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CEMENT and CONCRETE RESEARCH. Vo l. 6, pp. 77 3-7 82 , 1976. Per gam on Pr es s, In c.P r i n t e d i n t h e U n i t e d S t a t e s .

ANALYSIS OF CRACK FORMATION AND CRACK GROWTH IN CONCRETE BY

MEANS OF FRACTURE MECHANICS AND F IN ITE ELEMENTS

A H i l l e r b o r g , M M o d ~ e r a nd P -E P e t e rs s o nD i v i s i o n o f B u i l d i n g M a t e r i a l s

L u nd I n s t i t u t e o f Te c h n o l o g y, L u nd , Sw ede n

(Communicated by Z. P. Bazant)(Received August 24, 1976)

ABSTRACT

A m e t h o d i s p r e s e n t e d i n w h ic h f r a c t u r e m e ch an ic s i s i n t r o d u c e d i n t of in i t e e lement ana lys i s by means o f a mode l where s t ress es a re assumedto a c t ac ros s a c rack as long as i t i s na r r ow ly opened. T h is assump-t io n may be regarded as a way of expre ss ing the energy abs orp t ion Gin the energy ba lance approach , bu t i t i s a l so in agreement w i th re -s u l t s o f t e n s i o n t e s t s . A s a d e m o n s t r a t io n t h e m e t h od h as b e e n a p p l i e dto the bending of an un re in forc ed beam, which has l ed to an exp lana-t i o n o f t h e d i f f e r e n c e b etw ee n b e nd in g s t r e n g t h and t e n s i l e s t r e n g t h ,a nd o f t h e v a r i a t i o n i n b e n d i ng s t r e n g t h w i t h b ea m d e p t h .

Une m6thode es t p r es en t , e , pa r l aq ue l le l a m6chanique des rup ture s es ti n t r o d u i t e d an s l a n a l y s e d es ~l~ m en ts f i n i s ~ l a i d e d u n m o d u le , oOl e s c o n t r a i n t e s s o n t s up po s6 es d o p 6 r e r s u r l e s c 6 t 6 s d u n e f i s s u r e

t a n t q ue c e t t e f i s s u r e e s t ~ t r o i t e .Ce t te hypoth~se peu t e t re cons id~r~e comme un moyen d exp r ime r l a b -s o r p t i o n G d 6 n e r g i e en u sa n t l a p p r o c h e de l ~ q u i l i b r e d 6 n e r g i e .C e t t e h yp o th 6 se e s t a u s s i j u s t i f i 6 e p a r l e s r 6 s u l t a t s d es e s s a i s d et e n s i o n .

P ou r en p r o u v e r l a v a l i d i t Y , c e t t e m ~ th od e a ~ t ~ a p p l i q u ~ e au f l 6 c h i s s e -m e n t d u n e p o u t r e n on a r m ~ e e t f o u r n i t u ne e x p l i c a t i o n de l a d i f f 6 r e n c ee n t r e l a r ~ s i s t a n c e au m om en t de f l e x i o n e t l a r ~ s i s t a n c e ~ l e f f o r t det e n s i o n , a i n s i q ue d e l a v a r i a t i o n d e l a r ~ s i s t a n c e a u m o me nt d e f l e x i o ne n f o n c t i o n d e l a p r o f o n d e u r d e l a p o u t r e .

773

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774 Vo l . 6 , No. 6A . H i l l e r b o r g , M . M o d 4 er , P- E. P e t e r s s o n

I m p o r t a n c e o f c r a c k s a nd c r a c k g r o w t h

C r ac k f o r m a t i o n a nd c r a c k g r o w t h p l a y an i m p o r t a n t p a r t i nt h e p e r fo r m a n c e o f u n r e i n f o r c e d a nd r e i n f o r c e d c o n c r e t e . E x am p le s

o f t h i s a rec r a c k s p a c i n g a nd c r a c k w i d t h i n b e n d i n g

s h e a r c h ra c k s a nd t h e i r e f f e c t on s h e a r c a p a c i t y

c r a c k i n g m o me nt o f r e i n f o r c e d a nd u n r e i n f o r c e d beam s

m i c r o c r a c k s i n c o m p re s s io n a nd c o m p re s s io n f a i l u r e .

