Stress-Strain Curve for the Design of High-strength Concrete Elements

7/27/2019 Stress-Strain Curve for the Design of High-strength Concrete Elements

1/8

Mater ia ls and St ruc tures /Mat6r iaux et Const ruc t ions ,Vol. 33 , August-September 000, pp 411-418

Stress-strain curve for the design of high-strengthconcrete elementsI brahim A . E . M . Shehata 1, L idia C . D . Shehata and Tales S. M attos(1) C O PPE, UniversidadeFederal doRi o de aneiro, Ri o de aneiro, Brasil.(2) UniversidadeFederal Fluminense; CO PPE, UniversidadeFederaldo Rio deJaneiro, Rio deJaneiro, Brasil.(3) UniversidadeFederal Fluminense, Ri o deJaneiro, Brasil.

Paper received: November 3, 1999; Paperaccepted: March 23, 2000A B S T R A C T R i~ S U M i~

Although the great advances in concrete technologyhave led to the possibility of obtaining ready-mix concretewith compressive strength around 100 MPa, some nationaland international codes for concrete structures do notcover concrete strengths above 50 MPa. Many codes areunder revision, but some o f them (including the BrazilianCode) will still not inclu de high stren gth concretes.Due to the different characteristics of higher strengthconcrete some design procedures traditionally used innorm al strength concrete structures have to be changed.

Different types of stress-strain relationships for con -crete have been proposed for the non-linear analysis ofm e m b e r behavior and for the ultimate state analysis ofhigh strength concrete elements under com bined flexureand axial load.In this work comparisons are made bet ween prop osedstress-strain curves and between the axial load -m om entinteraction diagrams based on these curves. Compa risonsof test results with these diagrams, for columns subjectedto eccentric compression, give an idea of the differentdegrees of safety obtained using those curves.

Malgr~ les grands progr~s de la technologie en mati~re debSton, qui ont rendu possible l'obtention de bStons r&istant a unecompression d'environ IOOMPa, quelques codes nati onaux etinternat ionaux relatifs aux constructions en b~ ton ne couvrentpas des r&istances du bSton sup&ieures a 50 MPa. Plusieurscodes sont en cours de r&ision, mais beaucoup (dont le code br&i-lien) ne s'int&essent toujours pas a ux b~tons a haute r&istance.

En raison des diff&entes caract&istiques des b~tons a hauter&istance, certains procM& de dSfinition traditionnellement utili -s& pour les constructions en bSton normal doivent dtre mode&.

Diff&ents types de relations contraintes - d~formations ont ~t~propos& pour le bdton clans le cadre de l'analyse non-lin&ire deson comportement e t clans celui de l'analyse inale des ~l~ments enb&on arm~ en lexion combin& avec une charge axiale.

Ce t article compare les courbes contraintes - d~formations pro-pos&s et les diagrammes interaction charge axiale - mom ent quir&ultent de ces courbes. Les comparaisons effectu~es entre lesr&ultats exp&imentaux et ces diagrammes, pour des colonnessoumises a des compressions excentriques viennent illustrer les d!f-f&e nts degr& de s~curit~ obtenus grdce aces courbes.

1. INTRODUCTIONHigh-strength concrete can be the most cost-effectives o lu t ion fo r m any s t ruc tu ra l cas es wh i le providingimp rov ed durability. T his con crete is particularly advan ta-geous in compression members and the concrete stress-strain relationship is one of the main material propertiesrequired for the analysis of their behavior. A realistic c on -stitutive law for concrete un der compression makes it pos-

sible to do an accurate analysis of the member behaviorand can be used as a reference for the validation of simpli-fied and ap proximate calculation method s.

In recent years many stress-strain curves have beensuggested to cover different strength classes of concrete.

These curves were based on experimental results ofspecimens under axial compress ion, usual ly s tandardcyl inders. In determ ining the behavior o f a st ructuralelemen t under com bine d bendi ng and axial force, a keyquestion arises, that is whet her the stress-strain relation-ship obtained by testing a concrete cylinder can be used.Studies repo rted in [1] showed similarity betw een t heascending branch of the stress-strain curve for concretecolumns with rectangular cross sect ion submit ted toeccentric compression and that for cylinders tested inaxial compression, being c olum ns and cylinders cured inthe same laboratory environm ent. T his suggests that thecurves obtained from cylinders can be used for mem bersunder combined bending.

1359-5997/00 9 tLILEM 41 1


2/8

M a t e r i a ls a n d S t r u c t u r e s / M a t 6 r i au x e t C o n s t r u c t i o n s ,Vol. 33 , August-September 0 00S o m e o f t h e s e c u r v e s p r o p o s e d i n t h e l i te r a t u re a r ep r e s e n t e d h e r e a ft e r a n d c o m p a r e d w i t h e a c h o t h e r ( c o m -

p r e s s i v e s t r e s s e s a n d s t r a i n s a r e c o n s i d e r e d p o s i t i v e ) .A p p r o x i m a t e d e s i g n s t r e s s - s t r a i n c u r v e s a n d r e c t a n g u l a rs t r e s s b l o c k s a l r e a d y s u g g e s t e d f o r c a l c u l a t i n g t h e c o m -b i n e d b e n d i n g c a p a ci ty a re a l so g i v e n , w i t h e x a m p l e s o ft h e i r in f l u e n c e o n t h e a x ia l f o r c e - m o m e n t i n t e r a c t io nr es i s tance d iagr ams .

