torsion of compound bar 1960

8/2/2019 Torsion of Compound Bar 1960

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1Ht. d. Mech. Sci. Perg amo n Press Lt d. 1960. Vol. 1, 1717. 356 365. Pri nte d in Gre at Brita b)

T O R S I O N O F C O M P O U N D B A R S - - A R E L A X A T I O NS O L U T I O NJ . F . E L Y a n d O . C . Z I E N K IE W I C Z

N o r t h w e s t e r n U n i v e r s i t y , I l li n o is{Received 30 June 1959)

S u m m a r y - - T h i s p a p e r d e a ls w i t h a f in i te - d if f er e n ce s o lu t io n o f t h e t o r si o n p r o b l e m o fn o n h o m o g e n e o u s a n d c o m p o u n d p r i s m a t i c b a r s . G e n e r a l, g o v e r n i n g e q u a t i o n s f o r b o t hp r o b l e m s a r e d e v e l o p e d a n d t h e b o u n d a r y c o n d i t i o n s f o r a n i n t e r f a c e b e tw e e n p a r t sc o m p o s e d o f h o m o g e n e o u s b u t d i f f e re n t m a t e r i a l s a r e s ta t e d . T h e c a s e o f m u l t i p l yconnec t ed r eg ions i s d i s cussed and in t eg ra l cond i t i ons , ana logous t o t he cond i t i ons i nm u l t i p l y c o n n e c t e d h o m o g e n e o u s b a r s , a r e d e v e l o p e d .E x a m p l e s i l lu s t r a ti n g v a r i o u s t y p e s o f p r o b l e m s a r e w o r k e d o u t a n d t h e a c c u r a c y o ft h e m e t h o d d e m o n s t r a t e d b y c o m p a r i s o n w i t h s o m e k n o w n s o l u t i o n s .

xyzU V W

TXZ ~ Ty Z0

GT

s, n

N O T A T I O Nc o - o r d i n a t e sd i s p l a c e m e n t c o m p o n e n t s i n xyz d i r e c t i o ns h e a r s t r e s s e st w i s t p e r u n i t l e n g t hm o d u l u s o f r i g i d i t yP r a n d t l ' s s t r e s s f u n c t i o nt o r q u et a n g e n t a n d n o r m a l d i re c t io n s to c u r v er a t i o o f G 2 t o G ~

1. I N T R O D U C T I O NT H E i n c re a s i n g u s e o f g e o m e t r i c a l l y c o m p l i c a t e d c o m p o s i t e s e c t io n s i n p r a c t ic eh a s p r o m p t e d t h i s i n v e s t i g a t i o n o f t h e t o r s io n o f c o m p o s i t e b a r s . C o m p o s i t es e c t i o n s a r e u s e d in r e i n f o r c e d c o n c r e t e a n d a i r c r a f t , a s w e l l a s o t h e r m o r es p e c ia l iz e d a p p l ic a t i o n s , a n d t h e m e t h o d s p r e s e n t e d h e r e s h o u ld p r o v e u s e fu li n s o lv i n g t h e t o r s i o n p r o b l e m i n th e s e f i el ds .

S e v e ra l p r o b l e m s o n th e t o rs i o n o f c o m p o u n d p r i s m a t i c b a r s h a v e b e e n s o lv e da n a l y t i c a l l y b y M u s k h e l i sh v i l l i , 1 G o r g i d z e , 2 M i t r a , 3 T a k e y a m a , 4 S u h a r e v i k i , 5S h e r m a n , 6 C o w a n , 7 C r a v e n s a n d o t h e r s. T h e p r o b l e m s s o l v e d g e n e r a l l y d e a l tw i t h c r o s s - se c t io n a l s h a p e s t h a t c o u l d b e m a p p e d c o n f o r m a l l y , w i t h r e la -t i v e e a s e .