A r a t i o n a l d e s i g n i n t h e s e ca s e s o u g h t t o be b as ed o n r e a l i s -t i c t h e o r e t i c a l m o d e ls , w h ic h t a k e cr a c k f o r m a t i o n a n d c r a ck p r o p a -g a t i o n i n t o a c c o u n t . S o f a r n o s uc h m o de ls h a ve b e e n a v a i l a b l e . C o n -se qu en t ly the des ign me thods have had to be based on em pi r i ca l r e -s e a r c h , s u p p o r t ed b y s i m p l i f i e d m o d e ls .

R e ce nt a d v an c e s w i t h i n f r a c t u r e m e c h an ic s a nd f i n i t e e le m e n tm e th od s ( F E M ) h av e now g i v e n u s a p o s s i b i l i t y o f a n a l y s i n g c r a c kg r o w t h . F r a c t u r e m e c h a n ic s g i v e s t h e f u n da m e n t a l r u l e s f o r c r a c kp r o p a g a t i o n a nd FEM m a k e s i t p o s s i b l e t o a p p l y t h e s e r u l e s t o c om p -l i c a t e d c a s e s .

T he c a se s w e w i s h t o a n a l y s e a r e r a t h e r c o m p l i c a t e d , a s t h e yi n v o l v e d i v e r s e p h e n om e n a , s uc h as

f o r m a t i o n a nd p r o p a g a t i o n o f c r a c k s

t w o o r m o re p a r a l l e l c r a c ks

b e n t s h e a r c r a c k s

s h r i n k a g e s t r a i n s

i n t e r a c t i o n b etw ee n c o n c r e t e an d r e i n f o r c e m e n t

i n t e r a c t i o n b etw e en ce m en t m a t r i x a n d a g g r e g a t e .

I t i s t h e r e f o r e n e c e s s a r y t o u se FEM a nd a l s o t o t r y a nd f i n da m e th od w h i c h s i m p l i f i e s t h e a n a l y s i s a s m u c h a s p o s s i b l e .

Proposed approach

T h e re a r e m a n y m e th od s t o c h o o s e fr o m f r a c t u r e m e c h a n i c s , e . g .

t he s t r e ss i n t e n s i t y f a c t o r a pp ro ac h

the ene rgy ba lance approacht h e s t r i p - y i e l d m o d e l a c c o r d i n g t o D ug da le

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

The d i f f e r e n t m e th cd s a r e k n o w n t o g i v e c o h e r e n t r e s u l t s .

I n t h e s t re s s i n t e n s i t y f a c t o r a pp ro ac h th e s t re s s e s n e a r t h ec ra c k t i p a r e s t u d i e d . T hese s t r e s s e s t h e o r e t i c a l l y ag_ j~roach in f in i tya t t h e c r a c k t i p a c c o r d i n g t o t h e e x p r e s s i o n o = K / V 2 ~ r , w he re r i st h e d i s t a n c e f ro m th e c ra c k t i p a nd K i s a c o e f f i c i e n t , t h e s t re s s i n -t e n s i t y f a c t o r , d e p e nd in g on t h e l o a d , t h e c r a c k d im e n s i o n s , e t c . W henK r ea c he s a c r i t i c a l v a l u e Kc , t h e c r a c k p r o p a g a t e s .