2 . R E S E A R C H S I G N I F I C A N C ED u e t o t h e d i f f e r e n t c h a r a c t er i st i c s o f h i g h e r s t r e n g t h

c o n c r e t e , f o r d e s i g n i n g s t r u c t u r e s m a d e w i t h i t , s o m ed e s i g n p r o c e d u r e s t r a d it i o n a ll y u s e d i n n o r m a l s t r e n g t hc o n c r e t e s t r u c t u r e s h a v e t o b e c h a n g e d .D i f f e r e n t t y p e s o f s t re s s -s t ra i n r e l a t io n s h i p s f o r c o n -c r e t e h a v e b e e n p r o p o s e d f o r t h e n o n - l i n e a r a n a l y s i s o fm e m b e r b e h a v i o r a n d f o r t h e u l t i m a t e s t a te a n al ys is o fh i g h s t r e n g t h c o n c r e t e e l e m e n t s u n d e r c o m b i n e d f le x u r eand ax ial load .I n t h is w o r k c o m p a r i s o n s a re m a d e b e t w e e n p r o -p o s e d s t r e s s - s t r a i n c u r v e s a n d b e t w e e n t h e a x i a l l o a d -m o m e n t i n t e r a c ti o n d i a g r a m s b a se d o n t h e se c u r v e s.C o m p a r i s o n s o f t e st re s u lt s w i t h t h e s e d ia g r a m s , fo rc o l u m n s s u b j e c t e d t o e c c e n t r i c c o m p r e s s i o n , g i v e a ni d e a o f t h e v a l i d i t y a n d t h e d i f f e r e n t d e g r e e s o f sa fe t yo b t a i n e d u s i n g t h o s e c u r v e s .

3 . R E A L I S T I C S T R E S S -S T R A IN C U R V E S F O RC O N C R E T E

R e a l i s t i c s t re s s -s t ra i n c u r v e s f o r c o n c r e t e i n c o m p r e s -s i o n a re n e e d e d f o r t h e n o n - l i n e a r a n a ly s is o f s t r u c t u r a lm e m b e r s . T h e m a i n c h a r a c te r is t ic s o f th e s e c u r v e s a ret h e t a n g e n t m o d u l u s o f e l a st i c it y a t t h e o r i g i n , t h e s t ra i na t p e a k s t r e s s , w h i c h i n c r e a s e s w i t h t h e i n c r e a s e i n t h ec o n c r e t e s t r e n g t h , t h e s h a p e o f t h e s o f t e n i n g b r a n c h t h a tb e c o m e s s t e e p e r f o r c o n c r e t e w i t h h i g h e r s t r e n g t h a n dt h e u l t i m a t e c o n c r e t e s tr a in .T h e p r o p o s e d s t re s s -s t ra i n c u r v e s f o r t h e c o n c r e t e a r ee i t h e r c o n t i n u o u s f u n c t i o n s f o r t h e a s c e n d i n g a n dd e s c e n d i n g b r a n c h e s o r t w o s e p a r a t e f u n c t i o n s . T h r e es t r e s s - s t r a i n m o d e l s w e r e s e l e c t e d f o r t h e i n v e s t i g a t i o nr e p o r t e d h e r e i n t o r e p r e s e n t t h o s e t y p e s o f f u n c t i o n s .T h e a p p l i c at i o n o f th e s tr e ss -s tr a in c u r v e o f C E B - F I PM e 9 0 [2 ] i s l i m i t e d t o f& = 8 0 M P a a n d i ts c o n s t a n tva lue o f s t ra in a t peak s t re s s ( eco = 2 .2 % o) doe s no t agr eew i t h e x p e r i m e n t a l r es u lt s. T h e M e 9 0 e q u a t i o n fo r th ea s c e n d in g b r a n c h a n d t h e d e s c e n d i n g b r a n ch u p t o as tr ess l eve l % = 0 .5 f~ , an d th e eq ua t ion f o r the s t r a in ( 81)at thi s s tress level are:

( c l ( c l 2~__f._c \ 8 co I \ 8 co J ( 1 )

81 = ( 0 .2 5 A + 0 . 5 ) + 0 . ~ - ( 0 . 5 A + 1 ) 2 - 0 . 5 (2)8 c oa n d t h e e q u a t i o n f o r t h e r e s t o f t h e d e s c e n d i n g b r a n c h i sg i v e n b y :

, /]f c I ( 8 1 / ( g l 1 2 I ~ ,g c o + _ _ - ~ " (~ o co ( 3 )[ [ \ e c ~ \ ~ ' co J ; \ ~ ' ec ~ ;

w h e r e

4 . [ ( 8 1 / 2 . ( A - 2 ) + 2 . ( 8 1 I - A ][ \ 8 c o : \ e c ~ ] (4)

\ 8 c o ) JA = E c t = E c t 9 I~c~Ecs f c ( 5 )

- 0 . 3 3/a n d E c t a n d E c s a r e t h e t a n g e n t m o d u l u s o f e l a s ti c i ty a tt h e o r i g i n a n d t h e s e c a n t m o d u l u s o f e l as t ic i ty a t p e a kstress , respectively.I n C E B - 2 2 8 [ 3 ], f o r 5 0 M P a _< f c -< 1 0 0 M P a , t h ee q u a t i o n s f o r t h e a s c e n d i n g b r a n c h a r e t h e s a m e a s C E B -F I P M e 9 0 , b u t w i t h a s li g h t d i f fe r e n t v a l u e f o r E c t (E c t=2 2 0 0 0 [ f 1 0 ] ~ F o r t h e w h o l e d e s c e n d i n g b r a n c h o ft h e c u r v e a d i f f e r e n t e q u a t i o n a n d a v a l u e o f e~ o t h a td e p e n d s o n fc a r e p r o p o s e d a n d g i v e n a s f o l lo w s :

= 1 ( 7 )f c ( 8 c _ 1 " ~ 2l + / > j8 c o = 0 . 7 f e0 ' 3 1 " 1 0 3 ( 8 )

eco + 1Vl - - - (9 )g c oT h e v a l u e s o f (t ) a r e g i v e n i n t h a t r e f e r e n c e a s a f u n c -t i o n o f t h e c o n c r e t e s t r e n g t h ( s ee T a b l e 1 ) . I n t h is w o r ka n e q u a t i o n t h a t a p p r o x i m a t e s t h e v a l u e s o f T a b l e 1 a s af u n c t i o n o f t h e c o n c r e t e s t r e n g t h ( i n M P a ) is u s ed :t = 2.45 - 3 8 f & . 10 3 + 7.083f .1 0 -5 + 6.574fc 9 0 7 (10)

I t is w o r t h n o t i n g t h a t i n a l l t h e e q u a t i o n s o f t h eT a b l e 1 - V a l u e s o f ( t) a s g i v e n b y t h e C E B b u l l e t i n - 2 2 8 [ 3 ]fc k ( M P a ) 5 0 6 0 7 0 8 0 9 0 1 0 0

t ( % o ) 0 . 8 0 7 0 . 5 7 9 0 . 3 3 8 0 . 2 2 1 0 . 0 7 0 0 . 0 1 5

4 1 2


3/8

I . Shehata , L . Sheh ata , T . S . M at tos

d

v6

I O 0

s o M C ~ 9 0o

= " : : : : . . . . . . . - ' 7 " e eO 0 1 2 3 4 5Ir ( E - 3 )

l o o c [ ~ - f f T .