W h i l e t h e s e a n a l y t i c a l s o l u t i o n s o f t h e t o r s i o n p r o b l e m f o r b o t h h o m o -g e n e o u s a s w e ll as c o m p o s i t e b a r s a r e e x c e e d i n g l y i m p o r t a n t , i t is n e c e s s a r yt o u s e r e l a x a t io n m e t h o d s t o s o lv e t h e p r o b l e m o f e v e n a h o m o g e n e o u s b a rw h e n t h e g e o m e t r y o f t h e c r o s s -s e c t io n b e c o m e s c o m p l i c a t e d a s i n s t a n c e d b yn u m e r o u s p u b l i c a t i o n s b y S o u t h w e l l , 9 S h a w , 1 A l l e n , 11 D o b i e 12 a n d o t h e r s 13-16.S in c e t h e h o m o g e n e o u s b a r c a n b e e a s i l y t r e a t e d b y r e l a x a t i o n m e t h o d s , i t

356


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Torsion of compound bars--a relaxation solution 357seemed logical to attempt to extend these methods to cover the case of simplyand multiply connected composite bars when the geometry of the cross-section is complicated. Indeed, the t rea tm en t of this more general case alsoturns out to be simple.It should be pointed out that the purpose of the examples which werechosen for this paper is to illustrate how the method can be applied to thevarious general classes of problems, and not to illustrate the handling ofespecially complicated boundaries. More complex bou nda ry shapes can bedealt with as usual by using a finer mesh size which will require more timeto obtain the required solutions. It is anticipated tha t a ny digital computerprogrammes available for solution of the Poisson equation could be easilyadap ted to the t ype of problems discussed.

Equations will be developed for simply connected cross-sections, wherethe ma terial has a continuous ly vary ing G, and then for those tha t are com-posed of two distinct materials joined together, without slipping, at an inter-face. The addit ional conditions required in the case of a mul tipl y connectedcross-section will also be derived, and examples will be shown for each classof problem. Finall y, the m embrane analogy will be discussed in detail andother analogies pointed out.

2. GOVERNING EQUATIONS FOR CONTINUOUSLY VARYING GConsider Fig. 1 and a material which has a continuously varying G in

cross-section, but whose G is independent of z, i.e.G = G ( x , y ) . (1)

/

I~'XG. 1. Torsion of a bar. Co-ordinates used.Following the not ati on used by Timoshenko and Goodier, 17 and using

Prandtl's stress function , which defines the two stress components by8 8 (2)


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358 J . F . ELY and O. C. ZIENKIEWlCZa n d , w i t h t h e o t h e r s t r e s se s e q u a l t o z er o, w e f in d t h a t t h e e q u i l i b r i u m e q u a -t i o n s a re a u t o m a t i c a l l y s a ti sf ie d . I f , i n a d d it io n , w e ta k e t h e d i s p l a c e m e n tc o m p o n e n t s g i v e n b y u = - O y z , v = O x z ,

a w l 84 a w _ 1 8 4 O x (3 )8x - G Sy + Oy ' ~Y G Sx

a ll t h e r e l a ti o n s b e t w e e n s tr e s s e s a n d s t r a i n s a r e d e t e r m i n e d c o r r e c t l y . T oe n s u r e c o n t i n u i t y o f t h e a x i a l d i s p l a c e m e n t w , w e h a v e , b y d i f f e re n t i at i n g t h ef i n a l t w o e x p r e s s i o n s o f ( 3 ) , t h a t

O axYd+ =-2Go,o r ~ ( G ~ x ) + ~ ( G- - y ) 3 xy = - 2 0 .I n t h e c a s e o f c o n s t a n t G t h is r e d u c e s t o t h e f a m i l ia r P o i s s o n e q u a t i o n

a~5 824V 2 = ~ ~ y ~ - 2 GO .

