2 dimensional

8/6/2019 2 Dimensional

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l ~ t . J . M e c h . S c i . Per gam on Press Ltd . 1967. Vol. 9, pp. 143-155. Prin ted in Great Britain

E L A S T I C - P L A S T I C A N A L Y S I S O F T W O - D I M E N S I O N A LS T R E S S S Y S T E M S B Y T H E F I N I T E E L E M E N T M E T H O D

P . V . M ~ RC A L * a n d I . P . KI~CG~(Received 11 August 1966)

Summary--A method is described for the incremental elastic-plastic analysis of two-dimensional stress systems (plane stress, plane strain and axisy mmetri cally loaded bodyof revolution). The procedur e is based upon the finite element meth od of stress analysisand unifo rmly stressed triangular elements are used. The derivat ion of the increme ntalstress-strain relationship from the Prandtl-Reuss equations and the Von Mises yieldcriterion is presented. Numeric al examples are presented an d are compared withexperimental and theoretical results in the literature.

0"0, ~0O'r, ~raz, ez(Trz~ Erz

t( ~H"

YEG[am]{R}{v}

[g]

N O T A T I O Ncylindrical co-ordinate systemstress and strain in 0 co-ordinate directionstress and strain in r co-ordinate directionstress and strain in z co-ordinate directionshear stress and strain in rz planeequivalent stress and plastic straindeviatoric stressslope of 5 vs. ~v curvePoisson's ratioYoung's modulusshear modulusmatrix occurring in elastic-plastic stress-strain relationshipnodal force vectornodal displacement vectorstiffness ma tri x

I N T R O D U C T I O NI x A previous paper one of the authors suggested a stiffness method fore l a s t i c - p l a s t ic p r o b l e m s . 1 T h e m e t h o d e n a b l e d t h e d i ff e r e n ti a l e q u a t i o n sd e s c r ib i n g e q u i l ib r i u m t o b e e x p r e s s e d in t e r m s o f d i s p l a ce m e n t s . M o r e r e c e n t l y ,t h e m e t h o d o f f in i te e l e m e n t s h a s b e e n f o u n d t o b e a p o w e r f u l a p p r o a c h t os tr e s s a n a l y s is p r o b l e m s . P a r t o f i t s a d v a n t a g e s t e m s f r o m t h e a b i l i ty t oh a n d l e i r re g u la r sh a p e s o f b o u n d a r i e s a n d m i x e d b o u n d a r y c o n d i t io n s .

I t s e e m e d t o t h e a u t h o r s t h a t i t w o u l d b e u s e f u l t o m o d i f y t h e p a r t i a ls t if fn e s s m e t h o d d e v e l o p e d i n R e f . 1 f o r t h e f i n i te e l e m e n t s o l u t io n o f e l a st ic -p l a s t i c p r o b l e m s . T h e t e r m ' p a r ti a l s t i ff n e s s m e t h o d ' i s u s e d h e r e in o r d e r t oa v o i d c o n f u s i o n w i t h t h e t e r m ' s t i f f n e s s m e t h o d ' u s u a l l y u s e d i n t h e f i n i t eelement field to describe a force-displacement approach.

* Impe rial College of Science and Technology, Exhib ition Road, London, S.W.7.t Unive rsi ty of Wales. Form erly at : Berkeley Nuclear Laboratories, C.E.G.B.l0 143


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144 P.V. MARCALand I. P. KInGR E VI E W OF L I T E R AT UR E

Turner e t a l . 2 for mulat ed the stiffness app roa ch for a plane trian gular elementin plane stress. Sub seq uen tly ma ny investig ators, Clough 3, Argy ris 4, Melosh",Gallagher e t a l . 6 and Zienkiewicz:, have produced elements for different stressconditions and with more refinement, covering bending and three-dimensionalelements with triangles, rectangles, quadrilaterals and tetrahedra.