. . . .. . . . . . . . . . . . . g . . . . .

O0 i 2 3 4gr ( E - 3 )

f c = 20 MPaIc = 30 MPao. .fr = 50 MPafr = 70 MPa~.fc = 80 MPa

fc = 20 MPa*~f c =3 0 M Pa.. .f c = BO MF'afc = 70 M P af c = 90 MPao.fr = 100 M P a

zo o I ] *c =ZOMPa8 0 - - ~ C o ll in s e t a l [ fC =gMPa

~.-' ~. fc = 50 MPa

60 9 * ~ fc =~O'MPao.I. fc =90 MPa

. / . - , : 5 : . . . . - _

O0 1 2 3 4 58e ( E - 3 )

F i g . 1 - R e a l i s t i c s t r e s s - s t r a i n c u r v e s f o r c o n c r e t e , a c c o r d i n g t oC E B - F I P M C 9 0 , C E B b u l l e t in 2 2 8 a n d C o l l i ns et al.

10 080

60

, 0

2110

t. . '.,

, ~~I , o ~ t .

) 1 2 3 4 S~ r ( E - 3 )

CE~-22890

MC95~0228

S O

~ 2 ~ 0 2 2 8Collins e l e l

2 0

F i g . 3 - C o m p a r i s o n b e t w e e n t h e r e a l i s t ic s t r e s s - s tr a i n c u r v e s f o rc o n c r e t e s w i t h f c = 2 0 M P a , 5 0 M P a a n d 9 0 M P a .

C E B - M C 9 0 [2 ] a n d C E B - 2 2 8 [ 3] , t h e v a l u e o f fc isr e f e r r e d a s t h e a v e r a g e c o n c r e t e s t r e n g t h ( f c m = fck + 8M P a ) a n d n o t t h e c h a r a c t e r i s ti c o n e .

T h e e q u a t i o n o f t h e c u r v e s u g g e st e d b y C o l l i n s e t a l .[ 4] is t h e s a m e f o r b o t h t h e a s c e n d i n g a n d d e s c e n d i n gb r a n c h e s , b u t h a s a c o e f f i c i e n t t h a t i s c o n s t a n t f o r t h ea s c e n d i n g b r a n c h a n d v a r i es w i t h fc f o r t h e d e s c e n d i n gb ra n c h , so a s :

% ( e c I n ( t l )- f f = \ 7~co J n _ 1 + ( % 1 n k

\ gco J

( 1 2 /e c o = n - 1w he re k = 1 for gc - 8co , k = 0 .67 + ( fc /62) for ~co - co ,n = 0 .8 + ( fc /17)a n d E a = 3 3 2 0 ( fc )0.5 + 6 9 0 0 w i t h f c i n M P a .

E x a m p l e s o f t h e s e c u r v e s a r e g i v e n in F i g .1 . T h ec o m p a r i s o n m a d e i n F i g .3 b e t w e e n C o l l in s [ 4] an dC E B - 2 2 8 [ 3] s h o w s t h a t t h e g r e a t e s t d i f f e r e n c e b e t w e e nt h e m i s i n t h e d e s c e n d i n g p a r t o f t h e c u r v e s , w h i c h isa f f e ct e d b y m o r e p a r a m e t e rs t h a n t h e a s c e n d i n g o n e .

4 . D E S I G N S T R E S S - S T R A I N C U R V E S F O R rC O N C R E T E

A t u l t i m a t e l i m i t s t a t e , m o s t d e s i g n c o d e s u s e s i m p l i -f i e d s t r e s s- s t ra i n c u r v e s f o r c o n c r e t e i n c o m p r e s s i o n .

T h e p a r a b o l a - r e ct a n g l e s t re s s -s t ra i n c u r v e c o m m o n l yu s e d i n m a n y c o u n t r i e s ( i n c l u d i n g B r a z i li a n N B P , , [ 6] ),w h i c h w a s f ir s t s u g g e s t ed b y C E B , i s g i v e n b y :

a) f or ~c -< ~co = 2 % 0rS c 2 I g c ) ( g c / 2

0,8 5fc k "\ e ~ o : - \ e---~-o (13)b) fo r 2 % o _< eco -< Zcu = 3 .5 % 0

G c0 , 8 5 L ,- - ] ( 1 4 )T h e a b o v e - m e n t i o n e d e q u a t i o n s w e r e l a t e ly a d a p t e d

i n th e C E B - F I P M C 9 0 [ 2] f o r c o n c r e t e s w i t h 5 0 M P a


4/8

M a t e r i a ls a n d S t r u c t u r e s / M a t ~ r ia u x e t C o n s t r u c t i o n s , Vol. 33 , A ugust-September2 0 0 0

normal~force . I t ma y a l so be u sed for n on - l inea r ana lysi s,b u t c a n n o t p r e d i c t t h e p o s t - u l t i m a t e b e h a v i o r . T h e c o n -c r e t e s t r a i n a t t h e p e a k s t r e s s , t h e u l t i m a t e c o n c r e t es t ra i n , a n d t h e p e a k s tr es s o f th i s c u r v e d e p e n d o n f ck .A c c o r d i n g t o t h is c o d e :a ) for 0 < % < 0 ,6 f c n '

G c - - 8 cf c n F ' c n (20)

b) fo r 0,6 f~n < % < fcn,

t~c _ gcf e n 8 c n

w i t h

( ~ n o _ l / . ( : c C n - 0 " 6 i ~

ec 0.6a - % ' - ~- - -- - ( 2 2 )