T he s t r e s s i n t e n s i t y f a c t o r a p pr oa c h h as b e e n u se d a g r e a t d ea li n FEM a n a l y s i s . T he d i r e c t m e th od r e q u i r e s a FEM m e sh w i t h v e r y s m a lle le m en ts c lo s e to t he cr ac k t i p , w h ic h l i m i t s i t s a p p l i c a b i l i t y t o

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V ol . 6 , No. 6 775FRACTURE MECHANICS, ANALYSIS, CONCRETE

c o m p l i c a t e d p r o b l e m s . I n d i r e c t a nd s p e c i a l m e th od s p e r m i t th e us eo f g r e a t e r e l e m e n t s . T h e m e th od s ca n n o t e x p l a i n t h e f o r m a t i o n o fc r a c k s , o n l y th e p r o p a g a t i o n .

I n th e e n e rg y b a l a n c e a p pr o ac h i t i s a s su m e d t h a t a c e r t a i na m ou nt o f e n e rg y G i s a b s o rb e d by t h e f o r m a t i o n o f a u n i t a r e a o fc r a c k s u r f a c e . W hen a c r a c k p r o p a g a t e s a c e r t a i n a m o u n t o f s t o r e de n e rg y i s r e l e a s e d . T he c r a c k p r o p a g a t e s w h e n t h e r e l e a s e d e n e rg yi s equa l to o r g r ea te r than the absorbed ene rgy . FEM has bee n usedt o d e t e r m i n e t h e e n e rg y r e l e a s e r a t e i n t h e e n e rg y b a l a n ce a p p r o a c h ,s ee e . g . / 5 / . T h i s e n a b le s t h e u s e o f a FEM m e sh w i t h r a t h e r l a rg ee l e m e n t s . T he f o r m a t i o n o f c r a c k s c a n n o t b e e x p l a i n e d .

I n t h e D u gd ale m o d e l i t i s a s su m e d t h a t t h e r e i s a p l a s t i cz on e n e a r t h e c r a c k t i p a c c o r d i n g t o F i g . I . W i t h i n th e p l a s t i cz on e a s t r e s s e q ua l t o t h e y i e l d s t r e n g t h ~ y a c t s a c r o s s t h e c r a c k .T he B a r e n b l a t t m o de l i s s i m i l a r t o t h e D ug da le m o d e l, b u t t h e s t r e s si s a s s um e d t o v a r y w i t h t h e d e f o r m a t i o n . I t d oe s n o t s e em t o h av eb e en u se d i n f i n i t e e le m e n t a n a l y s i s .

T he b a s i c i d e a o f t h e m o d e l w e p r op o s e i s d e m o n s t r a t e d i n F i g .2 . I t i s i n s o m e r e s p e c t s s i m i l a r t o t h e B a r e n b l a t t m o d el. T he m od eli s d e s c r i b e d o n l y f o r m o de I ( t h e o p e n i n g m o d e) , b u t i t m ay a l s o b ea p p l i e d t o m o de s I I a nd I I I .

e l a s t i c c r o c k|

r e a l c r a c k ~ p i a s t i c z o n e

r t k l e n g t h

I

Fig . 1 The Dugda le mode l fo r

c ra ck t i p p l a s t i c i t y

F ig . 2 Proposed model

T he c r a c k i s a s su m e d t o p r o p a g a t e w h e n t h e s t r e s s a t t h e c r a c kt i p r ea c h es th e t e n s i l e s t r e n g t h f t W he n t h e c r a c k o p e n s t h e s t r e s si s n o t a s su m e d t o f a l l t o z e r o a t o n c e , b u t to d e c re a s e w i t h i n c r e a s -i n g c r a c k w i d t h w, f o r e x a m ple a c c o r d i n g t o F i g . 3 . A t t h e c r a c kw i d t h w t h e s t r e s s h as f a l l e n t o z e r o . F o r t h a t p a r t o f t h e c r a ckw h er e W < W l , t h e c r a c k i n r e a l i t y c o r re s p o n d s t o a m i c r o c ra c e dz on e w i t h s om e r e m a i n in g l i g a m e n t s f o r s t r e s s t r a n s f e r. A s t h e r e i sa s t r e s s t o b e o ve rc om e i n o p e n i n g t h e c r a c k , e n e rg y i s a b s o r b e d .T he a m o u n t o f e n e rg y a b s o rb e d p e r u n i t c r a c k a r ea i n w i d e n i n g t h ec rack f rom ze ro to o r beyond w i s

and corresponds to the area between the curve and the coordinateax is in Fig. 3.