( 4 )

(4a )A s t h e s h e a r s t r es s n o r m a l t o t h e e x t e r n a l b o u n d a r y m u s t b e z e ro , it

f o ll o w s f r o m t h e d e f in i ti o n (2 ) t h a t t h e s t r e s s f u n c t i o n o n t h i s b o u n d a r y m u s th a v e a c o n s t a n t v a l u e . T h i s c o n s t a n t c a n b e a r b i t r a r i l y f ix e d a s z e ro . T h es u f f i c i e n t a n d n e c e s s a r y b o u n d a r y c o n d i t i o n o n t h e e x t e r n a l b o u n d a r y Cb e c o m e s = 0 . (5 )

T o d e t e r m i n e t h e t o r q u e T a c t i n g o n t h e s e c t io n , w e i n te g r a t eT = f f a ( X - % z - Y ~ 'x z} d x d y ,

w h i c h , b y s u b s t i t u t i o n o f (2 ) a n d (5 ), y i e ld sT = 2 f f R d x d y . ( 6 )

F o r p u r p o s e s o f c o m p u t a t i o n i t i s c o n v e n i e n t t o r e w r i t e t h e a b o v e r e l at io n -s h ip s i n a n o n - d i m e n s i o n a l f o r m . S u b s t i t u t i n gx = x ' L , y = y ' L , G = G ' G o, = 4 ' ( G 00L2),

i n w h i c h L r e p r e s e n t s s o m e ty p i c a l d i m e n s i o n o f t h e s e c ti o n , a n d Go s o m et y p i c a l v a l u e o f G , w e h a v e , i n p l a c e o f (4 ) a n d (5 ),

' = 0 o r G . (8 )T h e s t r e s s e s a n d t h e t o r q u e a r e g i v e n b y

% . z = - ( G o O L ) 88n',, (9)T = 2G 0 0 L 4 f f d p ' d x ' d y ' . (10). )dR


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T o r s i o n o f c o m p o u n d b a r s - - a r e l a x a t io n s o l u ti o n 3 593. G O V E R N I N G E Q U A T I O N S F O R A D I S C O N T I N U O U S G

W e s h a l l n o w c o n s i d e r t h e s e c t i o n s h o w n i n F i g . 2 , w h e r e a d i s c o n t i n u i t yo c c u r s i n t h e v a r i a t i o n o f G a t t h e i n t e r f a c e C a, w h i c h s e p a r a t e s r e g i o n s R 1a n d R a. C l e a r l y , i f G 1 a n d G a r e f e r t o t h e v a l u e s o f G i n t h e r e s p e c t i v e r e g i o n s ,t w o f u n c t io n s , 1 a n d C a, s a t i s f y i n g e q u a t i o n (4 ) i n t h e i r d o m a i n s , a r e r e q u i r e d .

Z

FIc . 2 . Tor s i on o f a compo s i t e ba r .

JLX

O n t h e e x t e r n a l b o u n d a r y , w h e t h e r C 2 c u t s i t o r n o t , t h e b o u n d a r y c o n d i t i o n ( 5)s ti ll h a s t o b e im p o s e d . H o w e v e r , t o s p e c i f y t h e p r o b l e m c o m p l e t e ly , a d d i t i o n a lr e q u i r e m e n t s h a v e t o b e i m p o s e d o n t h e i n te r fa c e . T h e s e c o n d i ti o n s m u s te n s u r e t h a t : ( a) t h e s h e a r s t r e s se s n o r m a l t o t h e i n t e r f a c e a r e t h e s a m e i ne a c h r e g i o n ; a n d (b ) t h e a x i a l d i s p l a c e m e n t s a r e c o m p a t i b l e o n t h e i n t e r fa c e .T h e f i r st o f t h e s e c a n b e e x p r e s s e d a s

~1 828 s - 8 s- on C 2, (11 )w h i c h c a n b e s a t i sf ie d b y m a k i n g

1 = 2 o n C2, ( l l a )a s t h e a d d i t i o n o f a r b i t r a r y c o n s t a n t s d o e s n o t a f fe c t t h e r es u lt s .

T h e s e c o n d c o n d i t i o n i s s a t i s f i e d i f8 w 1 = 8 w a8 s ~ s o n C a, ( 1 2 )

w h i c h b y t h e u s e o f (3 ) c a n b e s h o w n t o b e e q u i v a l e n t t o1 8 1 = 1 88

o n C a, ( 1 2 a )G 1 8n G~ 8nn b e i n g t h e d i r e c t i o n o f t h e n o r m a l t o C a.

