Capurso_Limit Analysis of Continuous Media With Piecewise Linear Yield Condition

8/10/2019 Capurso_Limit Analysis of Continuous Media With Piecewise Linear Yield Condition

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L I M I T N L Y S I S O F C O N T I N U O U S M E D I W I T H

P I E C E W l S E L IN E R Y E L D C O N D I T I O N

M i c h e l e C a p u r s o

S O M M A R I O : Si affronta i l problema del l'anal is i l imite deim ez zi continui costituiti di ma teriale rigido perfettamente p la-stico ave nti snperficie di snervamen to di tipo poliedrico. Si de-finiscono in for m a unitaria, avvalendosi di noti con certi di pro-gram maz ione lineare, i principi di estrem o riguardan ti la valu-taz.ione della polenza specifica di dissipazione. S i deducono pod iprincipi fondam entali che governano il problenm del collassop/astico a ttraverso una semplice formulazione ba sata sul bennoto teorema cinem alico dell'analisi lim e. Si sn&gerisce inflne

mt procedim ento approsrim ato tatt'a~ratlo generate per la valu-/azione del carico di co l/asso di mt qualsiasi mezzo continuo.

S U M M A R Y : The prob lem of l imi t ana l-ys is o f continua Jormedb-y rigid-perfectO, plas lic ma terial w ith piecewise .yield stwfaceis discussed. IVith the use o f the known concepgs of linear pro-gram ming, the extremum principles concerning /he determinationof sp ecific dissipation power are defined in unitary for m . Thebasic principles governing the problem of plastic collapse are thenexpressed through a simp le formulation based on the well knownkinemaUcal theorem of limit anal),sis. Finally, an approximategeneral procedtwe is suegested or the calcMation of the collapse

load of continuous media.

1 . I n t r o d u c t i o n .

Th e l imi t ana lys i s o f rig id -p la s ti c con t inua w i th a s so -c i a t e d f l o w r u l e s i s b a s e d o n t h e t w o k n o w n t h e o r e m s o fs t ruc tu re p l a s t i c co l l apse [1 ] , [2 ] . However, t he sys t ema t i capp l i ca t ion o f such theorem s to r ea l s t ruc tu ra l p rob lem si s gene ra l ly qu i t e compl i ca t ed and canno t be gene ra l i zedto de te rmine un i t a ry p rocedures fo r numer ica l ca l cu la t ion .

On th i s sub jec t , many impor t an t o ld and r ecen t s tud ie s[ 3 t h r o u g h 11 ] r e fe r s u c h p r o b l e m s t o t h e k n o w n p r in c ip l e so f l in e a r p r o g r a m m i n g , b y c o n n e c t i n g t h e a p p r o p r i a t el inea ri za tion o f the y i e ld su r face wi th the d i sc re t iza t ionof the co n t inuu m, the reby o ffe ring a sys tema t i c p roce duref o r t h e n u m e r i c a l s o l u ti o n o f s u c h p r o b le m s . H o w e v e r,i n o u r o p i n i o n , t h e s i m u l t a n e o u s a p p l i c a ti o n o f b o t hd i sc re t i za t ion p rocedures p reven t s a fu l l eva lua t ion o fs u c h p r o c e d u r e s f o r t h e g e n e r a l f o r m u l a t i o n o f t h e p r o b l e m .

C o n t i n u u m d i s c re t iz a t io n - - w h i c h c a n b e a c h i e v e d b yseveral m~_thods finite difference s, f inite elem ents , seriesexpans ions e t c . ) - - on ly r ep resen t s a me an o f num er ica lca l cu la t ion to so lve an ac tua l s t ruc tu ra l p rob lem, wh i l ethe l inear iza t ion of the y ie ld surface impl ies the po ss ibi l i tyo f i n t ro d u c i n g , i n a u t o n o m o u s f o r m , t h e o r e ti c a l b a s ic

p r inc ip le s in add i t ion to the we l l known gene ra l p r inc ip l e s .Th e pu rpose o f th i s s tudy i s t o fo rm ula t e a gene ra l ap -

p roac h fo r t he l imi t ana lys i s o f r ig id -p la st i c bod ies h av ingpiecewise l inear y ie ld surface .