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Cr

t

0 w W

F i g . 3 A s su m ed v a r i a t i o n o fs t r e s s o w i t h c r a ckw i d t h w, g e n e r a l c a se

We now choose the cu rve in Fi g . 3 so th a t

7 d w = G I )C

0

which means th a t t he ene rgy absorbed pe r newformed u n i t c rack a rea i sthe same as in the ene rgy b a lance approach . The mode l o f F ig . 2 may

t h u s b e lo o k e d u p o n a s a w a y o f e x p r e s s i n g t h e e n e rg y b a l a n c e a p p r o a ch .At the same t ime the a s sum pt ion o f F ig . 2 may be looked upon as

a r e a l i t y . S t r e s s e s m ay b e p r e s e n t i n a m i c r o c r a c k e d z on e a s l o ng a st h e c o r r e s p o n d i n g d i s p la c e m e n t i s s m a l l. T h i s h as b e e n c l e a r l y d em o n-s t r a t e d i n te n s i o n t e s t s , u s in g a v e ry r i g i d t e s t i n g e q uip m e nt , e . g . ,b y E v an s a nd M a r a th e / 4 / ; c f . F i g . 5 .

B y t h e a p p l i c a t i o n o f t h e p r op o s e d m o d e l t h e c u r v e ~ w ) m a y bec ho se n i n d i f f e r e n t w a y s , e . g . a c c o r d i n g t o F i g s . 4 a , b o r c , w h i cha l l s h o w s im p l e m a t he m a tic a l r e l a t i o n s . F o r t y p i c a l y i e l d i n g m a t e r i -a l s , l i k e m i l d s t e e l , F i g . 4a s ee m s t o be t h e b e s t c h o i c e . I t c o r r e -s po nd s e x a c t l y t o t h e D u gd ale m o d e l w i t h f t = ~ y. a n d ~ I = COD a t i n i -t i a t i o n o f c r a ck g r o w t h . The d i s c o n t i n u i t y m ay g l v e r l s e t o s om e p r ob -le m s b y t h e a p p l i c a t i o n i n F E M , b u t t h e y a r e n o t s e r i o u s .

w w

l

L

W W W

b) C)

Fig . 4 Examples o f pos -s i b l e a s s u m p t i o n so f v a r i a t i o n o fs t r e s s ~ w i t h c r a c kw i d t h w i n p r a c t i c a la p p l i c a t i o n s

F o r c o n c r e t e i t s ee m s t h a t F i g . 4 c i s t h e b e s t c h o i c e as i tc o rr e sp o n ds r e a s o n a b ly w e l l w i t h te n s i o n t e s t r e s u l t s / 4 / , c f . F i g .5 . I t i s a l s o s i m p l e , c o n t i n u o u s a nd s u i t a b l e f o r FEM a n a l y s i s . F o rour purpose we have th e r e f o r e chosen F ig . 4c .

We then ob ta in

~ d w = f t w l / 20

o r f ro m I ) ,

w = 2Gc f tF o r o r d i n a r y c o n c r e t e G c / f t s e em s t o b e o f t h e o r d e r 0 . 0 0 5 -

2 )

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Vo l . 6 , No. 6 777FP~ACTURE ME CHA NICS , A N A LY S IS , CONCRETE

O.Ol mm, c f . / I / , and thus w o f the o rde r 0 .01 - 0 .02 mm. In thea p p l i c a t i o n we f u r t h e r a ss um e t h a t t h e c o n cr e te i s l i n e a r - e l a s t i cu n t i l f t i s re a ch e d.

