non-linear methods of structural analysis - argyris et al 1972, nuclear engineering and design

7/30/2019 Non-Linear Methods of Structural Analysis - Argyris Et Al 1972, Nuclear Engineering and Design

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N U C L E A R E N G I N E E R I N G A N D D E S IG N 1 9 ( 1 9 7 2 ) 1 6 9 - 1 9 7

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R e c e i v e d 2 7 D e c e m b e r 1 9 7 1

T h e f o u n d a t i o n s f o r t h e s o l u t i o n o f p h y s i c a ll y n o n l i n e a r p r o b l e m s in s t r u c t u r a l m e c h a n i c s a r e r e v i e w e d . B o t h

a s p e c t s, t h e m a t h e m a t i c a l c h a r a c t e r i s a t i o n o f t h e n o n l i n e a r b e h a v i o u r a n d t h e n u m e r i c a l s o l u t i o n w i t h i n t h e f r a m e

o f t h e f i n i t e e l e m e n t m e t h o d a r e d is c u ss e d . A n u m b e r o f e x a m p l e s a re p r e s e n t e d w h i c h a r e r e l e v a n t t o p r o b l e m s o f

r e a c t o r t e c h n o l o g y . I n p a r t i c u l a r p r o b l e m s o f n o n l i n e a r e l a s t ic i t y , v i sc o e l a s ti c i ty , c r e e p a n d e l a s t o -p l a s t ic i t y a r e

d e a l t w i t h .

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

O u r p r e c e d i n g p a p e r w a s c o n c e r n e d w i t h t h a t f i e ld

o f s t r u c t u ra l m e c h a n i c s i n w h i c h c a u s e a n d e f f e c t a r e

r e l a te d i n a l in e a r m a n n e r . T h e p r e s e n t p a r t d e a ls w i t h

n o n - p r o p o r t i o n a l , i r r e v e rs i bl e a n d h i s t o r y - d e p e n d e n t

p r o c e s se s , o f p a r t i c u l a r i n t e re s t i n t h e c o n t e x t o f

n u c l e a r r e a c t o r p r o b l e m s .

R e a c t o r c o m p o n e n t s a r e e x p o s e d t o s e v e r e e n -

v i r o n m e n t a l c o n d i t io n s u n d e r w h i c h t h e y a r e r e q u i r e d

t o p e r f o r m a c o m p l e x p a t t e r n o f o p e r a ti n g s c he d u le s .

T h e m e c h a n i c a l b e h a v i o u r is s tr o n g l y a f f e c t e d b y

t h e s e c o n d i ti o n s , f o r e x a m p l e , m o s t m a t e r i a l p r o p e r -

t ie s e x h i b it a t e le v a t e d te m p e r a t u r e s p r o n o u n c e d n o n -

l i n ea r i t y , s t r a i n r a t e s en s i t i v i ty , i r r ev e r s i b il i t y an d

m e m o r y e f f e c t s . T h e s u b s e q u e n t d i sc u s si o n is c o n -

ce r n ed w i t h t h e s e p h y s i ca l n o n l i n ea r i t i e s , s i n ce t h ey

p l a y a n e x c e p t i o n a l r o l e in r e a c t o r p r o b l e m s . T h i s

r e s t r i c t i o n s i mp l i f i e s co n s i d e r ab l y t h e ex p o s i t i o n

s i n ce a t t h e r e l ev an t s ma l l d e f o r ma t i o n s n o d i s t i n c t i o n

n e e d to b e m a d e b e t w e e n d e f o r m e d a n d u n d e f o r m e d

c o n f i g u r a t i o n a n d t h e l i n e a r s t r a i n - d i s p l a c e m e n t r e -

l a t io n s h i p s f r o m a p r o p e r m e a s u r e o f d e f o r m a t i o n

Mo r eo v e r , a t t en t i o n i s r e s t r i c t ed t o q u as i s t a t i c mo -

t i o n s , i n w h i ch i n e r t i a e f f ec t s r em a i n n eg l ig i b le .

T h e s o l u t i o n o f p h y s i c a l ly n o n l in e a r b o u n d a r y

* A l s o a t I m p e r i a l C o l l eg e o f S c i e n c e a n d T e c h n o l o g y ,

U n i v e r s i t y o f L o n d o n , L o n d o n S W 7 , U . K .

v a l u e p r o b l e m s d e p e n d s o n t w o f a c t o r s : o n t h e

d e v e l o p m e n t o f e f f e c ti v e n u m e r i c a l s o l u t i o n s c h e m e s

a n d o n t h e a p p r o p r i a t e c h a r a c t e r i s a t i o n o f t h e m e c h a -

n i ca l p r o p e r t i e s . O n l y r ecen t l y t h e r ap i d l y d ev e l o p i n g

c o m p u t e r t e c h n o l o g y i n c o n ju n c t io n w i t h m a t r i x o r

f i n it e e l e m e n t m e t h o d s h a s p r o v i d e d t h e m e a n s t o

co mb i n e t h e s e t w o d i s c i p l i n e s f o r t h e en g i n ee r i n g

a n a l y si s o f c o m p l e x n o n l i n e a r p r o b l e m s i n s t r u c t u r a l

m e c h a n i c s .

T h i s p a p e r i s c o n c e r n e d w i t h b o t h a s p e c t s . F i rs t ,

d i f f e r e n t c o n c e p t s o f n u m e r i c a l a n a l y s is a re r e v ie w e d

t o d e v e l o p a l g o r it h m s f o r s o lv i ng t h e n o n l i n e a r m e c h a -

n i ca l fi e ld p r o b l e m . S u b s e q u e n t l y , a n u m b e r o f c o n -

s t it u t iv e t h e o r i e s a r e s u m m a r i s e d a n d a p p l i e d t o m o d e l

v a r i o u s p h y s i c a l p h e n o m e n a i n t h e c o n t e x t o f r e a c t o r

p r o b l e m s . F o r i l l u s t ra t i o n , d i f f e r e n t n o n l i n e a r e x -

a m p l e s a r e p r e s e n t e d i n w h i c h t h e r e l e v a n t m e c h a n i c a l

c h a r a c t e r i sa t i o n is i m p l e m e n t e d . T h e c o n c e p t o f n o n -

l i n ea r e l a s t i c i t y i s u s ed t o d e s c r i b e t h e d e f o r ma t i o n

a n d c l ea v a g e b e h a v i o u r o f c y l i n d ri c a l c o n c r e t e t e s t

s p e c i m e n s u n d e r t r i a x ia l c o m p r e s s i o n . T h e t h e o r ie s o f

v i s co e l a s t i c i t y an d c r eep a r e i n v es t i g a t ed t o d e t e r mi n e

t h e t i m e r e s p o n s e o f a P C R V f o r a " g i v e n " c r e e p l a w .

T h e e l a s t o - p l a s t i c f o r mu l a t i o n i s r ev i ew ed an d ex -

t e n d e d t o a c c o u n t f o r t e m p e r a t u r e - a n d t i m e - d e p e n -

d en t ma t e r i a l e f f ec t s . T h i s co n s t i t u t i v e l aw i s t h en ap -

p l i ed t o s o l v e t h e p r o b l em o f a p r e s s u r e v e s s e l n o zz l e .



17 0 Z H. Argyris e t aL, Non- l inear m etho ds o f s t ructura l analysis

2 . N o n l i n e a r f o r m u l a t i o n

Bef o r e d i s cu s s i n g d i f f e r en t n u mer i ca l a l g o r i t h ms

t h e f o r m u l a t i o n o f th e n o n l i n e a r b o u n d a r y v a l u e p r o b -

l em i s b r i e f l y r ev i ew ed . Fo r a r ecen t s u r v ey t h e r ead e r

i s a l s o r e f e r r ed t o [1 4 ] w h i ch co n t a i n s a co m p r eh en -s iv e l is t o f r e f e r en c es t o t h i s s u b j ec t . F o r t h e an a l y s i s

o f d i s s ip a t i v e p r o ces s e s an i n c r em en t a l s t ep b y s t ep

f o r m u l a t i o n h a s to b e u s e d w h i c h a c c o u n t s f o r t h e

p a t h d e p e n d e n c y d u r i n g th e e v o l u t io n o f t h e p r o b l e m .

I f th e me ch an i ca l b eh av i o u r i s ch a r ac t e r i s ed i n d i f -

f e r en t i a l f o r m , a s is t h e ca s e e . g . i n h y p o e l a s t i c i t y o r

p l a s t i c i ty t h e f o l l o w i n g g en e r a l i s a t io n o f t h e p r i n c i p l e

o f v ir t u a l w o r k can b e u s ed t o d e s c r i b e t h e q u a s i s t a ti c

m o t i o n o f a s o li d , s e e e .g . [ 1 - 3 ] .

T h e v i r tu a l w o r k e x p r e s si o n ( 2 ) m a y n o w b e d is -

c r e t is e d w i t h t h e h e l p o f f in i te e l e m e n t d i s p l a c e m e n t

mo d e l s l e ad i n g t o t h e f o l l o w i n g i n t e r p o l a t i o n o f t h e

f i e ld v a r i ab l e s f r o m n o d a l v a l u e s

Ve l o c i t y f i e l d

tJ = ~ l i , (3)

T o t a l s t r a i n - ra t e f i e ld

= ~ +/ ! = V ml~, (4)

Effec t ive s t ress - ra te f i e ld

i t = E T ~ = E T ( ~ _ i l ) = E r ~ + ~ . (5 )

f f _ u t / d VV V

+ f _U tpo dS + f l}t jOudS.

So Su( 1 ) *

T h e i n f e r i o r b a r s r e f e r t o k i n e m a t i c a l l y c o m p a t i b l e

s t r a in an d d i s p l acem en t s t a t e s 1 ' an d _u w h i ch s a t i s f y

t h e k i n e m a t i c b o u n d a r y c o n d i t i o n s o n S u . O n t h e

o t h e r h an d , s u p e r i o r b a r s i n d i ca t e eq u i l i b r i u m s t r e s ss t a t e s # an d s u r f ace t r ac t i o n s f lu w h i ch s t a t i s f y s t a t ic

c o m p a t i b i l it y w i th t he b o d y f o rc e s l a n d t h e s u r fa c e

t r a c t i o n s P o . G i v e n th e r e l e v a n t s t r e s s - s t r a i n r a t e r e -

l a t i o n s h i p t h i s p r i n c i p l e r en d e r s t h e g o v e r n i n g f ie l d

e q u a t i o n s f o r q u a s i s ta t ic m o t i o n s , i f a p p r o p r i a t e i n i ti a l

co n d i t i o n s a r e s u p p l i ed . N o t e t h a t t h e t i me d e r i v a t i v e s

can ce l i n t h e ca s e t h a t t h e v i r t u a l w o r k ex p r e s s i o n r e -

m a i n s h o m o g e n e o u s i n t h e r a t e t e r m s y i e ld i n g d i r e c tl y

t h e w e l l - k n o w n i n c r e m e n t a l f o r m u l a t i o n .

E q . ( 1 ) may b e s p ec i a l i s ed t o t h e p r i n c i p l e s o f

v i r t u a l d i s p l aceme n t s an d v i r t u a l s tr e s s r e s p ec t i v e l y ,t h e f i r s t o f w h i ch s e r v es a s b a s is f o r t h e d i s p l acem en t

mo d e l s co n s i d e r ed b e l o w . I n t h i s c a s e w e h av e

T h e d i s p l a c e m e n t i n t e r p o l a t io n to f o r m s t h e b a si s f o rt h e s p a t ia l a p p r o x i m a t i o n o f t h e d i s p l a c e m e n t f ie ld

w i t h I ) d e n o t i n g t h e c o l u m n v e c t o r o f n o d a l v e l o c i t y

v a l u es . T h e V s y mb o l d e s c r i b e s t h e w e l l - k n o w n l i n ea r

d i f f e r en t i a l o p e r a t o r r e l a t i n g s t r a i n s t o d i s p l acemen t s .

N o t e t h a t o u r d i s c u ss i on i s r e s t ri c t e d t o p r o b l e m s o f

s m a l l d i s p l a c e m e n t s a n d d e f o r m a t i o n s f o r w h i c h t h e

s t r a i n - d i s p l a c e m e n t r e l a t i o n sh i p s r e m a i n l in e a r .

Fo r t h e t i me b e i n g t h e ma t e r i a l b eh av i o u r i s ch a r ac -

t e ri s e d b y a h y p o e l a s t i c f o r m u l a t i o n i n w h i c h E T d e -

s c r i b e s t h e i n s t an t a n eo u s s t a t e o f t h e s o l i d r e l a t in g

s t re s s - an d s t r a i n r a t e s i n a l i n ea r man n e r . T h e i n i ti a ls tr e ss a n d s t r a in c o n c e p t , i n t r o d u c e d w i t h i n t h e c o n -

t ex t o f s t r u c t u r a l an a l y s i s i n [1 ] an d e x t en d ed i n t h e

r e fe r e n ce s [ 4 - 1 2 ] , p r o v id e s an i de a l m e c h a n i s m t o

a c c o u n t f o r n o n - e la s t ic d e f o r m a t i o n s a r i si ng f r o m e n -

v i r o n m e n t a l c o n d i t io n s , s u c h a s t e m p e r a t u r e , m o i s t u r e ,

i r r ad i a t i o n , a s w e l l a s me ch an i ca l co n d i t i o n s , s u ch a s

p l a s t ic i t y a n d c r e e p . U s i n g t h e f u n d a m e n t a l l a w o f d e-

c o m p o s i t i o n , t h e t o t a l s t r a in r a t e m a y b e d i v i d e d i n to

e l a s t ic , s t re s s p r o d u c i n g s t r a i n r a t e s ~ an d i n t o i n i ti a l

s t ra i n r a t e s / I

T o t a l s t r a i n r a t e

"Y = i + / I ; ( 6 )

f s : t t h d V = f sc,dV+ f 6 i t t [ 9 o d S .V V S v

( 2 )

* Bold founts denote v ectors and m atrices, dots indicate timederivatives and the superscript t stands for the transpose o f amatrix or a vector.

E x a m p l e s f o r i n i ti a l s t r a in r a t e s



J. H. Argyris et al., Non.linear methods o f structural analysis 171

~ = ~ p

%

( n o n l i mu ~- p l a s t i c - s t r a i n r a t e ) +

(creep s t r a in r a t e ) +

( t h e r m a l d i l a t a ti o n r a t e ) +

( m o i s t u r e i n d u c e d d i l a t a t io n r a t e ) +

S u b s e q u e n t l y , t h e s e i n d i v id u a l e l e m e n t q u a n t i t i e s a r e

a s s e m b l e d , u s in g t h e b o o l e a n c o n n e c t i v i ty m a t r i x , i n t o

t h e d e s i r ed q u as i s t a ti c eq u i l i b r i u m eq u a t i o n s o f t h e

c o m p l e t e s t r u c t u r e

K( r , t ) ~ ( t ) = R ( t ) + R j ( t ) . ( 1 1 )

/ l i n ( i r r ad i a t i o n i n d u ce d d i l a t a t i o n r a t e ) . 3. Nonlinear solut ion me thods

W i t h in t h e f r a m e o f t h e g e n e r a li se d p r i n c i p le o f v i r tu a l

w o r k a l l m e c h a n i c a l c o n s t r a i n t s a r e t e m p o r a r i l y d is -

r e g a r d e d t o d e t e r m i n e t h e i n i ti a l l o a d s . E n v i r o n m e n t a l

i n it ia l d e f o r m a t i o n s s h o u l d b e c o n s i d e r e d p r e s c r ib e d

w h i l e i n s t an t an eo u s p l a s t i c s t r a i n s s h o u l d b e d e r i v ed

f r o m t h e t o t a l s t r a i n r a t e f i e l d an d t h e c r eep s t r a i n s

f r o m t h e g o v e r n i n g s t r e s s f i e l d . I t s h o u l d b e p o i n t edo u t t h a t f o r h i g h e r o r d e r e l e m e n t s , i n w h i c h t h e

s t re s s es v a r y w i t h in t h e e l e m e n t d o m a i n , c e r t a in s u b t l e

q u e s t io n s a r i se w i t h r e s p e c t t o t h e a p p r o p r i a t e r e p r e -

s e n t a t i o n o f " t a n g e n t i a l " m a t e r i a l la w s o r a n a l o g o u s

i n i ti a l s t r e ss o r s t r a i n d i s t r i b u t i o n s . Fo r a b r i e f d is cu s -

s io n o f t h e s e a p p r o x i m a t i o n p r o b l e m s th e r e a d e r i s

r e f e r r e d t o [ 1 6 ] .

U s i n g t h e f i n i t e e l emen t d i s c r e t i s a t i o n o n e o b t a i n s

v i a t h e v i r t u a l w o r k e x p r e s s io n t h e w e l l - k n o w n q u a si -

s t a t i c eq u i l i b r i u m e q u a t i o n s f o r a t y p i ca l f i n i t e e le -

m e n t

k ( p , t ) p ( t ) = P ( t ) + ) ( t ) , ( 7 )

w h e r e t h e e l e m e n t q u a n t i t i e s a r e d e f i n e d a s fo l lo w s

S t i f f n e s s

k = f v t o t E T V t O d V ,

V

N o d a l f o r c e s

( 8 )

e : f , ' f d 7 ÷ f , 'p a s ,V S

I n i t i a l l o ad s

v J = f v ,o ' r n dV = - f w ° t , d 7 .V V

(9 )

(lO)

T h e q u e s t io n s o f e x i s te n c e a n d u n i q u e n e s s h a v e

s t i m u l a t e d n u m e r o u s d i s c u s si o n s a m o n g t h e o r e t i c i a n s .

Wh i l e t h e ex i s t en ce i s n o r ma l l y a s s u r ed o n p h y s i ca l

g r o u n d s t h e u n i q u e n e s s o f t h e s o l u t io n c a n b e p r o v e n

but in spec ia l cases , e .g . fo r e l as t i c so l ids wi th a pos i -

t i v e d e f i n i t e s t r a i n en e r g y f u n c t i o n [ 1 3 ] . O n t h e o t h e r

h an d , i f o u r r e s u l t s d o s a t i sf y s t a t ic an d k i n em a t i cco mp a t i b i l i t i e s an d t h e r e l ev an t co n s t i t u t i v e r e l a t i o n -

s h ip s w e h a v e o b t a i n e d a s o l u ti o n w h i c h d e p e n d s f o r

d i ss ip a ti v e p r o c e ss e s o n t h e p a t h o f e v o l u t i o n . N o t e ,

h o w e v e r , th a t f o r g e o m e t r i c a ll y n o n l i n e a r p r o b l e m s

t h e s o l u t io n m a y b e n o n - u n i q u e .

B e f o r e r e v i e w i n g d i f f e r e n t n o n l i n e a r s o l u t io n t e c h -

n i q u es w e d i s cu s s b r i e f l y t w o r ecu r r i n g s u b p r o b l ems ,

t h e e v a l u a t io n o f t h e c u r r e n t f u n c t i o n v a l u e a n d t h e

a s s o c ia t e d f u n c t i o n g r a d i e n t. T h e f ir s t p r o b l e m c o n -

ce r n s t h e d e t e r m i n a t i o n o f eq u i l ib r a t i n g f o r ce s w h i le

t h e s eco n d d ea l s w i t h t h e l i n ea r i s a ti o n o f t h e s t r u c -t u r a l r e s p o n s e f o r a g iv en s t a t e o f d i s p l aceme n t s . F i g .