Th e f i r st pa r t o f th i s w ork e s t ab l ishes the ex t r em ump r in c i pl e s g o v e r n i n g t h e p r o b l e m o f d e t e r m i n i n g t h e s p e -c i fi c d i s s ipa t ion p ow er. In the seco nd pa r t , t he fundam en ta ltheo rem s o f l imi t ana lysi s a re d i scussed in un i t a ry fo rm .F ina l ly, t he th i rd pa r t o ff e r s some appropr i a t e numer ica l

p rocedures fo r t he ac tua l and sys t ema t i c app l i ca t ion o fsuch p r inc ip le s to the so lu t ion o f s t ruc tu ra l p rob lem s .

2 . E x t r e m u m p r i n c i p l e s fo r t h e ca l c u l a ti o n o fspe-c i f i c d i s s i p a t i o n p o w e r.

Consider a y ie ld surface , determined in the s t ress spaceo~s by m linear cond i t ions Fig . 1 ) :

n t s o i . s = cr a = 1 . . . . m) 2.1)

where nts= a re the d i r ec t ion cos ines o f t he c tth hype rp lane ,and a ~ is t he d i s tance f ro m the o r ig in o f such hyperp lane .

A gener ic s t ress v ecto r a = [cr is ] i s wi th in the y ie ldl imi ts I f :

o - n = = a l s n ~ s ~ < a ~ a = l . . . .. m ) 2 .2 )

tha t i s , i f i t i s con ta ined by the con vex do m in ion de f inedby the yie ld surface .

n o < [n < t0 , ( 7 ~ > t0 . (2 .4 )

Wi th such pos i t ion , Eq . (2 .3 ) can be wr i t t en :

+ - l + __ - x +

D ( ; , ~ ) = m a x { ( ( 7 , ~ - - o , ~ ) e ,j ,( ( 7 ,~ ( 7 , j) ,, ~ j< , ~ , ( 7 , > i O ,

( 77 j > i 0 } . ( 2 . 5 )

The t r anspos i t ion t ab leau then becomes :

>1 0 (7~1 (77s

a c t

X n U - - n U

> 1 ' ~ * J - - t

~/ 0 on the lef t of f i rs t ro w appl ies to a l l var iables ins u c h r o w ;

0 o n t h e r i g h t o f f ir s t r o w m u s t b e c o n s i d e r e d a si n s e r t e d b e t w e e n t h e l a s t c o l u m n v e c t o r a n d t h a t o b t a i n e db y m u l t i p l y i n g t h e f i r s t r o w v e c t o r t o t h e p a r t i t i o n e dmat r ix enc losed in the so l id hne r ec t angu la r f r ame .

W hen the dua l va r i ab le s 2 ~ ( a = 1 . . . . m) are in t rodu cedin the f ir s t co lum n, the dua l p rog ram can be r ead in t ab leau(2 .6 ) app ly ing the above mmt ioned ru l e s , bu t r ep lac ingr o w w i t h c o l u m n , l e f t w i t h t o p , r i g h t w i t hb o t t o m .

W i th such ru l e s , t he dua l min imu m pro gra m (2 .5 ), o r(2 .3) , becomes:

D( }o) = r a in {~) f ] , ,~2 = /> } , j , - - a te2= /> - - ~ q , f f / > 0 }

(2.7)

f o r w h i c h w e u s e t h e c o m m o n n o t a t i o n o f t e ns o r ia l s u m -m a t i o n a l s o f o r in d e x a . H o w e v e r, s i n c e t h e t w o c o n d i t i o n s :

m e a n t h a t :

n,~2~ = i~j (2.8)

p rogram (2 .7 ) can be a l so wr i t t en :

