Aggregate interlock

j.c.walraven

AGGREGATE INTERLOCK:

AGGREGATE INTERLOCK:

A theoretical and experimental analysis

BIBLIOTHEEK TU Delft P 1628 4187

1628

4 1 8

7

AGGREGATE INTERLOCK:

A theoretical and experimental analysis

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 8 oktober 1980

te 16.00 uur door

JOOST CORNELIS WALRAVEN

civiel ingenieur

geboren te Breda

Dit proefschrift is goedgekeurd door

de promotor: PROF.DR.IR. A.S.G. BRUGGELING

ACKNOWLEDGEMENT

The a u t h o r w i s h e s t o r e c o r d h i s thanks t o a l l t h e members o f t h e " C o n c r e t e S t r u c t u r e s Group" o f t h e S t e v i n L a b o r a t o r y who have c o n -t r i b u -t e d -t o -t h i s r e s e a r c h p r o j e c -t , i n p a r -t i c u l a r I r . E. V o s , who c a r r i e d o u t t h e e x p e r i m e n t a l p a r t o f t h e program, and P r o f . D r . - I n g . H.W. R e i n h a r d t , f o r t h e d i s c u s s i o n s on t h e s u b j e c t and h i s i n d i -c a t i o n s . F u r t h e r m o r e t h e a u t h o r i s g r e a t l y i n d e b t e d t o t h e CUR, t h e N e t h e r -l a n d s Committee f o r C o n c r e t e R e s e a r c h , f o r t h e i r f i n a n c i a -l s u p p o r t and encouragement. The f i n a n c i a l s u p p o r t o f t h e " S t i c h t i n g P r o f e s s o r B a k k e r f o n d s " f o r t h i s p u b l i c a t i o n i s g r e a t l y a p p r e c i a t e d .

CONTENTS 1 INTRODUCTION 1 1.1 M o t i v e s and scope o f t h e r e s e a r c h 1 1.2 Some a s p e c t s o f t h e r o l e o f t h e roughness o f t h e c r a c k s 3 i n c o n c r e t e s t r u c t u r e s 1.3 The f o r m u l a t i o n o f c r a c k c h a r a c t e r i s t i c s i n e x i s t i n g 11 n u m e r i c a l programs 1.4 Aim o f t h e r e s e a r c h program 17

2 STATE OF THE ART 19 2.1 A g g r e g a t e i n t e r l o c k 19 2.2 Dowel a c t i o n 27 2.3 A x i a l r e s t r a i n t s t i f f n e s s o f t h e r e i n f o r c i n g s t e e l 44 2.4 I n t e r a c t i o n o f components i n c r a c k s , c r o s s e d by r e i n - 49 f o r c e m e n t 2.5 C o n c l u s i o n s 52

3 A FUNDAMENTAL THEORY FOR AGGREGATE INTERLOCK 54

3.1 B a s i c a s s u m p t i o n s 54 3.2 G e n e r a l c o n s i d e r a t i o n s on t h e b a s i c v a r i a b l e s 60

3.3 D e t e r m i n a t i o n o f t h e r e l a t i o n between t h e p r o j e c t e d 65 c o n t a c t a r e a s A and A f o r a u n i t c r a c k a r e a on t h e one

x y

hand and t h e d i s p l a c e m e n t s between t h e c r a c k f a c e s on t h e o t h e r hand 4 EXPERIMENTS 90 4.1 G e n e r a l 90 4.2 Pure a g g r e g a t e i n t e r l o c k t e s t s 90 4.2.1 T e s t s p e c i m e n s , i n s t r u m e n t a t i o n and t e s t i n g p r o c e d u r e s 90 4.2.2 V a r i a b l e s 92 4.2.3 R e s u l t s 93 4.3 T e s t s on r e i n f o r c e d c r a c k s 95 4.3.1 Specimens, i n s t r u m e n t a t i o n and t e s t i n g p r o c e d u r e s 95

5 INTERPRETATIONS AND FURTHER ANALYSIS 110 5.1 A g g r e g a t e i n t e r l o c k 110 5.1.1 Comparison o f t h e o r y and r e s u l t s 110 5.1.2 A n a l y s i s o f some a s p e c t s o f a g g r e g a t e i n t e r l o c k on t h e 114 b a s i s o f t h e model d e v e l o p e d 5.2 T r a n s m i s s i o n o f f o r c e s a c r o s s r e i n f o r c e d c r a c k s 132 5.2.1 Components i n v o l v e d i n t h e t r a n s m i s s i o n o f f o r c e s 132 5.2.2 Comparison between r e i n f o r c e d and u n r e i n f o r c e d c r a c k s 136 5.2.3 Q u a n t i t a t i v e a n a l y s i s o f t h e b e h a v i o u r o f r e i n f o r c e d 143 c r a c k s 5.2.4 Specimens w i t h r e i n f o r c i n g b a r s i n c l i n e d t o t h e c r a c k 151 p l a n e 5.2.5 H y p o t h e s i s f o r t h e b e h a v i o u r o f r e i n f o r c e d c r a c k s s u b - 155 j e c t e d t o g e n e r a l c o m b i n a t i o n s o f e x t e r n a l l o a d s o r imposed d i s p l a c e m e n t s