1 s h o w s t h e g e o m e t r i c i n t e r p r e t a t i o n o f t h e f u n c t i o n

a n d f u n c t i o n g r a d i e n t f o r n o n l i n e a r s t r u c t u r a l r e s p o n s e

c o n d i t i o n s ; n o t e t h a t t h e m u l t i d im e n s i o n a l c o n f i g u ra -

t i o n s p ace o f t h e i d ea l i s ed s t r u c t u r e i s r ep r e s en t ed b y

a h y p e r s u r f a c e .

F o r a n a n a l y t i c f o r m u l a t i o n o f t h e n o n l i n e a r r e -

s p o n s e e q u a t i o n s t h e f u n c t i o n v a l u e s R ( r ) c a n b e

ev a l u a t ed b y d i r ec t s u b s t i t u t i o n . I n t h e ca s e o f s t r u c -

t u r a l p r o b l e m s it is m o r e c o n v e n i e n t t o t a k e t h e d e -

t o u r o v e r th e o u t - o f - b a la n c e f o r c e s R ~ ( r ) f o r a g iv e n

s t a te o f d e f o r m a t i o n . I n t h e c a se o f m a t e r i a l n o n -

l i n ea r it i e s t h e s e r e s i d u a l f o r ce s a r e ev a l u a t ed b e s t

us ing the c on ce p t o f in i t i a l s t resses o r s t ra ins , i l lus -

t r a t ed i n f ig . 2 f o r t h e u n i ax i a l c a s e . T h ey s e r v e a s i n -

p u t t o d e t e r m i n e t h e r e s i d u al lo a d s R ~ ( r ) o v e r t h e

m e c h a n i s m o f i n i ti a l lo a d s ( 1 0 ) .

T h e c o n c e p t o f t h e t a n g e n t ia l s t if f n e ss m a t r i x

p r o v i d es p h y s i ca l t r an s p a r en ce f o r t h e p i ecew i s e

l i n ea r i s a ti o n o f t h e n o n l i n ea r s t r u c t u r a l r e s p o n s e . Mo r e



172 J . 11. Ar ~,r i s e t a l . , Non-l inear met hod s o f s truc tural analysis

R 1

, 2 " !

i / [ I

f i r l . I

F i g . 1 . N o n l i n e a r s t r u c t u r a l r e s p on s e,

R [ r ) : E q u i l ib r i u m F o r c e s

R j~ ( } R e s i d u a l F o r c e s

K(r) : T a n g e n t i a l Stiffness

"7

t ic a l m a t h e m a t i c a l t r e a t m e n t o f n o n l in e a r e q u a t io n s

in severa l var i ab les the reade r i s re fer re d to [41 ] .

3 . 1. I n c r e m e n t a l m e t h o d s

A g r e a t n u m b e r o f n u m e r i c a l i n te g r a t io n s c h e m e s

h av e b een p r o p o s ed t o s o l v e co u p l ed f i r s t o r d e r d i ff e r en t i a l eq u a t i o n s s i mi l a r t o eq . ( 1 1 ) . T h e E u l e r -

Cau ch y a l g o r i t h m i s t h e s i mp l e s t an d b es t k n o w n

m e t h o d f o r c o n s tr u c t in g a s t e p b y s te p p r o c e d u r e

w i t h t h e h e l p o f a f o r w a r d d i f f e r e n c e s c h e m e i n c re -

m e n t i n g n o r m a l l y t h e p a r a m e t e r l oa d i n s te a d o f ti m e .

T h i s me t h o d , w h i ch i s k n o w n i n s t r u c t u r a l an a l y s i s

u n d e r t h e n am e o f t an g en t i a l s t if f n e s s , i s i l l u s tr a t ed i n

f ig . 3 . The fo l lowing a lgor i thm descr ibes th i s t echnique

f o r t h e i + 1 l o ad i n c r em en t

o " : E l a s t i c S l ~ e s s

r i , z : I n i l i a l $ f f a i n an d S l ; e s s

E 7 T a n g e n t i a l M a l e r i a l L a w

47

Fig. 2. Nonlinear stress-stra in relationship.

a b s t r a c t f o r m u l a t i o n s h a v e b e en p r o p o s e d t o c o n -

s t r u c t t h e f u n c t i o n g r ad i en t m a t r i x e .g . v i a p a r a m e t r i c

d i f f e r en t i a t i o n , b u t t h e s e r a t h e r o b s cu r e t h e p h y s i ca l

i n t e r p r e t a t i o n . T h e t an g en t i a l s t i f f n e s s K ( r ) i s de ter -

mi n ed f r o m eq . ( 8 ) v i a t h e t an g en t i a l ma t e r i a l l aw E T

c o r r e s p o n d i n g t o t h e c u r r e n t s t a t e o f d e f o r m a t i o n .

Bo t h f ig s. 1 an d 2 i l l u s tr a t e t h e p r o ced u r e o n t h e

co n s t i t u t i v e an d t h e s t r u c t u r a l l ev e l .

T w o a r e a s o f n u m e r i c a l a na l y si s p e r ta i n t o t h es o l u ti o n o f n o n li n e a r s t r u c tu r a l p r o b l e m s , m e t h o d s

f o r t h e s o l u t io n o f d i f f e r en t i a l eq u a t i o n s an d r o o t

f i n d in g s t e ch n i q u es . Bo t h t y p es o f p r o ced u r e s b a s e

t h e s o l u t i o n o n p i ecew i s e l in ea r i s a t io n o f t h e s t r u c -

t u r a l r e s p o n s e eq u a t i o n s , o n e u s i n g i n c r emen t a t i o n ,

t h e o t h e r i t e r a t i o n . E n e r g y s ea r ch t e ch n i q u es , n o r -

ma l l y u s ed f o r s t r u c t u r a l o p t i mi s a t i o n , h av e a l s o b een

p r o p o s ed , b u t t h ey h av e s o f a r p r o v en i n f e r i o r i n r e -

g a r d t o c o m p u t a t i o n a l e f f i c ie n c y [ 1 9 ] . F o r a th e o r e -

ri+ 1 = r i + r A ,

r~ x = K - l ( r i ) [Ri+ 1 - R i + R ~ ( r i ) ] .

0 2 )

N o t e , t h a t t h e r e s i d u a l t o ad v ec t o r R*_ ( r ) , d u e s p ec i -d

f i ca l t y t o n o n l i n ea r e f f ec t s , s h o u l d b e ad d ed t o each

l o a d in c r e m e n t i n o r d e r t o r e d u c e t h e a c c u m u l a t i o n

o f t r u n c a t i o n e r r o r s d u e t o l i n e a ri s a ti o n . T h e E u l e r -

Cau ch y a l g o r i t h m r eq u i r e s i n each l o ad i n g s t ep t h e

ev a l u a t i o n o f e l emen t a l s t i f f n e s se s , in i t ia l l o ad s , t h e i r

a s s e m b l y a n d t h e r e n e w e d d e c o m p o s i t i o n a n d b a c k -

s u b s t i t u t i o n o f t h e t an g en t i a l s t i ff n e s s ma t r i x . Mo r e -o v e r , t h e n ew s o l u t i o n o n l y s a t s if i e s eq u i l i b r i u m an d

co mp a t i b i l i t y b u t n o t t h e co n s t i t u t i v e l aw , w h i ch

co u l d b e co r r ec t ed f o r b y i t e r a t i o n o f t h e r e s i d u a l

f o r c e s. T h e c o n c e p t o f o u t o f b a l a n c e f o r c e s p r o v id e s

a n e x t r e m e l y u s e fu l m e a s u r e f o r t h e m a g n i t u d e o f e r -

/:?

. . . . . . . 4; . . . . . . . . I

R [ r . 1 ]

!

i i _r14 )

Fig. 3. Incremental m ethod w ith equilibrium corrections.



J. H. Argyris et aL, Non-linear me thod s o f structural analysis 173

r o r i n o u r s o l u ti o n [ 1 5 ] . I t s h o u l d b e m e n t i o n e d t h a t

t e c h n i q u e s f o r t h e p a r t ia l m o d i f i c a t i o n o f t h e s t r u c -

t u r a l s t i f f n e s s p r o v e v e r y e f f i c i en t f o r t r e a t i n g l o ca l i s ed

n o n l i n e a ri t ie s a s c o m p a r e d w i t h th e n e w f o r m a t i o n

a n d d e c o m p o s i t i o n o f t h e t a n g e n t ia l s t if f n e ss m a t r i x

a t e a c h l o a d i n c r e m e n t [ 1 7 ] . S u b s t r u c t u r i n g f u r t h e rr e d u c e s t h e c o m p u t a t i o n a l e f f o r t if n o n l i n e a r r e g i o ns

can b e i d en t i f i ed a p r i o r i in t h e s t r u c t u r e , e . g. in t h e

cas e o f l o ca l is ed p l a s t i c i t y n ea r s i n g u l a r p o i n t s , s u ch

as c r ack t i p s , e t c .

V a r i o us i m p r o v e m e n t s o f th i s s te p b y s te p m e t h o d

h a v e b e e n p r o p o s e d t o r e d u c e t h e o r d e r o f t r u n c a t i o n

e r r o r i n t h e E u l e r - C a u c h y a l g o r i t h m . A s i m p l e e x t e n -

s io n is b a s e d o n e x t r a p o l a t i o n t o t h e m i d s t e p o f t h e

n e w l o a d i n c r e m e n t w h e r e t h e r e l e v a n t s e c a n t st if f n e s s

i s e s t i ma t ed . E x t r eme cau t i o n i s c a l l ed f o r , a s d i s co n -

t i n u it i es i n t h e s t r e s s - s t r a i n l a w a n d t h e m u l t i d im e n -s i on a l b e h a v i o u r o f t h e s t r u c t u ra l r e s p o n s e e n d a n g e r

t h e s e e s t i ma t e s . N o t e t h a t t h e ab o v e s i n g l e s t ep a l g o -

r i t h m s m a y b e i n t e r p r e t e d a s s i m p l e c a s es o f t h e

N e w t o n - C o t e ' s i n t e g r a ti o n f o r m u l a e o f th e o p e n a n d

c l o s ed t y p e .

O t h e r t e ch n i q u es a r e av a i lab l e i n v o l v in g th e mu l t i -

p l e e v a lu a t i o n o f f u n c t i o n v a l u e s a n d f u n c t i o n g r a d i-

e n t s w i t h i n e a c h l o a d s t e p . T h e b e s t k n o w n m e t h o d s

a r e th e i m p r o v e d a n d t h e m o d i f i e d E u l e r te c h n i q u e s ,

b o t h r e q u i r e t w o e v a l u a t io n s o f t h e t a n g e n t i a l s ti f f-

n e s s to o b t a i n a b e t t e r e s t i m a t e f o r t h e g o v e r n i n g se -can t s t i f f n e s s w i t h i n each l o ad i n c r emen t . So p h i s t i -

c a t e d p r e d i c t o r - c o r r e c t o r a l g o r i t h m s o r R u n g e -

K u t t a t e c h n i q u e s h a v e b e e n p r o p o s e d i n a n a l o g y t o

t h e m o r e r e f i n e d n u m e r i c a l in t e g r a t i o n s c h e m e s f o r

t h e s o l u t i o n o f o r d i n a r y d i f f e r e n t ia l e q u a t i o n s ; b u t

t h e g r o s s l y en l a r g ed s t o r ag e r eq u i r em en t s an d t h e

m u l t i p l e e v a l u a t io n s o f f u n c t i o n s a n d f u n c t i o n g r ad i -

e n t s h a r d l y j u s t i f y t h e i m p r o v e m e n t in a c c u r a c y . I t is

m o r e t h a n d o u b t f u l t h a t t h e l o a d s t e p s c a n b e e x -

t e n d e d i n r e la t i o n t o t h e o r d e r o f th e i n t e g r a t i o n

m e t h o d i f w e c o n s i d e r t h e t i g h t s m o o t h n e s s r e q u ir e -

m e n t s n e c e s s a r y f o r s t a b i l it y . M o r e o v e r , e s t i m a t e s o f

t r u n ca t i o n e r r o r s , w h i ch a r e w e l l e s t ab l i s h ed f o r f u n c -

t i o n s w i t h s i n g l e v a r i ab l e s , p r o v i d e n o t h i n g b u t u p p e r

b o u n d s w h i c h a r e n o r m a l l y q u i t e c o n s e r v a t i v e ; t h is i s

p a r t i c u l a rl y t r u e f o r t h e l o w e r o r d e r m e t h o d s , w h i c h

a r e , i n g en e r a l , mo r e e f f i c i en t i n r eg a r d t o b o t h , s t o r -

a g e r e q u i r e m e n t s a n d c o m p u t a t i o n a l e f f i c i e n c y .

3 . 2 . I t e r a t i v e m e t h o d s

Wi t h i n t h e co n t ex t o f n o n l i n ea r s t r u c t u r a l an a l y s i s

t w o t y p e s o f r o o t f i n d in g t e c h n i q u e s h a v e e m e r g e d ,

t h e N e w t o n - R a p h s o n ' s m e t h o d a n d i ts m o d i fi c a ti o n ,

i n w h i ch t h e i t e r a t i o n i s c a r r i ed o u t w i t h o u t u p d a t i n g

t h e f u n c t i o n g r a d i e n t. T r u n c a t i n g t h e T a y l o r - s e ri e s e x -p an s i o n a f t e r t h e l i n ea r t e r m l ead s t o t h e f o l l o w i n g

i t e ra t iv e a l g o r i t h m w h i c h m a y b e w r i t te n i n e it h e r o f

t h e f o l l o w i n g fo r m s

ri+ 1 = r i + r& ,

N e w t o n - R a p h s o n

/l

r zx = K - 1 ( r i ) ( R i + I - R i ) + ~ K - l ( r / ) R ~ ( r l ) ,

/= 1( 1 3 )

M o d i f i e d N e w t o n - R a p h s o n

n

r A = K o 1 ( R i ÷ 1 - R i ) + ~ g ~ 1 R ; ( r j ) ,/=1 v

( 1 4 )

w h e r e i d e n o t e s t h e s t e p o f i n c r e m e n t a t i o n a n d ] t h e

cu r r en t cy c l e o f i t e r a t i o n ( f i g . 4 ) . T h e r e s i d u a l l o ad

v ec t o r R ~ ( r ) is ev a l u a t ed a s b e f o r e v i a i n i ti a l s t r e s s e s

o r s t r ai n s a c c o u n t i n g f o r t h e o u t - o f - b a l a n c e f o r c e s a t

each s t ag e o f i t e r a t i o n . E ach i t e r a t i o n cy c l e can b e

e i t h e r c a r ri e d o u t w i th a n e w , u p d a t e d , t a n g e n t ia l

s t i f fness K ( r ) o r w i t h t h e o r i g in a l s t i f f n e s K o . T h e

f ir st p ro c e d u r e i s o f t he N e w t o n - R a p h s o n t y p e , in -

v o l v in g at e a c h s t e p o f i te r a t i o n t h e e v a l u a t i o n o f t h e

r e s id u a l lo a d v e c t o r R ~ ( r ) , t h e d e t e r m i n a t i o n o f t h e

t an g en t i a l s t i f f n e s s , i t s t r i an g u l a r i s a t i o n an d t h e s u b -

s e q u e n t b a c k s u b s t it u t io n . T h e m o d i f i e d N e w t o n -

R a p h s o n m e t h o d r e q ui re s o n l y th e r e p e a t e d c o m p u t a -

t i o n o f t h e o u t - o f - b a l a n c e fo r c e s to g e t h e r w i t h t h e a s -

s o c i a t ed b ack s u b s t i t u t i o n w h i l e r e t a i n i n g t h e o r i g i n a l

s t i f fness g o in t r i angu lar i sed fo rm. In th i s case the

r a t e o f c o n v e r g e n c e m a y b e a c c e l e r a t e d c o n s i d e r a b l y

b y o v e r r e l a x a t i o n s c h e m e s w h i c h a c c o u n t a p p r o x i -

m a t e l y f o r t h e c h a n g e o f t a n g e n t i a l s t if f n e s s w h e n

d e t e r mi n i n g t h e i n i t i a l l o ad s . N o t e t h a t t h e mo d i f i ed

N e w t o n - R a p h s o n t e c h n iq u e i s a n a lo g o u s to i t e ra t io n

b y s u cces s i v e s u b s t i t u t i o n . T w o q u es t i o n s a l w ay s a r i s e

i n t h e c o n t e x t o f i te r a t iv e s c h e m e s , t h e r a n g e a n d t h e

r a t e o f co n v e r g en ce . A s mo s t t e ch n i q u es a r e o n l y l o ca l -



174 J. H. Argyris e t aL, Non- l inear methods o f s t ructura l analysis

R ~

/ ~ - . . - " / I ~ " , , I ,

R i / ~ l R ( r i - 0

f i f i f l + l

Fig. 4. Ini tial load i terat ion, Ne wton -Ra phso n scheme.

s o l u ti o n a l g o r i t h m s h o u l d c o m b i n e t h e a d v a n t a g e s o f

t h e i n c r e m e n t a l j st ep b y s t e p m e t h o d s w i t h t h o s e o f

t h e i n i t i a l l o ad i t e r a t i o n t e ch n i q u es i n o r d e r t o p r o -

v i de t h e m e a n s f o r r e d u c in g t h e o u t o f b a la n c e f o r c e s

t o a p r e s e t a c c u r a c y . A t o n e e n d o f t h e s p e c t r u m t h is

p r o ced u r e w o u l d d eg en e r a t e t o t h e t an g en t i a l s t i f f n e s sm e t h o d i n it s si m p l e s t f o r m ( i n c r e m e n t a l l o a d s c h e m e

w i t h o u t r e s i d u a l l o ad co r r ec t i o n s ) . O n t h e o t h e r en d

i t w o u l d y i e l d t h e d i r ec t l o ad i t e r a t i o n t e ch n i q u e w i t h -

o u t i n c r emen t a t i o n ( i n i t i a l l o ad i t e r a t i o n w i t h o r w i t h -

o u t u p d a t i n g o f t h e f u n c t i o n g r a d i e n t) .

4 . N o n l i n e a r d e f o r m a t i o n b e h a v io u r o f c o n c r e t e

l y c o n v e r g e n t t h e s u c c e ss o f t h e m e t h o d a l w a y s de -

p en d s o n t h e i n i ti a l g u ess . A d e t a i led d i s cu s s i o n o ft h e s e p r o b l e m s i s g i v en i n r e f . [ 1 2 ] w i t h i n t h e co n t e x t

o f in i t ia l l o ad t e ch n i q u es f o r t h e s o l u t i o n o f e l a s t o -

p l a s ti c p r o b l e m s .