Gallagher 6 and Argyris e t a l . s have extended the method to elast ic-plast icstress problems by ma king use of the so-called ther mal strain ap pro ach similarto th at suggest ed b y Mendelson a nd Manso n 9. Pop e 1 has s uggeste d a t an gen tmodulu s app roa ch for the solution of elastic-plastic probl ems by finite elements.In m an y respects our meth od is similar to tha t of Pope, the present ap proac hdiffering by the use of mea n parti al stiffness coefficients for elements whichyield in the next increment. Pope ad voc ate d the use of small increments ofload which just caused yield in the next element. It will be shown th at this maybe avoi ded b y the use of the m ean partial stiffness concept.

Swedlow n, by requiring th at the stresses a nd strains remained within certainlimits of the equ ival ent stress vs. equiv alen t plastic strain curve, has developeda tan gen t modulus app roac h to elastic-plastic plane stress problems. The twomethods referred to above are only applicable to work-hardening materials.

Summa rizin g the state of the li terature, i t appears t hat in general, in finitedifferences as well as in finite elements, there are two possible solutions ofelastic-plastic problems. One based on the thermal strain appro ach and theother on the tang ent m odulus approach. As yet i t seems pre matu re to decidewhich is superior. The for mula tion of a decision is no t helped by the lack ofdetailed information in the li terature on the c ompu tin g requirements ~ff eachmethod.

A ST I FF NE SS C ONC E PT FOR AN E L AST I C - PL AST I ( 'F I N I T E E L E M E N TIll this section we obtain the stiffness relations for a triangular ring element of a solid

body of revolution, see Fig. l, subjected to symmetric loading. Subsequently we shallshow in Appendix 2 how the plane stress and plane strain formulations may be obtainedt)y removing the relevant terms. We shall use an elastic-plastic material of the PrandtlReuss type which obeys the yon Mises yield criterion. The method is developed for amaterial with arbitrary work hardening but most of the examples are solved for anelastic-perfectly plastic material. As argued in Ref. 1, the elast ic-perfectly plastic materialforms the most stringent test of the method. Curiously this situation is reversed in formalanalysis.First of all, we obtain the incremental stress vs. strain relation for an elastic-plasticmaterial in terms of partial stiffness coefficients. The partial stiffness coefficients fi)r ageneral three-dimensional stress system have been given in Ref. 1; but for the sake ofcompleteness we briefly review here the equations appropriate to a solid body of rev()lul lollwith symmetrical loading.With symmetrical co-ordinates 0, r, z and using the conventional notation fin' 1hestresses, we have the Von Mises yield criterion :( a o - - a t ) 2 + (at - - a ~ ) ~ + ( a ~ - - a o ) 2 + 6a~z = 25 ~ (la)

This equation can be put in the form of an implicit differential :3 a ~ d a ~ + 3 a ~ d a ~ + 3 a ~ d ( r: + 6 (r r~ ( t ~ = 2 ~ ( 1~ ( I b )


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E l a s t i c - p l a s t i c a n a l y s i s o f t w o - d i m e n s i o n a l s t r e s s s y s t e m sw h e r e a ' d e n o t e s a d e v i a t o r i c s t r e s s , e . g .

a~ = ~ ( 2 a 0 - a . - - . )a n d t h e p r e f ix d i s u s e d t o d e n o t e c h a n g e s o v e r a s m a l l i n c r e m e n t o f lo a d .

z

1 4 5

I iZ~

~ r

F IG . 1. R o t a t e d t r i a n g u l a r e l e m e n t w i t h d i m e n s i o n s .A t e n s i l e o r c o m p r e s s i v e t e s t o n t h e m a t e r i a l c a n b e u s e d t o p r o v i d e a r e l a t i o n b e t w e e n

t h e e q u i v a l e n t s t r e s s 5 a n d t h e e q u i v a l e n t p l a s t i c s t r a i n ev . H e n c e a t a p a r t i c u l a r v a l u eo f e q u i v a l e n t p l a s t i c s t r a i n w e h a v e

d 5 = H " d ~ vw h e r e H ' i s t h e s lo p e o f t h e e q u i v a l e n t s t r e s s 5 v s . t h e e q u i v a l e n t p l a s t i c s t r a i n e v c u r v e .

