmodulus f o r p la n e s p a r a l l e l t o t h e xy p l a n e ; B 1 and B , a r e c o e f f i c i e n t s w hich / 1c h a r a c t e r i z e e l o n g a ti o n d ue t o t h e e f f e c t of t a n g e n t i a l s t r e s s Xi n d i f f e r e n t t e n s , she a r due t o nor ma l s t resses X and Y .

o r , expressedY'X Y

It f o l l o w s f r o m t h e r e l a t i o n s h i p ci t h a ti k = aki_ -

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For an i sotropic homogeneous p l a t e of c o n s t an t r i g i d i t y , MI, M an d H, as2is known, hav e th e foll owi ng form:

where

i s t h e c y l i n d r i c a l r i g i d i t y of t h e p l a t e .The con di t i on of equi l ib r ium of t h e p l a t e i n t h e s ec on d s t a t ew i l l have the fo l lowing form:

obvious ly ,

where A i s t h e work of t h e e x t e r n a l f o r c e s i n t h e s e c o nd s t a t e .2I n t h e t heo ry o f buck l i ng , we a r e i n t e r e s t e d i n t h e equa t i on wh ich co r res -

ponds t o th e moment of t r a ns i t io n of th e p l a t e from t h e f i r s t s t a t e t o t h e s e c -ond. T h is t r a n s i t i o n of t h e f i r s t s t a t e t o t he second ev ide nt ly can be accom-p l i s h e d a t t h e moment t h a t t h e f o r c e s a c t i n g i n t h e p l a n e o f t h e p l a t e have ex-ceeded a c e r t a i n l i m i t , a l b e i t by a s m a l l q u a n t i t y .

W e s h a l l c a l l t h e s y st em o f f o r c e s t h a t c o rr e sp o nd s t o t h i s l i m i t t h e c r i t -i c a l system, and w e s h a l l c a l l t he s t a t e of t h e p l a t e t h a t c o rr e sp o nd s t o t h i ssystem of f o r c e s t h e c r i t i c a l s t a t e .

A t t he moment o f t he c r i t i c a l s t a t e ( a t t h e b ou nd ar y of s t a b i l i t y ) t h ep l a t e can assume bo th a plane and a d e f l e c t e d s h a p e , i . e . , b o th t h e f i r s t andthe s econd s t a t e of t h e p l a t e i s equa l l y p robab l e . At t h a t moment, th e work oft h e e x t e r n a l f o r c e s of b o t h t h e f i r s t and t he second s t a t e w i l l b e e q u a l t oe a c h o t h e r [ 1 7 , 191.out , and w e s h a l l ob ta in th e sought Brian-Timoshenko equa t ion :

All terms t h a t c o n t a i n u and u i n t h i s case w i l l cance l

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_ _ --- _or- r e p l a c i n g & e v a l u e s of M1' 2(71, we f i n a l l y o bt ai n:

M and H i n e qua t ion (11) w i t h t h e i r e x p re s s io n s /l

I f , however, w e r e p l a c e t h e v a l u es o f M M and H i n ' e q u a t i o n ( 11 ) by t h e i r1' 2cornon e x p r e s s i o n s ( 9 ) ' w e s h a l l ob ta in the well-known Brian-Timoshenko equ ati on[161.

3. S ta b le , Uns ta b le , -a N e u t r a l - E q u i l i b r i u mBoth equation (11) and ( 1 2 ) a r e t h e o n l y n e c e s s a ry c o n d i t i o n o f e q u i l i b r iu m .The n e c e s s a r y an d s u f f i c i e n t c o n d i t i o n of a n e l a s t i c s y st e m whose f o r c e s h a ve a

p o t e n t i a l , as w e know, i s t h a t t h e p o t e n t i a l e ne rg y i n t h e e q u il i br i u m p o s i t i o nhave a s t a t i o n a r y v a lu e, i . e . ,

or, e xpr e sse d i n a no the r m a nner ,

~.6l-I=0,

-( i = 1 , 2 , 3 , . . .,n),

where II is t h e t o t a l p o t e n t i a l e n er gy of t h e s y s te m and qc oor d ina te .