F ig . 5 shows a compar i son be tween our a s sum pt ions w i th G c / f t= 0 .0 1 m m, ~ / f t = I 0 0 00 an d a t e n s i o n t e s t f ro m / 4 / w i t h a g a g el e n g t h o f 1 (2 5 m m ) . T h i s c o r re s p o n d s t o a t h e o r e t i c a l a v e r a g ee l o n g a t i o n o v e r t h e rg a g e l e n g t h w h en ~ r e ac h e s O , i . e . w = W l , o f2 . 0 . 0 1 / 2 5 = 8 0 0 - I 0 - . The a s s um p t io n s s ee m t o a g re e r e a s o n a b l y w i t ht h e t e s t r e s u l t . A l o w e r v a l u e o f E / f t w o u l d ha ve im p r ov e d t h e a g r e e -m e n t , b u t f r o m t h e p o i n t o f v i e w o f t h e e n e rg y b a la n c e a p p r o a ch t h eE - v a l u e c o r r e s p o n d i n g t o u n l o a d in g i s m o s t i m p o r t a n t a n d t h i s j u s t i -f i e s t h e c h o i c e E / f t = I 0 0 0 0 .

F i g . 5 A t e s t r e s u l tfr om a t e n s i l et e s t a c c o r d i n gto Evans Ma ra-t h e / 4 / , c om p ar edt o a c o r r e s p o n d i n ga s s u m e d r e l a t i o nb y t h e a n a l y s i s

ft

o r ~ .

0 200 ZOO 600 BOO St rai n x 106( 1 g a g e l e n g t h )

A s p e c i a l f e a t u r e o f th e p ro p os e d m eth od i s t h a t i t e x p l a i n sn o t o n l y t h e g ro w t h o f e x i s t i n g c r a c k s , b u t a ls o t h e f o r m a t i o n o f newc r a c k s , a s i t i s a s s um e d t h a t c r a c k s s t a r t f o r m i n g w h en t h e t e n s i l es t r e s s r ea ch e s f t , i . e . , t h e sam e c r i t e r i o n i s used f o r f o r m a t i o n a ndp r o p a g a t i o n o f c r a c k s .

T he a n a l y s i s c an be p er fo r m e d w i t h a r a t h e r c o a r s e m e sh ,a s i n th e ex am p le b e l o w, b ec au se t h e r e a r e n o s t r e s s s i n g u l a r i t i e sa nd t h e a m o un t o f a b s o r b e d e n e rg y i s n o t v e r y s e n s i t i v e t o t h e m es hs i z e . T he p o s s i b i l i t y o f u s i n g a c o a rs e m es h m e an s t h a t r a t h e r c om p -l i c a t e d p ro b le m s c an a ls o b e t r e a t e d w i t h o u t u s i n g t o o m a n y e l e m e n ts .

A p p l i c a t i o n t o an u n r e i n f o r c e d b ea m i n b e n d in gI n o r d e r t o s tu d y t h e a p p l i c a b i l i t y o f th e m eth od t h e f o l l o w -

i n g c a s e h as b e e n a n a l y s e d / 2 / .

An u n r e i n f o r c e d c o n c r e te be am w i t h a c o n s t a n t r e c t a n g u l a r c r o s s -se c t io n i s l oaded by a pu re bend ing moment M acc ord in g to F i g . 6 . Whent h e b e n d i n g m o m en t r e a ch e s a v a l u e M t h e t e n s i l e s t r e s s i n t h e b o t t o mf i b r e r ea c he s f t - A s w e a ss um e t h a t t h e c o n c r e t e c a n n o t ta k e h i g h e rt e n s i l e s t r e s s e s t ha n f t , c r ac k s w i l l fo rm a nd s t a r t o p e nin g w he n M i sinc re ase d above M . We w i l l now s tu dy how these c rac ks g row when theb e n d in g m o me nt i n c r e a s e s . I n o r d e r t o s i m p l i f y t h e c a l c u l a t i o n s w ea ss um e t h a t o n l y o n e c r a c k o p e n s , a n d t h a t t h i s h ap pe ns a t t h e s e c t i o n

o f s y m m e t r y.The f i n i t e e le m e n t m e s h u se d f o r t h e c a l c u l a t i o n i s s h o w n i n

