OF Ultimate Strength Analysis Prestressed Concrete Pressure Vessels

 

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 I G N 7 ( 1 96 96 8 ) 3 3 4 - 3 4 4 . N O R T H - H O L L A N D P U B L I S H I N G C O M P . ,

U L T IM

TE

OF PRESTRESSED

STREN GTH CO NCRE TE Y

R

N

AMSTE RDAM

L Y S IS

PRESSUR E

V E SS SS EL S

RASHI D

Gulf General Ato mi c Incorpo rated

San Diego

California

U SA SA

R e c e i v e d 1 9 M a r c h 1 9 68 68

A m e t h o d of o f a n a l y s i s o f p r e s t r e s s e d c o n c r e t e p r e s s u r e v e s s e l s s u b j e ct c t e d to t o p r e s s u r e o v e r l o a d is is p r e s e n t e d . T h e m e t h o d i s b a s e d o n t h e fi f i n i te te e l e m e n t v a r i a t i o n a l p r o c e d u r e . A c r a c k i n g c r i t e r i o n f o r c o n c r e t e i s d i s c u s s e d i n d e t a i l. l. T h e s t e e l e l e m e n t s a r e a s s u m e d t o b e e l a s t i c / p e r f e c t l y p l a s t i c . V o n - M i s e s y i e l d c r i t e r i o n a n d P r a n d t l - R e u s s f lo l o w eq e q u a t i o n s d e f in i n e th th e b e h a v i o r o f t h e liner in the range of plastic d eformations. T h e c o n s t i t u t i v e a n d e q u i l i b r i u m e q u a t i o n s o f th t h e t h r e e t y p e s o f e l e m e n t s u t i l i z e d i n th th e s t r u c t u r e , n a m e l y , c o n c r e t e , u n i a x i a l s t e e l a n d l i n e r e l e m e n t s , a r e d e r i v e d . T h e s e e q u a t io i o n s a r e v a li li d t h r o u g h o u t the entire range of deformations. The case conside red is one which involves two-dim ensional (axisym m e t r i c a n d pl p l a n e) e ) s t a t e s o f s t r e s s o n ly l y . A s t e p - b y - s t e p s o l u t io io n p r o c e d u r e , b y w h i c h t h e p r e s s u r e i s a p p l i e d i n c r e m e n t a l l y , i s ad a d o p t e d. d . T h e u t il i l i t y o f t h e m e t h o d i s d e m o n s t r a t e d b y a n ex ex a m p l e : a n d a l i m i t e d experimental correlation is given.

1. I N T R O D U C T I O N It is well known that conc rete cracks at a tensile stress level of about ten percen t of its ultimate compressive strength. This troublesome property, which has so conveniently been circumv e n t e d th r o u g h r e i n f o r c e m e n t a n d p r e s t r e s s in g , plays an impo rtant role in the ultimate strength analysis of prestresse d concre te reactor vessels ( P C R V s ) . T h e a n a l y s i s p r e s e n t e d h e r e w i ll ll s h o w that failure com pressive stresses, if they develop at all in a PCR V under pressu re overloa d, are of limited e xtent and can be ignored. Con seq u e n t ly , l i m i t a n a l y s i s m e t h o d s b a s e d o n t h e c o m pressive block concept, which have served us so w e l l in t h e u l t i m a t e s t r e n g t h a n a l y s i s o f b e a m type structures, are of question able validity when applied to PCRVs. In conne ction with the ultimate strength analysis of PCR Vs, one requires full know ledge of t h e m a t e r i a l f a i l u r e c h a r a c t e r i s ti c s u n d e r m u l t i axial stress states. Of particular interest in this analysis is that part of the failure surface on which e ach point has at least one positive coordinate. Unfo rtunately, the art of materia l charac terization of concre te at failure, especially in that region of the failure surface, is not suffic i e n t ly d e v e l o p e d . H o w e v e r , b a s e d o n e x i s t i n g data, a reasona ble approx imation of the failure s u r f a c e u n d e r i s o t h e r m a l c o n d i t io n s c an b e m a d e for present

purposes.