6 A MATHEMATICAL FORMULATION OF THE RELATION BETWEEN 159 STRESSES AND DEFORMATIONS OF CRACKED REINFORCED CONCRETE,

TAKING INTO ACCOUNT THE CRACK PROPERTIES

6.1 The s t r e s s - d i s p l a c e m e n t r e l a t i o n f o r a s i n g l e c r a c k 159 6.2 The r e l a t i o n between s t r e s s e s and d i s p l a c e m e n t s i n 162

c r a c k e d r e i n f o r c e d c o n c r e t e 7 RETROSPECTIVE VIEW 168 8 SUMMARY 171 9 REFERENCES 177 10 NOTATION 188 11 APPENDICES 191

1 INTRODUCTION

1.1 M o t i v e s and scope o f t h e r e s e a r c h

A f u n d a m e n t a l demand i n t h e d e s i g n o f s t r u c t u r e s i s t h a t s u f f i c i e n t s a f e t y a g a i n s t f a i l u r e i s o b t a i n e d . W i t h r e g a r d t o c o n c r e t e s t r u c -t u r e s i -t c a n be s -t a -t e d -t h a -t numerous e x p e r i m e n -t s have been c a r r i e d out i n o r d e r t o e n a b l e t h e u l t i m a t e r e s i s t a n c e o f many t y p e s o f s t r u c t u r a l members under v a r i o u s l o a d i n g c o n f i g u r a t i o n s t o be e s t i -mated. However, w h i l e i t i s t r u e t h a t a t p r e s e n t many e q u a t i o n s a r e

a v a i l a b l e f o r p r e d i c t i n g b e a r i n g c a p a c i t i e s i n adequate agreement w i t h a v a i l a b l e t e s t r e s u l t s , i t c a n n o t be c l a i m e d t h a t t h e t h e o r i e s a l w a y s t r u l y e x p l a i n t h e c a l c u l a t e d s t r e n g t h s . L a c k o f f u n d a m e n t a l knowledge i n s e v e r a l r e s p e c t s means t h a t i t i s o f t e n i m p o s s i b l e t o be c o n f i d e n t i n e x t e n d i n g e x i s t i n g e q u a t i o n s t o s t r u c t u r e s w i t h a more complex b e h a v i o u r . P a r t i c u l a r l y i n t h e l a s t decade an i n c r e a s e i n s c a l e and c o m p l e x i t y of new s t r u c t u r e s has o c c u r r e d . Problems e n c o u n t e r e d i n t h e d e s i g n of o f f s h o r e s t r u c t u r e s s u b j e c t t o w i n d , waves, d r i f t i n g i c e and c o l l i s i o n s w i t h o t h e r f l o a t i n g o b j e c t s go f a r beyond t h e l e v e l o f the a c t u a l l y a v a i l a b l e e x p e r i m e n t a l knowledge. The same c a n be s a i d of t h e c o n s t r u c t i o n o f r e a c t o r v e s s e l s , r e q u i r i n g a h i g h degree o f s a f e t y a g a i n s t any p o s s i b l e damage caused by e a r t h q u a k e m o t i o n s o r f l y i n g o b j e c t s . A l s o s e a r e s i s t i n g s t r u c t u r e s , such as t h e s u r g e t i d e b a r r i e r i n t h e E a s t e r n S c h e l d t ( O o s t e r s c h e l d e ) i n t h e N e t h e r -l a n d s , w h i c h has t o p r o t e c t thousands o f p e o p -l e from a n o t h e r f -l o o d d i s a s t e r , c o n f r o n t the d e s i g n e r w i t h problems w h i c h he c a n n o t s o l v e by s i m p l y c o n s u l t i n g t h e l i t e r a t u r e . S m a l l - s c a l e t e s t s on s u c h s t r u c t u r e s c o u l d p r o v i d e a b e t t e r i n s i g h t i n t o t h e s e s i t u a t i o n s , but o f t e n t h e i n c o m p l e t e knowledge about s c a l e l a w s , p a r t i c u l a r l y f o r c o n c r e t e s t r u c t u r e s , l e a d s t o new i n c e r t a i n t i e s . There i s e v i -d e n t l y a nee-d f o r improve-d metho-ds o f a n a l y s i s .