D(} q) = min {~ff ln ,~2 = = ~ , j , 2 = /> 0} . (2 .9 )

0 e t

W e mus t take in to con side ra t io n that , i f (7~j and ~.0

respec t ive ly rep resen t two op t ima l se ts o f t he p r ima l (3)a n d d u a l (9 ) p r o g r a m s , f o r t h e k n o w n p r o p e r t i es o f t h el i n e a r p r o g r a m m i n g , w e m u s t h a v e :

a t o

),~ -----0 if nq ao < (7~

a 0

no n o = (7= if )~ > 0 .

(2.10)

However, s ince the opnmal se t mus t a l so sa t i s fy thel imi t at ions o f p ro gra m (2 .9 ) , we ob ta in :

~,j = a,~2~' (2. 11 )

w h e r e :

~ 0 = ( 7 =

(2.12)

Tak ing in to acc oun t Eqs . (2.1 ) o f t he y i e ld p l anes ,we f ind that Eqs . (2 .11) and (2 .12) are jus t the p las t icf low ru le s a s soc ia t ed wi th the chosen y ie ld su r face . The2 = therefo re be com e


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D e n o t i n g b y ;t= any sys t em o f p l a s t ic m ul t ip l i e rs s a t i sfy ingthe cond i t ion :

n71;t~ = } , j , 2 = > /0 ( 2 . 1 4 )

f rom (2 .12 ) we h ave :

( a ~ - o , jn , j) ( 2 - - 2 ; ) i >0 . ( 2 . 1 5 )

From Eq . (2 .11 ) and f rom f i r s t Eq . (2 .14 ) we ob ta in :

=

n , ] ( f f - - 2 ,, ) = 0 (2 .16 )

a n d , c o r r e s p o n d i n g l y :

x z~

a ~ n ~ ( 2 - -20) = O. (2.17)

Co mp ar ing E q . (2.17 ) w i th Eq . (2 .13 ) and in cons ide ra t ionof (2 .9 ), w e ob ta in the inequa l i ty :

a ~ f > i a ~ ) . o = D ( ~ , j ) (2.18)

w h i c h f u r t h e r c o n f i r m s , i n a u t o n o m o u s f o r m , E q . ( 2 . 9 ) .

3 . R e m a r k s o n t h e c a l c u l a t i o n o f t h e t o t a l d i s s i p a t i o np o w e r .

L e t u s c o n s i d e r a b o d y h a v i n g v o l u m e V d e l im i t e db y S . We r e f e r t h is b o d y t o a s y s t e m o f c ar te s ia n o r t h o g o n a laxes x , ( i = 1 , 2 , 3 ). W e sha ll no w app ly to each po in to f s u c h c o n t i n u u m a g e n e r ic s y s t e m o f d i s p la c e m e n trates :

t }, = , } , ( x , , x . , , x a ) ( i = 1 , 2 , 3 ) ( 3 . 1 )

con t inuous wi th p i ecewlse con t inuous de r iva t ives in thev o l u m e V.

D e n o t i n g b y :

1 o ; , ,[ 'u = ~ \ c-~xj a x , / = c9 ;0~ (3.2)

the s t ra in ra tes associa ted wi th the d isplacement ra tes3 . 1 ) ,w e w a n t t o d e t e rm i n e t h e t o ta l p o w e r o f d is s i p a ti o n :

P ( i t , ) = f v D ( ' e ' s ) d V (3.3)

F o r t h i s p u r p o s e , d e n o t i n g b y :

a , i = a i j ( x l , x 2 ,x a ) ,

) ~ = 2 a x ~ , x ~ . , x s )

(3.4)

the in t e rna l s t r e s ses and the p l a s t i c mul t ip l i e r s wh ichcan be a s soc ia t ed , po in t by po in t , w i th the s t r a in r a t es(3 .2 ) t h ro ug h the ex t r em um pr inc ip le s o f p rev ious sec t ion ,w e o b t a i n :