I t s h o u l d b e m e n t i o n e d t h a t n o i n c r e m e n t a l p r o c e -

d u r e is r eq u i r ed f o r p a t h i n d ep en d en t r ev e r s i b le p r o -

ce s s es ; h en c e , t h e t o t a l l o ad can b e ap p l i ed d i r ec t l y

f o r p r o b l ems i n n o n l i n ea r e l a s t i c i t y , o r f o r t h e d e f o r -

m a t i o n t h e o r y o f p l a s ti c i ty , o r in t h e f i e l d o f t h e r m o -

e l a s ti c it y w i t h a t e m p e r a t u r e s e n s it iv e m e d i u m [ 1 6 ] .

A s l o n g a s t h e r a n g e o f c o n v e r g e n c e is n o t e x c e e d e d ,

t h e i t e r a t i v e p r o ces s co n v e r g es t o a s o l u t i o n w h i chs a ti s fi e s e q u i l ib r i u m , c o m p a t i b i l i t y a n d t h e s t r e s s -

s t r a i n r e l a t i o n s t o p r e s e t a ccu r acy . T h e i t e r a t i o n h i s -

t o r y a n d t h e t y p e o f c o n v e r g e n c e c r i te r i o n i s o f n o i n -

f l u en ce b ecau s e t h e s o l u t i o n i s u n i q u e i f t h e s t r a i n

en e r g y d en s i t y r ema i n s p o s i t i v e d e f i n i t e . I n co n t r a s t ,

d i s s i p a t i v e p r o ces s e s r eq u i r e i n c r emen t a l p r o ced u r e s t o

m i n i m i s e t h e d e v i a t i o n f r o m t h e t r u e p a t h d u r i n g t h e

e v o l u t i o n o f t h e p r o b l e m . I n t h i s c a se p a r t ic u l a r c a r e

h as t o b e ex e r c i s ed i n r eg a r d t o t h e co n v e r g en ce c r i t e r -

i o n an d t h e ch o i ce o f t h e s t ep s i ze . S t r i c t l y , f o r d e t e r -

mi n i n g a s o l u t i o n i t i s n o t s u f f i c i en t t o r ed u ce e . g . av e c t o r n o r m o f t h e r e s i d ua l l oa d s t o a p r e s e t a c c u r a c y ;

i t i s r a t h e r n eces s a r y t o s eek s a t i s f ac t o r y co n v e r g en ce

o f t h e co mp o n en t s o f i n i t i a l s t r e s s e s o r s t r a i n s . O n t h e

o t h e r h a n d , t h e r e i s n o g u a r a n t e e t h a t o u r s o l u t i o n is

t h e c o r r e c t o n e i f t h e p a t h h i s t o r y h a s n o t b e e n a p -

p r o x i m a t e d i n t h e l i n e a r is a t io n p r o c e s s b y s u f f i c i e n t ly

s ma l l i n c r emen t s ; h en ce i n p r ac t i c e l e s s s t r i n g en t co n -

v e r g e n c e c ri te r i a a r e m o r e a p p r o p r i a t e .

I n co n c l u s i o n o n e can s ay t h a t a f l ex i b l e n o n l i n ea r

I n t h e f o l lo w i n g , c e r ta i n c o n c e p t s o f n o n l in e a r

e l a s ti c s o l id s a r e b r i e f l y r ev i ew ed . Fo r a d e t a i l ed d i s-cu s s i o n t h e r ead e r i s a l s o r e f e r r ed t o [ 1 8 - 2 0 ] ; a t t en -

t i o n is d r a w n t o t h e a n a l o g y w i t h t h e d e f o r m a t i o n

t h e o r y o f p l a s t ic i t y . T h e se i d e a s ar e a p p l i e d t o m o d e l

t h e n o n l i n e a r d e f o r m a t i o n a n d c l e a v a ge b e h a v i o u r o f

co n c r e t e t y p e ma t e r i a l s u n d e r t r i ax i a l co n d i t i o n s . T h i s

ch a r ac t e r i s a t i o n o f t h e ma t e r i a l b eh av i o u r i s f i n a l l y

u s ed t o an a l y s e a cy l i n d r i ca l s p ec i men s u b j ec t ed t o

a x i al c o m p r e s s i o n .

4 . 1 . N o n l i n e a r e l a s t i c s o l id s

A s o l i d i s s a id t o b e e l a s t i c i f t h e s t r a i n en e r g y f o ra s ta t e o f d e f o r m a t i o n i s i n d e p e n d e n t o f h o w t h is s t a te

i s r each e d , i . e. i f t h e r e ex i s t s an e l a s ti c p o t en t i a l U

w h i c h is i n d e p e n d e n t o f t h e d e f o r m a t i o n p a t h . T h i s

p o t e n t i a l f u n c t i o n c a n t h e n b e u s e d t o d e s c r ib e t h e

e l a s t i c co n s t i t u t i o n a s f o l l o w s

~ U* = - - ( i s )~ ¢ '

w h e r e

u = u ( c ) .

Fo r i s o t r o p i c s o l i d s t h e ma t e r i a l p r o p e r t i e s a r e i n d e -

p e n d e n t o f th e o r i e n t a t i o n . C o n s e q u e n t l y t h e e l a s ti c

p o t en t i a l i s t h en f u l l y d e s c r i b ed b y t h e t h r ee i n v a r i -

a n t s o f t h e s t r a in t e n s o r , h e n c e t h e s t r e s s - s t r a i n l a w

can b e w r i t t en i n t h e f o l l o w i n g f o r m



Z 1-1.Argyris e t a l ., Non-l inear me thods o f s t ruc tural analysis 175

a Uc r = ~ - ~ 1 e 3 , 3

w h e r e

a u a u

+ 2 0 -~ 2 t + 3 ~ 3 t t t , ( 1 6 )c o n c e p t s d e g e n e r a t e t o t h e w e l l -k n o w n m a t e r ia l r e la -

t i o ns f o r i so t r o p ic so l ids

o o = 3 K e o , a D = 2 G E D . ( 2 0 )

U = U ( S l , J 2 , J 3 ) a nd e 3 , 3 = { 1 1 1 0 0 0 } .

No te t h a t t h e r e a r e two so u r ce s f o r no n l inea r i t i e s i n

t h e s t r e s s - s t r a i n l a w , t h e f u n d a m e n t a l f o r m o f th e

e l a s t i c p o ten t i a l and th e qu ad r a t i c ex p r e ss io n in t . I t

i s o f t e n p o ss ib l e t o de sc r ib e th e b eh a v io u r o f r ea l

m a t e r ia l s b y p o t e n t i a l s w h i c h a r e o n l y f u n c t i o n s o f

th e f i rs t two s t r a in inv a r ian t s . I n t h e se ca se s we o b -

t a in t h e f o l lo wing ex p r e ss io n in wh ich th e no n l inea r i t y

a r ise s so l e ly du e to t h e f o r m o f t h e e l a s t ic p o ten t i a l

a u a u* = O ) 1 e3,3 + 2 ~ 2 t , ( 1 7 )

w h e r e

U = U ( J 1 , J2)"

D e c o m p o s i n g t h e s t r a in i n t o h y d r o s t a t i c a n d d e v ia -

t o r i c c o m p o n e n t s a n d a s s u m in g t h a t t h e r e i s n o c o u p l -

i n g b e t w e e n b u l k a n d s h e a r d e f o r m a t i o n , t h e s t r e s s -

s t r a in r e l a t i o nsh ip s im p l i f ie s f u r th e r t o

( a U 1 aU 2. s , )

a v2+ 2 D, (18 )

w h e r e

v = U l ( J 1 ) + u 2 ( 7 2 )

a n d 7 2 d e n o t e s t h e s e c o n d i n v a r ia n t o f t h e d e v i a t o r i c

s t r a in t enso r ED. ( T h e su b sc r ip t D deno te s dev ia to r i c

c o m p o n e n t s ) . T h e e l as t ic m a t e r i a l p r o p e r ti e s a r e n o w

s p e c i fi e d b y t w o i n d e p e n d e n t f u n c t i o n s o f t h e e l a st ic

p o t e n t i a l w h i c h c a n b e d e t e r m i n e d d i r e c t l y f r o m t h e

v o l u m e t r i c a n d d i s t o r ti o n a l d e f o r m a t i o n b e h a v i o u r v ia

t ang en t i a l b u lk and sh ea r m o du l i

a u , a u 2

- K T , r o = - f f T - 3 " o = G T 3 " o . ( 1 9 )° o - a j 1

o # 2

% , r o deno te t h e o c t ah ed r a l s t r e s se s and e o, 3"0 t h e

o c tah e d r a l s t r a in s . I n t h e ca se o f l i nea r e l a s t ic i t y , t h e se

4 .2 . D e s c r i p t i o n o f n o n l in e a r d e f o r m a t i o n b e h a v i o u rB a s e d o n t h e p r e v i o u s c o n c e p t s w e a t t e m p t n o w

t o c h a r ac t e ri s e t h e d e f o r m a t i o n b e h a v i o u r o f c o n c r e t e

ty p e m a te r i a l s . T h e r e i s v i r t u a l ly no ex p e r im e n ta l evi -

dence av a il ab le f o r t h e n o n l inea r b eh av io u r o f co n -

c r e t e u n d e r t r i a x ia l c o n d i t i o n s , h e n c e o u r f o r m u l a t i o n

h a d t o b e b a s e d o n t r i a x ia l t e s t d a ta o f m o r t a r c u b e s

[ 2 1 ] .

T h e ex p e r im en ta l r e su l t s c l ea r ly ind ica t e t h a t t h e

s h e a r a n d b u l k d e f o r m a t i o n d o n o t u n c o u p l e , h e n c e

t h e b u l k a n d s h e a r m o d u l i a r e a f u n c t i o n o f v o l u m e t r ic

a s we ll as d i s to r t i o na l de f o r m a t io n . Mo r eo v e r , twop h e n o m e n a c a n b e o b s e r v e d w h i c h e x c e e d t h e s c o p e

o f e l a s ti c b eh av io u r . T h e m o r t a r sam p le s ex h ib i t h y s -

t e r e si s e f f ec t s u n de r cy c l i c h y d r o s t a t i c p r e ssu r e , wh ich

a r e i l l u s t r a ted in fi g. 5 ; t h ~ a l~o sh o w a p r o no u n ced

l imi ta t ion in shear res is ta r lce mainly as funct ion of

h y d r o s t a t i c p r e s s u r e , s ee f i g. 6. B o t h p h e n o m e n a m a y

b e a c c o u n t e d f o r i n t h e t a n g e n ti a l b u lk a n d s h e a r

m o du l i u sed to de sc r ib e th e s t r e s s - s t r a in r e l a t i o nsh ip

i n a n i n c r e m e n t a l f o r m

~ = ( 3 K T + 2 G T ) e oa e 3 ,3 + 2 G T ~ D Z x , ( 2 1 )

wh e r e th e t ang en t i a l m a te r i a l p r o p e r t i e s a r e de f ined b y

B u l k m o d u l u s

K r = K r (eo , k ) = o 0 J 3 e o a , ( 2 2 )

S h e a r m o d u l u s

a T = a T (3 '0 , ° o , rma x) = ( % : , / 3 ' o , , ) f ( % , rmax)"

( 2 2 )

T h e lo ad ing p a r am e te r k de sc r ib e s t h e h y s t e r e s i s e f-

f e c t s s im p l y b y k e e p i n g tr a c k o f t h e p a t h o f l o a d in g

w h i l e th e p a r a m e t e r r m ax d e f i n e s t h e m a x i m u m s h e a r

r e s i s t ance o f t h e m a te r i a l a s f u nc t io n o f t h e o c t ah ed r a l

no r m a l and o c t ah ed r a l sh ea r s t r e s s .

4 .3 . E x a m p l e

T o i l l u s t r a t e t h e p r ev io u s co ncep t s a cy l ind r i ca l

t e s t sp ec im en is ana ly sed wh ich i s su b jec t ed to an in -



1 7 6 J . H . A r g y r i s e t a l. , N o n - l in e a r m e th o d s o f s t r u e tu r a l a n a ly s i s

1 0 0 0

i G ' I

_ . G" - -ew

E~ 1 = G ' 2 = G ' 3

o~ 6 0 0 . . . . . . . . .. . . . . T . . . . .

4 0 0 _ _

A ' . # ~ , ; ; ' / , < ' / ," 7 , " 1 " /

0

/ . / , , , / / ~ . / / , ' . ' ,; , ' ~ " i . . . i.' , ' - , ,

, , / _ _ _ , , , , f , . / / .~ 1 ' , " i " i - l . ' , ,

' i I 1 . ,' " J . / "5 0 0 0 I 0 0 0 0 1 5 0 0 0

t. O C t ( x l 0 - 6 )

F i g. 5 . S t r e s s - s t r a i n r e l a t i o n s h i p u n d e r h y d r o s t a t i c s t r e s s s t a t e .

t j

E 2 0 0 ~ , £ ~ 3 0 " i

f f c c t t k m 2 I

2 6 0 0

o I I0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 1 0 0 0 0

/ " o ~ ( ~ l o - 6 )

F i g . 6 . O c t a h e d r a l s h e a r r e l a t i o n s h i p f o r d i f f e r e n t h y d r o s t a t i c

p r e s s u r e s .

c rea s i n g ax i a l l o ad . F i g . 7 p re s en t s an o v e ra l l v i ew o f

t h e g e o m e t r y a n d t h e a x i s y m m e t r i c f i n i t e e l e m e n t

i d e al is a ti o n. F o r s y m m e t r y o n l y t h e u p p e r h a l f n e e d

b e c o n s i d e r e d . A t t h e u p p e r m o s t n o d e s , t h e v e r t i c a l

d i s p l a c e m e n t s a r e a s s u m e d t o b e i d e n t i c a l w h i l s t t h e

l a t e r a l d i s p l a c e m e n t s a r e s u p p r e s s e d a l t o g e t h e r . T h e s e

b o u n d a r y c o n d i t i o n s s h o u l d s im u l a t e t h e e f f e c ts o f

t h e l o a d p l a t e n s o n t h e s p e c i m e n u n d e r a c t u a l t e s t

c o n d i t i o n s .

B a s e d o n t h e t e s t d a t a o f r e f. [ 2 1 ] , t h e f o l l o w i n g

fo u r p o s s i b i l i t i e s a r e co n s i d e red t o ch a rac t e r i s e t h e

z i

/

/ /

/ /

/ /

: / /// /

/ /

D = G i n

5 6 T R I A X E l e m e n t s 1 , . - -

2 3 8 D e g r e e s o f F r e e d o m

F i g . 7 . C y l in d r i c a l t e s t - s p e c i m e n , s t r u c t u r e a n d i d e a l i s a t i o n .

m at e r i a l p ro p e r t i e s , s ee a l s o f i g s . 5 an d 6 . In t h e f i r s t

c a s e a s ta n d a r d l i n e a r a n a l y si s is c a r r ie d o u t i n w h i c h

t h e i n i t i a l m a t e r i a l p ro p e r t i e s d e s c r i b e t h e l i n ea r e l a s -t i c r e s p o n s e

K T = K o = 6 6 7 0 0 k p / c m 2 ,

G T = G o = 5 0 0 0 0 k p / c m 2 .

( 2 3 )

I n t h e s e c o n d c a s e t h e n o n l i n e a r s h e a r d e f o r m a t i o n

b e h a v i o u r is a c c o u n t e d f o r b y a s s u m i n g t h a t t h e s h e a r

m o d u l u s d e p e n d s o n l y o n t h e o c t a h e d r a l s h e a r s tr e ss ,

w h i l e t h e i n f l u e n c e o f t h e h y d r o s t a t i c s t a t e o f st r es s is

a s s u m e d t o r e m a i n n e g l i g ib l e [ 2 2 , 2 3 ]

G T = G O = 1 0 0 0 / ( - 0 . 0 2 5 + 0 . 0 0 0 5 t o )

fo r r o > 9 0 . (2 4 )

I n t h e t h i r d c a s e t h e n o n l i n e a r v o l u m e t r i c b e h a v i o u r

is a l s o c o n s i d e r e d , b u t t h e c o u p l i n g b e t w e e n b u l k a n d

s h e a r d e f o r m a t i o n is s ti ll o m i t t e d . H y s t e r e s i s e ff e c t s

n e e d n o t b e c o n s i d e r e d a s l o n g as t h e s p e c i m e n is



Z H. Argyris et aL, Non-linear methods of structural analysis 1 7 7

Pk p/ c rn 2

10 0 0

50 0

/ / / L I

/ / . .. . zx . . . . . . . . . . . . . . .

/

K = K o G = GO

L [ ~ _ _ _ K = K o G = G ( 'r )

- - . - - K = K ( % ) G = G ( r o )

. . . z~ . . . K = K ( G o ) G = G ( r o , r m a x )

0.61 0.6 2 0.63 0.6/., 0.6 5 ~zzz

P

kp/cm z

1000

50 0

f 11 '~ S

l / f / / / f f l A B <~- -F

t < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 .0 1 0 . 6 2 0 . 6 3 0 6 4 o . o s ~ , z

F i g .8 . C y l in d r ic a le s t - s p e c i m e n ,o a d - d e f o r m a t io n ia g ra m .

loaded monotonically

K o, 0 ~< tro ~< 400;

KT(Oo) = ~ K o, 400 ~< tro ~< 630; (25)

\-~K o, 630 ~< oo.

Finally, in the last case, the limited shear resistance isincorporated in addition to the other sources of non-

linearities in form of

~'max = 110 + 0.4 oo . (26)

The tangential stiffness method is applied to implement

these alternative material models for axial load in-

crements of pa = 50 kp/cm 2. To account for the vary-

ing stress distribution within each TRIAX 6 element a

linear interpolation scheme is used to describe the

spatial variation of the tangential material law which

is determined from the current state of stress. If the

state of stress exceeds the shear capacity of the materi~

al the tangential shear modulus is simply reduced to a

small value in order to avoid singularities.

Fig. 8 illustrates the load-deformation diagram for

the different material formulations. One observes that

each refinement in the material characterisation leads

to p ronounc ed differences in the structural response.

Figs. 9 and 10 shows the distribution of normalised

axial stresses for the four cases of material descriptions,

These results demonstrate very clearly the distortion

of the intended homogeneous stress distribution due

to nonlinear and frictional effects; hence, an appropri-

ate analysis of the test specimen should always be

carried out in conjunction with the construction of

nonlinear material models from test data. To this end,

the finite element method provides not just a con-

venient tool for stress analysis, but can also serve toidentify material properties from experimental results

[201.

5 . Creep o f concre t e

In this sect ion certain concepts of viscoelastic solids

are briefly reviewed as they form the basis of the time

variable behaviour of concrete. For a detailed discus-

sion the reader is referred to [24-26]. Subsequently,

these ideas are applied to the numerical solution of

concrete structures in the presence of creep, using the

effective modulus method, the rate o f creep method

and the method of superposition; the validity and short-

comings of these three theories are briefly discussed to-

gether with their numerical implementation. In con-

clusion the effects of these different interpretations o f

a "given" creep law are illustrated on two examples, a

thick-walled cylinder and a PCRV.