F o r t u n a t e l y , a l s o i n t h e f i e l d o f c a l c u l a t i o n t e c h n i q u e s new p e r s p e c t i v e s were opened by t h e development o f t h e computer. The i n t r o -d u c t i o n o f t h e f i n i t e element metho-d, by means o f w h i c h t h e

behav-i o u r o f c o m p l behav-i c a t e d s t r u c t u r a l systems c o u l d be a n a l y s e d , a l t h o u g h i n i t i a l l y o n l y on the a s s u m p t i o n o f l i n e a r e l a s t i c m a t e r i a l p r o p e r t i e s , was an i m p o r t a n t s t e p f o r w a r d . However, f o r c o n c r e t e s t r u c -t u r e s , w h i c h e x h i b i -t a s -t r o n g l y n o n - l i n e a r c h a r a c -t e r due -t o -t h e f o r m a t i o n o f c r a c k s and p l a s t i f i c a t i o n o f m a t e r i a l s , more s o p h i s -t i c a -t e d programs had -t o be d e v e l o p e d , i n v o l v i n g -t h e p o s s i b i l i -t y o f r e d i s t r i b u t i o n o f f o r c e s . T h e r e f o r e n o t o n l y t h e n u m e r i c a l programs had t o be i m p r o v e d , b u t a l s o c l o s e a t t e n t i o n had t o be p a i d t o a c -c u r a t e f o r m u l a t i o n o f t h e p r o p e r t i e s o f t h e -c o n s t i t u e n t m a t e r i a l s . To g i v e added impetus t o f u r t h e r development i n t h i s d i r e c t i o n , t h e r e s e a r c h p r o j e c t " C o n c r e t e M e c h a n i c s " was s t a r t e d . I t was e x p e c t e d t h a t t h e b e s t r e s u l t s w o u l d be o b t a i n e d by a c o l l a b o r a t i o n o f i n -v e s t i g a t o r s i n t h e f i e l d s o f computer a n a l y s i s and m a t e r i a l r e s e a r c h . P a r t i c i p a t i n g i n t h e p r o j e c t were on t h e one hand t h e R i j k s w a t e r -s t a a t , a d i v i -s i o n o f t h e N e t h e r l a n d -s M i n i -s t r y o f T r a n -s p o r t and P u b l i c Works, c o n c e n t r a t i n g on t h e n u m e r i c a l p a r t o f the r e s e a r c h program, and on t h e o t h e r hand t h e T e c h n o l o g i c a l U n i v e r s i t i e s o f D e l f t and E i n d h o v e n and t h e I n s t i t u t e f o r A p p l i e d S c i e n t i f i c Re-s e a r c h on B u i l d i n g M a t e r i a l Re-s and B u i l d i n g S t r u c t u r e Re-s (IBBC-TNO), f o c u s i n g on t h e m a t e r i a l a s p e c t s . The o v e r a l l p r o j e c t " C o n c r e t e M e c h a n i c s " was f i n a n c i a l l y s u p p o r t e d and o r g a n i z e d by t h e CUR,

the r e s e a r c h committee i n t h e N e t h e r l a n d s C o n c r e t e A s s o c i a t i o n . A s u r v e y o f t h e s t r u c t u r e o f t h e p r o j e c t i s p r e s e n t e d i n F i g . 1.1. e x p e r i m e n t a l v e r i f i c a t i o n f u n d a m e n t a l knowledge d e s i g n r u l e s m i c r o - m o d e l macro-model c r a c k a r e a bond a r e a l i t e r a t u r e F i g . 1.1 S t r u c t u r e o f t h e o r g a n i z a t i o n o f t h e p r o j e c t "Concrète M e c h a n i c s "

The n u m e r i c a l p a r t o f t h e p r o j e c t has meanwhile r e s u l t e d i n two n o n - l i n e a r f i n i t e element programs, t h e m i c r o and t h e macro-model. The m i c r o - m o d e l i s a b l e t o d e s c r i b e t h e f o r m a t i o n of d i s c r e t e c r a c k s i n a c o n c r e t e s t r u c t u r e ^ I n t h i s way t h e d i s p l a c e m e n t s a t a c r a c k can be d e t e r m i n e d and t h e e f f e c t o f t h e s e d i s p l a c e m e n t s on t h e i n -t e r n a l s -t r e s s e s c a n be -t a k e n i n -t o a c c o u n -t . ' S i n c e -t h i s model c a n make a dominant c r a c k d i s t i n c t l y d i s c e r n i b l e , i t i s e s p e c i a l l y s u i t a b l e f o r t h e a n a l y s i s o f complex s t r u c t u r e s and r e i n f o r c i n g d e t a i l s ( G r o o t e n b o e r ^ ^ 27 3 ) •Tne o t h e r model,(^the macro-model, i s based on a d i f f e r e n t c o n c e p t ^ T o c a l c u l a t e t h e b e h a v i o u r under l o a d i n g a s t r u c t u r e i s d i v i d e d i n t o i m a g i n a r y l a y e r s o f c o n c r e t e and s t e e l , h a v i n g d i f f e r e n t p r o p e r t i e s The e f f e c t o f c r a c k i n g i s t a k e n i n t o account by m o d i f y i n g t h e s t i f f n e s s c h a r a c t e r i s t i c s o f a l a y e r . ¡The c r a c k s a r e as such "smeared o u t " (Blaauwendraad e t a l .