%0 1~ c n

~c n- f~" (23)Ec.In these equa t ions , the n om ina l m od ulus o f e la s ti c ity ,t h e n o m i n a l c o m p r e s s iv e s tr e n g t h o f c o n c r e t e , th e c o n -c r e t e d e f o r m a t i o n a t p e a k s t r e ss a n d t h e u l t im a t e d e f o r -m at ion a re g iven as :Ec,1 = 10000 fcn~ (24)f c n= 0 . 7 0 f ck + 2 . 8 fo r f ck -< 4 4 M P a ( 2 5- a)

f e n = 0 . 5 6 f c k + 8 . 9 6 f o r 4 4 M P a _< f c k -< 9 4 M P a (25-b)gco = (1.9+ 0.00 4 f~n)%0 (26)~cu = 2,5 eco - 1.5 (fcn/Ecn), (27)wh ere f& and fcn a re in MPa .The s t re s s - s t ra in d iagrams based on these equa t ions ,for di fferent values of fck and safety coeff ic ient 3 'c takenequa l to uni ty , a re shown inF i g . 2 . T h e N B R [ 6 ] d i a -g r a m s a r e s i m i l a r t o t h eM c g 0 o n e s f or f c k < - 5 0M P a . I t c a n b e s e e n f r o mF i g . 4 t h a t t h e d i f f e r e n c e sb e t w e e n t h e d i a g r a m s a r eg r e a t e r f o r h i g h e r c o n c r e t es t rengths . The part ia l factorof s a fe ty for the ma te r i a l Yci s e q u a l t o 1 . 5 f o r t h eM C 9 0 a n d 1 . 4 f o r t h e N Sa n d t h e N B R ; f o r t h e C E Bbul l e t in 22 8 curv e , 7c i s 1 .5w h e n fc k -< 5 0 M P a a n d[1.5/(1.1 - fck/500)], with fck

i n M P a , w h e n f ck > 5 0 M P a .I f t h e s e v a l u e s a r e i n t r o -d u c e d i n t o t h e d e s i g n c u r v ee q u a t i o n s , f u r t h e r d i f f e r -

402 0 - - -

O0 0 , 5 1 1 ,5 2 2 ,5 3 3 ,5 4e c ( E - 3 )

o ~ 3 , U - - " 4~;c (E-3 )

0 0 , 5 1 1 , 5 2 2 5 3 3 , 5 4E:~" e -3 )

F i g . 2 - Design stress-stra in c u r v e s f o r c o n c r e t e , a c c o r d i n g t oC E B - F I P M C 9 0 , C E B bul l e t i n 2 2 8 a nd N S 3 4 7 3 ( Yc = 1 ) .

e n c e w i l l b e n o t i c e d .In the des ign o f s t ructural m em bers , a rectangu lar s t ressb lock for the concre te is a l so norm al ly used . I t can predic tt h e r e s i s t a n t m o m e n t a n d a x i a l f o r c e w i t h a r e a s o n a b l eaccurac y, bu t i t s greate s t advantag e, s implici ty, i s no t soimportant nowadays , s ince des ign by hand calculat ions isra re ly done . Di f fe rent va lues o f s tre ss and he ight have bee iasugges ted for this s impli f ied s t ress block, and examples o fthe m are given in Table 2. This table includ es a lso the val-

T a b l e 2 - V a l u e s p r o p o s e d f o r s tr e ss a n d h e i g h t o f t h e c o n c r e t e r e c t a n g u l a r s t re s s b l o c kA u t h o r o r c o d e S t re s s H e i g h t O t h e r d e s ig n d a t aI

C E B - F IP M C 9 0 [ 2 ] I 0 . 8 5 ( 1 - f c i J 2 5 0 ) f ed ~ cu = ( 4 - 0 . 0 2 f ck )% OX ~ s u = 1 0 % 0f ed = f c k / l . 5 f ck < 8 0 M P a

C A N 3 - A 2 3 . 3 [ 7 ] ( 0 . 8 5 - 0 . 0 0 1 5 f c ) f e > 0 . 6 7 f c ( 0 . 9 7 - 0 . 0 0 2 5 f c ) x >_ 0 . 6 7 x e cu = 3 . 5 % 0f c k -< 8 0 M P a

0 . 8 5 x f o r f c_ < 2 8 M P aA C 1 3 1 8 [ 8 ] 0 . 8 5 f c % u = 3 % 0

[ 0 . 8 5 - 0 . 0 5 ( f c - 2 8 ) / 7 ] x >_ 0 . 6 5 xf o r f c > 2 8 M P a

I br ah im a n d ( 0 . 8 5 - f j 8 0 0 ) f c- > 0 . 7 2 5 f c ( 0 . 9 5 - f c / 4 0 0 ) x >_ 0 . 7 x % u = 3 % 0M a c G r e g o r [ 9 ]

0 . 8 5 f ck % u = 3 . 5 % o ( b e n d i n g )N B R 6 1 1 8 [ 6 ] % u = 2 . 0 % 0

fe d = f c k / 1 . 4 0 . 8 x ( a x i a l c o m p r e s s i o n )~ su = 1 0 % 0

f c , f c k a n d c d i n M P a x = n e u t r a l a x i s d e p t h

4 1 4


5/8

I. Sh ehata, L. Shehata,T. S. Mattos

~176

t 1! i0 *0 ~ 5 1 ~ 5 2 Z 5 3E c (E .3 )

ii. i3. 5

Fig . 4 - C o m p a r i s o n b e t w e e n t h e d e s i g n s tr e s s- s t ra i n c u r v e s f o rc o n c r e t e s w i t h f ck = 2 0 M P a , 5 0 M P a a n d 9 0 M P a .

u e s o f u l t i m a t e s t ra in c o n s i d e r e d f o r t h e c o n c r e t e a n d t h es te e l. T h e d a t a o f t h e c u r r e n t B r a z i li a n c o d e [6 ] w i l l b ekep t in i ts r ev i sed ve r s ion and a r e the sam e as those of thep r e v io u s C E B - F I P M o d e l C o d e .