1 7 8 J . H . A r g y r i s e t a l ., N o n - l i n e a r m e t h o d s o f s t r c u t u r a l a n a l y s is

) 1 1 2/ 11

., / 1 0

/

io

K = K 0 O = GO

l o

/K = K ( a o) G = G(T " )

p = 5 0 0 k p / c m 2

-L-L_LII

~ //'°K = K o o = O %)

0 8I t , I . ~

1 2 I~oIO

)90 8

09

11

io

,oK = K ( ¢ o ) G = O ( v o , ' r m a x )

Fig . 9 . Cylindrica l te s t -sp ecim en, d is tribution of azz /p =

5 0 0 k p / c m 2 .

5 . 1 . V i s c o el a s t i c s o l i d s

I n g en e r a l, m a t e r ia l s p o s s e s s m e m o r y , t h a t m e a n s

t h a t a t a g iv e n in s t a n t o f t i m e t h e m a t e r ia l r e s p o n s e

i s a fu n c t i o n o f t h e c u r r e n t i n p u t a s w e l l a s t h e h i s -

t o r i e s o f i n p u t a n d r e s p o n s e . F o r s i m p l i c i t y a t t e n t i o nis n o w r e s t r ic t e d t o u n i a x ia l c o n d i t io n s ; t e m p e r a t u r e

a n d o t h e r e n v i r o n m e n t a l e f f e c t s a r e a l s o o m i t t e d a t

t h e m o m e n t . B a s e d o n t h e p r i n cip l e o f s u p e r p o s i ti o n ,

t h e c r e e p r e s p o n s e o f a n a g e i n g m a t e r i a l is d e s c r ib e d

b y t h e f o l l o w i n g l i n e a r i n t e g r a l e x p r e s s i o n u s i n g t h e

s t e p f o r m u l a t i o n

1 4 1I O l o

j ,

D9 f - ~ (i / I

//

K = K 0 G = G O

11

/K = K ( ¢ o ) G = G ( % )

1 or . . . . . - / ' o 9

¢ ' i

i 1 / 'l ,¢, !

/ , i/ :

/ i

J i

1 o ~ I o

K : K o G : G (r o)

p = 1 0 0 0 k p l c m 2

07 07

08~0910111

(/

K = K ( ~ o ) G = G('ro,r'max)

Fig . 10 . Cyl indrica l te s t -spe cim en, d is tribution of O zz/p =1 0 0 0 k p / c m 2 .

7 ( t ) = O ( t o ) I E ~ o ) + C ( t l , t O ) }

+ f E-(6+ ( 2 7 )

T h e t o t a l s t r a i n 3 '( t ) d e f i n e s t h e c u r r e n t s t a t e o f d e -

f o r m a t i o n , c ( t , r ) d e n o t e s t h e u n i t c r e e p f u n c t i o n , i . e .

t h e c r e e p s t r a i n a t t h e t i m e t d u e t o a u n i t s t r e s s i n p u t

a t t h e t i m e r , a n d E ( r ) d e s c r i b e s t h e t i m e v a r ia b l e

e l a s t i c r e s p o n s e o f t h e m a t e r i a l . F i g . 1 1 i ll u s t r a t e s t h e

s t e p f o r m u l a t i o n f o r d i s c r e t e t i m e s t e p s . I n t e g r a t in g



J . H . Argyr i s e t aL , Non- l inear m e tho ds o f s t ruc tura l analys is 17 9

I n p u t ,

t o t 1 t 2 t t i m e

R e s p o n s e

to t, t2

f ' 2 { ~ 2 + , c , , , 2 , }

~ , , [ - k ÷ , c , , , , } t

~°o { -~o+ ° l ,, ,o , }

t i m e

F ig . 1 1 . S t ep fo rm u la t i o n o f v isco e l as t i c so l i d , me th o d o f

su p erp o s i t i o n .

eq . ( 2 7 ) b y p a r t s l e ad s t o t h e f o l l o w i n g i mp u l s e f o r -

m u l a t i o n

_ o ( t ) t . o ( 1 }7 ( 0 E ( t ) f o ( r )~ - ~ r ~ ( r ) + C ( t ,' r ) d r , ( 2 S )

+t o

t o a c c o u n t f o r v a ri a b le e n v i r o n m e n t a l c o n d i t i o n s ,

s u c h a s t e m p e r a t u r e o r m o i s t u r e . T h e y a r e k n o w n a s

t h e d i r e c t f u n c t i o n a l d e p e n d e n c e a n d t h e " t i m e - s h i f t "

p r i n c i p l e . T h e l a t t e r s i mp l i f i e s t h e ma t e r i a l i d en t i f i c a -

t i o n t o t h e ch a r ac t e r i s a t i o n o f t h e b eh av i o u r a t e . g . a

r e f e r e n c e te m p e r a t u r e w h i l e a c c o m m o d a t i n g v a ri a b let e m p e r a t u r e c o n d i t io n s b y t r a n s f o r m a t i o n o f th e r e al

t i me ax i s [4 2 , 4 3 ] .

I t s h o u ld b e m e n t i o n e d t h a t t h e p r i n c ip l e o f s u p e r-

p o s i t io n a s s u m e s th a t t h e r e i s n o c o u p l i n g b e t w e e n t h e

s in g le r e s p o n s e b e h av i o u r s f r o m i n d i v i d u a l s t ep i n p u t s .

T h i s a s s u m p t i o n h o l d s r a t h e r w e l l f o r c o n c r e t e u n d e r

mo n o t o n i c l o ad i n g co n d i t i o n s a s l o n g a s t h e s t r e s s

s t a te s r e m a i n w i t h i n 5 0 % o f th e u l t i m a t e s t re s s [ 2 7 ] .

Bas ed o n r h eo l o g i ca l mo d e l s a l t e r n a t i v e d i f f e r en -

t ia l f o r m u l a t i o n s h a v e b e e n p r o p o s e d [ 2 5 ] . I n t h e

c a s e o f t i m e v a r i a b le p r o c e s s e s t h e y i n v o lv e d i f f e r e n -t i a l eq u a t i o n s w i t h v a r i ab l e co e f f i c i en t s an d h en ce

t e n d t o c o m p l i c a t e th e m a t e r i a l i d e n t i f ic a t i o n a n d t h e

r e s u l t i n g m a t h e m a t i c a l m o d e l .

5 . 2 . D e s c r i p t i o n o f c r e e p b e h a v io u rT o d a t e , t h r e e m e t h o d s a r e a v a il a b le t o d e s c r i b e

t i me v a r i ab l e p r o ces s e s o f q u as i s t a t i c n a t u r e , t h e e f -

f e c t iv e m o d u l u s m e t h o d , t h e r a te o f c r e e p m e t h o d a n d

t h e m e t h o d o f s u p e r p o s i ti o n . A ll t h r e e m e t h o d s y i e l d

i d en t i ca l r e s u l t s f o r s t re s s d i s t r i b u t i o n s i n v a r i an t w i t h

t i m e , b u t t h e y l e a d t o p r o n o u n c e d d i f f e r e n c e s u n d e rmo r e r ea l i s t i c co n d i t i o n s , s u ch a s PCRVs a r e s u b -

j e c t e d t o .

w h e r e t h e k e r n e l f u n c t i o n d e f i n e s t h e m e m o r y o f t h e

m a t e r i a l . A n a l o g o u s r e s p o n s e f u n c t i o n a l s c a n b e c o n -

s t r u c t e d t o d e s c r i b e t h e r e l a x a t i o n b e h a v i o u r o f t h e

s o l i d , b u t i n g en e r a l ex p e r i men t a l d a t a a r e av a i l ab l e

o n l y f r o m c r e e p t e s t s ; h e n c e t h e e x p r e s s i o n f o r s t r e ss

r e l a x a t i o n h a s t o b e d e t e r m i n e d n u m e r i c a l ly b y a

c o s t l y " i n v e r s i o n p r o c e d u r e " i n v o l v in g th e t i m e i n t e -

g r a t io n o f t h e t o t a l d e f o r m a t i o n h i s t o r y .T h e e x t e n s i o n t o m u l t ia x i a l c o n d i t io n s r e q u i r e s

t h e m u l t ip l e f o r m u l a t i o n o f r e s p o n s e f u n c t i o n a l s; f o r

e x a m p l e , t w o i n d e p e n d e n t i n te g r a l l a w s d e s c r ib e t h e

t i m e - d e p e n d e n t v o l u m e t r i c a n d s h e a r b e h a v i o u r o f a n

i s o t r o p i c so l id . F o r c o n c r e t e i t is c o m m o n l y a s s u m e d

t h a t Po i s s o n ' s r a t i o is t h e s ame f o r e l a s ti c an d c r eep

b eh a v i o u r ; i n t h is c a s e t h e u n i ax i a l t e s t r e s u l t s a r e

s u f f ic i e n t t o d e s c r i b e c r e e p a n d r e l a x a t i o n u n d e r t ri -

a x i a l c o n d i t i o n s . T w o m e t h o d s a r e c o m m o n l y a p p l i e d

5 . 2 . 1 . E f f e c t i v e m o d u l u s m e t h o d

T h i s m e t h o d f u r n is h e s a s i m p l e m e a n s t o o b t a i n

d i r e c t ly t h e m a t e r i a l r e s p o n s e f o r a g i ve n i n s t a n t o f

t i me . A n a l o g o u s t o ag e i n g e l a s t i c i t y , t h e c r eep b eh av i -

o u r i s i n co r p o r a t ed i n a t i me v a r i ab l e ma t e r i a l l aw , r e -

l a t i n g t h e t o t a l d e f o r ma t i o n 7 t o t h e e l a s t i c s t r e s s o

I n v e r s io n y i e ld s t h e d e s i r e d " e f f e c t i v e " m a t e r i a l l aw

o ( 0 = E e f r ( , t o ) 7 ( ) , ( 3 0 )

w i t h

E ( t )Eeff = 1 + ~o(t , t o )



1 8 0 ,L H . A r g y r i s e t a l ., N o n - l in e a r m e th o d s o f s t r u c tu r a l a n a ly s i s

I n p u td I o ' = I

I It o T t t i m ~

5 . 2 . 2 . R a t e o f c r e e p m e t h o d

I n c o n t r a d i s t in c t i o n t o t h e e f f e c t i v e m o d u l u s

m e t h o d , t h e r a t e o f c r e e p m e t h o d ( t i m e h a r d e n i n g

m e t h o d ) d e s c r i b es t h e c r e ep d e f o r m a t i o n s b y a n i nt e -

g r al e x p r e s si o n w h i c h a c c o u n t s f o r t h e v a r i a t i o n o f

s t r e ss e s w i t h t i me

Response c { r , tO} Rate of Creep

I I L - - - " - . . . . c o , ~t o " r t t i m e

F i g. 1 2 . E n g i n e e r i n g c r ee p f o r m u l a t i o n , r a t e o f c r e e p a n d e f -

f e c ti v e m o d u l u s m e t h o d .

a n d

= E ( t ) c ( t , t o ) .

T h i s ma t e r i a l l aw o n l y d e s c r i b e s i n s t an t an eo u s

p h e n o m e n a a n d d o e s n o t c o n s i d e r t h e h i s t o r y d e -

p e n d e n c e o f s tr e ss e s a n d d e f o r m a t i o n s . A f t e r r e m o v -

i ng t h e l o a d i n g t h i s f o r m u l a t i o n a l w a y s y ie l d s c o m -

p l e t e c r eep r e co v e r y , h en ce i t is u n s u i t ed t o d e s c r i b e

p e r m a n e n t c r e e p d e f o r m a t i o n s , s ee fi g . 12 .

T h e e f fe c t iv e m o d u l u s m e t h o d h a s b e e n f a v o u r e d

i n t h e p a s t b e c a u s e o f i ts s i m p l e n u m e r i c a l i m p l e m e n -t a t i o n i n v o l v i n g o n l y an e l a s ti c s t r e s s an a l y s i s f o r a

g i v en i n s t an t o f t i m e

K (t) r ( t ) = R ( t ) . ( 3 1 )

A t t h e t i me u n d e r i n v es t i g a t i o n t h e " t an g en t i a l s t i f f -

n e s s " i s d e t e r mi n ed u s i n g t h e r e l ev an t e f f ec t i v e

m o d u l u s E e f f ( t , t o ) . D u e t o t h e t e m p e r a t u r e s e n si ti v-

i t y o f t h e u n i t c r e e p f u n c t i o n t h e m a t e r i a l p r o p e r -

t ie s v a r y s p a t ia l ly w i t h t h e t e m p e r a t u r e d i s tr i b u t io n

i n t h e s t r u c t u r e . N o t e t h a t i n g en e r a l a r ed i s t r i b u t i o n

o f s t re s s es o c c u r s d u e t o t h e r m a l c r e e p e x c e p t f o r

c r e e p f u n c t i o n s w h i c h m a y b e u n c o u p l e d f o r t i m e

a n d t e m p e r a t u r e e f f e c t s . S t e a d y - s t a t e t e m p e r a t u r e

c o n d i t i o n s l e a d t h e n t o s t re s s s ta t e s w h i c h r e m a i n

c o n s t a n t i n t i m e a s lo n g a s th e r e a r e n o p r e s c r ib e d

d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s a n d a s lo n g as

P o i s s o n ' s r a t i o f o r c r e e p r e m a i n s c o n s t a n t . T h e c r e e p

d e f o r m a t i o n s c a n t h e n b e s i m p l y e v a lu a t e d f r o m t h e

t i m e d e p e n d e n c e o f t h e u n i t c re e p f u n c t i o n .

a(to ) t aC (to,T)7 ( 0 = e( t ) + r / c( t = E - - ~ 0 + f o ( r ) ~ dT".

t o ( 3 2 )

T h i s f o r m u l a t i o n d o e s n o t c o n s i d e r th e h i s t o r y o f d e -

f o r m a t i o n , h e n c e n o c r e e p r e c o v e r y c a n b e d e sc r i b e d

o n u n l o ad i n g . F i g . 1 2 i l l u s t r a t e s t h e s h o r t co mi n g s o f

t h e r a t e o f c r e e p a s w e l l a s o f th e e f f e c t iv e m o d u l u s

m e t h o d i n c h a ra c t e r is i n g t h e d e f o r m a t i o n r e s p o n s e t oa uni t s t ress pu l se .

T h e r a t e o f c r e e p m e t h o d is i m p l e m e n t e d i n t h e

a n a ly s is s i m p l y b y a s t e p f o r w a r d i n t e g r a ti o n s c h e m e

a c c o r d i n g to s o m e f i n i te d i f f e r e n c e a p p r o x i m a t i o n . I n

t h e s i mp l e s t c a s e t h e c r eep s t r a i n i n c r emen t s a r e d e t e r -

mi n ed a s s u mi n g t h a t t h e e l a s t i c s t r e s s e s r ema i n co n -

s t a n t w i t h in t h e t i m e s t e p . T h e n t h e y a r e c o n v e r t e d

i n t o f o r ce s v ia t h e i n i ti a l l o ad t e ch n i q u e l e ad in g t o t h e

f o l l o w i n g i n c r em en t a l s ch em e o n t h e s t r u c t u r a l l ev e l

K ( t ) r a ( t ) = R a ( t ) + R j a ( t ) . ( 3 3 )

Fo r n o n - ag e i n g ma t e r i a l s t h e s t r u c t u r a l s t i f f n e s s r e -

m a i n s t h e r e b y u n a l t e r e d ; h e n c e t h e t i m e m a r c h i n g

a l g o r i th m r e d u c e s t o r e p e a t e d s o l u ti o n s f o r n e w l o a d

i n c r e m e n t s , a r a t h e r i n e x p e n s iv e o p e r a t i o n , e s p e c i a l ly

i f o n e c o n s id e r s t h a t t h e t o t a l h i s t o r y o f s t r u c t u r a l b e -

h a v i o u r is o b t a i n e d i n c o n t r a s t t o t h e e f f e c t iv e m o d u -

l u s m e t h o d . I t s h o u l d b e m e n t i o n e d t h a t t h e c h o i c e

o f t i m e s t e p i s c r u c ia l f o r t h e s u c ce s s o f t h e m e t h o d .

E x p e r i e n c e s h o w s t h a t n o s t a b i li t y p r o b l e m s o c c u r

a n d c o n v e r g e n c e p r o p e r t i e s a r e s a t i s f a c to r y a s l o n g a s

t h e m a x i m u m c r e e p s t r a i n i n c r e m e n t r e m a i n s s m a l l e r

t h an t h e cu r r e n t e l a s t i c s tr a i n . T h i s o r an a l o g o u s c r i-

t e r ia c a n b e u s e d t o s e l e c t a u t o m a t i c a l l y t h e n e x t t i m e

s t ep b y e x t r a p o l a ti o n w h i c h m a y b e c o r r e c t e d i f t h e

e s t i m a t e p r o v e s t o o l a rg e . A l t e rn a t e f o r m u l a t i o n s h a v e

b e e n p r o p o s e d t o c o m p u t e t h e n e w i n c r e m e n t s o f in i-

t i al l o ad s v i a M ax w e l l o r Ke l v i n s o l id s ( s t r e s s r e l ax a -

t i o n o r c r e e p d e f o r m a t i o n ) b u t t h e y d o n o t o f f e r a

g r e a t a d v a n t a g e o v e r t h e s i m p l e r f o r w a r d d i f f e r e n c e

s c h e m e [ 3 5 ] .



J. H. Argyris et al., Non-linear metho ds o f structural analysis 181

5 . 2 . 3 . M e t h o d o f s u p e r p o s i ti o n

T h e h e r e d i t a r y f o r m u l a t i o n o f t h e p r e v i o u s s ec t i o n

a c c o u n t s f u l l y f o r t h e s tr es s a n d d e f o r m a t i o n h i s t o ri e s

t h r o u g h t h e c o n c e p t o f m e m o r y f u n c t io n s . F o r t h e i r

e v a l u a ti o n a c o n s i d e ra b l e a m o u n t o f i n f o r m a t i o n i s

r e q u i r e d f r o m t h e e x p e r i m e n t a l i s t ; h e n c e t h e r e d u c -t i o n o f d a t a a n d t h e m a t h e m a t i c a l m o d e l l in g b e c o m e s

a n e x t r e m e l y i m p o r t a n t d i sc i pl in e w h i c h s h o u l d b e

s t r o ng ly a l li ed w i th t h e a s so c i a t ed so f tw a r e dev e lo p -

m e n t s .