6 ^ ) / Both programs e n a b l e n o n - l i n e a r m a t e r i a l p r o p e r t i e s t o be . • \

i n s e r t e d i n an a p p r o p r i a t e way. j

The m a t e r i a l r e s e a r c h was s u b d i v i d e d i n t o t h r e e p a r t s : a s t u d y o f the mechanisms d i r e c t l y r e l a t e d t o t h e c r a c k a r e a , an a n a l y s i s o f the bond a r e a and a s t u d y o f l i t e r a t u r e t o p r o v i d e r e m a i n i n g c h a r a c t e r i s t i c s , such as t h e b e h a v i o u r o f c o n c r e t e under b i a x i a l l o a d -i n g .

The p a r t o f t h e p r o j e c t r e p o r t e d i n t h i s document i s c o n c e r n e d w i t h the s t u d y o f t h e c r a c k a r e a and was c a r r i e d o u t a t t h e D e l f t U n i -v e r s i t y o f T e c h n o l o g y . An adequate u n d e r s t a n d i n g and f o r m u l a t i o n o f t h e t r a n s m i s s i o n o f f o r c e s a c r o s s c r a c k s i n c o n c r e t e i s n e c e s -s a r y t o t a k e f u l l a d v a n t a g e o f t h e p o -s -s i b i l i t y , i n c l u d e d i n t h e n u m e r i c a l programs, t o d e s c r i b e r e d i s t r i b u t i o n s o f f o r c e s . 1.2 Some a s p e c t s o f t h e r o l e o f t h e roughness o f t h e c r a c k s i n c o n c r e t e s t r u c t u r e s P r i o r t o d i s c u s s i n g some examples i l l u s t r a t i n g t h e r o l e o f t h e roughness o f t h e c r a c k s , i t may be u s e f u l t o p o i n t o u t two i m p o r -t a n -t f e a -t u r e s i n h e r e n -t i n -t h e b e h a v i o u r o f c r a c k s i n r e i n f o r c e d and p r e s t r e s s e d c o n c r e t e .

d i r e c t i o n of the l o c a l p r i n c i p a l t e n s i l e s t r e s s e s i n the con-c r e t e i n the u n con-c r a con-c k e d l o a d i n g s t a g e , they do not n e con-c e s s a r i l y a l s o open p e r p e n d i c u l a r l y t o t h e i r d i r e c t i o n . T h i s i s m a i n l y caused by the f a c t t h a t the s t r e s s - s t r a i n r e l a t i o n s o f c r a c k e d r e i n f o r c e d o r p r e s t r e s s e d c o n c r e t e d i f f e r e s s e n t i a l l y from those i n the u n c r a c k e d s t a g e . Whereas i n the u n c r a c k e d s t a g e t h e i n -f l u e n c e o-f the r e i n -f o r c e m e n t on the s t r e s s - s t r a i n r e l a t i o n s i s s m a l l and the b e h a v i o u r i s n o t f a r from b e i n g i s o t r o p i c (De J o s s e l i n de Jong |~ 33 J and Walraven [] 80 ] ) , i n the cracked s t a t e g e n e r a l l y an a n i s o t r o p i c type of b e h a v i o u r i s d e v e l o p e d , r e s u l t i n g i n shear s t r e s s e s p a r a l l e l t o the c r a c k s . I n an e a r -l i e r p u b -l i c a t i o n [_ 81 J this subject was discussed. Apart from t h i s argument, i t must a l s o be n o t e d t h a t i n n o n - p r e s t r e s s e d c o n c r e t e s h r i n k a g e and t e m p e r a t u r e s t r e s s e s as w e l l as p r e v i o u s l o a d s may p r o d u c e a c r a c k o f any d i r e c t i o n , e x i s t i n g b e f o r e the u l t i m a t e l o a d i s i n t r o d u c e d . The e x i s t e n c e o f c r a c k s i n o t h e r d i r e c t i o n s t h a n would d i r e c t l y r e s u l t f r o m the l o a d s i n the u l t i m a t e l o a d i n g s t a t e i t s e l f , i s sometimes even a n e c e s s a r y a s -s u m p t i o n f o r d e -s i g n : an example o f t h i -s i -s t h a t c u r r e n t d e -s i g n c r i t e r i a i n the U n i t e d S t a t e s f o r c o n v e n t i o n a l l y r e i n f o r c e d nu-c l e a r nu-c o n t a i n m e n t v e s s e l s £ ] ~2 r e q u i r e that the s t r u c t u r e be d e s i g n e d t o w i t h s t a n d the s i m u l t a n e o u s o c c u r r e n c e of i n t e r n a l p r e s s u r e and the i n e r t i a f o r c e s g e n e r a t e d by a s t r o n g m o t i o n e a r t h q u a k e . Crack p l a n e s f o r m i n the c o n c r e t e w a l l i n the h o r i -z o n t a l and v e r t i c a l d i r e c t i o n s ( F i g . 1.2) due to i n t e r n a l pres-s u r e . The c y c l i c pres-s h e a r i n g f o r c e pres-s due t o a pres-s e i pres-s m i c e x c i t a t i o n must t h e n be t r a n s m i t t e d a l o n g t h e s e p l a n e s ( L a i b l e e t a l . [] 40 ]). / / / / ' / I