5 . T E S T R E S U L T SS o m e e x p e r i m e n t a l s t re n g t h s o f st o c k y c o l u m n sr e p o r t e d i n t h e l i t e r a t u r e [ 1 0 , 1 1 ] , w i t h s y m m e t r i c a lr e i n f o r c e m e n t t e st e d u n d e r e c c e n t r i c c o m p r e s s i o n , w e r e

c h o s e n t o b e c o m p a r e d w i t h t h e t h e o r e t ic a l o n e s b as e do n d i f f e r e n t c o n c r e t e s t re s s -s t ra i n r e l a t i o n s h i p s .A p a r t f r o m t w o , t h e c o l u m n s t e s t e d b y I b r a h i m a n dM a c G r e g o r [ 1 0 ] w e r e s u b j e c t e d t o a x i a l c o m p r e s s i v ef o r c e s w h o s e e c c e n t r i c i t y w a s a d j u s t e d t o m a i n t a i n z e r os t r a in a t o n e f a c e o f t h e c o l u m n s ( t r ia n g u l a r s t r a in d i a -g r a m ) . T h e c o l u m n s w i t h r e c t a n g u l a r c ro s s s e ct i o n h a dc o n c r e t e s t r e n g t h v a r y in g f ro m 5 9 M P a t o 1 2 8 M P a ,l o n g i t u d i n a l r e i n f o r c e m e n t r at i o e q u a l t o z e r o o r 1 . 3 3 %a n d , e x c e p t f o r t w o t h a t h a d v o l u m e t r i c t r a n s v e r s e r e i n -f o r c e m e n t r at io o f 3 . 9 % , c o n f i n e m e n t r e i n f o r c e m e n tr a ti o s in t h e r a n g e o f 0 t o 1 . 1 % . A l l s p e c im e n s h a d r e i n -f o r c e m e n t w i t h y i e l d s tr es s a r o u n d 4 3 0 M P a . T h e l o wc o n f i n e m e n t r e i n f o r c e m e n t r a t i o s h a d n o n e o r l i t t l ee f f e ct o n t h e b e h a v i o r o f t h e s p e c i m e n s . T h e t w oc o l u m n s w i t h h i g h e r c o n f i n e m e n t r e in f o r c e m e n t a n dd i s t ri b u t e d l o n g i t u d i n a l s t e el s h o w e d i m p r o v e d b e h a v i o r.F o s t e r a n d A t t a r d [ 1 1 ] t e s t e d s q u a r e c r o s s s e c t i o nc o l u m n s w i t h c o n c r e t e s t r e n g t h s f r o m 4 0 M P a t o 9 3M P a a n d l o a d e c c e n t r i c i t i e s v a r y i n g b e t w e e n 0 . 0 5 a n d0 : 3 3 t i m e s t h e d i m e n s i o n o f th e c r o ss s e c t io n . T h ec o l u m n s h a d e i th e r 2 % o r 4 % l o n g i t u d i n a l r e i n fo r c e -m e n t r a ti o ( y i el d st re ss o f 4 8 0 M P a ) a n d v o l u m e t r i c la t -e r al r e i n f o r c e m e n t r a ti o i n t h e r a n g e o f 1 . 0 % t o 3 . 7 %( y ie l d st re s s o f 3 6 0 M P a o r 4 6 0 M P a ) .

6 . C O M P A R I S O N B E TW E E N E X P E R IM E N T A LA N D T H E O R E T I C A L R E SU LT S

T h e b e a r i n g c a p a c i t y o f s e c t i o n s w i t h a g i v e n d i s t ri b -u t i o n o f r e i n f o r c e m e n t u n d e r m o m e n t a n d a x ia l f o rc ec a n b e c l e a rl y p r e s e n t e d i n i n t e r a c t i o n d i a g ra m s . W i t ht h e a id o f t w o c o m p u t e r p r o g ra m s , o n e f o r t h e n o n - l i n -

p .

0,15 I If c = ,7 2 . 8 M P a [i0,1

..~ ., :...:2:...... " ~ .' ~ ' ~ . ~

o i , "':i':-.. \0 0,2 0,4 0,6 0,8

W0 , 1 5 i

fc = 9 8 . 8 M P a. _ / _ - 2 . " - > -A , . . * . ' 5 ,, % , ~ ~ ,

O 0 0,2 0,4 0,6 0, 8V

0,15 II f c 1 2 4 . 8 M P a ] I=

I

o.0s ~ :::'=::::::'"" " " ""...,.::.~::.;::~ \O0 0 ,2 0 ,4 0,6 0, 8

NS

M Cg O~te a

1 " , s t ; F r~(io)[m m

~.ocou,m tCE B

T e s t s F ~

.L_1 m m

NB ~MC'JO

CEBr e s ~ F , ~

L1 m m

Fig. 5 - I n t e r a c ti o n d i a g r a m s fo r t h e c o l u m n s w i t h o u t r e i n f o r c e -m e n t c o n s i d e r i n g d i f f e r e n t d e s i g n s t r e ss - s tr a i n c u r v e s f o r t h ec o n c r e t e .

e a r an a ly s is a n d t h e o t h e r f o r t h e r e c t a n g u l a r s tr es s b l o c k ,t h e t h e o r e t i c a l a x i a l l o a d - m o m e n t i n t e r a c t i o n d i a g r a m sf o r t h e t e s t e d c o l u m n s w e r e o b t a i n e d . T h e c a l cu l a ti o n sw e r e b a s e d o n t h e f o l l o w i n g a s s u m p t io n s :9 p l a n e s e c t i o n s r e m a i n p l a n e9 s t r a i n s i n t h e s t e e l a r e t h e s a m e a s t h e s t r a i n s i n t h ea d j a c e n t c o n c r e t e9 t e n s i le s t r e n g t h o f c o n c r e t e i s n e g l e c t e d

9 s t r es s -s t r a in r e la t ion sh ip f or s tee l is idea l ized as e las to-plas t ic9 s t r e s s -s t ra i n r e l a t i o n s h i p f o r c o n c r e t e i s g i v e n b y d i f f e r -e n t p r o p o s a l s9 e v e n t u a l c o n f i n i n g e f f e c t o f t h e l a te r a l r e i n f o r c e m e n t i si g n o r e d .