I t c a n b e s e e n t h a t t h e s t r e ss f u n c t i o n w i ll b e c o n t i n u o u s a c r o ss t h e i n t e r -f a ce , b u t i ts d e r i v a t i v e s w i ll i n g e n e r a l b e d i s c o n t i n u o u s t h e r e . T h e p r o b l e mt h u s b e c o m e s v e r y s i m i la r t o t h o s e e n c o u n t e r e d i n p o r o u s m e d i a f lo w , o r in


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360 J. F. ELY an d O. C. ZIENKIEWICZthe determination of magnetic fields, when the permeabilities differ abruptly.It is well known that phenomena of refraction of the flow lines occur insuch instances.

Considering the case of a bar made of two materials, each having constantelastic properties, and using the non-dimensional notation of the previoussection, we have V 2 ~ = - 2 in R1, (13)

V 2 ~ = - 2 a in R2, (14)with ' = 0 on external boundary and

; = ~, (15)

on interface, in which arbitrarilye 1 ~ e 0

G2and ~ = G~"These relations will be used in the illustrative examples shown later.It should be noted that the methods of the previous section, using a

continuously varying G, could be used to solve this problem of a discontinuous Gat an interface, by approximat ing to the d iscontinuit y in G by a continuousbut steep, variat ion. This is particul arly easy to do in finite-difference treat-ment, and will be illust rated later. The results obtained by using this pro-cedure, however, are not as accurate (for a given mesh size) as those obtainedby assuming the G discontinuous, as does the method of this section.

4. M U LTIPLY C O N N EC TED C R O SS-SEC TIO N SIn the problems examined so far the condition of single-valued displace-

ments is automatically satisfied, as could be shown by integration of theexpressions (3). If, however, referring to Fig. 2, the region enclosed by C~were empty , an additi onal condition would be required. Such a condition,usually expressed in a form of a line integral, is well known for the problemof torsion of multiply connected, homogeneous sections (see Timoshenko andGoodier17). For the prob lem on hand, where G1 ma y be variable, an equivalentcondition has to be established. We shall derive this b y taki ng the case of ahole as a limiting case of the problem discussed in the previous section. Nowwe shall take G 2 as being equal to a c ons tan t k in R2, and let thi s constanttend to zero.

Clearly, one of the conditions on the "interface" now is that 1 is equalto a constan t. Condit ion (12a) becomes insufficient in the limit, mere ly giving~ 2 / ~ n = O , which is already implicit, no condition being imposed on ~ 1 / ~ n .However, integrating (12a) around the interface C2, we have


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Torsion of compound bars--a relaxation solution 361where G2 is a consta nt, while G1 is a funct ion of x and y. Applyin g Green'sTheorem in two dimensions to the right-hand side, we obtain

1 01 1 f ( I 0 2 2 + a 2 , t d x d y ,c , G1 an - G 2 JJAo , [~x~x2 aY~wher e Ac, is th e ar ea inside the curve C9.. Since G2 is const ant , e quati on (4a)applies inside and on C2. Therefore we obtain after substitut ion

1 01c, G1 ~n

1 a1c, Gx an

- - - - = 1 (~" (j j ~v , -- 2G~.O) dx dy

- _ 2 0 A t , . (16)It can be seen th at this e quatio n reduces to the familiar form when G 1 isa constant; namely,

= - - 2 G O A v .1C, ~nA sufficient number of boundary conditions is now available for a uniquesolution of multiply connected sections.

It can be observed th at as G2 tend s to zero, 2 mus t become c ons tant inregion R~ to sat isfy the gove rning equa tion (4), and the a nalo gy of the " f l o a t i n gdisc" is again applicable, as it is in the case of homogeneous sections.