Tak ing in to cons ide ra t ion tha t , a s :

a~s = m j ( x , , x = , x s ) ,

2 = = 2 = x l, x . . , x s )(3.6)

are respect ively any s t ress s ta te wi thin ymld l imi ts andany sys t em o f non-n ega t ive k inemat i ca lly admiss ib le p l a s t icmul t ip l iers , the fundamental inequal i t ies

0

a=2 = > a=ag

(3.7)

a re sa t i s f i ed po in t by po in t , we can wr i t e

= i f V I

V

However, t he ac tua l ca l cu la t ion , t h rough Eq . (3 .8 ) ,o f d is s ip a te d p o w e r P is v e r y c o m p l ic a t ed . A l t h o u g h , w ec a n c al cu l at e a l o w e r b o u n d o f P t h r o u g h t h e f ir s t E q .(3.8) , by d iv id ing the vo lu me V in to a f imte num ber o fs u b - v o l u m e s V n ( h = 1 . . . . M ) , a n d a s s u m i n g f o r e a c hs u b - v o l u m e :

h

a~j(x t , x . , , xa) = au = cons t , in V n. (3 .9)

O n t h i s a s s u m p t i o n , b e i n g :

. j ' = f , ; , v d.}/j dV 3.10)

and deno t ing by ah = the va lues o f t he l imi t s t re s ses f o rthe gene r i c p l a s t i c p l ane a , cons tan t i n each sub-vo lume(bu t poss ib ly va r i ab le f rom one to ano the r ) , t he f i r s tE q . ( 3 .8 ) y ie ld s as b e s t lo w e r b o u n d P , o f to t a l p o w e rP, t h e q u a n t i t y :

h h I ~ h : c ~

P, , ---- m ax taoet,,im/r,v ~ 1 0 .(3.12)

By compar i son wi th second Eq . (3 .8 ) , t he dua l va r i ab le sF tn= can b e ob ta ined f ro m re l a t ions :

- fu ,, = a = d V (3.13)V h

keep ing in m ind tha t Eq . (3 .12 ) can be cons ide red equ iv -a l en t t o se cond E q . (3 .8 ) i f and on ly i f :

( ;l ~ ) = a , F u d V = a : ' 2 ~ d V. (3.5) P, ( ' u , ) = P(u, ) (3 .14)

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tha t i s , i f the c lass of func t ions (3 .9) conta ins a t leas t oner e al s o l u t i o n o f t h e o p t i m i z a t i o n p r o b l e m e x p r es s e d b yfirst Eq. (3.8).

Log ica l ly, a s t he subd iv i s ions o f vo lu me V inc rease ,Eq. (3 .14) can be sa t is f ied wi th an ever smal ler d i fferenceand , a t h i s r eason ing , we can expec t fo r any d i s t r ibu t ionof to ta l p las t ic mul t ip l iers /zh~ sa t i s fy ing the cond i t ions :

x o; .hl l U ~ h : e t j

otI~1~>~ 0

( i , j = 1 , 2 , 3 ; h = l . . . .. M )

a = l . . . . . m ; b = l . . . . . A )

(3.15)

to sa t i s fy the fundamen ta l i nequa l i ty :

cr=tz, , >1 P( 'uO (3.16)

which wi l l be used fo r t he approx ima te numer ica l ca l cu -l a t ion o f the con t in ua s t a t i c co l lapse load .

4 . F u n d a m e n t a l t h e o r e m s o f t h e l i m i t a n a ly s i s .