T h e n o n - d i m e n s i o n a l i n t e r a c t i o n d i a g r a m s(b t = M /bdef c and v = N /bdf c ) s e en in F igs . 5 , 6 , 9 and 10w e r e g e n e r a t e d c o n s i d e r i n g t h e d e s i g n s t r e s s - s t r a i nc u rv e s o f N B R 6 1 1 8 [6 ], C E B - F I P M C 9 0 [2 ], C E B b u l -l e t i n 2 2 8 [ 3 ] , N S 3 4 7 3 [ 5 ] a n d C o l l i n s e t a l . [ 4] . T h e l a t -t e r i s t h e r e a l i s t i c c u r v e w i t h t h e c o n c r e t e s t r e s s m u l t i -p l i e d b y ( 0 . 6 + 1 0 /f ~) < 0 . 8 5 , w i t h fc i n M P a . I n o r d e r t oa v o i d d i s t o r ti o n i n t h e i n t e r a c t io n d i a g ra m s , t h e u l t i m a t ec o n c r e t e s t r a i n a d o p t e d f o r t h i s c u r v e w a s 3 . 0 % o f o rc o n c r e t e w i t h f c < 5 0 M P a a n d ( 2 . 6 + 0 . 0 0 8 f c) % 0 e c o , w h e n fc > 8 7 . 5 M P a( M e g 0 ) o r fc > 1 1 0 M P a ( b u l l e t i n 2 2 8 ) .

4 1 5


6/8

M a t e r i a ls a n d S t r u c t u r e s / M a t 4 r i a u x e t C o n s t r u c t i o n s , o l : 3 3 , A u g u s t -S e p t e m b e r 2 0 0 0

0 , 2 tf c = 7 1 , 6 M P a

I ~ o .1 i " " . ' . ' . ~ ./ t-: . , \o , c ~ 9 I" ; J : ,

O " " " %0 0,2 0,4 0,6 0,8 Iv

0, 2

0 ,, . , . . . . . . i ; ' . . . . . : '9%. I0,05 ~'.'~ !"~~

0 0 0,2 0,4 0,6 0,8 1V

0,2 I0,15 ~ ~ . ~ f c = 12 5 5 MP a

~.t 0,1 ~ ~ " ' ~ 'o.o~ ":7",,~ . . 9

0 '%"%0 0,2 0,4 0,6 0,8 1v

1,2

NBRMCgOCEB

1, 2 2~m~m 0

i N SNORMC90CEB(I0)

S~ 01,2 mmF i g . 6 - I n t e r a c t i o n d i a g r a m s f o r c o l u m n s w i t h 1 .3 3 % s t e e l r a t i oc o n s i d e r i n g d i f f e r e n t d e s i g n s t r e s s - s t r a i n c u r v e s ( c o l u m n s w i t hl o w l a te r a l s t e e l r a t i o ).

0 2

0,15~LL 0.i

0,05o ;

O 2

O,15~, O,1

0,051010

0, 2

0 , 1 5

~ 0 , 1

0,05

0 0

.< ...~.:...:;.:.... __I__/ / % ..\ \ 9 ,

~ , " 9 "% .;%%..9 %

0,6 0~8 1VI

. , ' : : 1 " : : ~ : ' : - . , . t f c = 8 3 . 8 M P aI . - ~ - .. - . .. ~ i : ~ ; . ,) : ' ' " , , . ' , . .~ - " " ~ % ,

%~ " .: . 50 ,2 0 ,4 0 ,6 0/3 1

v

0 ,2 0 ,4

t f c =1 25 .5 MPa " '~ ' ~ '~ " ~%%"% ~, %..,. % % .%'9 .,%k %

/ ..'.j~. 9; ~ . . . ' - - . . " . . ' . . . . . ; . . ; ,

NB~C~CAACT,~ F( C

E1 ,2

0 ,2 0 ,4 0 ,6 O~ 1v

1 ,2

NBRM C 9 0

CA~ACl

)01 2 i

F i g . 8 - I n t e r a c t i o n d i a g r a m s f o r c o l u m n s w i t h 1 .3 3 % s t e e l r a t i oc o n s i d e r i n g d i f f e r e n t r e c t a n g u l a r s tr e s s b l o c k s ( c o l u m n s w i t h l o wl a t e r a l s t e e l r a t i o ) .

o , 15 I ==

fc = 72 .8 MP a M~QO; .: .. _ - = ., ] . ~ - ' ~~ ~ ' ' ' " i, Ao" , ~ i 9 - , (,o). ~ 9

0 , 05 / \ , ,, .. % " . ,,, *',,%,. "~00 0 2 0, 4 0 , 6 O ~ 1 mm

vO,15 I ~f c = 98 8 M Pa " ' 9Mcgo

' - - ' ; . . '; . . '~ . . . . . . - . - 7

o / , i \ " , ~ . / " , .0 02 0,4 0,6 O~

P

0,15 if c = 1 2 4 . 8 M P a " ~ - -MCgO

#0 .~ 9 k %'0'.- ( 1 0 )0,0 5/ ~ ~ 9 %..;.....%~ %,, ~9 " \ . ~ [--'-,,

0 02 O,4 0,6 0,8 1V

F i g . 7 - I n t e r a c ti o n d i a g ra m s f o r th e c o l u m n s w i t h o u t r e i n f o r c e -m e n t c o n s i d e r i n g d i f f e r e n t r e c t a n g u l a r s t r e s s b l o c k s 9

0,350, 3

0,25P ' 0 . 2 /

0,150, 1

0,050 0

P

0,250, 2

0,150, 1

0,050

0

I

f c = 4 2 . 0 M P a

"~

"-'k.-.N.,0,2 0,4 0,6 0,8 I 1.2

v

, . f c = 7 4 . 3 M P a. - - : : . : - . -

.... . ~ ,. ~%, "~.~I % 9 'N0 , 2 0 . 4 0 , 6 0 . 8 1 1 , 2v9 _ _ j _ _

9 [ f c = 9 2 . 3 M P a

! . : : . : { _/ - " ' ' . I " : : : ' k

~ " < . . . . ,0 , 2 0 , 4 0 , 6 0 . 8 1 1 , 2v

NSNBRMC~OCd$1nset(CEBT e ~ b * F r ~0 1 )[E150 x 1~ b

1. 4

NSN BRMCm. .CEB(11)

D150 x 1511 ,4

NSN BRmMC'~

C ~ i n s e t l dCEB

r ~ t ~Fr ~N '

D150 x 151,1, 4

F i g . 9 - I n t e r a c t i o n d i a g r a m s f o r c o l u m n s w i t h 2 % s t e e l c o n s i d e r -i n g d i f f e r e n t d e s i g n s t r e s s -s t r a i n c u r v e s f o r c o n c r e t e .