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g. 1.2 Cracking of a v e s s e l under i n t e r n a l pressure and seismic l o a d i n g [ 40 ]

b. As a r e s u l t of i t s rough s t r u c t u r e the c r a c k s can t r a n s m i t f o r c e s i n the normal and the p a r a l l e l d i r e c t i o n i f the c r a c k f a c e s a r e s h i f t e d i n o p p o s i t e d i r e c t i o n s . Three mechanisms can c o n t r i b u t e t o t h i s t r a n s m i s s i o n :

- A g g r e g a t e i n t e r l o c k : t h i s mechanism i s d i r e c t l y r e l a t e d t o the way i n w h i c h a c r a c k i s formed i n c o n c r e t e . Because the s t r e n g t h o f the hardened cement p a s t e i n most c o n c r e t e s i s lower than the s t r e n g t h of the a g g r e g a t e p a r t i c l e s , c r a c k s i n t e r s e c t t h e cement p a s t e b u t r u n a l o n g the edges o f t h e a g g r e g a t e p a r t i c l e s . So the a g g r e g a t e p a r t i c l e s , e x t e n d i n g from one o f the c r a c k f a c e s , " i n t e r l o c k " w i t h the o p p o s i t e f a c e and r e s i s t s h e a r d i s p l a c e m e n t s ( F i g . 1.3.a).

Dowel a c t i o n : t h i s term d e n o t e s the r e s i s t a n c e of a r e i n f o r c -i n g b a r , c r o s s -i n g a c r a c k , t o s h e a r d -i s p l a c e m e n t ( F -i g . 1.3.b). I t i s g e n e r a l l y assumed t h a t dowel a c t i o n i s i n f e r i o r t o ag-g r e ag-g a t e i n t e r l o c k f o r the r e i n f o r c e m e n t r a t i o s w h i c h a r e used i n p r a c t i c e .

- A x i a l f o r c e s i n r e i n f o r c i n g b a r s i n c l i n e d t o t h e c r a c k p l a n e ( F i g . 1.3.c).

a b c

F i g . 1.3 Aggregate i n t e r l o c k (a), dowel a c t i o n (b) and a x i a l s t e e l f o r c e ( c )

I n the f o l l o w i n g some examples a r e g i v e n , i l l u s t r a t i n g the r o l e of the t r a n s m i s s i o n of f o r c e s i n c r a c k s and e m p h a s i z i n g the need to have a b e t t e r u n d e r s t a n d i n g of t h i s phenomenon.

F i g . 1.4 shows the c r a c k i n g p a t t e r n o f a beam w i t h o u t s h e a r r e i n -f o r c e m e n t , s u b j e c t e d t o a 4 - p o i n t l o a d i n g t e s t , a t about 80% o-f the u l t i m a t e l o a d Q 82 ] .

F i g . 1.4 Measurements on c r a c k s i n a beam w i t h o u t s h e a r r e i n -f o r c e m e n t Q 82 ~2

D u r i n g t h i s t e s t some s t r a i n gauge r o s e t t e s were s t u c k on the c r a c k s , i m m e d i a t e l y a f t e r o b s e r v a t i o n ( a t a c r a c k w i d t h of w = 0 . 1 0 - 0 . 1 5 mm), so t h a t the d i s p l a c e m e n t s c o u l d be measured. I t i s seen t h a t an i n c r e a s i n g amount of s h e a r d i s p l a c e m e n t was r e c o r d e d , so t h a t s h e a r f o r c e s can be e x p e c t e d t o be t r a n s m i t t e d a c r o s s the c r a c k s (see a l s o F i g . 1.3.a). E s t i m a t e s o f the c o n t r i b u t i o n of a g g r e g a t e i n t e r l o c k i n the c r a c k s on the b a s i s of e x p e r -i m e n t s c o n d u c t e d on s -i n g l e c r a c k s (see s e c t -i o n 2.1), v a r -i e d between 40 and 90% ( T a y l o r [_ 75 ] , F e n w i c k C1 8D » Swamy Q 73 ] ) .

The r e s e a r c h p r o j e c t f r o m w h i c h t h e r e s u l t s p r e s e n t e d i n F i g . 1.4 were b o r r o w e d , r e f u t e d a p e r s i s t e n t m i s u n d e r s t a n d i n g about a g g r e -g a t e i n t e r l o c k . A s t u d y of the phenomenon t h a t the s h e a r r e s i s t a n c e o f s u c h t y p e s o f beam does n o t i n c r e a s e l i n e a r l y w i t h the s c a l e of the beam, b u t l e s s , was u n d e r t a k e n . T h i s was g e n e r a l l y b e l i e v e d to be due t o the f a c t t h a t i n e x p e r i m e n t s the s i z e of t h e a g g r e g a t e p a r t i c l e s was the o n l y v a r i a b l e t h a t was n o t s c a l e d p r o p e r l y , w h i c h would l e a d t o d i f f e r e n c e s i n a g g r e g a t e i n t e r l o c k . I n the i n v e s t i -g a t i o n a s e r i e s o f d i f f e r e n t l y s c a l e d beams, made o f -g r a v e l con-c r e t e , was con-compared w i t h an i d e n t i con-c a l s e r i e s , made of l i g h t w e i g h t c o n c r e t e . I n s p i t e of the f a c t t h a t l i g h t w e i g h t c o n c r e t e can o n l y