L e t u s c o n s i d e r a b o d y h a v i n g v o l u m e V a n d b o u n d a r i e sS s u b je c t i n V t o b o d y f o r c es A ~ , i n th e p o r t i o n S r o ft h e b o u n d a r y t o s u r fa c e f o rc e s T ~ a n d i n t h e c o m p l e m e n -t a ry po r t ion 3 ,, o f S to con s t r a in t s ann u l l ing a ll t he com -p o n e n t s o f d i sp l a c e m e n t u ,.. D e n o t i n g b y :

1/, = t}i( :q, x o, xa) (4.1)

any sy s t em o f d i sp lacemen t r a t e s , t he loads m ul t ip l i e r kde f ined by r e l a t ion :

D ( e l j ) d Vk - - (4 .2)

f X,i , ,dV + f sp T, i , ,dS

is called kinem a/ica//y admissible if k,~r ep resen t t he s t r a inra t e s a s soc ia t ed wi th the d i sp lacemen t r a t e (4 .1 ) and thesel a t t e r comply wi th the ex te rna l compa t ib i l i t y cond i t ions :

u l= 0 in S , , . (4.3)

A s s u m i n g :

f .A'~i,,dV + f s r T/m dS = l (4.4)

Eq . (4 .2 ) becomes :

k = f D ( 'e, /)dV.V

(4.5)

The k inemat i ca l ly admiss ib l e mul t ip l i e r t he re fo re co in -c ides wi th the to ta l p ow er o f d iss ipat ion P (see (3 .3)),a s soc ia t ed wi th s t r a in r a t e s (4 .1 ) . The k inem, t i c theo rem

of l imi t ana lys i s s ta t es h ow eve r tha t t he co l l apse mul t i -p l ier 6 i s the minimum kinemat ica l ly admiss ible coeff ic ient .Ta k i n g i n t o c o n s i d e r a t i o n t h e e q u i v a l e n c e b e t w e e n t h ek inemat i ca l ly admiss ib l e coe ff i c i en t and the to t a l powero f d i s s ipa t ion , f rom Eq . (3 .8 ) we ded uce tha t t he co l l apse

mul t ip l i e r ~ can be ca l cu la t ed us ing the cond i t ion :

6 = rain (}l ,) ma x(mj) f ~r,,,e.,/dV O)V

sub jec t t o cons t r a in t s :

(4.6)

i 0 - =cgta'O~

u t = O

X t ; M V + ff V 8 T1

o r e v e n u s i n g t h e c o n d i t i o n :

a ~/m1 ~< in V

in S ,

Tt}~dS = 1

(4.7)

6 = m in (t}0 ra in (2=) f v a=2=dV (4.8)


n,~2= = c g . ; t / ) , 2 ~ > / 0 i n V

ue = 0 in Su

f v A ~it ,d V + f s T T,i , dS= 1 .

(4.9)

Eqs . (2 .4 ) and (4 .8 ) a re bo th ana ly t i ca l fo rmula t ions o fthe k inemat i ca l t heo rem . I t i s ea sy to d emo ns t r a t e tha tthey a l so inc lude , a s pa r t i cu la r a spec t , t he s t a t i c t heo rem.In f ac t , i f we deno te by s a gene ri ca l mul t ip li e r o f ex te rna lloads a nd res t r ic t the c lass of s tresses a~j app ear in g inEq. (4 .6) to the c lass which sa t is f ies the equi l ibr ium con-d i t ions :

c)f~ri/+ sA 'j = 0 in V (2) (4.10)

~rqni = sT / i n S r

whe re n l ( i = 1 , 2 , 3 ) a re the d i r ec t ion cos ines o f t he n o r-m a l t o t h e e x t er n a l s u r fa c e in S T , w e o b v i o u s l y h a v e :

f v s (4 .11 )yO~ijdV

fo r wh icheve r sys t em o f ve loc i t ie s u~ chosen wi th r e spec tto cond i t ions :

e,~ = O(,}ti}, in V ,

ut = 0 in S ,

f ,, X[ u ,d V + f , T [mdS= 1 .

(4.12)

To the fo rmula t ion expres sed by Eqs . (4 .6 ) and (4 .7 )we the re fo re r ep lace the cond i t ion :

6 = m ix (~ro) s (4.13)

(l) Th e symbols in parenthesis indicate th e variables f orwhich the relative extremum condition is valid.