S u b s t i t u t i n g f o r d 5 i n e q u a t i o n ( l b ) ,' a " ' 3 - U ' d ' 3 a ' * - H ' =32 aOd~+~5 ~da,_ 2 5 a . 5 da ,: d ~ , 0 (2 )

T h e P r a n d t l - R e t L u s s t r e s s - s t r a i n r e l a t i o n s t o g e t h e r w i t h e q u a t i o n (2 ) f o r m a l i n ea rs y m m e t r i c m a t r i x r e l a t i n g t h e s t r e s s i n c r e m e n t s t o t h e s t r a i n i n c r e m e n t s .

dso

d e r

d e , =

d e r ,

0

- l v v 0E Ev 1 v 0v v 1E E E 0

1o o o

3 a a 3 a , 3_ 2 5 2 5 2 ~ a

3 a~ -2(}3~253 ~~25

3~ ~(Y

daoda~d o z

d a , ~

d ~

( 3 a )


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146 P . V . MARCAL an d I. P. Kt~GWe new use mat r ix nota t ion and le t the above equa t ion be represented by

{de~ = [a , , ] {de} (:~b)where [a,,,] is the 5 x 5 sym met r ic ma tr i x and {da} and {ds} are the eoh mm vectors of s tress(wi th the except ion of d~ ) and s t ra in .

Mult iplying bot h sides by th e inverse mat r ix [a, , ] 1 we haw,{da} = [a,,,] -~ (de} (4)

and because of the l inear i ty of the e quat ions we see th at the eleme nts of the inverse real r ix[a,,~] -1 m a y be wr it te n a s

~a~ for i ,j = O , r , z , r z , pHere for ease of repr esen tat i on a~ is used to repre sent ~ and s~ equals zero. Thus the lastcolumn of the inverse mat r ix m ay be neglec ted. Because of i t s phys ica l in te rpre ta t ion, ~erefer to the ?a~/~si coefficients as th e part ial stiffness coefficients.

T H E T R A N S I T I O N R E G I O NThe so lut ion of the p robl em is carr ied out in inc reme nts of load in order to take

adv ant age of the l inear i ty of equ atio n (4). Howe ver , complicat i ons ar ise when the elast i( .par t of the s t ruc ture wi th s t resses near y ie ld becomes plas t ic wi th the next increment ofload. This region which is usual ly adja cen t to t he elast ic-p last ic interface has been cal ledthe t ransi t ion region in Ref . 1.

The technique used for deal ing with each element in the t ransi t ion region is to ( )btab~an es t imate of the s t ra in in crement f rom the result s of the previous load increment . Thises t imate of s t ra in i s then a ssumed to ac t in the fol lowing mann er as the load is appl ied .Ini t ia l ly a pro por t io n of this s tra in acts e last ical ly so as to cause yield at a p ar t icu lar ra t ioof s t resses. Subseque nt ly the b ehaviour i s assumed to be e las t ic -plas t ic and the par t ia lst i f fness coeffic ients for this s tra in ing arc obt aine d on t he basis of the above-m cntion e( lratio of stres ses at yiehl. Hen ce we can define a mea n stiffness coefficient of the eh,me~,tby wei ghting of the elast ic and elast ic- plast ic s t i f fnesses in the rat io

~'J/mean ~ \ 8 f f e l as~ ic \~Ei/elastic-plasticwhere m = strai n required to cause yiel d/es t ima ted strain.

The f i rst est i mate of the strains is obvious ly not going to be correct , but the use of t h~.resul t ing mean s t i ffnesses for the solut ion of the equi l ibr ium equa t ions descr ib ing thepro ble m wil l provi de a c loser est i mat e of the strains . In fact t he st ra in est imat( ,s a t( ,usual ly quite good and because the t ransi t ion region at any stage is qui te small comparedwith the whole structu re , t he me an st i f fness coeff ic ients converge rapidly. In pra ct ice anadeq uate appro ximat ion to convergence of the me an s t i ffness coef f ic ients i s obta ined a f te rtwo i terat ions. We no te here tha t the m ean st i f fness coeff ic ients are the only st i f fnesscoeff ic ients to ch ange dur i ng th e solut ion for an in crem ent of load and, in order to get thebes t out of th is approach, we mu st use a numer ica l approach for the solut ion of theequi l ibr ium equa t ions which can take adva ntag e of the previous resul t s tha t a re a l readyc lose to the cor rec t answer . In th is paper we use the me tho d of sys temat i c over - re laxa t ionwhich sa t i sf ies the above requi rements .