C o n c r e t e c r a c k i n g c r i t e r io n , l i n e r p l a s t i c it y and yield criterion will be discusse d in detail. The plasticity of the uniaxia l steel m emb ers, n a m e l y , b o n d e d r e i n f o r c e m e n t a n d p r e s t r e s s in g cables, is elementary and will not be given special consideration. The stress-strain relations for the concre te a n d l i n e r , v a l i d th r o u g h o u t t h e e n t i r e r a n g e o f d e f o r m a t i o n s , a r e d e r i v e d . T h e s e e q u a t io n s a r e then used to derive the equilibrium equation s of t h e in d i v i d u a l e l e m e n t s t h r o u g h t h e m e c h a n i s m of the virtual work principle. An increm ental sol u t i o n t e c h n i q u e i n w h i c h t h e p r e s s u r e i s in creased linearly within each step is adopted. The e q u i l i b r iu m e q u a t i o n s a r e s o l v e d i n e a c h i n c r e m e n t by t h e o v e r - r e l a x a t io n i t e r a ti v e m e t h o d ( s e e r e f . [1 [ 1 ]) ]) . E x a m p l e a n a l y s e s a n d c o r r e l a t i o n with experiment are presented.

2.

CON CRETE

CRACKING CRITERION

In discussin g concre te cracking w hen a mu ltiaxial stress state exists, one is actually referring to that part of the stress failure surface on which each point has at least one positive c oordina te ( s e e f i g . 1 ). ) . T h e t a n g e n t a t a n y p o i n t o n th th e s u r face determ ines the surface of the crack for the stress state defined by the coordinates of that point. Existen ce

of the failure su rface

s h o w n i n f ig ig . 1

 

ANALYSIS ANALY SIS OF PRES TRESS ED CONCRETE PRESSU PRESSURE RE VESSE VESSELS LS

cr

335

lack of expe rimen tal data, it is not possib le to t a k e t h e s e tw tw o v a r i a b l e s , n a m e l y t i m e an an d t e m p e r a t u r e , i n to to a c c o u n t a t t h i s ti ti m e . I n f a c t , s u f f i c i e n t d a t a to to d e f i n e a s h o r t - t i m e c r a c k i n g c r i terion hardly exist. I n v i e w o f th th e t w o - d i m e n s i o n a l n a t u r e o f t h e a n a l y s i s p r e s e n t e d h e r e a f a i l u r e s u r f a c e o f t he he general shape shown i n fi g. 1 cannot be accommodat ed. Thi s i s due to to t he fact t hat t he fai l ure surface tangent and consequently the crack surface, are of general three-dimen sional orienta t io io n . In In a x i s y m m e t r i c a n a l y s i s , h o w e v e r , c r a c k s can exist only in two-directions: in a plane parall el to to and cont ai ni ng the the axi s of sy mm et ry , and i n c l o s ed ed c o n i ca ca l s u r f a c e s . T h e a x i s y m m e t r i c s t r e s s failure surface , ther efor e, takes on the shape shown in fig. 3. In this f ig ur e, cr3 is in the hoop direction and ~1 and cr2 cr2 a r e . i n t h e plane. In t he anal ysi s t hat fol l ows t he dynami cs of t he c r a c k i s i g n o re re d . F u r t h e r m o r e , t he he m a t e r i a l is is assumed to remain elastic for any stress state bel ow t he cracki ng st rengt h. z

Fig. 1. 1. Failure s urface of concrete. implies isotropic behavior for all stress states within the region enclosed by the failure surface. U n l ik ik e p e r f e c t l y p l a, a, ~t ~t ic ic y i e ld ld s u r f a c e s , w h i c h c h a r a c t e r i z e a c l a s s o f d u c t i le le m a t e r i a l s t he he liner for example), this failure surface is not r e p r o d u c i b l e u p on on r e l o a d i n g f o l l o w i n g u n l o a d in in g from a failure str ess state. In other words, once a failure st res s state is reach ed along a virgin path the failure surface immediately collapses to a l ower sur fac e of t he type type shown i n fi g. 2. 2. Col l a p s e of of t he he t e n s i o n - c o m p r e s s i o n r e g i o n s c o n t i nues unt i l t he fai l ure surface i s bounded by t he zero principal stres~s planes. The failure surface of concrete is, in general, t i m e a n d t e m p e r a t u r e d e pe pe n d e n t. t. H o w e v e r , f o r