T h e m e t h o d o f s u p e r p o s i ti o n h a s o n l y b e e n i m -

p lem en ted nu m er i ca l ly in a f ew ca se s f o r sp eci a l fo r m s

o f t h e ke r ne l f u n c t io ns . T h i s i s no g r ea t su r p ri se a s

v e r y g e n e ra l c r e e p o r r e l a x a t io n f u n c t i o n s o v e r t a x t h e

c a p a c i t y e v e n o f t h i r d g e n e r a t i o n c o m p u t e r s . F o r a ge -

ing v i sco e la s t i c m a te r ia l s o ne o b ta ine s t h e f o l lo wing

se t o f l i nea r in t eg r a l equ a t io ns o n th e s t r u c tu r a l l ev e l

t

to K ( t , r ) r ( r ) d r = R ( t ) • ( 3 4 )

We r eca l l t h a t t h e f o r m a t io n o f tim e - v a r iab l e e l em en t

s t i ff ne ss m a t r i ce s r equ i r e s t h e ev a lu a t io n o f s t re s s r e -

l a x a t i o n f u n c t i o n s f r o m c r e e p f u n c t io n s . D e p e n d i n g

o n t h e f o r m o f th e c r e e p k e rn e l a n " i n v e r s i o n " , in -

v o l vi n g t h e t o t a l d e f o r m a t i o n h i s t o r y , m i g h t b e n e c e s -

s a ry f o r e a c h p o i n t o f t h e b o d y ( s a y th e p i v o t p o i n tso f t h e n u m e r i c a l i n t e g ra t i o n s c h e m e o f e a c h e l e m e n t ) .

T h e s o l u t io n o f e q . (3 4 ) y i e l d s t h e n t h e n o d a l p o i n t

d i sp l acem e n t h i s to r y f r o m w h ich th e s t r a in and s t re s s

h i s to r ie s m a y b e c o m p u t e d . N o t e t h a t a t e a c h t i m e

o f c o n s i d e r a t i o n d i f f e r e n t " m e m o r y " s t if fn e s se s h av e

to b e ev a lu a t ed wh ich r e l a t e t h e p r e sen t r e sp o nse to

t h e h i s t o r y o f d e f o r m a t i o n l e a di n g to a h i s t o r y d e -

p e n d e n t l o a d c o r r e c t i o n . A l t e r n a ti v e l y , t h is c o r r e c t i o n

m a y b e de t e r m ined d i r ec t ly u s ing th e in i t i al l o ad co n -

cep t y i e ld ing th e f o l lo wing se t o f s t r u c tu r a l r e sp o n se

e q u a t i o n s

t a

K ( t ) r ( t ) = R ( t ) + f -~r R j ( t , r ) d r .

t o

( 3 5 )

T h i s ex p r e ss io n h as t h e g r ea t adv an tag e th a t t h e c r ee p

f o m u l a t i o n ( 2 7 ) c a n b e u s e d d i r e c t ly f o r c o m p u t i n g

in i ti a l s t ra in s w i th o u t t h e de to u r v i a t h e s t r e ss r e l ax a -

t i o n f u nc t io ns . M o r eo v e r , f o r no n -.ag e ing m a te r i a l s t h e

s t i f f ne ss r em a ins co ns t an t w i th t im e and h ence t r i -

ang u la r i sa tio n i s need ed o n ly o nce . I n g ene r a l , b o th

ty p es o f i n t eg r a l equ a t io n s a r e so lv ed b y s t ep f o r w ar d

i n t e g r a ti o n p r o c e d u r e s s i nc e t r a n s f o r m t e c h n i q u e s

a r e r e s t r i c t ed to t im e inv a r iab l e p r o cesse s . T h e nu m er -

i ca l sch em e co ns i s t s o f ex p ans io ns in to a se ri es o ft i m e i n c r e m e n t s w h e r e i n t e g r a ti o n s a r e p e r f o r m e d a c-

c o r d i n g t o s o m e f i n it e d i f f e r e n c e a p p r o x i m a t i o n .

No te ag a in th a t th i s m e th o d r equ i r e s a l l p r ev io u s

s o l u t io n s , h e n c e a n e x t e n si v e a m o u n t o f i n f o r m a t i o n

h as to b e h and led in o r de r t o o b ta in so lu t io ns o v e r

e x t e n d e d t i m e p e r i o d s . It s h o u l d a ls o b e m e n t i o n e d

th a t t h e s i ze o f t im e s t ep i s g o v e r ned b y th e ch a r ac -

t e r i s t ic t im e o f t h e f o r c ing f u nc t io n o r t h e m a te r i a l

b eh av io u r ( r e l ax a t io n and r e t a r da t io n t im es ) , t h e l a t -

t e r o f wh ich i s no r m a l ly c r i t i c a l. As a co nseq u ence ,

a n d i n c o n s t ra s t t o t h e r a t e o f c r e e p m e t h o d , t h e t i m einc r em en t s m a y no t b e en l a r g ed wi th p r o g r e ss ing ev o -

l u t i o n o f t h e p r o c e ss .

A l l t h e se f ac to r s co m p l i ca t e co ns ide r ab ly th e ap -

p l i c a ti o n o f t h e s u p e r p o s i t io n m e t h o d t o a r b i t r a r y

m a te r i a l r e sp o nse f u nc t io na l s . Fo r t h i s r ea so n , m o s t

o f t h e p r ev io u s inv es t ig a tio ns we r e r e s t r i c t ed to sp ec ia l

f o r m s o f k e r n e l f u n c t i o n s [ 2 6 , 2 8 , 3 0 - 3 2 ] .

5 .3 . Exam pl e s

T w o ex am p le s a re g iv en b e lo w in o r de r t o i l l u s tr a t e

th e d i f f e r e n t c r eep f o r m u la t io ns . F i r s t , a th i ck - wa l l edcy l inde r i s su b jec t ed to in t e r na l p r e ssu r e , r ad i a l and

ax ia l p r e s t re s s ing and a t em p e r a tu r e g r ad ien t . Fo r

th i s ca se t h e e f f ec t iv e m o du lu s m e th o d , t h e r a t e o f

c r e e p m e t h o d a n d t h e m e t h o d o f s u p e r p o si t io n a r e

im p lem en ted , u s ing th e sp ec i f ic c r eep f u nc t io n o f f ig .

1 3 . T h i s m a th em a t i ca l m o de l was dev e lo p ed in r e f .

[ 3 3 ] f r o m t h e c o n c r e t e t e s t d a ta o b t a i n e d a t t h e

T a y l o r W o o d r o w C o . , L o n d o n . T h e r e a f t e r , t h e e f f ec -

t iv e m o d u l u s m e t h o d a n d t h e r a t e o f c r e e p m e t h o d

a r e ap p l ied to ana ly se th e l o ng t e r m b eh av io u r o f t h e

1 :5 s c ale T h o r i u m H i g h T e m p e r a t u r e R e a c t o r ( T H T R )

m o de l u n de r a r ea li s t ic l o ad ing h i s to g r am , inv o lv ing

p r e s t r e s s , t em p e r a tu r e and p r e ssu r e lo ads . Ag a in th e

sam e c r eep law i s u sed to de sc r ib e th e t im e d ep en den t

b e h a v i o u r o f t h e m a t e r ia l .

5 .3 .1 . T h ick - wa l l ed cy l inde r

T h i s academ ic ex am p le i s ch o sen to i l l u s t ra t e t h e

a p p l i c a ti o n o f t h e e f f e c t i v e m o d u l u s m e t h o d , t h e r a t e

o f c re e p m e t h o d a n d t h e m e t h o d o f s u p e r p o s it i o n t o



18 2 J . 1 t . A r g y r i s e t a l . , N o n - l i n e a r m e t h o d s o f s t r u c t u r a l analys is

2 0

19

18

17

76

15

14

13

12

I1

10

D e h n u n g [ 10 ' ]f o r o = I [ k p l c m ~ ] ~ _ .

65

6O

5O

45

4O

the c r e ep l aw o f f i g . 13 . F i g . 14 s hows the f i n i t e e l e -

m e n t i d e a l i s a t i o n a n d t h e l o a d i n g c o n d i t i o n s o f t h e

t h i c k - w a l l e d c y l i n d e r w h i c h h a s b e e n a l s o u s e d in r e f.

[ 3 4 ] . T h e d i s t r ib u t io n o f c i r c u m f e r e n t i a l s t re s s e s is

s h o w n i n fig . 1 5 . T h e r e s u l ts o f t h e t h r e e c r e e p a n a l -

y s i s m e t h o d s a r e g iv e n f o r d i ff e r e n t i n s t a n t s o f t im e .N o t e t h a t t h e s o l u t i o n o f t h e s u p e r p o s i t i o n m e t h o d

f al ls b e t w e e n t h o s e O f t h e e f f e c t iv e m o d u l u s a n d

r a t e o f c r e e p m e t h o d . T h i s f a c t c a n b e o b s e r v e d a l s o

i n f ig . 16 , wh i c h i l l u s t r a t e s the t i m e v ar i a t i on o f the

c i r c u m f er en t i a l s t r e s s e s an d r ad i a l d i s p l a c em en t s a t

t h e i n s id e o f t h e c y l i n d e r . T i m e s t e p s o f t a = 0 . 2 d a y s

h a d t o b e u s e d f o r t h e m e t h o d o f s u p e r p o s it io n t o

i n s u re s t a b il it y , h e n c e t h e s o l u t i o n w a s t r a c e d o n l y u p

t o t = 1 0 d a y s b e c a u s e o f t h e e x c e s s i v e d a t a h a n d l in g

a n d c o m p u t e r t i m e r e q u i r e m e n t s . N o t e t h a t t h e r e-

s u l t s o f t h e r a t e o f c r e e p m e t h o d a n d t h e e f f e c t i v em o d u l u s m e t h o d s a p p r o a c h e a c h o t h e r w i t h p r o g re s s-

i n g t i m e , b o t h s o l u t i o n s s h o u l d i n th e e n d c o i n c i d e i f

t h e c r e e p c a p a c i t y o f t h e m a t e r ia l is li m i t e d .

l , , , , I = ~ t i n , H I = I l l i l = l l ~ i , I . . . . l _

• ' 2 ' ' ' 5 1 0 5 0 1 0 0 ~ 0 0 I 0 0 0 5 0 0 0 I O O O O

tue* l [Tage]

Fig. 13. Uniaxial cre e p l aw [ 3 3 ] .

5 . 3 .2 . T H T R 1 :5 s ca l e m o d e l

T h i s re a l is t ic e n g i n e e r in g e x a m p l e c o n c l u d e s t h e

i n ve s t ig a t io n o n c r e ep o f c o n c r e t e . T h e g e o m e t r y o f

t h e T H T R 1 :5 s c a le m o d e l h a s b e e n p r e s e n te d i n o u r

c o m p a n i o n p a p e r t o ge t h e r w i t h t h e a x i sy m m e t r ic

T e m p e r a t u re 7 . _ _ 8 0 o cDistribution

,01

r i : 2 0 in

P r e s t r e s s F o r c e s :

p z = 1 0 k s i

po =1.0 k s i

I n t e r n a l P r e s s u r e :

P i = 0 . 2 k s i

p z t i i t i t i i i i i f i i i t t t t t i

m, I

r o =/-,0 in -- ~-- '.

1 6 . I R I A X 6 E L e m e n t s

8 1 D e g r e e s o f F r e e d o m

. ] T o= 3 0 ° C

Po

Fig. 14. Thick-w alled cy l i nde r , i d e a l i s a t i on and l oad i ng cond i t i on s .



J . H . A r g y r i s e t aL , Non - l in ear m e thods o f st r u c tu r a l ana l ysi s 183

~ttk s l

- 3

- 2

-1

o AB

i -t = o d a y s

~ttks i

G tt i

k s i

- 3

- 2

-1

- 3

- I

0A B

t =10 a y s

• ig . 15. Thick-w alled c ylind er, circum ferential stress

distribution at different t im es .

t = 0 . 5 d a y s

I~r A~B

- - S u pe r po s lt io n M e th o d

- - - - - R a t e o f C r e e p M e t h o d

~ . - - E f f e c t i v e M o d u t u s M e t h o d

-50

-~O~

- 3 . 0 ~

- 2. 0

- 1 . 0

0 1 1 1 0 1 0 0 1 0 0 0 5 0 0 0

C i r c u m f e r e n t i a l S t r e s s

I

t d a y s

t O 0 0 5 d O 0

0.5

0 .1 0 1 0 0

0 5 - " ~ " ~ ' ~ "

- 1 . 0 ~ ' ~ . .

- 1 , 5 ~ M e t h o d o f S u p e r p o s i t i o n

- 2 . 0 - - - - - - R a t e o f C r e e p M e t h o d

- - . - - E f f e c t iv e M o d u l u s M e t h o d

- 2 . 5 -

- 3 . 0

~ - d a y s L I

R a d ia l D i s p l a c e m e n t

~ " % .

\ ' \

\

0 0 1 8 ~

Fig . 16 . Thick-wai led cy l inder , t ime varia tion of s tress and

disp lacement a t po int A.

16

a t t J

ks i

I

Fig . 1 7 . TH TR 1 :5 s ca l e m ode l ax i s ym m etr i c ana l ys is , d i s -

p l a c e m e n t s a t t im e t = 0 .

f in i t e e l e m e n t id e a l i s a t i o n . F i g . 1 7 il l u s t r a t e s t h e e l a s -

t ic d i s p l a c e m e n t s o f t h e c o n t a i n m e n t v e s se l s u b je c t e d

t o p r e s t r e s s , d e a d w e i g h t , t e m p e r a t u r e g r a d i e n t a n d

p r e s s u r e l o a d i n g . B o t h e f f e c t i v e m o d u l u s a n d r a t e o f

c r e e p m e t h o d s a r e i m p l e m e n t e d , u s i n g a ga in t h e c r e e p

l a w o f f i g . 13.



184 J . H . A r g y r i s e t a l. , N o n - l in e a r m e th o d s o f s t r u c tu r a l a n a ly s i s

0.1774

i

I !

0 2 16 1

01637

P r es t r ess and Dead W eigh t

Internal Pressure P i = , '-0 kp/cm 2

T em per a t u r e G r ad ien~

A'T= 20%

-- - 0~005

G eom et r y

0 10 50 cm

Displacements

0 0 0 5 0 1 0 c m

Ef f ec t ive M od u lus M et h od

Results at t = 30 years

II i

0 1 0 8 0

F i g. 1 8 . T H T R 1 : 5 s ca l e m o d e l a x i s y m m e t r i c a n a l y si s, c r e e p -

d i s p l a c e m e n t s a f t er 3 0 y e a r s .

01981

I iJ J

= r es t r ess and O ead W eigh t

I n t e rna l P r es s ur e P i = 4 0 ) / c m 2

T em per a t u r e G r ad ient

T : 2 0 ° C

G eom et r y

0 " ~ 50cm

Displacements _

0 Y ~ O , O ~ m

~ E f f ec t ive M od u lus M et h odResults at t = 30 years l

(Steel not Incorporated )

7

01&22

II

F i g . 1 9 . T H T R 1 : 5 sc a l e m o d e l a x i s m m e t r i c a n a l y s i s , c r e e p -

d i s p l a c e m e n t s a f t e r 3 0 y e a r s .



J. H. Argyr is et al . , Non-linear m etho ds o f structural analysis 185

T h e c r e e p d i s p l ac e m e n t s , as o b t a i n e d w i t h t h e e f -

f ec t iv e m o du lu s m e th o d , a r e p r e sen ted in f ig . 1 8 . Fo r

co m p ar i so n , t h e sam e c r eep ana ly s i s i s p e r f o r m ed

neg lec t ing th e s t i f f ne ss o f t h e s t ee l co m p o n en t s , su ch

as li ne r , p r e s t re s s ing cab le s and r e in f o r cem en t b a r s .

T h e r e su l t s a r e sh o w n in f i g . 1 9 ex h ib i t ing an inc r ea seo f c r e e p d e f o r m a t i o n s b y 3 0 % t o 5 0 % . T h e t i m e v ar ia -

t i o n o f s t r es se s is i l lu s t r a t ed in f i g . 2 0 a t a p o in t o n th e

o u t s ide su r f ace o f t h e v e sse l f o r t h e g iv en lo ad ing h i s to -

g r am . B o t h e f f e c ti v e m o d u l u s a n d r a te o f c r e e p

m e t h o d s a r e i m p l e m e n t e d s u c h t h a t t h e o r i gi n o f t h e

c r e e p f u n c t i o n c o i n c i d es w i t h t h a t o f t h e r e le v a n t ty p e

o f l o ad ing ; t h e co m b ined c r eep r e sp o nse i s t h en o b -

t a ined b y su p e r p o s i t i o n . T h e r e a r e co ns ide r ab le d i f -

f e r en c e s b e t w e e n t h e r e s u lt s o f t h e e f f e c ti v e m o d u l u s

a n d t h e r a t e o f c r e e p m e t h o d n e a r t h e d i s c o n t i n u it i e s

o f l o ad ing wh ich d im in i sh w i th p r o g r e ss ing t im e .S i m i la r p h e n o m e n a c a n b e o b s e r v e d a t t h e o t h e r

p o in t s o f t h e v esse l no t sh o w n h e r e . F ig . 2 1 i l l u s tr a t e s

f o r t h e sam e lo ad ing h i s to g r am th e s t r e s s r e l ax a t io n

b e tw een th e t im e t = 0 and t = 30 y ea r s a t t h e sec t io n

A - B w h i c h is f o r th e r a t e o f c re e p m e t h o d a lw a y s

l a rg e r t h a n f o r t h e e f f e c t i v e m o d u l u s m e t h o d . F i n a l l y ,

f ig . 2 2 sh o ws th e s t r e s s red i s t r i b u t io n wi th t im e a s o b -

t a i n e d f r o m t h e r a t e o f c re e p m e t h o d f o r a ti m e i n -

v a r i abl e t em p e r a tu r e l o ad ing . No te t h a t t h e s t r e s s d is -

t r i b u t i o n a t t h e s e c t io n A - B r e la x e s c o m p l e t e l y w i t h

p r o g r e ss ing t im e .

I n c o n c l u s i o n i t s h o u l d b e m e n t i o n e d t h a t t h e r e

a r e a n u m b e r o f q u e s ti o n s i n t h e c o n t e x t o f c o n c r e t e

c r e e p w h i c h h av e n o t b e e n t o u c h e d u p o n h e r e , s i n ce

th i s p r e l im ina r y inv es t ig a tio n was p r im ar i ly co nc e r ned

w i t h c r e e p a n a ly s is p r o c e d u r e s . T h e r e m a i n i n g p r o b -

l em s l ie m a in ly in t h e f i e ld o f r e f in ing m a th em a t i ca l

m o d e l s t o d e s c r ib e c r e e p p h e n o m e n a u n d e r t r ia x i a l

co nd i t i o ns f o r l o ad ing and u n lo ad ing , e sp ec i a l ly u nd e r

t r ans i en t co nd i t i o ns , t ak ing in to co ns ide r a t io n th e

nu m er i ca l s im p l i f i c a t io ns o f c e r t a in c r eep l aws .

6 . Diss ipat ive phenom ena in m e t a l s

B a se d o n t h e f l o w t h e o r y o f p la s t i c it y t h e r m o -

e l a s to - p l a s t i c s t r e s s - s t r a in r e l a t i o ns a r e f o r m u la t ed

a n d t h e n e x t e n d e d t o i n c o r p o r a t e c r e e p . T h e i n it ia l

l o ad t ech n iqu e i s r ev i ewed f o r t h e i t e r a tiv e so lu t io n

o f p l a s t ic i t y p r o b l e m s a n d a d a p t e d t h e r e a f t e r f o r t h e

ana ly s is o f c r eep p r o b lem s . I n co nc lu s io n an ex am p le

i s p r e sen ted to i l l u s tr a t e t h e se t e ch n iqu es o n a r ea li s t ic

eng inee r ing p r o b lem , a sp h e r i ca l sh e l l w i th no zz le

wh ich i s su b jec t ed to in t e r na l p r e ssu r e a s we l l a s t em -

p e r a tu r e g r ad ien t s .