E L E M E N T S T I F F N E S SIn order to obta in t he el ement st i f fnesses ( i.e. forces at nodes caused by unit displace-

men ts a t the nodes), we have used the formula t ion of Clough and R ashid TM for a toroidalr ing of t r ian gula r cross-sect ion. The el ement st i f fnesses in Ref . 12 were obtai ned b y vir tualwork pr inc iples and th is i s s t r ic t ly not appl icable to an e las t ic -plas t ic s i tua t ion. However ,because we have l in ear ized the stress vs. s t ra in relat ions for an in crem ent of load, we haw,argue d t ha t this co nver ts the non -l inear e last ic-pl ast ic prob lem to a ser ies of successiv(.elastic pro ble ms wi th ch ang ing coefficients, so tha t we are jtmtified in using a v irtu al w(~rk


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E l a s t i c - p l a s t i c a n a l y s i s o f t w o - d i m e n s i o n a l s t re s s s y s t e m s 14 7f o r m u l a t i o n . M o r e r e c e n t l y , b e c a u s e o f p o s s i b le o b j e c t i o n s t o t h e u s e o f v i r t u a l w o r k f o re l a s t i c - p l a s t i c p r o b l e m s , w e h a v e o b t a i n e d a n a l t e r n a t i v e f o r m u l a t i o n b y m e a n s o f f o rc ec o n s i d e r a t i o n s . T h i s p r o d u c e d t h e s a m e v a l u e s f o r t h e s t i f f n e ss co e f f ic i e n ts a n d t h e s a m er e s u l t s fo r t h e e x a m p l e o f a t h i c k c y l i n d e r u n d e r i n t e r n a l p r e s s u r e . T h i s l a s t f o r m u l a t i o n i sd e s c r i b e d in A p p e n d i x 1 . W e q u o t e h e r e t h e m a i n r e s u l t s g i v i n g t h e s t i f f n es s [k ] f o r a ne l e m e n t [ k ] = [ F ] [ a , , ] ~ I V ] ( 5 )w h e r e [ V ] i s a d i s p l a c e m e n t t o s t r a i n t r a n s f o r m a t i o n m a t r i x , [ F ] i s a s t r e s s t o n o d a l f o r cet r a n s f o r m a t i o n s m a t r i x a n d [ ~ ,, ] -1 is t h e s t r e s s v s . s t r a i n m a t r i x g i v e n b y t h e p a r t i a ls t if f ne s s m a t r i x d e s c r i b e d a b o v e .

T h e n o d a l p o i n t s t i f f n e ss e s a r e t h e n a s s e m b l e d t o o b t a i n t h e s t r u c t u r a l s t i f f n es s m a t r i x[ K ] .

M E T H O D O F S O L U T I O NT h e r e a r e 2 N l i n e a r e q u a t i o n s o f e q u i l i b r i u m f o r t h e N n o d a l p o in t s . F o r a n i n c r e m e n t

o f l o ad , t h e s e m a y b e w r i t t e nN{ d R } = ~,K,~dv~ f o r n = 1 , N (6 )J - 1

w h e r e { dR } is t h e 2 N x 1 m a t r i x g i v i n g t h e a p p l i e d l o a d s a t t h e n o d a l p o i n t s , { dv} i s t h e2 N 1 m a t r i x g i v i n g t h e d i s p l a c e m e n t s a t t h e n o d a l p o i n t s a n d [ K ] i s t h e s t r u c t u r a ls t i f f n e s s m a t r i x .