3. STRESS- STRAIN RELATIONS FOR CRACKED E LEM ENT Consi d er an el e men t , shown i n fi g. 4, cra cke d in the pl ane at an angl e a wi t h t he hori zont al a x i s . F o r o p e n, n, i . e . s t r e s s f r e e , c r a c k , t h e a x e s m u s t b e p r i n c i p a l a x e s . B e f o r e we we d i s c u s s the stres s state of the cracked e lement we consider the relationships between the principal stresses and strains and the coordinate stresses and strains. Specifically we write z

z

o-FF o-

/j

f

\~

• ~

fJ j j

J

f

7/

J

Fig. 2. Collapsed failure surface of concrete,

Fig. 3. Failure surface of concrete for axisymm etri c str ess es.

 

336

Y.R. Y. R. RASHID Conservation of energy gives cT~ = ~T ~T~ ~ = TR T~ R E : ETH~ ETH~..

6)

where RT R T is the tra nsp ose of R, etc. Fr om (6 (6), ), H = R T/ ~R .

T h e m a t r i c e s o f e l a s t i c c o e f f i c i e nt nt s , n a m e l y H a n d H , a r e n o t r e s t r i c t e d t o t he he i s o t r o p i c c a s e but, for the purp ose of this analy sis , involve the ortho tropi c constants . The axes of orthotropy are r, z and 8. At this point we propose to treat the influence of a crack on a continuous elemen t as a mech an i s m t h a t c h a n g es es t h e e l e m e n t s b e h a v i o r f r o m isot ropi c to orthotro pic. In this new state the element will have no stiffness in a direction norm a l to to t h e c r a c k s u r f a c e . T h i s s i m p l y m e a n s t h at at t he he m a t e r i a l c o n s t a nt nt s in in t he he s t r e s s - s t r a i n r e l a t i o n s , w h i ch ch a r e i n t h e m s e l v e s s t i f f n e s s c o e f f i c i e n t s , a r e i d e n t i c a l l y z e r o i n th th e d i r e c t i o n

~r Fig. 4. Cracked element.

1 1 1 31 3 1 ° s 2 in 2 ° s n 2 1 t 2

= sin2a

cos2a

0

0

1

-s in 2 0

azz

,

(la)

to ~

t < ti

|

i=1,2,3

a n d z e r o o t h e r w i s e . E x p r e s s i n g g( g( t ) i n t e r m s o f the unit step function h(t), h(t), we obtain

le ll

e3

of the normal to the crack. O n t h is is b a s i s , t he he s t r e s s - s t r a i n r e l a t i o n s f o r a c racked ele men t can be convenien tly defined. Consid er the pie cewis e continuous function g(t) shown in fig. fig. 5. Thi s functi on has a valu e of unity in the intervals

~aO0~

and

e2

(7)

=

i

coi2a

sin2a

0

½sin2a

si 2a

cos2a

0

~½s in2 a

0

1

0

ezz ,

Ib)

°°t

lerzl

w h e r e a l , e l e t c . a r e p r i n c i p a l q u a n ti ti t i e s. s. W r i t ing eqs. (la) and (lb) symbolically *, ff = P a ,

(2)

~-= ~= Re.

(3)

t

tO e t)

and to

The elastic stress-strain relations in the two coordinate systems are cr = He

4)

and = He

;

5)

g(t)

tl

T. to

tl

Unless needed needed for clarit y, the use of brackets to identify mat ric es will be dispens ed wi with th..

Fig. 5. Element crack ing histo ry.

t

 

ANALYSIS OF PRESTRESSED CONCRETE PRESSURE VESSELS g t) = h t) - eC t),

ca)

where t i)i ) ,

e(t) = h (t-

i = 1, 2, 3 ,

(9)

and I t ( t)t ) =

l lfort >0 0fort <

;

h (t -

li) =

llfort>ti

0

0fort