5. MEMBRANE AND OTHER ANALOGIESIt has been pointed out th at the torsion problem of a homogeneous bar is

analogous, mathematic ally, with several other physical problems. Theseinclude the mem bran e un der constant pressure, the problem of viscous flowof fluid in a tube , flow of a cu rre nt in a co ndu cto r of variab le thickness, a nd ingeneral any of the physical problems which satisfy Poisson's or Laplace'sequation. These analogies can be extend ed to the case of a composite bar intorsio n with re wardin g results. We shall discuss only one of these, namely ,the m embr ane analogy. It is well known th at the deflections of a membr anesubjected to a constant pressure and tension satisfy the equation

V2z = qT(in which q is a pressure, T th e mem bra ne tension a nd Z the deflection). It caneasily be shown that in the case of a memb rane with variable tension theequivalen t expression is

( b z \ O / ~ z )ax

The analogy with equat ion (4) is now obvious if1T = ~ , q = 2 0 , z -= .

24


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362 J . F . ELY an d O. C. ZIENKIEWICZD i s c o n t i n u o u s v a r i a t i o n o f G c a n b e i n t e r p r e t e d a s a d i s c o n t i n u o u s v a r i a t i o no f t h e m e m b r a n e t en s io n a n d b e c a us e

- A q~zT~n=

b y s t a t i c s , a l l t h e o t h e r r e l a t i o n s h i p s f o r i n t e r f a c e c o n d i t i o n s , a s w e ll a s fo rm u l t i p l y c o n n e c t e d r eg io n s , c a n be d e d u c e d . I t s h o u l d b e o b s e r v e d t h a t t h ev a r i a t i o n o f T i m p l i e s e x i s t e n c e o f d i s t r i b u t e d f o r ce s p a r a l le l t o t h e x y p l a n e s - -w h i c h i n t h e c a s e o f a n i n t e r f a c e b e c o m e s a li n e f o r c e . A s a p o s s i b l e e x p e r i -m e n t a l s o l u t i o n t h e m e m b r a n e a n a l o g y d o e s n o t a p p e a r p r a c t ic a b l e .

6. E X A M P L E ST h e g o v e r n i n g e q u a t i o n s a n d t h e i r r e s p e c t i v e b o u n d a r y c o n d i t i o n s c a n b e

e a s i ly t r a n s f o r m e d i n t o s u i t a b l e f i n it e - d if f e re n c e r e l a t io n s h i p s a n d a s o l u t i o nc a n t h e n b e o b t a i n e d b y a p p l i c a t io n o f r e l a x a t i o n m e t h o d s . A s t h e t e c h n i q u e so f b o t h a r e w e l l k n o w n a n d d e s c r i b e d i n m a n y t e x t s ( e.g . A l l e n n ) i t is n o t

YL2

~2oo'-- 400~ ~

1134- -l O G O 1 ~ 7

'% \ / I1462 i 13

Y

T - 0 . 1 3 8 8 ( G ~ L 4 )

J-X

Fro . 3 (a ). Ex am pl e A . Tor s i on o f a squa re compos i t e ba r ; a = l ;va lues of ' 104.p r o p o s e d t o e l a b o r a t e o n t h e s e m a t t e r s h e r e . T h e g e n e r a l t r e a t m e n t o fP o i s so n ' s e q u a t i o n is n o w v i r t u a l l y s t a n d a r d f o r t h e u s u a l N e u m a n n a n dD i r i ch l e t b o u n d a r y c o n d i t i o n s a n d t h e m a i n p r o b l e m s e n c o u n t e r e d h e r e ar et h o s e p e r t a i n i n g t o t h e in t e r f a c e b e t w e e n t h e t w o m a t e r i a l s . A g a i n , as t h ec o n d i t i o n s a t t h e i n t e r f a c e a r e e s s e n t i a l l y s i m i l a r t o t h o s e e n c o u n t e r e d i n t h et r e a t m e n t o f s e e p ag e o r m a g n e t i c f ie ld p ro b l e m s i n m e d i a o f d i f f e r en t p e r m e -a b i h ti e s , t e c h n i q u e s o r r e la x a t i o n t r e a t m e n t a r e w el l k n o w n . T h e s e , u s e d f o rt h e f i r st t i m e b y C h r i s t o p h e r s o n , TM a r e e x c e l l e n t l y d e s c r ib e d i n A l l e n ' s t e x t ,a n d n o s p e c ia l d i f f i c u l ty w a s e n c o u n t e r e d i n t h e i r a p p l i c a t i o n i n t h e e x a m p l e ss o l v e d .