s u p p l y a v e r y reduced a g g r e g a t e i n t e r l o c k component, as a r e s u l t o f t h e low s t r e n g t h o f t h e l i g h t w e i g h t a g g r e g a t e p a r t i c l e s p e r m i t -t i n g -t h e c r a c k -t o r u n b o -t h -t h r o u g h -t h e hardened cemen-t p a s -t e and the p a r t i c l e s ( F i g . 1.5), t h e s e n s i t i v i t y t o t h e i n c r e a s e o f t h e s c a l e d i d n o t d i f f e r from t h a t o b t a i n e d f o r g r a v e l c o n c r e t e . Only the s h e a r r e s i s t a n c e was l o w e r o v e r t h e f u l l range o f t e s t s . I t was d e m o n s t r a t e d t h a t n o t a g g r e g a t e i n t e r l o c k , b u t t h e i n f l u e n c e of t h e s t r a i n g r a d i e n t on t h e f l e x u r a l t e n s i l e s t r e n g t h o f t h e con-c r e t e , i s r e s p o n s i b l e f o r t h e s con-c a l e e f f e con-c t . T h i s may i l l u s t r a t e t h a t c a u t i o n i s n e c e s s a r y i n t e s t i n g s t r u c t u r e s on model s c a l e i f no f u l l u n d e r s t a n d i n g o f a g g r e g a t e i n t e r l o c k i s a v a i l a b l e .

i g l j p i

lightweight concrete F i g . 1.5 A g g r e g a t e i n t e r l o c k i n g r a v e l c o n c r e t e ( i n t e r m e d i a t e o r low s t r e n g t h ) and l i g h t w e i g h t c o n c r e t e The c a p a b i l i t y o f a g g r e g a t e i n t e r l o c k t o t r a n s m i t s h e a r f o r c e s was even more c l e a r l y d e m o n s t r a t e d i n an i n v e s t i g a t i o n i n t o t h e i n f l u e n c e o f a x i a l t e n s i l e f o r c e s on t h e s h e a r r e s i s t a n c e o f sym-m e t r i c a l l y r e i n f o r c e d beasym-ms w i t h o u t s h e a r r e i n f o r c e sym-m e n t (Regan

L 61 J). P r i o r to shear loading, the beams were subjected to a x i a l

t e n s i l e l o a d s , p r o d u c i n g c r a c k s p e r p e n d i c u l a r t o t h e beam a x i s . These t e n s i l e l o a d s were k e p t c o n s t a n t d u r i n g l o a d i n g , a t v a l u e s between 0 and 130 kN. The l o a d i n g arrangement i s p r e s e n t e d i n F i g . 1.6.

The i n f l u e n c e o f a x i a l l o a d i n g on t h e s h e a r s t r e n g t h was found t o be v e r y s m a l l . I n a l l beams f a i l u r e was caused by t h e f o r m a t i o n o f a s i n g l e w e l l - d e f i n e d c r a c k , w i t h an i n c l i n a t i o n n o t a b l y f l a t t e r t h a n t h a t o f any e a r l i e r c r a c k s .

750 1500 750 2P p 2P bars 016 moment shear

•cracks prior to testing by axial tension cracks during testing

tension

F i g . 1.6 Beam subjected to combined shear and a x i a l loading | _ 6 l ]

I t was apparent f r o m t h e b e h a v i o u r o f beams c o n t a i n i n g v e r t i c a l c r a c k s o v e r t h e i r w h o l e d e p t h t h a t shear was b e i n g t r a n s m i t t e d by a g g r e g a t e i n t e r l o c k and dowel a c t i o n , e s p e c i a l l y as such c r a c k s near p o i n t s o f c o n t r a f l e x u r e were n e v e r even p a r t i a l l y c l o s e d by the f o r m a t i o n o f c o m p r e s s i o n zones due t o b e n d i n g . A s p e c i a l beam was t e s t e d t o i n v e s t i g a t e t h e p o s s i b l e s i g n i f i c a n c e o f dowel a c t i o n . T h i s beam had a s m o o t h - s i d e d c r a c k p r e f o r m e d i n i t a t t h e s e c t i o n of c o n t r a f l e x u r e . The c r a c k was formed by a s h e e t o f a c e t a t e p a p e r . The beam f a i l e d a t a v e r y low l o a d w i t h t h e f o r m a t i o n o f dowel c r a c k s a t t h e l e v e l s o f t h e t o p and b o t t o m r e i n f o r c e m e n t . H i g h r e l i a b i l i t y i s f u r t h e r imposed on t h e c a p a c i t y o f c r a c k s t o t r a n s m i t shear f o r c e s i n t h e a p p l i c a t i o n o f t h e o r i e s based on t h e p r i n c i p l e o f p l a s t i c b e h a v i o u r . I n t h e s e t h e o r i e s i t i s assumed t h a t s h e a r d i s p l a c e m e n t s o f t h e c r a c k f a c e s a r e c o m p l e t e l y p r e -v e n t e d by a g g r e g a t e i n t e r l o c k , i m p l y i n g t h a t t h e c r a c k s c a n open o n l y i n t h e p e r p e n d i c u l a r d i r e c t i o n . W i t h r e g a r d t o beams w i t h s h e a r r e i n f o r c e m e n t , i t i s assumed t h a t t h e r e i s an upper and a lower s t r i n g e r , w h i c h a r e b o t h i n f i n i t e l y s t i f f i n t h e a x i a l d i -r e c t i o n and have no f l e x u -r a l s t i f f n e s s a t a l l : as a -r e s u l t t h e f l e x u r a l moment i s f u l l y r e s i s t e d by t h e s e s t r i n g e r s and t h e shear