i s a g e n e r a l i z e diThe e qu i l ib r ium of a m a t e r i a l system, as w e know, c a n be s t a b l e , uns t a b le ,

OJC n e u t r a l . I f t h e p o t e n t i a l e ne rg y h a s a minimum, i . e . ,

i'I-I >0,Ithe equilibrium w i l l be s t a b l e ,

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It is has a rnzximum, i.e., 1t h e e q u i l i b r i u m w i l l b e u n s t a b l e ; and i f i t h a s n e i t h e r a minimum no r a maximumi . e . , -

E1l-I=Oand B r I = O , ' (17)t h e e q u i l ib r i u m w i l l b e n e u t r a l .

Neu t ra l equ i l i b r i um de te rmined by equa t i ons (17) w i l l obvious ly correspondt o t h e c r i t i c a l s t a t e of equi l ib r ium.

4. Express ion ---f t h e T o t a 1 , P o t e n t i a l E ne rg y of a P l a t eTo f i n d th e c r i t i c a l s t a t e , i t i s obv ious ly neces sa ry t o de r i ve an exp res -

s i o n f o r t h e t o t a l p o t e n t i a l en er gy . I n o ur c as e , t h e l ef t- h an d s i d e o f equa-t i o n ( 1 2 ) may be considered as t he t o t a l p o t e n t i a l e n e rg y of a p l a t e su b je c t t obuckl ing .

I

I n de e d, t h e f i r s t do ub le i n t e g r a l i n e q u at i on ( 1 2 ) i s ' n o n e o t h e r t ha n t h ework of t he i n t e r na l bend ing fo rces Ai, w h i l e t h e s ec o nd , i f i t i s r e a d w i t h aminus s ign, i s t he work o f t he . ex t e r na l fo r ces AC on se qu en tl y, t h e t o t a l p o t e n t i a l e n er gy f o r t h e case of b u c k li n g c an b e w r i t t e ni n t he fo l l owing form:

(boundary and volume forces).e

' . n =A, - A io r

5. D i f f e r e n t i a l Eq ua ti on o f t h e E l a s t i c S u r f a c e o f a P l a t e-nd Boundary ConditionsWe s h a l l u s e co n d i t io n ( 13 ) t o o b t a i n t h e d i f f e r e n t i a l e q u at i on and t h e

boundary condi t ions .By i n s e r t i n g t h e v a l ue of II from ( 1 9 ) i n t o (13) and pe r forming va r i a t i on

and some t ransformations, w e ob ta in :7


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S i d z d y ( - - - f + 2 - +++ - -'M ~ P H a w a [ T - - + S - 3 f -w aw d [ T z d y + S z ] ) 2 w -w &t-(20)z axay ay dz l a z a y d yai- ds {Q&v-- M --n }-2 Si (I?) Ew (s,) ,

i=

where

ddS (21)- (M, ,)OS ( n , ) C O S (n, y)- COS' (n,5 )- OY'. ( n , )) ]+-F [ T I g + S $ ] COS (n,z)+ [T a u+ S F ]r c o a ( n , y ) } ?_ _

M =M , os' (n +2 H co3 (n 5) os ( n )+nl os' (n ) ,N = M ,- , ) 00s (n, ) os (n, y )- [oos' ( n, )- os' (n,341,. .-

i n which 1 S . (N) d e n o t es t h e sum o f t h e d i s c o n t i n u i t i e s wh ic h t h e f u n c t i o n Ni=undergoes a t t h e c o r n e r p o i n t s sb et we en t h e e x t e r n a l d i r e c t i o n of t h e n o rm al t o t h e c o n t o u r a n d t h e z and y ax-es; t h & q u a n t i t i e s M

of t h e c o n t o u r ( n , z ) , an d ( n , y ) a re t h e a n g l e siM and H h av e t h e i r p r e v i ou s v a l u e s ( 7 ) .