E q u a t i o n (6 ) i s s o l v e d b y t h e a c c e l e r a t e d G a u s s - S e i d e l i t e r a t i o n w i t h a n a c c e l e r a t i o nf a c t o r b e t w e e n 1 .8 a n d 1 .9 . T h i s i t e r a t i v e p r o c e d u r e i s w e l l s u i t e d t o t h e p r e s e n t a p p r o a c h ,b e c a u s e a f t e r a f e w c y c le s o f i t e r a t i o n , a n i m p r o v e d a p p r o x i m a t i o n t o t h e s o l u t i o n isa v a i l a b l e a n d m a y b e u s e d t o m o d i f y t h e m e a n p a r t i a l s t i f f ne s s c o e f f i c i en ts . A s n o t e de a r l i e r o n l y t h e m e a n p a r t i a l s t i f f ne s s es c h a n g e d u r i n g t h e s o l u t i o n f o r a n i n c r e m e n t o f l o a d .T h e f lo w se q u e n c e o f t h e p r o g r a m d e s c r i b e d b e lo w i n d i c a t e s t h e s t e p s t a k e n i n t h e s o l u t i o no f a n e l a s t i c - p l a s t i c p r o b l e m .

F L O W S E Q U E N C E1. C a l c u l a t e p a r t i a l { f u l ly e l a s t i c ) s t if f n e s s co e f f ic i e n t f o r u n i t l o a d a n d e l a s t i c b e h a v i o u r .2 . F o r m f o r c e - d i s p l a c e m e n t r e l a t i o n s a n d s o l v e b y s y s t e m a t i c r e l a x a t i o n .3 . S c a l e a l l e l a s t i c v a l u e s i n o r d e r t o c a u s e y ie l d a t t h e p o i n t o f m a x i m u m e q u i v a l e n t

s t r e s s . L e t L r e p r e s e n t t h e l o a d a t y ie l d .4 . A p p l y a n i n c r e m e n t o f l o a d ( s a y 0 .1 L ) .5 . S t o r e s tr e s s e s f r o m p r e v i o u s l o a d i n g a n d e s t i m a t e s t r a i n i n c r e m e n t s t h a t w i ll b ec a u s e d b y t h e n e x t i n c r e m e n t o f l o a d .6 . C a l c u l a t e p a r t i a l s t i f f n e s s c o e f f i c i e n ts Oa~/~j f o r e a c h e l e m e n t a f t e r d e c i d i n g w h e t h e r

a p o i n t i s e l a s t i c , e l a s t i c - p l a s t i c o r i n t h e t r a n s i t i o n r e g i o n .7 . F o r m f o r c e - d i s p l a c e m e n t r e l a t i o n s a n d s o l ve f o r e q u i l i b r i u m .8 . R e p e a t 6 a n d 7 t h r e e t i m e s .9 . C a l c u l a t e i n c r e m e n t o f e q u i v a l e n t p l a s t i c s t r a i n s

~ -d ~ = ~ d ~ i, j = O,r,z, rzi f d ~ v n e g a t i v e , s t o p ; i f p o s i t i v e o r z e r o , c o n t i n u e .

1 0. C a l c u l a t e s t r e s s i n c r e m e n t s a n d a d d t o p r e s e n t s t r es s e s . C a l c u l a te a n d s t o r e s t r a i ni n c r e m e n t s a s e s t i m a t e f o r n e x t i n c r e m e n t o f l oa d .

1 1. O u t p u t d i s p l a c e m e n t s a t e a c h n o d e , s t r a i n s a n d s t r e s s e s a t e a c h e l e m e n t .1 2. R e t u r n t o 4 f o r a n o t h e r i n c r e m e n t o f l o a d i f t o t a l l o a d n o t a p p l i e d .