4 1 6


7/8

I . S h e h a t a , L . S h e h a t a , T . S . M a t t o s

0, 4Q 3 50, 3

Q 2 50 , 2 - -

0 , 1 50, 1

0 , O 50 0

0 ,30 ,2 5 - -g,2

~L[ 0,150,1

q050

_- q 2 5

q 1 5 , ~ , ,, I ' - , , ( " . : : . " X jo o

9 I . 1 NSI -9 " f c = 4 1 . 0 M P a ~ " ~............... ...:.~.,-2--,. I .... :; '". . . . . . . . . . . . 9- . ~ i ~ ;~ . . , - , , , . . . . . . . ." ; - . 9 O_t)

1 5 0 x 1 5 10 , 2 0 , 4 Q,6 0 ,8 1 1, 2 1 ,4 1 ,6

V

~ f ' L /

V

MC~calm, eta

CEBo ,

1 5 0 x 1 5 C0 , 2 0 ,4 0 , 6 0 , 8 1 1 , 2 1 , 4 1, 6

V

F i g . 10 - In t e r a c t i o n d i a g r a m s f o r c o l u m n s w i t h 4 % s t e el r a t i oc o n s i d e r i n g di f f e re nt de s ign s t r e s s - s t r a in c urve s for c o n c r e t e .

T h e i n t e r a c t i o n d i a g r a m s i n F i g s . 7 , 8 a n d 1 1 w e r ed e t e r m i n e d u s i n g th e r e c t a n g u la r d i s t ri b u t i o n s o f c o n -c r e te s t r es ses d e f in ed in Tab le 1 .F o r t h e c o m p r e s s i v e s t r e n g t h o f th e c o n c r e t e a n d t h ey i e l d s t r e n g t h o f t h e s t e e l t h e m e a n v a l u e s g i v e n i n [ 10 ,1 1] w er e ad o p ted . Co ef f ic ie n ts Y c an d y.~ w e r e co n s id e r ede q u a l t o o n e .A s c o u l d b e e x p e c t e d , f o r p u r e b e n d i n g o r c o m -

p o u n d b e n d i n g w i t h l a r ge e c c e n t r i c it i e s , t h e c o n c r e t es t re s s -s t ra i n c u r v e u s e d m a d e a l m o s t n o d i f f e re n c e i n t h ec o m p u t e d s t r e n g t h o f r e c t a n g u l a r c r o ss s e c ti o n s . O n t h eo t h e r h a n d , f o r s e c t i o n s w i t h p r e v a i l i n g c o m p r e s s i o n , t h es t r e s s - s t r a i n c u r v e h a d a s t r o n g i n f l u e n c e o n t h e c a l c u -la ted s t r en g th .F r o m t h e i n t e r a c t i o n d i a g r a m s i n F ig s . 5, 6 , 9 a n d 1 0a n d t h e e x p e r i m e n t a l r e s u l t s p l o t t e d i n t h e m , i t c a n b ec o n c l u d e d t h at :9 t h e r e i s n o t m u c h d i f f e r e n c e b e t w e e n t h e r e s u l tso b t a i n e d u s i n g t h e t w o C E B d e s i g n s tr e s s- s tr a in c u r v e s .9 t h e d e s i gn c u rv e s o f t h e N o r w e g i a n C o d e a n d t h o s e o fC o l l i n s e t a l . l e a d t o t h e m o s t c o n s e r v a t i v e re s u lt s .9 A p a r t f r o m t h e N B R c u r v e , w h i c h i s sa fe o n l y fo r l o ws t r en g th co n cr e te , in g en er a l , a l l th e d es ig n s t r e s s - s t r a inc u r v e s t e n d t o l e a d t o s a fe a n s w e r s .I n th e d iag r am s o f F ig s. 7 , 8 an d 1 1 i t can b e o b s e r v e dth a t :9 t h e A C I a n d N B R r e c t a n g u la r s tr es s bl o c ks , i n g e n -e r al , l ead to ca lcu la ted s t r en g th s c lo se to each o th e r .9 t h e r e c t a n g u l a r s tr es s d i a g r a m s o f th e C a n a d i a n c o d e

I 0 , I '9 t c = 4 2 . 0 M P ao.,~ , , " ' Q ; ' < " , , . . .

1% ~ ,o , 1 % ' C "

O 0 0 , 2 0 , 4 0 , 6 0 , 8 1 1 , 20 ,3

O , g 50 , 2 i

P- o aS /o ,10 , 0 5

1 , 4 1 , 6

o o

9 ] i i ~ - - - N ,.R9 f c = 7 4 . 3 M P a ~ " '* ". -. = '~ ' ~

. . . . . . . . . . .. . t o, . . ~ : - - , ,-a:a.~ 9 - . : . . . . ~' "9 - . . . . . . . T % , ,; =% . , ~ " ," , ' , . . ~ , .

. " ~ 1 5 0 x 1 5m m0 , 2 0 , 4 0 , 6 0 , 8 1 1 , 2 1 ,4V

o , 2 5 1 t ' 2 ~ . .~ I9 I tT~=md ~1: 2 L ': f c = ~ 3 M P . , ~ o ;

o.1~ " ' ~ , - , [ b . L . [ I I ~ : ITesst Fr ~(11} Ial , . . "

O , O 5 " :1 5 0 x 1 5 0

0 " , rn m0 , 2 0 , 4 0 , 6 0 , 8 1 1 , 2 1 , 4% ,

F i g . I 1 - I n t e r a c t i o n d i a g r a m s f o r t h e r e i n f o r c e d c o l u m n s w i t h2 % s t e e l r a t i o c o n s i d e r i n g d i f f e r e n t r e c t a n g u l a r s t r e ss b l o c k s .

a n d o f I b ra h i m a n d M a c G r e g o r t e n d t o g i v e s i m il a rr e s u l t s , w i t h l i t t l e d i f f e r e n c e f o r h i g h e r s t r e n g t h c o n -c r e tes .9 t h e s t r e n g t h s o b t a i n e d w i t h t h e M c g 0 s tr es s b lo c k a rem u c h l o w e r t h a n t h e o n e s o b t a i n e d w i t h t h e o t h e rs , p a r-t i c u la r l y f o r c o n c r e t e s w i t h h i g h e r s t r e n g t h .9 t h e r e su l ts o f th e A C I a n d N B R r e c t a n g u l a r st re ssb l o c k s t e n d t o b e o n t h e u n s a f e s i d e f o r h i g h - s t r e n g t hc o n c r e t e s , w h i l e t h e d i a g r a m s b a s e d o n I b r a h i m a n dM a c G r e g o r s tr es s b l o c k t e n d t o b e s a f e a n d t h o s e b a s e do n t h e M C 9 0 a r e a lw a y s e x t r e m e l y c o n s e r v a t i v e .