From ( 2 0) ' w i t h t h e u s u al c o n s i d e r a t i o n s , w e ob ta in an Euler-Lagrange equa-1' 2

t i o n of o u r v a r i a t i o n p r o b l e m :

o r , by s u b s t i t u t i n g t h e v a lu e s o f M M and H i n ( 23 ) by t h e i r e x p r e ss i o n s ( 7 ) ,1' 2

where X and Y a re the components of t h e r e s u l t a n t v e c t or of th e volume for ces-i

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a p p l ie d t o a u n i t area of t h e p l a t e .Equa t ion ( 23 ) a l so c a n be ob ta ine d ge om e t r i c a l ly

The s t a t i c e q ua t io nThe equa tio n obta ine d (24) makes it p o s s i b l e t o w r i t e d i f f e r e n t i a l equa-

W e s h a l l c i t e s e v e r al p a r t i c u l a r cases of d i f f e r e n t i a l e q ua t io n ( 2 4 ).

a f t e r c o n s i d er i n g t h e e q u i l i b r i u m of a deformed e lement of t h e p l a t e .12 = 0 a l s o g i v e s t h e s o u gh t eq u a t i o n ( 23 )* .

t i o n s a l s o f o r p l a t e s w i t h o t h e r v a l u e s o f a i k ( z J y ) and J ( z , y ) .

1. To obta in a d i f f e r e n t i a l e qu a ti o n f o r a n o r th o go n al a n i s o t r o p i c p l a t eof v a r i a b l e t h i c k n e s s , w e m ust p l a c e t h e f o l l o w i n g i n e q u a t io n ( 2 4 ) :a,,=a,, = Q,, 5 a,, t

-

which i s e q u i v a l e n t t o -p , = p , = o ,

T h e r e m a i n i n g c o e f f i c i e n t s w i l l t he n t a k e on the f o l lowing f or m :

' .2. Equa t ion ( 24 ) f o r the c a se of a homogeneous or thog ona l an is o t ro pi c /1

p l a t e of c o n s t a n t t h i c k n e s s , t a k e s o n t h e form:

a

where

B = EsJL - ,V, '

v E JI E s J 2 1 4 GJ .2c=- ' i - VIV,

*Equa t ion (23) and a l s o e x p r e s s i o n s ( 2 1 ) and (22) obv iou s ly a re i n v a r i -a n t s b o t h w it h respect t o t h e s t r u c t u r e of t h e p l a t e m ate r i a l and a l s o w i thr e g ar d t o t h e v a r i a t i o n o f i t s r i g i d i t y .

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3 .e qua t ion ( 2 4 ) assumes th e follow ing form:In an i s o t ro pi c and homogeneous medium, but w i t h va r i ab le th i ckn ess ,

2where D = EJJ1 - V i s t h e c y l i nd r i c al r i g i d i t y of t h e p l a t e .4. When D = c ons t , e qua t ion ( 2 5 ) becomes a d i f f e r e n t i a l e qu at io n f o r a n

i s o t r o p i c p l a t e w i t h c o n s t a nt t h i ck n es s :

ti& . equa t ion was f i r s t d e r i v ed by R e i s sn e r [15].e qua t ion (26) , I f w e s t a t e t h e f o l l o w i n g i n. .-T, cons1, T,= onst and 8= o n s t , ,

- --,we w i l l o b t a i n a Saint-Venant e qua t ion [ 2 0 ] .