6 . 1 . E l a s t o - p l a s t i c s o l i d s

T h e g e n e ra l c o n s t i tu t i v e t h e o r y o f t h e r m o - e l a s t o -p l a s t i c b eh av io u r h a s b een l a id o u t i n [36 ] and

adap ted f o r s t r u c tu r a l analy s is i n [ 7 ] . T h e f o l lo wing

d i scu ss io n is co nce r n ed wi th an ex t ens io n o f t h e v o n

M i se s y i e l d c o n d i t io n a n d t h e P r a n d t l - R e u s s f l ow

r u le t o n o n - i s o th e r m a l c o n d i t i o n s w i t h t e m p e r a t u r e

d e p e n d e n t h a rd e n i n g . T h is ti m e i n d e p e n d e n t s t r e s s -

s t r a in r e l a t i o n i s t h en g ene r a l ised to inco r p o r a t e c r eep

n e g l ec t in g t h e h i s t o r y d e p e n d e n c e o f c r e e p d e f o r m a -

t io ns . No te t h a t f o r cy c l i c l o ad ing co nd i t i o n s , P r ag e r' s

m e t h o d o f k i n e m a t i c h a r d e n i n g o r Z i e g le r 's m o d i f i c a-

t i o n o f t h i s t h eo r y co u ld ea s i ly b e inco r p o r a t e d to ac -co u n t f o r t h e B au sch ing e r e f f ec t . I n t h e ca se o f no n -

m e ta l l i c m a te r i a ls t h e y i e ld co nd i t i o ns and f lo w r u l e

m a y b e a d j u st e d t o i n c lu d e t h e e f f e c t s o f h y d r o s t a t i c

p r e ssu r e s im i l a r t o D r u cke r ' s f o r m u la t io n f o r m a te r i a l s

wi th in t e r na l f r i c t i o n and co h es io n . Fo r a m o r e de t a i l ed

p r e se n ta t io n th e r eade r i s a l so r e f e r r ed to [ 1 2 ] and

[ 1 6 ] .

F o r d u a l i t y o f t h e t r a n s f o r m a t i o n l aw s t h e c o r re -

sp o nd ing s t r e s s and s t r a in v ec to r s a r e ex p r e ssed a s

f o l lo ws

* = {o~x Oyy Ozz x/ % xy v~ oy z x/%~x},• 1 1 1 , (36 )

= ,y y V zz- y - yz

T h e f o r m u l a t i o n i s r e s tr i c te d t o i s o t r o p ic c o n d i t i o n s

f o r w h ich th e l i nea r e la s t i c m a te r i a l b eh av io u r i s de -

sc r ib ed b y th e m a te r i a l s t i ff ne ss m a t r ix

w h e r e I d e n o t e s t h e u n i t m a t r i x a n d w h e r e t h e v e c t o r

e 3 ,3 is de f ined in eq . ( 1 6 ) . Us ing th e f u nda m en ta l l aw

o f d e c o m p o s i t i o n th e t o t a l d e f o r m a t i o n r a t e ~ c a n b e

w r i t t e n a s

desc r ib e s t h e e l a s t i c , s t re s s p r o du c ing c o m p o n en t s

a n d q p , / 1 0 , / I c t h e p la s t ic , t h e r m a l a n d c r e e p c o m -

p o n e n t s r e s p e c ti v e ly , o f t h e t o t a l d e f o r m a t i o n r a t e .

Om i t t i ng th e de t a i l s o f t h e de r iv a t io n th e p l a s t i c f l o w



186 J. H. Argyris e t al., Non-linear m etho ds o f structural analysis

i [ Load,r~g Histogram ] -]

w internal Pressurep = 40 kp/cm

Temperature d T= 20 °C

iPrestressing and D ead Weight

~o ~do 3 & 665 100[ )0 "days

kplcm2

- 4 0 B

~20 f ~ o

I !I i 1000 _~

0 10 160 3(~5 665 ' 1000 0 days

kp/cm 2

-90 -- Rate of Creep Methodo Ef fect Modulus Method

-7 0

50

- 3 0 - ~]0 100 365 1000 10000 days

%;

- t6 o ; • o

-40 , --~10 100 365 665 10000 days

Fig. 20. THTR 1:5 scale mod el, axisymm etric analysis, stress-redistribution due to creep at point B.

r u l e can b e w r i t t en a s

• 14p=~pS = - ( s s t ~ , + ~ s o , ( 3 9 )

w h e r e

3S = - - 2 ~ ~D and ~2 _~-- o to ~D"

Fo r me t a l s , t h e en e r g y d i s s i p a t i o n d u e t o p l a s t i c i t y i s

d e f i n ed b y t h e s ca l a r f u n c t i o n s f f an d ~ , t h e v an Mi se s

eq u i v a l en t s t r e s s an d s t r a i n , w h i ch d es c r i b e t h e d ev i a -t o r i c e n e r g y u n d e r m u l t ia x i a l c o n d i t io n s b y e q u i v a le n t

u n i a x ia l c o m p o n e n t s . T h e p h y s ic a l m e a n i n g o f th e

h a r d e n i n g p a r am e t e r s ~ an d ~o i s i l l u s tr a t ed i n f i g . 2 3 .

N o t e t h a t ~0 = 0 in t h e ca s e o f i s o t h e r m a l co n d i t i o n s .

T h e t h e r m a l d e f o r m a t i o n i s d e s c r ib e d b y t h e f o l -

l o w i n g e x p r e ss i o n w h i c h m i g h t b e n o n l i n e a r i n t e m -

p e r a t u r e

n o = r e ( T - T O ) = otO. ( 4 0 )

! L !

0 - 2 0 - 4 0 ~ 6 0~rr kP/crn2

0 - 2 0 - 4 0G zz p / cm 2

A . . . . . . . ~ - ~ / 7

! I / z zf i

B A ] / z z

] / ~ / - - R a t e f Creep M e t h o d/ ! I - - i E f f ec t ive M od u lus M et h od

i / ' l | - - - - E l a s t i c R e su l ts fo r t = 0

u - 2 0 - 4 0 - 6 0 - 8 0 - 16 0 - 12 0~ r k p /c m 2

I f = 3 0 y e a r s

Fig. 21. THT R 1:5 scale mod el axisymm etric analysis, stressdistribution a t section A -B for given loading history.

T r e f e r s t o t h e a b s o l u t e t e m p e r a t u r e a n d T O t o t h e

r e f e r e n c e t e m p e r a t u r e o f t h e s o l id . F o r i s o t r o p i c c o n -

d i t ions

m = a te3 ,3 ,

w h e r e a d e n o t e s t h e c o e f f i c i e n t o f t h e r m a l e x p a n s i o n .

I n a n a l o g y t o t h e t h e o r y o f p l a s ti c i ty t h e v a n M i se s

y i e l d c r i t e ri o n a n d t h e P r a n d t l - R e u s s f l o w ru l e c a n

b e u s ed t o d e s c r i b e t h e en e r g y d i s s i p a t i o n d u e t o c r eep

b y eq u i v a l en t u n i ax i a l q u an t i t i e s

ilc : ~?cs . (41)

W i t h in t h e e n g i n e e ri n g t h e o r y o f c r e e p w e a s s u m e

t h a t t h e r e e x i s ts a n e q u a t i o n o f s t a t e w h i c h d e f i n e s

t h e c r ee p r a t e o f t h e s o l i d a s a f u n c t i o n o f s t a t e v a r i -

ab l e s s u ch a s s t r e s s , t emp e r a t u r e an d t i me o r an eq u i v -

a l e n t h a r d e n i n g p a r a m e t e r a c c o u n t i n g f o r h i s t o r y e f -

f e c t s [ 1 8 ] . T h i s t h e o r y e x h i b i t s s h o r t c o m i n g s s im i l a r

t o t h e d e f o r m a t i o n t h e o r y o f p l a s ti c i ty w h i c h s u g g e st

t h a t t h e m e m o r y t h e o r y o f n o n li n e a r v is c o e la s ti c

s o li d s s h o u l d b e u s e d t o f o r m u l a t e t h e c r e e p r e s p o n s e

o f m e t a l s [ 3 9 ] . F o r t h e s o l u t i o n o f e n g i ne e r i n g p r o b -

l e m s a si m p l e r a p p r o a c h i s n e c e s s a r y , h e n c e a n u m b e r

o f c u m u l a t i v e c r e e p t h e o r i e s h a v e b e e n p r o p o s e d t o



J . H. Argyris e t al . , Non. l inear methods of s tructural analys is 1 8 7

0

2

3()

20 0

3 0 2 s 2 o ~ s ~ 0 s 0 - g - ~ b - i s - 2 '0 - 2 s 0 - g -

~ t t k p / c m 2 ( ~ r r k p / c m 2

3 s 3 0 2 s 2 0 ~ s ~ 0 s 0 - s - ~ '0 - 1 's - 2 ' 0 - ~ s - 3 ' 0 - 3 s

az z kp. lcm 2

F i g . 2 2 . T H T R ! : 5 s c a le m od e l ax i sy m m et z i c ana l ys i s, s tr ess r ed i s t r i bu t i on und er tem per a tu r e l oad i ng .

equlvalem ~rain ~ p temperotur e

a) D~ ini t ion of ~ b) Def ini t ion of tp

F i g . 2 3 . W or k - haz d en i ng paz am et er sat d i f f e r ent t em per a t u r es .



188 J. H. Argyris et al., Non-linear methods of structural analysis

B y~ strainhorden,rig

11 t ime

a} Vo.riab[e Stress Condit ions

~r2

r~

t1 t ime t

b) Var io.ble Tem perature C ondit ions

Fig. 24. Transient effects on creep rate.

e x t e n d t h e c r e e p l aw c o n s t r u c t e d f r o m c o n s t a n t

s t re s s and t em p er a tu r e t e s t s t o t im e v a r y ing p ro cesse s .

I n t h e f o l lo wing , o n ly th e t im e - and s t r a in - h a r den ingr u le s a re co ns ide r ed .

T i m e h a r d e n i n g

T h i s m a te r ia l l aw f u r n ish es a l i nea r r e l a t i o nsh ip b e -

twee n s tr es s- , t o t a l s t r ain - and t em p er a tu r e - r a t e s .

Ho wev e r , it i s s t r i c t l y v a l id in t h e d i f f e r en t i a l senseo n l y a s F , G a n d s v a r y w i t h t i m e . T h e " t a n g e n t i a l "

m a te r i a l p r o p e r t i e s a r e g iven b y

~c = f l ( o , 0 , t ) , ( 42 )

S t r a in h a r den ing

~e = f2(° ,O,r le) •

N o t e t h a t f o r s e c o n d a r y cr e e p , i n w h i c h t h e c r e e p

r a t e i s i n d e p e n d e n t o f t im e , b o t h f o r m u l a t i o n s y ie l diden t i ca l ex p r e ss io ns . F ig. 2 4 i l lu s t r a te s t h e d i f f e r ence

o f b o th h a r d en ing r u l e s f o r v a r y ing s t re s s and t em p er a -

tu r e co nd i t i o ns a t a sp ec i f i ed t im e . A l th o u g h ex p e r i -

m en ta l da t a t end to su p p o r t t h e s t r a in - h a r den ing r u l e ,

t h e m e th o d m ay l ead to d i f f i cu l t i e s i n de sc r ib ing

c r eep du r ing u n lo ad ing i f t h e accu m u la t ed c r eep

st ra in i s la rge . Hence , the s impler t ime-hardening

m e th o d i s g ene r a l ly f av o u r ed f o r nu m er i ca l c a lcu l a-

t i o ns invo lv ing p r im ar y c r eep . Su b s t i t u t io n in to eq .

( 38 ) l e ads t o t h e f o l lo wing ex p r e ss io n f o r t h e b eh av i -

o u r o f an i so t r o p ic so l id ex h ib i t i ng e l a s t i c , p l a st i c ,t h e r m a l and c r eep e f f ec t s

gl = E - l ( ! + 2 ~ + (~°s + ot)O + S~c" ( 4 3 )

Pa r t ia l i nv e r s io n y i e ld s t h e de s i r ed r e l a t i o nsh ip f o r t h e

e las t ic s t ress ra tes

6 = E t = F : t + GO - Es ~ c . ( 4 4 )

~ [ + 2 G '~2 G s s t) ' G = _ _ t ~ ~ ~ o s } .F = E ( I f + 3 G

(45)

No te th a t t h e f i r s t te r m s o n th e r i g h t h and s ide s r e -

p r e sen t t h e l i nea r e la s t ic p o r t i o n o f t h e co n s t i t u t iv e

l aw w h i l e t h e s e c o n d t e r m s f o r m t h e c o r r e c t io n s d u e

to p l a s t i c i t y . It sh o u ld b e em p h as i sed th a t t h e c r eepr a t e c o m p o n e n t s h a v e t o b e c o n s i d e r e d p r e sc r i be d

s ince th ey a r e de t e r m ined f r o m th e s t a t e o f s t re s s ,

t e m p e r a t u r e a n d t i m e b u t d o n o t i n v o lv e t h e i r t im e

der iva t ives .

6.2 . In i t ia l load technique

I n t h e p a s t , two nu m er i ca l t e ch n iqu es h av e b een

ap p l i ed to t h e so lu t io n o f p l a s t i c i t y p r o b lem s , t h e

t ang en t i a l s t if f ne ss and th e in i ti a l l o ad m e th o d . I n

th e f o l lo wing we co n s ide r o n ly th e in i ti a l l o ad i t e ra -

t i o n in wh ich th e s t r u c tu r a l s t i f f ne ss p r o p e r t i e s nee dn o t b e c h a n g e d f o r e a ch l o a d in c r e m e n t . T w o a l te r -

n a t iv e f o r m u l a t i o n s o f t h e f l o w r ul e m a y b e u s e d f o r

th e ev a lu a t io n o f t h e in i t i a l lo ads .

6 .2.1 : In i t ia l s t ra in m eth od

I n th is m e th o d th e p l a s t i c s t r a in inc r em en t s a r e

de t e r m ined u s ing th e f l o w r u l e i n eq . ( 39 ) i nv o lv ing

th e e l a s t ic s tr e s s and t em p e r a tu r e inc r em en t s



Z H. Argyr is et aL, Non-linear me thods o f structural analysis 189

' ~ - - - - ~ GA G ive n S t r e s s I n c r e m e n t

I D ~ / ~ De r ive d I n i t ia l S t ra in I n c r e m e n t

E

T o t a l S t r a i n ~S t r a i n A p p r o a c h) I n i t i a l

/ )T o t a l S t r a i n

b } I n i t i a l S t r e s s A p p r o a c h

Fig. 25. Initial load m ethods.

1qpA =-( ss ta A + ~°sOA. ( 4 6 )

T h e i n it ia l l o a d i n c r e m e n t J a o f e q . ( 1 0 ) c a n n o w b e

c o m p u t e d l e a d in g t o a n i t e ra t iv e p r o c e d u r e , s i n ce t h e

cu r r en t e l a s t i c s t r e s s i nc r em en t s a r e no t kno wn a

p r io r i . F ig . 2 5 a i l lu s t r a t e s t h is f o r m u la t io n f o r t h e

o ne - d im en s io na l c a se . I f co nv e r g ence can b e a ssu r ed

a t a l l , t h i s m e th o d p r o v ides b e t t e r r a t e s o f co nv e r g ence

th an th e a l t e r na t iv e in i t ia l s t re s s ap p r o ac h desc r ib ed

b e l o w ; o n t h e o t h e r h a n d , t h e i n i t ia l s tr a in m e t h o d

b r eaks do wn in t h e ca se o f no n - h a r den ing m a te r i a l s ,

w he re ~" = 0.

6 .2 .2 . I n i ti a l s t re s s m e th o d

I n th i s m e th o d th e p l a s t i c s t r a in inc r em en t s a r e

de t e r m ined u s ing an a l t e r na t iv e f o r m o f t h e f l o w r u l e

i n vo l vi n g t h e t o t a l s t r a in a n d t e m p e r a t u r e i n c r e m e n t s

_ 2 G 2 G 0q p a ~ + 3 G S s t ' fa + ~ s a" ( 4 7 )

T h e i n i ti a l lo a d i n c r e m e n t J A o f e q . ( 1 0 ) c a n n o w b e

c o m p u t e d l e a di n g t o a n i t e ra t i v e p r o c e d u r e , s in c e t h e

c u r r e n t t o t a l s t ra i n i n c r e m e n t s a r e n o t k n o w n a p r io r i,

F ig . 2 5 i l lu s t r a t e s th i s f o r m u la t io n f o r t h e o ne - d im en-

s io nal c a se . T h i s p r o c edu r e was f i r st p r o p o se d in t h e

f o r m o f i n i t ia l s t re s se s in r e f . [ 11 ] . A m o r e co nc i se

p r e sen ta t io n i s dev e lo p ed in re f . [ 1 2 ] wh ich co n ta in s

a ls o a r ig o r o u s t r e a t m e n t o f t h e p r o b l e m o f c o n v e r -

g e n c e . N o t e t h a t e x p r e s s i o n ( 4 7 ) f o r t h e f l o w r ul e

a l so ap p l ie s t o no n - h a r de n ing m a te r i a l s , wh e r e ~" = 0 ,

en la rg ing co ns ide r ab ly th e r ang e o f co n v e r g ence in

co m p ar i so n to t h e a s so c i a t ed in i t i a l s t r a in ap p r o ach .

I t i s i n t e re s t i n g t o n o t e t h a t t h e d i f f e r e n c e s b e t w e e n

the in i t ia l s t ress and in i t ia l s t ra in schemes ar ise so le ly

f r o m t h e a l t e r n at i v e d e f i n it i o n s o f q p a .

I n t h e f o l lo wing , t h r ee m e th o ds a r e d i scu ssed f o r

t h e n u m e r i c a l t r e a t m e n t o f t h e i n i ti a l l o a d f o r m u l a -

t i o n . T h e y a r e d e n o t e d i n s h o r t b y D I M , t h e d i r e c t

i n c r e m e n t a l m e t h o d , N I M , t h e n o r m a l i t e r a ti v e m e t h -

o d , a n d V I M , t h e i m p r o v e d i t e ra t i v e m e t h o d . D I M

o p e r a t e s s o l el y o n l o a d i n c r e m e n t a t i o n w i t h o u t i te r a-

t i o n wh e r e th e m a g n i tu des o f t h e e l a s t i c al ly su p p r e ssed

p la s t i c s t r a in s a r e e s t im a ted d i r ec t ly f r o m th e p r ev io u s

lo ad s t ep . NI M inco r p o r a t e s w i th in each lo ad inc r e -

m en t an i t e r a t iv e sch em e b ased o n su ccess iv e su b s t i t u -

t i o n w i t h o u t u p d a t i n g t h e i n it ia l f u n c t i o n g r a d ie n t .