E x a m p l e s A a n d B ( F ig s . 3 ( a - c ) a n d 4 ) s h o w p r o b l e m s in w h i c h s t r a i g h ta n d c u r v e d i n t e rf a c e s b e t w e e n t w o m a t e r ia l s o c c u r r e s p e c t i v e l y . T h e a c c u r a c y


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Tors ion o f com pound baxs - - a r e l axa t ion so lu t ion 363o f t h e r e l a x a t i o n s o l u t io n c a n b e s e e n f r o m t h e t a b l e b e l o w , in w h i c h t h es t if fn e s s o f t h e b a r s o f e x a m p l e A o b t a i n e d f r o m a re l a ti v e ly c o a r se m e s hs o l u t i o n a re c o m p a r e d w i t h v a l u e s c o m p u t e d b y a se ri es s o l u t io n d e v e l o p e db y M u s k h e l i s h v i l l i L

200~ .4~0""-"-"S /600~640 95"~

I l l l / / /

~ . ~ 1 0 4 4 ~ 9 2 1~ 12 0 0 " ' - -~ 1 4 0 0 ~

~ 1 8 0 ' ' ~

2000

L

J603 .I

I FIG. 3(b). Exam ple A. Tors io n of a squa re compos i te bar ; a =L ,

I l lI

E -

i

JIL2i

. ~ L ~ x

T = 0.1941 (GOL4) ; val ue s of ' x 104 .

(1710) (2623) (2837) (2388) (1989) (1500) (86 3)FIG. 3(c) . Exam ple A. Tors ion of a square comp osi t e bar ; a = 3 ;

T = 0"235 8 (G oOL4) ; va hms of~ ' x 104.

LT

~ X

T = D ( G o 8 L 4 )Tors ional r ig id i ty , D

V alues o f B y r e l axa t ion T im oshe nko M uskhe l i shv il l i * ( % d i sc repancy )a = 1 0-138 8 0-1406 0-1407 1.28= 2 0.194 1 0-1972 1.57a = 3 0.2 358 0.23 99 1.71

* Only two ter ms of the series solution were used, and thes e values are always greater th an thetrue value. The error increases with increasing a.


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364 J . : F . E L Y a n d O . C . Z I E N K I E V i I C ZT o t e s t t h e a l t e r n a t i v e m e t h o d o f s a t i s fy i n g t h e i n t e r f a c e c o n d i t i o n s , i n

o n e o f t h e e x a m p l e s [ s ee F i g . 3 ( c ) ], G w a s a s s u m e d t o v a r y i n a c o n t i n u o u sw a y f r o m G 1 t o G 2 w i t h i n a d i s t a n c e o f o n e m e s h l e n g t h . T h e f in i te d i f f e re n c ee x p r e s s i o n o f e q u a t i o n (4 ) c a n n o w b e u s e d d i r e c t l y a t a l l m e s h p o i n t s . V a l u e si n p a r e n t h e s e s , [ F ig . 3 ( c) ], r e f e r t o a n a s s u m e d c o n t i n u o u s v a r i a t i o n o f G , w h i l et h e o t h e r s a r e fo r a d i s c o n t i n u o u s v a r i a t i o n o f G .

FIG. 4 . Ex am ple B . Tor s ion o f a squa re ba r wi th a c i r cu la r i n se r t ;= 1 0; v a l u e s o f ~ ' 104T h e c o a r s en e s s o f t h e m e s h r e s u l t e d i n a n e r r o r o f a b o u t 6 p e r c e n t i n t h e

f in a l s t if f n e ss a s c o m p a r e d w i t h t h e e x a c t s o l u t i o n , i .e . i t p r o v e d t o b e l es sa c c u r a t e t h a n t h e m e t h o d e m p l o y i n g t h e c o r re c t i n t e r f a c e c o n d i t i o n s . T h ep r o c e d u r e m a y , h o w e v e r , be a d v a n t a g e o u s w h e n n u m e r o u s c u r v e d in t e r-f a c e s o c c u r .