f o r c e i s f u l l y c a r r i e d by the web of the beam. Due t o t h e s e c o n -d i t i o n s s h e a r s t r e s s e s a r e provoke-d i n the c r a c k s , r e s u l t i n g i n c o m p r e s s i o n d i a g o n a l s w i t h a f l a t t e r i n c l i n a t i o n t h a n t h e c r a c k s t h e m s e l v e s . The " p l a s t i c t r u s s " l e a d s to a more e c o n o m i c a l d e s i g n of the s h e a r r e i n f o r c e m e n t t h a n the o r i g i n a l Morsch t r u s s ( F i g .

1.7), w h i c h has c o m p r e s s i o n d i a g o n a l s , by d e f i n i t i o n i n c l i n e d a t 45 degrees t o the beam a x i s , and w h i c h i s g e n e r a l l y known t o o v e r -e s t i m a t -e t h -e f o r c -e s i n t h -e s h -e a r r -e i n f o r c -e m -e n t c o n s i d -e r a b l y . 1 Mörsch truss

1

'plastic' truss F i g . 1.7 Morsch t r u s s a n a l o g y and p l a s t i c t r u s s a n a l o g y

However, t o g e t good agreement between the p r e d i c t i o n o f the p l a s t i c t h e o r y and the r e s u l t s o f e x p e r i m e n t s , m o s t l y c o r r e c t i o n c o e f -f i c i e n t s a r e n e c e s s a r y , w h i c h a r e n o t the same -f o r a l l c a s e s o -f l o a d i n g and cannot be p h y s i c a l l y e x p l a i n e d i n a s a t i s f a c t o r y way. To o b t a i n a b e t t e r i d e a o f the m e r i t s and l i m i t a t i o n s o f the p l a s -t i c -t h e o r y and -the way i n w h i c h improvemen-ts c o u l d be a c h i e v e d , i -t i s n e c e s s a r y t o have a b e t t e r u n d e r s t a n d i n g o f the most f u n d a m e n t a l a s s u m p t i o n o f the t h e o r y : the f u l l p r e v e n t i o n o f s h e a r d i s p l a c e -ments i n c r a c k s by a g g r e g a t e i n t e r l o c k . F i g . 1.8 shows some measure-ments o f the o p e n i n g d i r e c t i o n of c r a c k s i n a p a r t i a l l y p r e s t r e s s e d beam, t e s t e d i n an e a r l i e r i n v e s t i g a t i o n ( B r u g g e l i n g e t a l . [_ 8 ~~\ ) . I t i s seen t h a t the a s s u m p t i o n o f f u l l y p r e v e n t e d s h e a r d i s p l a c e -ment does n o t h o l d t r u e . However, i t i s f e l t t h a t t h e o c c u r r e n c e of shear d i s p l a c e m e n t s i s a b e t t e r argument f o r t h e p r e s e n c e o f s h e a r s t r e s s e s i n the c r a c k s than an o b s e r v a t i o n o f p e r p e n d i c u l a r

c r a c k o p e n i n g w o u l d be (see a l s o F i g . 1. 3 . a ) . F i g . 1.8 Measurements on c r a c k s i n a p a r t i a l l y p r e s t r e s s e d beam w i t h o u t s h e a r r e i n f o r c e m e n t \_ 8 ~J The l a s t example i s c o n c e r n e d w i t h t h e g e n e r a l l y u n d e r e s t i m a t e d r o l e t h a t a g g r e g a t e i n t e r l o c k p l a y s i n t h e d e s i g n o f mesh r e i n -f o r c i n g s y s t e m s , w h i c h i s a -f u n d a m e n t a l p r o b l e m -f o r c o n c r e t e s h e l l s , s h e a r w a l l s , box g i r d e r s and v e s s e l s , p r e s t r e s s e d as w e l l as u n p r e s t r e s s e d , and has been s t u d i e d i n t e n s i v e l y ( [ 2 , 7, 39, 4 l ] ) . G e n e r a l l y , the r o l e o f a g g r e g a t e i n t e r l o c k has been d i s r e g a r d e d , a r g u i n g t h a t t h e f r i c t i o n i s h i g h l y v a r i a b l e and, t o be on t h e s a f e s i d e , s h o u l d be n e g l e c t e d . R e c e n t l y Bazant V4 J pointed out, that t h i s argument i s f a l s e . The r e a s o n i s t h a t d u r i n g subsequent t a n -g e n t i a l d i s p l a c e m e n t t h e wed-gin-g e f f e c t o f s u r f a c e a s p e r i t i e s i n c o n t a c t causes f u r t h e r r e l a t i v e n o r m a l d i s p l a c e m e n t s , w h i c h i s m a n i f e s t e d by an o v e r a l l volume i n c r e a s e due t o shear ( d i l a t a n c y )