We s h a l l o bt ai nc o nt o ur i n t e g r a l a ndp e r f e c t l y f r e e, thening :

t h e bounda ry c ond i t ion s f o r th e p l a t e by c ons id e r ing theth e sum i n the e q ua t ion (20) ., by usin g the well-known rea son ing , we o bt ai n the follow-

-If he e dge of t h e p l a t e i s

03

i-I. -

where &, M an d N ha ve th e i r f or me r va l ue s (21), (221, (22a) .Thus, f o r e xa mple , f o r the f r e e s i de y = b of a r e c ta n g u l a r p l a t e , t h e /I

boundary condi t ions ( 2 7 ) assume t h e fo l lowing form:

.10


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, . .I x = O Y y = b a n d x = a , y = b a t t h e c o r n er s of t h e p l a t e :

~

The boundary cond i t i on s f o r an i s o t r op ic homogeneous p l a t e of constax:I

~

r i g i d i t y w i l l b e o b t a i n e d i ft h e i r common exp res sio ns ( 9 ) .t a n k u l a r p l a t e , t h e b o un da ry c o n d i t i o n s t a k e o n t h e f o l l o w i n g f orm :

ins t e a d o f K1' M and B i n ( 2 7 ) , w e s u b s t i t u r aI n p a r t i c u l a r , f o r th e f r e e s i d e y = b of a rec -

x = 0, y = b and x = a, y = b a t t h e co r n e rs of t h e p l a t e :

si(5)w (s,) =0a t dy

(28a)

(?Sa')

The usua l K i r c hhof f boundar y c on d i t i ons e nc oun te r e d i n a l l cour ses on thet h e o r y of e l a s t i c i t y w i l l be ob taine d i f w e place (T2)y=b = (S)y=b = 0w hi ch c o r r e sp o n d s t o t h e case of the a bsenc e of e x te r n a l f o r c e s on th e con tour.But s ince cases a re e nc ou nt er ed i n t h e l i t e r a t u r e where Kirchhof f ' s boundaryc o n d i t i o n s a re extended t o i n d i v i d u a l p a r t i c u l a r p r ob l em s an d t o l o ad e d s i d e sof a p l a t e , which n a t u r a l l y l e a ds t o e r r o r s , w e t h e n m us t c o n s id e r t h i s ques-t i o n i n g r ea t e r d e t a i l and c i t e an example.

i n ( 2 8 a )

A s a n example, l e t u s co ns id er t h e s t a b i l i t y of t h e r od shown i n F igu r e 1w i t h t h e b oundary c o n d i t i o n s g i v e n i n t h i s a r t i c l e , andn two v a r i a t i o n s :w i t h Kir c hhof f ' s bounda r y c ond i t ions .

The d i f f e r e n t i a l e q ua t io n of t h e e l a s t i c l i n e of the rod w i l l be ob ta ine das a p a r t i c u l a r case from ( 2 4 ) :-

d @ w ( 2 9 )- yl E J 2) T , =0.~ _ -

The boundar y c ond i t ion s f o r t h e f r e e loa de d e nd of the Lr o d w i l l be obta ined as a p a r t i c u l a r case from (28a):

~ - -_.(30)- E J ~ + Trw e] = o ,'dy y = b -_11


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, .Kir c h hof f ' s bounda ry c ond i t ion s f o r the same f r e e end o f th e r od ha ve the

- - - - _ol lowing form:

The b o m d a r y c o n d i t i o n s f o r t h e f a s t e n e d e nd , w h ic h a r e g ive n here , andKi r c hhof f ' s bounda r y c ond i t ions , c o inc ide :-.

Now, s o l v i n g the problem fo r a rod under boundary condi t ions (30) an d (32)and boundary condi t ions (31) and (321, w e o b t a i n the f o l l o w i n g t r a n s c e n d e n t a le q u a t i o n f o r d e t e rm i n in g t h e c r i t i c a l f o r c e i n t he f i r s t case:

-

a nd t h e f o l l o w i n g t r a n s c e n d e n t a l e q u a t i o n f o r the second case:-- -_sin 'Ab+cosV,b=O. ( 3 3 ' )


14/15

, . ' ..The boundary conditions could be derived also for the case of elasticfastening of the edge of a plate, and also for the case of support of the edgeof a plate on an elastic contour, but in the presence of the fundamental bound-ll49ary conditions given above, a description of these boundary conditions is notdifficult; therefore, we shall not dwell on them.