R E S U L T SW h e n t h e c o m p u t e r p r o g r a m w a s d e v e l o p e d i t w a s u s e d to i n v e s t i g a t e t h e f o l lo w i n g

t h r e e c a s e s . T h e f i r s t c a s e s t u d i e d w a s t h a t o f a t h i c k c y l i n d e r s u b j e c t e d t o i n t e r n a lp r e s s u r e b e c a u s e i t w a s p o s s i b le t o c o m p a r e t h e r e s u l t s w i t h t h e l i t e r a t u r e ( e.g . H o d g e a n d


6/13

1 48 P . V . M A RO AL a n d I . P . K I ~ ( ~W h i t e la ) . T h e s e c o n d c a s e s t u d i e d w a s t h a t o f a f i a t t e n s i o n s p e c i m e n w i t h a c e n t r a l h o le .T h i s e x a m p l e w a s s t u d i e d b y m e a n s o f a b i r e f r i n g e n t c o a t i n g t e c h n i q u e b y T h e o c a r i s a n dM a r k e t o s . 14 F i n a l l y t w o s h a p e s o f n o t c h e d t e n s i o n s p e c i m e n s w e r e s t u d i e d i n c o n d i t i o n so f p l a n e s t r es s , p l a n e s t r a i n a n d a x i s y m m e t r i c l o a d i n g o f a b o d y o f r e v t ) l u t io n . T h e f l a nkn o t c h e s w e r e o f s e m i - c i r c u l a r a n d 9 0 i n c l u d e d a n g l e r e s p e c t i v e l y a n d a r e s i m i l a r i n sh a l ) et o t h o s e s t l l( t ie d b y A l l e n a n d S o u t h w e l l . '~( i t T h i e t " c y l i n d e r w i t h i n t e r n a l p r e s s u r e

T h e w r i t e r s h a v e o b t a i n e d s o m e r e s u l t s f o r t h e t h i c k c y l i n d e r u n d e r i n t e r n a l p r e s s u r ( ~w i t h v a r i o u s e n d c o n d i t i o n s , i n o r d e r t o m a k e a c o m p a r i s o n w i t h t h e m a n y e x a m p l e s g iv eHi n t h e l i t e r a tu r e . H o w e v e r , s o m e d i f f ic u l t y w a s m e t w i t h i n t r y i n g t o i m p o s e t h e s a m e e n dc o n d i t i o n s a s t h e e x a m p l e s i n t h e l i te r a t u r e . T h e m a i n e a tt se o f t h i s w a s t h a t i n a l l t h ee x a m p l e s t h e a s s u m p t i o n w a s m a d e t h a t t h e c y l i n d e r d e f o r m e d i n a p l a n e - s t r a i n c o n d i t h mi n a l o n g i t u d i n a l d i r e c t i o n (e~ = c o n s t a n t ) . I n g e n e r a l s u c h a c o n d i t i o n d o e s n o t e x i s t a m lt h e f in i te e l e m e n t r e s u l t s do n o t s h o w i t. T h e o n l y p l a n e - s t r a i n e n d c o n d i t i o n t h a t c o u l db e im p o s e d o n t h e c y l i n d e r b y t h e f i n it e e l e m e n t p r o g r a m is t h a t o f z e ro s t r a i n i n t h el o n g i t u d i n a l d i re c t i o n a n d t h i s c o r r e s p o n d s t o t h e c a s e t r e a t e d b y H o d g e a n d W h i t e i)lR e f . 1 3. T h e b r o k e n l i n e in F ig . 2 s h o w s t h e r e s u l t f o r t h e p r e s e n t a n a l y s i s f o r a c y l i n d e rw i t h ~ d i a m e t e r r a t i o o f 2 t o 1 . T h e r e s u l t s o b t a i n e d b y H o d g e a n d W h i t e a r e p l o t t e d a s af u ll l in e i n F i g . 2. T h e r e i s a r e a s o n a b l e a g r e e m e n t b e t w e e n t h e r e s u l t s . A s i m i l a r a g r e e m e n tw a s o b t a i n e d f o r t h e t h i c k c y l i n d e r w i t h o p e n - e n d c o n d i t i o n w h e n c o m p a r e d w i t h t h e) 'e s u lt s o f M a r e a l ~6. n o t w i t h s t a n d i n g t h e s l i g h t l y d i f f e r e n t l o n g i t u ( l i n a l e n d c o n (t it io n .~ .

I-0

0.g

0-6 S"o-( / ' 4

2 dimensional

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