F i g . 6 . T he b e n d i n g m o m en t M i s a p p l i e d a s a c o u p l e o f f o r c e s a t t h e

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778 V ol . 6 , No. 6A . H i l l e r b o r g , M . M o d # e r , P -E . P e t e r s s o n

M

4 J4 2414 0J 93 8J.163 5

3 33 23 t

F i g . 6 B e n t r e c t a n g u -l a r b e am a nda c o r r e s p o n d i n gF E M r e p r e s e n t a -t i o n

l e f t e nd o f t h e b e a m . T he c r a c k i s a ss um ed to o pe n a t t h e s e c t i o n t ot h e r i g h t , w h i ch i s t h e s e c t i o n o f s y m m e tr y.

M o i s t h e m o m e n t w h i c h g i v e s ~ 31 = f t , w h e r e ~ 31 i s t h e s t r e s sa t p o i n t 3 1. Mo w o u ld be t h e f a i l u r e m o m e nt i f t h e m a t e r i a l w e re e l a s -t i c an d p e r f e c t l y b r i t t l e . W hen M i s r a i s e d a b ov e M t h e c r a c k s t a r t so p e n i n g a t p o i n t 3 1 . A t t h a t p o i n t we i n t r o d u c e a f o r c e c o r r e s p o n d i n gt o t h e r e l a t i o n b etw ee n s t r e s s ~ and c ra c k w i d t h w a c c o r d i n g t o F i g .4 c . Wi th t h i s new f i n i t e e l e m e n t s y st em we c an c a l c u l a t e t h e s t r e s sa t p o i n t 32 an d w e c an d e t e r m i n e t h a t v a l u e M = M , w h i c h g i v e s as t r e s s 0 3 2 : f t We c an n ow i n t r o d u c e a n o t h e r f o r c e a t p o i n t 3 2 a ndc a l c u l a t e a m om en t M = M , g i v i n g 0 33 = f t e t c . B y p r o c e e d i n g i n t h e

s am e w a y w e g e t a r e l a t i o n b e tw e e n c r a c k d e p t h a nd a p p l i e d m o m en ta c c o r d i n g t o F i g . 7 .

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W hen t h e c r a c k g ro w s t h e c o r r e s p o n d i n g b e n d i n g m o m e n t r e a c h e s amaximum value Mmax w h e re u p on i t s t a r t s d e c r e a s i n g . A s t h e m a xim u m v a l u ei s r ea c h ed th e s t r u c t u r e b e c om e s u n s t a b l e i f M i s k e p t c o n s t a n t , a ndi t f a i l s s u d d e n l y a s t h e c r a c k p ro p a g a t e s .

T he r e l a t i o n M m ax/M i s t h e s a m e a s t h e r e l a t i o n b e tw e e n b e n -d i n g s t r e n g t h a nd t e n s T Te s t r e n g t h , a s M i s t h e m o m en t w h i c h m ak est h e m ax im um b e n d i n g s t r e s s i n t h e u n c r a c k e d s e c t i o n e q u a l t o t h e t e n -s i l e s t r e n g t h .

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I t c an be s h o w n t h a t t h e b e h a v i o u r o f t h e b e a m d ep en ds o n th ep a r a m e t e r H / I , w h ere H i s t h e b e am d e p t h an d 1 i s a c r i t i c a l l e n g t h ,d e f i n e d b y c c

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F i g . 8 T h e o r e t i c a l v a r i a -t i o n o f r a t i o b e-tween bending andt e n s i l e s t r e n g t hw i t h b ea m d e p t h Ha nd I c = K c / f t ) ~ =

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