(8) In fact, the solution of the problem un dou btedly belong sto the class satisfying Eq . (4.10) with s = &

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O , a ~ + s A ~ = O ,

m~ a t j 0 } . 5 . 1 0 )

i t b e i n g u n d e r s t o o d t h a t th e c o m m o n n o t a t i o n o f te n s o r ia ls u m m a t i o n i s e x t e n d e d t o i n d e x k . T h e f u n c t i o n s u ~~ shal lb e l i n e a r l y i n d e p e n d e n t a n d w i l l b e c h o s e n i n s u c h a w a y

that each one sa t is f ies the in ternal compat ib i l i ty, tha t i srespects the m ater ia l cont in ui ty o f the bo dy , and sa t is f iesthe ex te rna l compa t ib i l i t y wi th the cons t r a in t s , so tha t :

ku ; = 0 i n S ,, ( i = 1 , 2 , 3 , k = l . .. . . N ) . ( 5. 3)

O n t h e o t h e r h a n d , t h e c o n t i n u i t y in t h e w h o l e v o l u m eo f th e s t r ai n c o m p o n e n t s :

Assuming a s independen t va r i ab le s the pa ramete r s qXand the to t a l p l a s t i c mul t ip l i e r s /~= , Eq . (5 .8 ) becomes :

= = ~t ~* .k hk ~ ok .I ,~ . = r a i n { a , , m , l , , , ~ m , = q ~ , J . ~ , , , /- - 0 , ~ q = 1 } 5 . 1 1

i t b e i n g u n d e r s t o o d t h a t t h e m i n i m u m c o n d i t i o n r e f e r sto q~ as wel t as to n =.

O n t h e o t h e r h a n d , w e c a n d e t e r m i n e t h e m a x i m u mcond i t ion b y no t ing tha t , i f i n Eq . (5.8 ) w e l imi t t he f ie ldo f sea rch fo r a t fl ' t o the c l a s s sa ti s fy ing the cond i t ions :

k ke u = O , l O ~ ( i , j= 1, 2 , 3 ; k = 1 . . . . N) (5 .4)

h hk Iatjeo = ~ (5.12)

wi l l no t be s t r i c t ly r equ i red and p iecewise con t inu i tywi l l be su ff i c i en t . Fo r the s t r e s s componen t s a~ j (x , , x2 ,x 3 ), o n c e d i v id e d v o l u m e V i n t o a fi n it e n u m b e r o f s u b -vo lum es Vn (h = -- 1 . . . . M ) , w e wi l l a s sume tha t fo r eachs u b - v o l u m e :

h~, j (x l , x . ,_ , xa) = a 'S = cons t , in Vt , (5 .5)

a n d a o n w il l b e a s s u m e d a s u n k n o w n s .O n th i s a s sump t ion , t he in t eg ra l appea r ing in Eq . (4 .6 ) , i f :

~ f o , ,, ; , , v 5 . 6 )U ~ V h

and in co nsid era t io n o f Eqs . (5 .2) , sat is f ies the re la t ion :

fV h .k hkj ~ M V = a ~q eui t (5.7)

where s i s an a rb i t r a ry coe ff i c i en t , we have :

t~ ,k hk t-~k'hafjq ets = ~ q = s (5.13)

then Eq . (5 .8 ) fu rn i shes :

h h k k ~ h o~6, = m ax {s aoe b = ~ , , t j crt i ~< ah} (5.14)

s i n c e t h e m a x i m u m i s v a l i d o b v i o u s l y o n l y f o r t h e s o l erem ainin g var iables , tha t i s a t j n . Th e tw o pr inciples (5 .11)a n d ( 5 . 1 4 ) , b o t h i n c l u d e d i n t h e l i n e a r p r o g r a m m i n g p r o -

c e d u r e s , c o r r e s p o n d o n e t o a n o t h e r a s p r i m a l a n d d u a lp r o g r a m a n d t h e r e f o r e m a k e i t p o s s i b l e t o d e t e r m i n e ,in add i t ion to the a pprox im a te va lue o f t he co l l apse mul -t ip l i e r, t he co l l apse mechan i sm(s ) and co r re spond ings ta t i c cond i t ions .