T h i s m e t h o d a c c o u n t s f o r th e e r r o r m a d e d u e t o t h e

f i n it e si ze o f t h e l o a d i n c r e m e n t . V IM i s a N e w t o n -

R a p h s o n t y p e p r o c e d u r e i n v o lv i ng t h e r e p e a t e d e v al u -

a t io n o f t h e f u n c t io n g r ad ien t f o r t h e p l a s t i c v a r iab l e s

i n o r d e r t o i m p r o v e t h e c o n v e r g e n c e p r o p e r ti e s o f

NI M. A sch em a t i c r ep r e sen ta t io n o f t h e se in i t ia l l o ad

techniques i s g iven in f ig . 26 . For a de ta i led d iscus-

s io n o n th e se i t e r a t iv e in i ti a l lo ad t ech n iqu es , t h e

r eade r m ay co nsu l t r e f s . [ 9 , 1 0 ] and [1 2 ] , t h e l a s t

One espec . ia l ly fo r quest io ns reg arding the range and

r a t e o f c o n v e r g e n c e .

I n t h e f o l lo wing th e r e su l t s o f two e l a s to -p l a s t ic

p r o b lem s a r e p r e sen ted to i l l u s t ra t e t h e se i t e r a t iv e

sch em es . T h e f i r s t ex am p le dea l s w i th t h e e l a s to -

p l a s t i c analy s is o f t h e s t a t i c a l ly inde te r m in a te t r u ss i n

f ig . 2 7 , wh i l e t h e sec o nd o n e inv o lv es t h e r ec t ang u la r

m em b r ane wi th cen t r a l c r ack in f i g . 2 8 . Fo r b o th ca se s

a m o d i f ic a t io n o f t h e R a m b e r g - O s g o o d e q u a t io n d e -

sc r ib e s t il e u n iax ia l s t r e s s - s t r a in l aw



190 J. H. Argyris e t a l . , Non- l inear methods o f s t ructura l analysis

r l p a = f ( ~ ? p , j }

- - - - VIM Newton Raph~on

. . . . . . . . . . . . . . NIM Succesive Substi tutio n

A DI~4 No Iteration

I

r ~ p d ? ~ d t r i p d

F ig . 2 6 . Res i du a l l o ad i t e r a t i o n .

R R

9 N o d a l P o i n t s

1 5 F L A 2 E l e m e n t s ,.Z

F ig . 2 7 . In de t e rmina t e t ru ss .

1" 1 aY I ( l - ~ . a y ) m ( l ~ ) m l~ P - m E - "

( 4 8 )

T h e m a te r i a l i s a s su m ed an a lu m in iu m a l lo y 2 02 4- T 3

wi th th e ch a r ac t e r i s t i c p r o p e r t i e s : E = 1 1 .4 X 1 06

lb f / in 2 , v =0 .3 , O y = 34 5 00 lb f / i n 2 , and m = 1 0 . An

e la s ti c so lu t io n d e t e r m ines t h e l o ad ing a t wh ich lo ca l

y i e ld ing o ccu r s f i r s t wh i l e t h e su b sequ en t l o ad inc r e -

m en t s a r e de sc r ib ed b y th e lo ad f ac to r X wh ich i s i n-c r em en ted b y hz~ = 0 .1 .

D i s c u s s i o n o f r e s u l t s. T ab le s 1 and 2 su m m ar i se

th e r e su l t s o f t h e e l a s to - p l a s ti c ana ly se s o f t h e s t r u c -

tu r e s sh o w n in f ig s . 2 7 and 2 8 . T h e t ab l e s ind ica t e

t h e n u m b e r o f i te r a t i o n s n e c e ss a r y t o o b t a i n a r e la -

t i ve dev ia t io n o f l es s t h an 1 0 - 4 f o r t h e equ iv a l en t

p l a s ti c - ra t e s i n su b sequ e n t i t e r a t io ns . Bo th i t e r a t iv e

p r o cedu r e s , NIM an d V I M , a r e ap p l i ed to t h e in i t i a l

s t re s s and in i t ia l s t r a in f o r m u la t io n .

y,

\Z ,

C r a c k '

I

3 49 Nod a l P o i n t s

156 T RIM 6 E l e m e n t s ~

b = 6 0 i n

F ig . 2 8 . M emb ran e wi th cen t r a l c r ack , g eo met ry an d i dea l i sa .

t i o n .

T ab le 1 p r e sen t s t h e r e su l t s f o r t h e s im p le t ru ss

co n f i r m ing th e th eo r e t i c a l p r ed ic t io ns o f r e f . [ 1 2 ] i nr e g ar d t o t h e c o n v e r g e n c e p r o p e r t i e s o f t h e n o r m a l

i t e r a tiv e m e th o d . T h e r ang e o f co nv e r g ence o f NIM i s

l a r g er f o r t h e in i ti a l s t re s s ap p r o ac h th an f o r t h e in i ti a l

s t r a in m e th o d wh ich d iv e r ges a l r eady a t ~ = 1 .5 . Ho w -

ever , i f i t conv erges a t a l l, the in i t ia l s t ra in fo rm ula-

t i o n c o n v e rg e s fa s t er . M o r e o v e r , V I M , t h e N e w t o n -

R a p h s o n m o d i f i c a t io n o f N I M , i m p r o v e s c o n s id e r a b l y

th e r a t e o f co nv e r g ence i f ap p l i ed to e i t h e r t h e in i t ia l

s t re s s o r i n i ti a l s t ra in ap p r o ach . I t sh o u ld b e no te d

th a t i n t h e ca se o f V IM and in i ti a l s tr a in , t h e r ang e o f

co nv e r g enc e is a lso co ns ide r ab ly en la r g ed in co m p ar i -so n to NI M.

T ab le 2 r ep r o du ces t h e co r r e sp o nd ing da t a f o r

t h e r e c t a n g u la r m e m b r a n e w i t h c e n t r a l c r ac k . T h e r e -

su i ts f o r t h i s s t r u c tu r e ag r ee v e r y we l l w i th t h e t h eo r e -

t i c a l p r ed ic t io ns o f [ 1 2 ] . T h e NI M - in i ti a l s t re s s m e th -

o d n o w r equ i r e s f a r l es s i t e r a t io ns t h an in t h e ca se o f

th e t r u ss . On ly a t r e l a t i v e ly h ig h lo ads can th e nu m b er

o f i t e r a ti o n s b e r e d u c e d m a r k e d l y b y t h e a p p l i c a t io n

of VIM. Whils t NIM -ini tia l s tress converges mo re



J . H. Argy ris e t al ., Non . l inear m etho ds o f s tructural analys is

Table 1

Indeterminate truss, number of iterations for difference in equivalent plastic strain < 10 -4.

191

MaximumLoad-factor

effective stress

Omax [lbf/in2 ]

Initial stress approach Initia l strain approach

NIM VIM NIM VIM

1.0 34500 . . . .

1.1 37730 10 4 6 3

1.2 40740 12 3 6 31.3 43460 15 3 9 3

1.4 45890 20 3 17 31.5 48120 26 3 diverg. 2

1.6 50240 35 3 21.7 52370 53 3 2

1.8 51620 79 3 3

1.9 57040 116 3 3

2.0 59610 163 3 4

Table 2Membrane with central crack, number of iterations for difference in equivalent plastic strain < 10 -4.

MaximumLoad-factor

effective stress

Omax [lbf/in2 ]

Initia l stress approach Initial strain approach

NIM VIM NIM VIM

1.0 34500 . . . .

1.1 36930 5 4 22 5

1.2 38980 5 4 diverg. 4

1.3 40690 7 4 5

1.4 42180 6 5 51.5 43450 5 4 5

1.6 44550 5 5 5

1.7 45520 5 5 5

1.8 46390 7 5 5

1.9 47170 8 5 5

2.0 47880 7 5 5

quickly when the num ber of constraints is greater,

the converse is true for NIM-initial strain. Even for

?, = 1.1 t here are 22 it eratio ns necessary and diver-

gence occurs already for ~. = 1.2. Note t hat VIM-initi-

al strain requires approximately the same nu mber of

itera tions as VIM-initial stress.

In conclusion one can state tha t VIM improves

considerably the range of convergence of the initial

strain approach as well as the rate of convergence for

structures with few kinematic constraints. In the case

of co nti nuu m problems the initial stress formu lation

is suited best for autom atic comp utatio ns. It should

be mentioned that the improved iterative method

VIM is considerably more costly per iterati on step

than NIM depending on the numb er of Newton-vari-

ables involved. An opt imum approach in regard to

compu tation al effort is achieved if Newton variables

are considered only when they exceed a certain thres-

hold.

The application of the initial load tec hnique to

the solution of creep problems is but a special case of

the elasto-plastic algorithm. Foll owin g the DIM proce-

dure the creep strain increments are compute d and

converted into initial loads but now for the subse-

que nt time step instead of the subsequ ent load incre-

ment. In this case it is tacitly assumed that the stresses



19 2 J . 1 1 . A r g y r i s e t a l., N o n - l i n e a r m e t h o d s o f s t r u c t u r a l a n al y si s

5 0 ° 6 0 °7 0

8 0 ° 9 0 ° 10 0 ° 150 - -

i

A m b i e n t T e m p e r a t u r e T = 2 0 ° C 250 °

2 2 0 c

I

o 0 o /

I n s i d e T e m p e r a t u r e o f V e s s el T = 2 5 0 ° C

10 12515

~J

S t e a d y S t a t e T e m p e r a tu r e L o a d i ng

E q u i v a l e n t S t r e s s # k p / m m 2

~ / // // ~ R e g i o n o f P l a s t i c D e f o r m a t i o n

T i m e t = 0 d a y s

Fig . 29 . Spher ica l shel l w i th nozz le , s teady- s ta te t emper a tu r e

d is t r ibu t ion .

Fig. 31. Spher ical shel l with no zzle , equiva lent stress d is tr ibu -

t i o n .

G ~

k p /m m 2

f

2 o lf

i

10

0

- - r = 0 %

- - 2 0 0

4 0 0

6 0 0

oi , o;2 o13 o'4 o 's % p

Fig. 30. U niaxia l s tress-plas tic s train pro per t ies .

r e m a i n c o n s t a n t w i t h i n e a c h t i m e i n c r e m e n t . T h e

p r o b l e m o f t h e a p p r o p r i a t e c h o i c e f o r t h e t i m e s t e p

h as b een d i s cu s s ed i n s ec t i o n 4 w i t h i n t h e co n t ex t o f

t h e r a t e o f c r e e p m e t h o d . F o r f u r t h e r d e t a i ls s e e a ls o

[3 7 ] an d [3 8 ] . F o r an an a ly s i s o f e l a s to - p l a s t i c p h en o -

m e n a i n t h e p r e s e n c e o f la r ge d i s p l a c e m e n t s a n d

d y n a m i c e f f e c t s th e r e a d e r i s r e f e r r e d t o [ 4 4 ] .

6.3. Example

I n co n c l u s i o n a r ea l i s ti c t h e r m a l s t r e s s an a l y s is is

p e r f o r m ed o n a s p h e r i ca l p r e s s u r e v e ss e l w i t h a rad i a l

n o zz l e u s i n g th e i n i ti a l lo ad t e c h n i q u e . T h e i n s i d e o f

t h e v e s se l i s s u b j ec t ed t o s u d d e n t em p e r a t u r e r i s e fo r

w h i c h t h e s t e a d y s t a t e t e m p e r a t u r e d i s t r ib u t i o n i s

f i r s t d e t e r mi n ed . Su b s eq u en t l y , an e l a s t o - p l a s t i c s t r e s s

an a l y s i s i s c a r r i ed o u t a n d t h e i n s t an t an eo u s s t r e ss d is -



J. H. Argyr is et a l . , Non- l inear methods o f s t ructura l analysis 193

Steady State Temperature Load i ng

Eq u i va lent S t r es s # k p/ m m 2

/ t // / ~ R eg i on o f P l as t i c Deformation

. F ig. 32. Spherical shell with nozzle, equivalent stress distribu-tion.

t r i b u ti o n i s c o m p u t e d f o r t e m p e r a t u r e d e p e n d e n t

m a t e r i a l p r o p e r t i e s . T h e r e a f t e r , a c r e e p a n a l y s is i s

p e r f o r m e d t o t r a c e t h e s t r es s r e l a x a t io n a s a f u n c t i o n

o f t i m e . F i n a l l y , t h e s t r e ss d i s t ri b u t i o n i s d e t e r m i n e d

f o r a n i n t e r n a l p r e s s u r e u n d e r w h i c h l i m i t a n a ly s i s

w o u l d p r ed i c t p a r t i a l co l l ap s e .

T h e g eo m e t r y an d i d ea l i s a ti o n o f t h e v e s s e l i s i ll us -t r a t e d i n t h e c o m p a n i o n p a p e r o n l i n e a r m e t h o d s o f

s t r u c t u r a l a n a ly s i s ; 2 3 0 a x i s y m m e t r i c T R I A X C 6 e l e-

m e n t s a r e u s e d t o d i s cr e ti s e th e s t r u c t u r e w i t h 5 5 1

n o d a l p o i n t s f o r b o t h t h e t h e r m a l a n d t h e m e c h a n i c a l

a n a l y si s . N o t e t h a t t h e v a r i a t io n o f t e m p e r a t u r e - ,

p l a s ti c - a n d c r e e p s t r a in s is a p p r o x i m a t e d w i t h i n e a c h

e l e m e n t b y l i n e a r i n t e r p o l a t i o n o f t h e i n it ia l s t ra i n s.

F o r t h e t e m p e r a t u r e a n a l y s is t h e f o l l o w i n g b o u n d -

a r y c o n d i t i o n s a r e i m p o s e d . T h e i n s i de s u r f a c e o f t h e

o C

2 0 0

I ° ° 1

k p /m m 2

3 0

2 0

I 0

o iott

k p f m r n 2

10

(

- I 0

-2 0

-3 0

- z . o r

- - - L i n e a r T h e r m o - E l a s t i c S o l u t i o n

T h e r m o - E t a s t o - P l a s ti c S o l u t i o n

- - - S o l u t i o n a f t e r C r e ep fo r 1 81 D a y s\

Fig. 33. Sp herical shell with nozzle, stresses at sec tion A -C .

v e ss e l i s s u b j e c t e d t o a p r e s c r ib e d t e m p e r a t u r e o f

2 5 0 ° C w h i l e t h e a m b i e n t a i r t e m p e r a t u r e o n t h e o u t -s i d e r ema i n s a t 2 0 ° C . T h e u p p e r f l an g e r i n g an d t h e

b o t t o m r i m a r e in s u l a te d a g a i n s t h e a t f l o w . T h e t h e r -

m a l m a t e r i a l p r o p e r t i e s a r e a s f o l lo w s

T h e r m a l c o n d u c t i v it y

Sp ec i f i c h ea t

H e a t - t r a n s f e r c o e f f i c i e n t

D e n s i t y

k = 0 . 3 9 8 k c a l / c m h ° C ,

c = 0 . 1 1 4 k c a l / k g ° C ,

h = 0 . 0 4 2 6 k c a l / c m 2 h ° C ,

p = 7 . 9 X 1 0 - a k g / c m 3 .

T h e r e s u l t in g s t e a d y s t a t e t e m p e r a t u r e d i s t r i b u t io ni s i l l u s t r a ted i n f ig . 2 9 i n f o r m o f c o n t o u r l i n es .

T h e m e c h a n i c a l p r o p e r t i e s o f m i l d s te e l a re t a k e n

t o b e

E l a st ic p r o p e r t i e s E = 2 1 0 0 0 k p / m m 2 , u = 0 . 3 0 ,

T h e r m a l e x p a n s i o n ct = 1 . 7 - 0 . 7 ( 1 - 0 / 7 2 0 ° C ) 1 0 - s ,

( 4 9 )



194 J . 1-1. Argyris e t aL, Non - l inear m etho ds o f s tructural analysis

r~ m~

Fig. 34. Spherical shell with nozzle, equ ivalent stress distribu-tion.

P l a s t i c p r o p e r t i e s

R a m b e r g - O s g o o d s t r e s s - s tr a i n l aw o f e q . ( 4 8 )

w i t h O y = 2 o [ 1 - ( 0 / 8 5 0 ° C ) 2 ] k p / m m 2 , m = 2 0.

T h e t e m p e r a t u r e d e p e n d e n c e o f t h e s t r e s s - s tr a i n l a w

is i l lus t ra ted in f ig . 30 .

Cr eep p r o p e r t i e s ( f o r d e t a i l s s ee r e f . [4 0 ] .)

( 2 ~ 2 ) 6 .5 t11c 'a 3 .9 k p /m m 10 9 sec

- - s . ( 5 0 )

T h e i n c r e m e n t a l p r o c e d u r e s t a rt s f r o m a s tr e s s f r e e

r e f e r en ce s t a t e an d ap p l i e s a f r ac t i o n o f th e s t ea d y

s t a te t e m p e r a t u r e d i s t r i b u t i o n f o r e a c h l o a d in g s t e p .