7. C O N C L U D I N G R EM A RK ST h i s w o r k i n v e s t ig a t e d t h e i n f l u e n c e o f t h e a s s u m e ds t re s s -s t ra i n c u r v e f o r c o n c r e t e o n t h e p r e d i c t i o n o f t h es t r e n g t h o f c o n v e n t i o n a l a n d h i g h s t r e n g t h c o n c r e t ec o l u m n s u n d e r e c c e n t r i c ax ia l l o a d .I t ha s b e e n s h o w n t h a t , i n t h e c a s es o f p r e v a i l i n gc o m p r e s s i o n , t h e s t r e s s - s t r a i n d i a g r a m a d o p t e d f o r t h ec o n c r e t e h a s a g re a t i n f l u e n c e o n t h e c a l c u l a t e d s t r e n g t h .T h e d e s i g n st r es s -s t ra i n d ia g r a m s o f C E B - F I P M C 9 0 ,

C E B b u l l e ti n 2 2 8, N S 3 4 7 3 a n d C o l l in s e t a l . , i n g e n e r a l ,g a v e s a f e r e s u l t s f o r t h e 6 8 c o l u m n s w i t h r e c t a n g u l a rc r o s s s e c t i o n t e s t e d u n d e r e c c e n t r i c c o m p r e s s i o n a n a -l y z e d . T h e t r a d i t i o n a l p a r a b o l a - r e c t a n g l e s t r e s s - s t r a i nr e l at i o n sh i p o f t h e N B R 6 1 1 8 ( th e s a m e as t h e C E B - F I P

4 1 7


8/8

Mater ials and St ruc tu res /Mat6r iaux et Const ruc t ions ,Vol. 33 , August-September 000

MC 90 for fck < 50 MPa) lead to unsafe results when usedfor high strength concrete. As for the simplified rectan-gnlar stress blocks studied, the ones that were not cali-b ra ted fo r h igh s t reng th concre tes (o f ACI and N BRcodes) tended to overestimate the strength o f columnswith these concretes, while the ones proposed in theC an ad ian co d e an d b y I b r ah im an d MacGr eg o r [ 9 ]tend ed to g ive safe answers . The CE B-M C9 0 s t ressblock always lead to extr emel y conservative results.

LIST OF SYMBOLSA Relatio n betwe en Ect and Ecs (see equation (5)Ect tangent m odulu s o f elasticity at origin for concret eEcs secant mod ulu s o f elasticity at peak stressEcn secant modu lus of elasticity at nominal strength fcnfc compressive strength of concretefcd design compressive strength of concret efck characteristic compressive strength of concre tefc~ average compressive strength of concretefcn nomi nal compressive strength of concret e accordingto NS 3473 Ek coefficientn coefficientt coe f f i c ien tix coefficient~c con cret e strainecn nomi nal co ncrete strain at fcneco con cret e strain at peak stressecu ultimate co ncrete strainesu ultim ate steel strained concre te strain on the falling branch at a stress = 0.5 fcq coefficient

bt norma lized flexural strength (M/fc.b.d2)v norma lized norm al strength (N/fc.b.d)coeff icient

REFERENCES[1] Ibrahim, H. H. H. and MacGregor, J. G., 'Flexural behavior oflaterally reinforced high-strength concrete sections' , AC I

StructuralJournal 93 (6) (1996) 674-684.[2] Comitd Euro-International du B&on, 'CEB-FIP Model Code1990', Bulle t in d ' Information no 213/214 (Lausanne,Switzerland, 1993) 437p.[3] Comitd Euro-Internat ional du B&on, 'High PerformanceConcre te ' , Bu l le t in d ' In form a t ion no 228 (Lausanne,Switzerland, 1995) 46!0.[4] Collins, M, P., Mitchell, D. and MacGregor,J. G., 'StructuralDesign Considerations for High-Strength Concrete', ConcreteInternational: Design and Construction15 (5) (1993) 27-34.[5] Norwegian Council for Building Standardization, 'ConcreteStructures Design Rules-NS3473 E', (Stockhohn, 1992) 78p.[6] Brazilian Technical Standards, 'Project and Execution ofReinforced Concrete Structures - NBR 6118', only available nPortuguese (ABNT, Rio deJaneiro, Brazil, 1980) 76p.[7] Canadian Stan&rd Association, Design of Concrete StructuresforBuilding- CAN3-A23.3-94', Concrete Design Handbook(Canadian Portland Cement Association,Ottawa, Canada, 1995).[8] American Concrete Institute, 'Building Code Requirements forStructural Concret e and Commentary - ACI 318-95', (ACI,Detroit, 1995) 369p.[9] Ibrahim, H. H. H. and MacGregor,J. G., 'Modification of theACI Rectangular Stress Block for High-Strength Concrete ',A C IStructuralJournal 94 (1) (1997) 40-48.[10] Ibrahim, H. H. H. and MacGregor, . G., 'Tests of EccentricallyLoaded High-Strength Concrete Columns', ACI StructuralJournal 93 (5) (1996) 585-594.[11] Foster, S. J. and Attard, M. M., 'Experimental Tests onEccentrically Loaded High-Strength Concrete Columns', A C IStructural ournal 94 (3) (1997) 295-303.

418

Documents

Stress-Strain Curve for the Design of High-strength Concrete Elements