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6 . C o n c l u s i o n s .

T h i s w o r k h a s d e m o n s t r a t e d h o w t h e r i g i d - p l a s t i cm a t e r i a l s w i t h p i e c e w i s e l i n e a r y i e l d c o n d i t i o n s c a n b es t u d ie d in an a u t o n o m o u s f o r m b a s e d o n k n o w n m a t h e -m a t i c a l p r i n c ip l e s o f l in e a r p r o g r a m m i n g . O f p a r t ic u la ri m p o r t a n c e a r e t h e u n i t a r y f o r m u l a t i o n s o f t h e e x t r e m u mp r i n c ip l e s c o n c e r n i n g t h e s p e c if ic p o w e r o f d i s s i p a ti o n

and o f t he p r i nc i p le s co n ce rn i n g t he s t a t ic c o l l ap se mu l t i -p l ie r o f th e c o n t i n u a o n w h i c h i s b as e d t h e p r o p o s a l o f anumer i c a l app r o a c h f o r a s t r i c t l y gene ra l c a l c u l a t i on . Theapp l i c a t i on o f t h i s p r o c e du re t o a c t u a l c a se s o f t e chn i ca li n te r e s t w i l l b e t h e s u b i e c t o f o t h e r s t u d i e s t o b e p u b l i s h e din t h e n e a r f u tu r e .

Received 15 June 1970.

REFERENCES

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[2] A .A. GVOZDEV The determination of lhe value of lhe collapseload fo r statical/), indeterminate O,stems tmdergoing plaslic de-formation, Int. Jour. Mech. Sci . Vol. I 1960.

[3] W. S. DORN and H. S. GREENBEaG Linear programmingand plaslic lim il a naO,sis of slructures,T.R. No. 7 Dept.of Math. Carnegie Inst. of Technology 1955.

[4] A. CHartNES C. E. LEX~KE and O. C. ZXENKZEWlCKZVi rln al w ork, linear programming and plastic l imi t analysis,

Proc. Roy. Soe. A Vol. 251 1959.[5] W. Pa^G~rt Lin ear Ungleichungen in der Ba usla lik,Schwei-zerishe Bauzeitung No. 19 1962.

[6] C. CEaAI~INZ and C. GavaazNz Calcolo a rotlura e program-magione lineare,Giornale del Genio Civile No. 1 1965.

[7] D .C.A . KOOVMaN and R. H. L ~NCE On linearprogrammin~

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andplaslic lira# anal,sis,Jou rn. Mech. Phys. Solids Vol. 131965.

[8] C. GAVARZNZ Plastic ana lysis of strnctnres and duali O, inlinear programming, Meccanica Vol. 1 1966.

[9] R.H. LANCE D ua li O, in the finite-difference method of plasliclimO analysis,T.R.NS F GK 687/2 Dept. Theor. Appl.Mech. 1967.

[10] C. CERADINZ and C. GAVARIrqX Calcolo a rolttn'a e program-ma~ione lineare. Conlintd bi e lridimensionali. Nora I. Fonda-menli teorici,Giornale del Genio Civile No. 2-3 1968.

[11] C. CErtADINX and C. GXVARZNI Caleolo a rollura e program-ma~ione lineare. Contimd b i e lridimensionali, lola II . A p -p/ica~ioni a piastre e a vo/te di rivo/tt~ione,Giornale del Genio

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Universit y Press 1963.[14] L. MtSRACCmN L Programma~ione matemalica,Ed. UTET

Torino 1969.

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Capurso_Limit Analysis of Continuous Media With Piecewise Linear Yield Condition

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