T h e f i n a l t e m p e r a t u r e c o n d i t i o n i s r e a c h e d a f t e r 1 3

i n c r e m e n t s . S u b s e q u e n t l y , t h e s t e p w i s e c a l c u l a ti o n i s

c o n t i n u e d t o a c c o u n t f o r s e c o n d a r y c r e e p e f f e c t s . T h e

t i me i n t e r v a l i s au t o ma t i ca l l y ad j u s t ed i n s u Ch a man -

n e r t h a t t h e c h a n g e o f th e m a x i m u m e q u i v a l e n t s tr e s s

i n th e v e s se l n e v e r e x c e e d s 5 % o f t h e c u r r e n t m a x i m u m

e q u i v a l e n t s tr e ss . T h e p r o c e d u r e w a s t e r m i n a t e d a f t e r

2 1 i n c r e m e n t s w h i c h c o r r e s p o n d s t o a t i m e o f 1 8 1

d ay s . F i g s . 3 1 , 3 2 , 3 3 an d 3 4 i l l u s t r a t e t h e eq u i v a l en t

s t r e s s d i s t r i b u t i o n s a t d i f f e r en t i n s t an t s o f t i me . O b -

s e r v e t h a t t h e p l a s t i c zo n es ( s h ad ed a r ea s ) d ev e l o p

f i r s t a t t h e i n s i d e o f t h e f l an g e . T h i s o ccu r s d u e t o t h e

r e l a ti v e ly lo w t e m p e r a t u r e a t t h e o u t s id e o f t h e f la n g e

w h i c h p r e v e n t s t h e e x p a n s i o n o f t h e h o t i n n e r p o r t i o n ,

t h u s i n d u c i n g l a r g e co mp r es s i v e h o o p s t re s s e s . N o t e

t h a t i n t h e c a s e o f a t e m p e r a t u r e d e p e n d e n t y i e ld s u r-

f ace t h e ma x i m u m eq u i v a l en t s t r e s s i s n o t t h e o n l y

f a c t o r c a u s i ng p la s t ic f l o w , b u t r a t h e r a c o m b i n a t i o n

o f t h e s t a te o f s t r es s a n d t e m p e r a t u r e . F i g s. 3 3 a n d 3 4

i l lu s t r a t e t h e s t r e ss r e l ax a t i o n d u e t o c r eep cau s i n g a

s i g n i fi c an t r ed i s t r i b u t i o n o f s t r e s s es in t h e f l an g e.Fo r co mp ar i s o n , t h e s p h e r i ca l s h e l l w i t h n o zz l e i s

a l s o s u b j ec t ed t o a mo n o t o n i ca l l y i n c r ea s i n g p r e s s u r e

w h e r e b y t h e t e m p e r a t u r e a n d c r e e p e f f e c t s a r e di sr e -

g a r d ed . F i g . 3 5 s h o w s t h e eq u i v a l en t s t r e s s d i st r i b u -

t i o n f o r a p r e s s u re p = 3 3 . 4 k p / c m 2 a t w h i ch p l a s t i f i-

c a t i o n b eg i n s t o o ccu r . T h e i n t e r n a l p r e s s u r e i s f u r t h e r

i n c re a s e d t o 9 0 . 2 k p / c m 2 f o r w h i c h t h e t h e o r y o f

l i mi t an a l y s is p r ed i c t s a co l lap s e mec h an i s m a t t h e

j u n c t i o n b e t w e e n t h e n o z z l e a n d t h e s p h e r e . T h e s t re s s

d i s t r i b u t i o n s h o w n i n f ig . 3 6 i s a t t a i n ed a f t e r 1 7 l o ad -

i n g i n c r em en t s . I t i n d i ca te s ag a in t h a t l i m i t an a l y s isp r o c e d u r e s y i e l d b u t a c o n s e r v a ti v e lo w e r b o u n d f o r

t h e l i mi t p r e s s u r e . I n co n c l u s i o n o n e may s t a t e t h a t

a n i m p o r t a n t c l a s s o f t h e r m o m e c h a n i c a l p r o b l e m s c a n

b e s o l v ed b y t h e i n i t i a l l o ad t e ch n i q u e , a cco u n t i n g

f o r p l a s t i c i t y a s w e l l a s f o r c r eep w i t h o u t h a v i n g t o

u p d a t e t h e s t i f f n e s s p r o p e r t i e s w i t h i n c r ea s i n g l o ad

o r t i me s t ep s .

7 . C o n c l u d i n g r e m a r k s

A c l o se l ia i s o n b e t w e en t h e d i s c ip l i n e o f n u m er i ca l

an a l y s i s an d ma t e r i a l id en t i f i c a t i o n i s i mp e r a t i v e f o r

a s u cces sf u l an a l y s is o f p h y s i ca l l y n o n l i n ea r p h en o -

m e n a .

I n a f i n i te e l e m e n t a n a ly s is a c o m b i n a t i o n o f i n-

c r emen t a l an d i t e r a t i v e s o l u t i o n t e ch n i q u es , i n v o l v i n g

t h e t an g en t i a l s t i f f n e s s an d i n i t i a l l o ad me t h o d , f u r -

n i s h es a v e r y g en e r a l n u mer i ca l t o o l t o d ea l w i t h mo s t

n o n l i n e a r p r o b l e m s in r e a c t o r t e c h n o l o g y .

I n t h e f i e l d o f ma t e r i a l i d en t i f i c a t i o n t h e r e a r e a

n u m b e r o f m a t h e m a t i c a l m o d e l s f o r d e s c ri b in g d i ff e r -

e n t a s p e c t s o f m a t e r i a l b e h a v io u r :

T h e t h e o r y o f n o n l i n e a r e l a s t ic i t y f u rn i s h e s a n a p -

p r o p r i a t e f o r m u l a t i o n f o r t h e n o n l i n e a r d e f o r m a t i o n

b e h a v i o u r e s p e c i a l ly i f i t is s u p p l e m e n t e d b y t h e t h e o r y

o f f r a c t u r e t o a c c o u n t f o r li m i t e d s t re s s a n d d e f o r m a -

t i o n s t a t e s in t h e m a t e r i a l . T h e f l o w t h e o r y o f p l a s ti c -

i t y p r o v i d e s a m a t h e m a t i c a l m o d e l s u i te d t o d e s c r ib e

p a t h d e p e n d e n t a n d i r r e ve r s ib l e p ro c e s s es . T h e h e r e -

d i t a r y v i s co e l a s t i c f o r mu l a t i o n an d t h e en g i n ee r i n g



J. H. Argyris et aL, Non-linear methods of structural analysis 195

Internal Pressure

Eq u i va lent S t r es s ~ k p l m m 2

17.520

/

internal Pressure

Eq u i va lent S t r es s # k p/ m m 2

/ . // / // Region of P last ic Deform at ion

75

7S

/ " 0

12.5

15

/ 175

"~ 70

72.5

2S

27.5

32 5

22 30 35 325

Fig. 35 . S pherical shell with nozzle, equivalent stress distribu-tion.

Fig. 36. Sph erical shell with nozzle, equivalent stress distribu-tion.

t h e o r y o f c r e e p c a n b e u s e d t o d e s c r ib e t i m e d e p e n .

d e n t e n e r g y d is s ip a t io n w i t h o r w i t h o u t m e m o r y e f -

f ec t s .

I t s h o u l d b e e m p h a s i z e d t h a t t h e f i n i te e l e m e n t

m e t h o d is b a s e d o n e n e r g y p r in c i p le s , h e n c e t h e

c h o s e n c o n s t i t u t iv e m o d e l s h o u l d p r o v i d e a n a p p r o p r i -

a t e r e p r e s e n t a t i o n o f t h e e n e r g y d i s t ri b u t i o n w i t h i n

e a c h e l e m e n t a d d i n g a n o t h e r a p p r o x i m a t i o n t o t h e

d i s c r e t i s a t i o n o f t h e g e o m e t r y a n d t h e d e f o r m a t i o n

b e h a v i o u r .

A ck n ow l ed gemen t s

A s u r v e y s u c h a s a t t e m p t e d h e r e i n v o lv e s e x te n s i v e

t e a m w o r k . T h e a u t h o r s w o u l d l i k e t o t h a n k a l l t h e

m e m b e r s o f t h e I S D , in p a r t i c u l a r H . B a l m e r , J. S t .

D o l t s in i s , G . F a u s t a n d J . S z i m m a t w h o s e d e d i c a t e d

e f f o r t la i d th e f o u n d a t i o n s f o r t h e p r e s e n t p a p e r .

P a r t o f t h e w o r k r e p o r t e d i n t h is p a p e r i s s p o n -

s o r ed b y t h e B u n d e s m i n i s t e r i u m f 't ir B i l d u n g u n d

W i s s e n s c ha f t ( F S r d e r u n g s v o r h a b e n S B B 4 ) , w h i c h i s

g r a t e f u l ly a c k n o w l e d g e d .

References

[1] J. H. Argyris, Energy Theorems and Structural Analysis,Aircraft Eng. 26 (1954) and 27 (1956). A lso as book(Btttterworths London, 1960).

[2] K. W ashizu, Variational Methods in ElastieiW andPlasticity (Pergam on Press, 1968).



196 J . t t . A r g y r i s e l al .. N o n - l in e a r m e th o d s o f s t r u c tu r a l a n a ly s i s

[ 3 ] V . V . N o v o z h i lo v , T h e o ry o f E la s t i c i ty (P e rg a m o n P re s s ,

O x f o r d 1 9 6 1 ) .

[ 4 ] R . H . G a l l a g h e r , J . P a d lo g a n d P . P . B i j l a ard , S t re s s

A n a l y s i s o f H e a t e d C o m p l e x S h a p e s ,J . A m . R o c k e t S o c .,

3 2 ( 1 9 6 2 ) .

[ 5 ] J . H . A .rgy ri s, R e c e n t A d v a n c e s in Ma t r ix Me th o d s o f

S t r u c t u r a l A n a l y s i s ( P e r g a m o n P r e s s, L o n d o n 1 9 6 4 ) .[ 6 ] J . H . A rg y r i s , C o n t in u a a n d D is c o n t in u a , P ro c . 1 s t C o n f .

M a t r i x M e t h o d s S t r u c t . M e c h ., W r i g h t P a t t e r s o n A i r

F o r c e B a s e , O h i o ( 1 9 6 5 ) .

[ 7 ] J . H . A rg y r i s , E la s to -P la s t i c Ma t r ix D i s p la c e m e n t A n a ly s i s

o f T h r e e - D i m e n s i o n a l C o n t i n u a , J . o f t h e R o y . A e r o n .

S o c . 6 9 ( 1 9 6 5 ) .

[ 8 ] J . H . A rg y r i s , E la s to -P la s t i c A n a ly s i s o f T h re e -D im e n -

s i o n a l M e d i a , A c t a T e c h n i c a A c a d e m i a e S c i e n t i a r u m

H u n g a r i c e 5 4 ( 1 9 6 6 ) .

[ 9 ] J . H . A rg y r i s , D . W . S c h a rp f a n d J . B . S p o o n e r , D ie

e l a s t o p la s t i s c h e B e r e c h n u n g y o n a U g e m e i n e n T r a g w e r k e n

u n d K o n t i n u a , I n g e n i e u r A r c h i v 37 ( 1 9 6 9 ) .

[ 1 0 ] J . H . A rg y r i s , D . W . S c h a rp f a n d J . B . S p o o n e r , T h eE l a s to - P l a st i c C a l c u l a ti o n o f G e n e r a l S t r u c t u r e s a n d

C o n t i n u a , P r o c e e d in g s , 3 r d C o n f e r e n c e o n D i m e n s i o n -

i n g , B u d a p e s t ( 1 9 6 9 ) .

[ 1 1 ] O . C . Z ie n k ie w ic z , S . Va l l i a p a n a n d I . P . K in g , E la s to -

P la s t i c S o lu t io n s o f E n g in e e r in g P ro b le m s , In i t i a l S t r e s s ,

F i n i t e E l e m e n t A p p r o a c h , I n t . J . N u m . M e t h . E n g . 1

( 1 9 6 9 ) .

[ 1 2 ] J . H . A rg y r i s a n d D . W . S c h a rp f , Me t h o d s o f E la s to -

p la s t i c A ia alys is , P ro c . o f t h e IS D - IS S C S y m p . o n

F i n i t e E l e m e n t T e c h n i q u e s , I n s t i t u t f i ir S t a t i k u n d

D y n a m i k d e r L u f t- u n d R a u m f a h r t k o n s t r u k t i o n e n ,

S t u t t g a r t ( 1 9 6 9 ) , a l s o t o b e p u b l i s h e d i n a n e x t e n d e d

f o r m i n Z A M P .[ 1 3 ] Y . C . F u n g , F o u n d a t i o n s o f S o l i d M e c h a n i c s ( P r e n t i c e -

H a l l In c . , 1 9 6 5 ) E n g le w o o d C l i f f s , N . I .

[ 1 4 ] J . T . O d e n , F i n i t e E l e m e n t A p p l i c a t i o n s i n N o n l i n e a r

S t r u c t u r a l A n a l y s i s, P ro c e e d . S y m p . A p p l . F i n i t e E l e m t

Me th o d s in C iv i l E n g . , N a s h v i l l e , T e n n . , (1 9 6 9 ) Va n d e r -

b i l t U n i v . / A S C E .

[ 1 5 ] J . H . A rg y r i s a n d D . W . S c h a rp f , B e re c h n u n g v o rg e -

s p a n n t e r N e t z w e r k e , B a y e r i s c h e A k a d e m i e d e r W i s s en -

s c h a f t e n , S o n d e r d r u c k 4 a u s d e n S i t z u n g s b e r i c h t e n

( 1 9 7 0 ) .

[ 1 6 ] J . H . A rg y r i s , H . B a lm e r , J . S t . D o l t s in i s a n d K. W i l l a m ,

F i n i t e E l e m e n t A n a l ys is o f T h e r m o m e c h a n i c a l P r o b -

l e m s , P a p er p r e s e n t e d a t A i r F o r c e T h i r d C o n f e r e n c e o n

M a t r i x M e t h o d s i n S t r u c t u r a l M e c h a n i c s , D a y t o n , O h i o

( 1 9 7 1 ) .

[ 1 7 ] J . R o y , A l lg e m e in e M o d i f ik a t io n s v e r fa h re n fi i r d i e

l i n e a re t r o d n i c h t l i n e a r e B e r e h c n u n g y o n T r a g w e r k e n

u n d K o n t i n u a m i t d e r M a t r i z e n v er s c h i eb u n g s m e t h o d e ,

D r . l n g . T h e s is , S u b m i t t e d a t t h e U n i v e r s i t y o f S t u t t g a r t

( 1 9 7 1 ) .

[ 1 8 ] Y . N . R a b o t n o v , C r e e p P r o b l e m s i n S t r u c t u r a l M e m b e r s

( N o r t h - H o l l a n d P u b l . C o ., 1 9 6 9 ) .

[ 1 9 ] J . T . O d e n , F i n i t e E l e m e n t F o r m u l a t i o n o f P r o b l e m s o f

F i n i t e D e f o r m a t i o n a n d I r r ev e r s i b le T h e r m o d y n a m i c s

o f N o n l i n e a r C o n t i n u a , U . S . - J a p a n e s e S y m p o s i u m o n

R e c e n t A d v a n c e s i n M a t r i x M e t h o d s o f S t r u c t u r a l

A n a l y s is a n d D e s i gn , T h e U n i v e r s i ty o f A l a b a m a P r e s s

( 1 9 7 1 ) .

[ 2 0 ] K . T . K a v a n a g h a n d R . W . C l o u gh , F i n i t e E l e m e n t A p -

p l i c a t i o n s in t h e C h a r a c t e r i s a t i o n o f E l a s t ic S o l i d s, I n t .

J . o f S o l i ds a n d S t r u c t u r e s 7 ( 1 9 7 1 ) .[ 2 1 ] T . K a w a m o t o , K . T a n i t a a n d M . A k i m o t o , C h a r a c t e r is t i c s

o f D e f o r m a t i o n o f R o c k - l i k e M a t e r i al s u n d e r T r i a x i al

C o m p r e s s i o n , P a p e r 2 - 2 P r e s e n t e d a t t h e I n t e r n a t i o n a l

C o n f e r e n c e o n R o c k M e c h a n i c s , B e l g r ad ( 1 9 7 0 ) .

[ 2 2 ] C . V . G i r i j av a l l a b h a n a n d K . C . M e h t a , S t r e s s - s t r a i n

r e l a t i o n s h i p f r o m C o m p r e s s i o n T e st s o n N o n l i n e a r

M a t e r i al s , P r o c e e d . S y m p . A p p l . F i n i t e E l e m e n t M e t h o d s

in C iv i l E n g . , N a s h v i l l e , T e n n . , Va n d e rb i l t U n iv . /A S C E

( 1 9 6 9 ) .

[ 2 3 ] M . N . K h a n a n d B . S au g y , E v a l u a t i o n o f S o m e C o n c r e t e

C h a r a c t e r i st i c s o n N o n - L i n e a r B e h a v i o r o f a P C R V , A C I

S e m i n a r o n C o n c r e t e f o r N u c l e a r R e a c t o r s , B e r l i n (1 9 7 0 ) .

[ 2 4 ] J . L . S a c k m a n n , C r e e p i n C o n c r e t e a n d C o n c r e t e S t r u c -t u r e s , P r oc e e d i n g s o f t h e P r i n c e t o n U n i v e r s i t y C o n L o n

S o l id M e c h ., P r i n c e t o n ( 1 9 6 3 ) .

[ 2 5 ] Z . P . B a z a n t , P h e n o m e n o l o g i c a l T h e o r i e s o f C r e e p o f

C o n c r e t e B a s e d o n R h e o l o g i c a l M o d e l s, A c t a T e c h n i c a

C z . A c a d . S c i . 1 (1 9 6 6 ) .

[ 2 6 ] R . L . T a y lo r , K . S . P i s t e r a n d G . L . G o u d re a u , T h e r m o -

m e c h a n ic a l A n a ly s i s o f Vi s c o e la s t i c S o l id s , In t . J . fo r

N u m . M e t h o d s in E n g . 1 ( 1 9 7 0 ) .

[ 2 7 ] A . M. N e v i l l e , C re e p o f C o n c re t e : P l a in R e in fo rc e d a n d

P r e s t re s s e d ( N o r t h - H o l l a n d P u b l i s h in g C o ., A m s t e r d a m ,

1 9 7 0 ) .

[ 2 8 ] Y . R . R a s h id , U l t im a te S t r e n g th A n a ly s i s o f P re s t r e s s e d

C o n c re t e P re s s u re Ve s s e l s, N u c le a r E n g in e e r in g a n dD e s ig n 7 (1 9 6 8 ) 3 3 4 .

[ 2 9 ] H . C e d e r b e r g a n d M . D a v i d , C o m p u t a t i o n o f C r e e p E f -

fe c t s in P re s t r e s s e d C o n c re t e P re s s u re Ve s s e l s U s in g

D y n a m i c R e l a x a t i o n , N u c l e a r E n g i n e e r i n g a n d D e s ig n

9 ( 1 9 6 9 ) 4 3 9 .

[ 3 0 ] O . C . Z ie n k ie w ic z a n d M. W a ts o n , S o m e C re e p E f fe c t s

i n S t r e s s A n a l y s is w i t h P a r t i c u l a r R e f e r e n c e t o C o n -

c re t e P re s s u re Ve s s e ls , N u c le a r E n g in e e r in g a n d D e s ig n

4 ( 1 9 6 6 ) 4 0 6 .

[ 3 1 ] L . G . S e in a , C re e p , C ra c k in g , a n d S h r in k a g e in C o n c re t e

F r a m e S t r u c t u r e s , J o u r n a l o f t h e S t r u c t u r a l D i v is i o n

9 5 ( 1 9 6 9 ) .

[ 3 2 ] Z . P. B a z a n t , N u m e r i c a l l y S t a b l e A l g o r i t h m w i t h I n -c re a s in g T im e S te p s fo r In t e g ra l -T y p e A g e in g C re e p ,

F i r s t I n t e r n . C o n f . S t r u e t . M e c h a n i cs R e a c t o r T e c h n o -

lo g y , p a p e r H 2 /3 .

[ 3 3 ] V . H a n s s o n , W e r k s t o f f a n n a h m e n z u m K r i e c h e n tr o d

S c h w i n d e n d e s B e t o n s i n S p a n n b e t o n - R e a k t o r d r u c k -

b e h ~ i l te rn , B e r i c h t N r . 3 d e r F o r s c h u n g s g ru p p e S B B 6 ,

I n s t i t u t f . K o n s t r u k t i v e n I n g e n i e u r b a u , B o c h u m ( 1 9 7 0 ) .

[ 3 4 ] G . L . E n g l a n d a n d M . P h o k , T i m e D e p e n d e n t S t r es s es

in a L o n g T h ic k C y l in d r i c a l P re s t r e s s e d C o n c r e t e Ve s s e l

S u b j e c t e d t o a S u b s t a n t i a l T e m p e r a t u r e C r o s s fa l l,

N u c l e a r E n g i n e e r in g a n d D e s i g n 9 ( 1 9 6 9 ) 4 8 8 .



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