I n e x a m p l e C ( F ig . 5 ) t h e c a s e o f a m u l t i p l y c o n n e c t e d r e g i o n is c o n s i d e r e di n w h i c h t h e i n t e r f a c e b e t w e e n t w o m a t e r i a l s c u t s t h e h o le b o u n d a r y . T h ep r o c e d u r e f o l l o w e d h e r e w a s e s s e n t ia l l y t h e s a m e a s d e s c r i b e d b y S h a w 1 f o rt h e t r e a t m e n t o f m u l t i p l y c o n n ec t e d , h o m o g e n e o u s s h a f ts . S o l u t io n s w i t h ac o n s t a n t a n d a r b i t r a r y v a l u e o f t h e s t re s s f u n c t i o n o n t h e h o le b o u n d a r y a r ef ir s t o b t a i n e d u s i n g t h e g o v e r n i n g e q u a t i o n (4) a n d e x c l u d i n g t h e n o n -h o m o g e n e o u s t e r m . T h e n a s o l u t i o n o f t h e fu l l e q u a t i o n w i t h a z e ro v a l u e o f


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Torsion of compound ba rs -- a relaxation solution 365the s t r es s func t i on of the bo le boundary i s compu t ed , and the f ina l s o lu t ionob ta ined by a l inea r combina t ion o f these . Th i s combin ed s o lu t ion s at i sf i esthe in tegra l condi t ion (16). Agai n no specia l d i f f icul t ies were encou nter ed .

FIG. 5. Exa mpl e C. Torsion of a composite bar with a circula r hole.R E F E R E N C E S

I. N. I. MUSKHELISHVILLI, ome Basic Problems of the Mathematical Theory of Elasticityp. 561. Noordhoff , Groningen (1953).2. A. YA. GORGIDZE,Trudy Tbiliss Math. Inst. Razmadze 17, 95 (1949).3. D. . MITRA, Bull. Calcutta Math. Soc. 47, 191 (1955).4. HISAO TAKEYAMA,J. Soc. Appl. Mech., Japan 2, 88 (1949).5. I. V. SUHAREVIKI,Inzh. 8born. Akad. Naulc SS SR 19, 107 (1954).6. D. I. SHERMAN and M. Z. NARODETSKII, nzh. sborn. Akad. Nauk SSSR 6, 17 (1950).7. H. J. COWAN,Appl. Sci. Res., Hague (A) 3, 344 (1952).8. A. H. CRAVEN, Mathematika 1, 96 (1954).9. R. V. SOUTHWELL,Relaxation Methods in Theoretical Physics p. 85. Clarendon Press,Oxford (1946).10. F. S. SHAW, The Torsion of Solid and Hollow Prism s in the Elastic and Plastic Rangeby Relaxation Methods. Austral. Counc. Aeronaut. Rep. ACA-11,38 (1944).11. D. N. DE G. ALLEN, Relaxation Methods p. 198. McGraw-Hil l, Lo ndon (1955).12. W. B. DOBIE an d A. R. GENT,Struct. Engr. 39, 203 (1952).13. H. J. COWAN,Civ. Engng. Lond. 48, 567, 827, 950 (I953).14. W. B. DOBYE,Struct. Engr. 30, 34 (1942).15. T. J. HIGGINS, J. Appl. Phys. 14, 469 (1943).16. L. E. PAYNE, Iowa St. Coll. J. Sci. 23, 381 (1949).17. S. TIMOSUENI~O and J . N. GOODIER, Theory of Elasticity. McGraw-Hill, New York(1951).18. D. G. CHRISTOPHERSON and R. V. SOUTHWELL,Proc. Roy. Soc. A 168, 317 (1936).

torsion of compound bar 1960

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