-1.2 .

3 .

4 .5.6.7.

8.9.

REFERENCES .Mikhlin, S. G.: PZoskaya deformatsiya v anizotropnoy srede. (PZane St ra inLekhnitskiy, S . G.: Some cases of a plane problem in the theory of elasti-in an Anisotropic Medium.) Moscow and Leningrad, 1936.city of an anisotropic body. In: Eksperimental'nyye metody opredeZe-niya napryazheniy i defomats i y v uprugoy i pZasticheskoy zomkh.(Experimental Methods of Determining Stresses and Strains i n EZasticand PZastic Zones.)centrated forces on stress distribution in an anisotropic elasticmedium. Prikladnuya Matematika i Mekhanika, Vol. 3, No. 1, 1936; Stress

in an unbounded anisotropic plate weakened by an elliptic hole. DokZadyAkademii Nauk SSSR, Vol. 4 (131, No. 3 , p . 107, 1936; A plane staticproblem in the theory of elasticity of an anisotropic body. PrikZadnayaMatematika i Mekhanika, Vol. 1, No. 1, 1937; On certain questions re-lated t o the theory of bending of thin plates. Prikladnaya Matematikai Mekhanika, Vol. 2 , No. 2, 1938; Some cases of elastic equilibrium ofa homogeneous cylinder with arbitrary anistropy.i Mekhanika, Vol. 2, No. 3, 1938.mizotrophoho seredovyshcha.1938.bodies. Journ. de Math. Pures e t AppZ . , Vol. 10, Liouville, 1865.On the elastic behavior of a stressed cylindrical body.v . d . KanigZ. Ges. des Wiss. u. G . A . Univ. Zu Gtrttingen, No. 16, 1886.ed Applicata, Vol. 2 0 , No. 2, 1892.Concrete Plate s Taken i n t o Account. 1921, (in Polish); An applicationof the theory of bending of orthotropic plates. Zeitschr. f. angew.Math. u. Mech., Vol. 6 , No. 3, 1926; A n Engineering Problem of t heS t a t i c s of Orthotropic Plates. (in Polish), Warsaw, 1929.sheet as an anisotropic plate. Trudy T sAG I , No. 76, 1931.FZota, No. 9, 1937.

ONTI, 1936; On the question of the effect of con-

PrikZadnaya MatematikaSavin, H. M.:'

Osnovna pZoska statychna zadacha t e o r i i pmizhnosti dl ya(FundamentaZ PZane S t a t i c Problem i n\ the Theory of EZastici ty f o r an An iso tro pic Medium.) Arv UkrSSSR, Kiev,

Saint-Venant:Voigt:Somigliana: Study on strain in a cylindrical body. Annuli d i Mat. PuraHuber, M. T.:

An account of the various kinds of homogeneity of solidNachricht.

The Theory of RectanguZar Aniso tropic Pla tes wi th Reinforced-

Sekerzh-Zen'kovich, Ya. I.: Calculation of the stability of a plywoodBalabukh, L. I.: The stability of plywood plates. Tekhnika Vozdushnogo

10. Rostovtsev, G. G.: On the question concerning the reduced width of an /15orthotropic plate. Tekhnika Vozdushnogo FZota, No. 1 0 , 1937.11. Serensen, S. V.: Osnovy tekhnicheskoy t e o r i i upru gosti. (Fundamentalsof the Engineering Theory of EZasticity.11934.f u r de reine und angew. Math., Vol. 4 0 , Crelle, 1850.

-ONTI, NKTP, Kharkov and Kiev,. 12. Kirchhoff: On the equilibrium and motion of an elastic plate. Journ.

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. 1 -.13 .14.

15.16.17.18.

19.20.

ASA TT F-11,346Lyav, A . : Matematicheskaya teoriy a upru go st i. (The Mathematical TheoryLur'ye, A . I.:

cf E%a s t

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