( F i g . 1.9). Due t o t h i s d i l a t a n c y t h e r e i n f o r c e m e n t c r o s s i n g t h e c r a c k i s t e n -s i o n e d . I f t h i -s e f f e c t i -s n o t t a k e n i n t o a c c o u n t , i t may happen t h a t t e n s i l e y i e l d i n g o f t h e r e i n f o r c e m e n t o c c u r s b e f o r e t h e ap-p l i e d l o a d comap-ponent n o r m a l t o t h e c r a c k a l o n e i s g r e a t enough f o r 10

-a c h i e v i n g i t . F u r t h e r m o r e i t m-ay o c c u r t h -a t t h e c r -a c k w i d t h i n t h e s e r v i c e a b i l i t y s t a t e i s g r e a t e r t h a n e x p e c t e d ( F i g . 1.9), w h i c h would be r a t h e r u n d e s i r a b l e i n , f o r i n s t a n c e , b r i d g e s o r n u c l e a r s t r u c t u r e s . C o n s e q u e n t l y , t a k i n g a g g r e g a t e i n t e r l o c k i n t o a c c o u n t g e n e r a l l y l e a d s t o h e a v i e r r a t h e r t h a n l i g h t e r r e i n f o r c e m e n t and thus a n e g l e c t o f t h i s mechanism i s s e e n t o be g e n e r a l l y on t h e u n s a f e s i d e . F i g . 1.9 Shear d i l a t a n c y i n c r a c k s The f o r m u l a t i o n o f c r a c k c h a r a c t e r i s t i c s i n e x i s t i n g n u m e r i c a l programs E a r l i e r n u m e r i c a l programs d i s p l a y a g r e a t v a r i e t y o f p r o p e r t i e s , a t t r i b u t e d t o the c r a c k s . N g o / S c o r d e l i s Q54 ] , N i l s o n Q 55 ] and Stauder £ 71 ] i n t r o d u c e d c r a c k s i n t h e i r n o n - l i n e a r f i n i t e element programs by d i s c o n n e c t i n g the nodes between the e l e m e n t s . I n t h i s way d i s c r e t e c r a c k s were o b t a i n e d , each s i d e o f w h i c h was c o n s i d e r e d t o be an i n d e p e n d e n t e x t e r n a l boundary, u n a b l e t o t r a n s m i t s h e a r s t r e s s e s ( F i g . 1.10). The d i s a d v a n t a g e o f t h i s method i s t h a t t h e c r a c k d i r e c t i o n s a r e r e s t r i c t e d t o the d i r e c t i o n o f the element edges. A n o t h e r , more g e n e r a l l y a p p l i e d method, w h i c h i s n o t s u b j e c t t o t h i s r e s t r i c t i o n , i s t o "smear o u t " the e f f e c t o f c r a c k i n g o v e r the e l e m e n t s : i t i s t h e n assumed t h a t , i f t h e t e n s i l e s t r e n g t h o f the c o n c r e t e i n an element i s r e a c h e d , an i n f i n i t e number o f i d e n t i c a l c r a c k s w i t h t h e same d i r e c t i o n i s formed ( F i g . 1.11).

F i g . 1.10 C r a c k r e p r e s e n t a t i o n F i g . 1.11 by means o f d i s c o n n e c t -i n g nodes ^ 5 4 , 5 5 , 71 ] C r a c k r e p r e s e n t a t i o n by means of smeared-out c r a c k f i e l d s The c r a c k e d elements a r e c o n s i d e r e d as c o n t i n u a w i t h an a n i s o t r o p i c b e h a v i o u r . The a s s u m p t i o n t h a t t h e s e c r a c k s behave as smooth s u r -f a c e s , u n a b l e t o t r a n s m i t s h e a r s t r e s s e s , was made by Cervenka Q 13 3 ana Loov C 4 ° H • I n h e r e n t i n t h i s assumption i s t h a t t h e

p r i n c i p a l s t r e s s d i r e c t i o n s a f t e r c r a c k i n g a r e f i x e d i n , and p e r p e n d i c u l a r t o , t h e d i r e c t i o n o f t h e c r a c k s : as a r e s u l t a r e d i s t r i -b u t i o n o f f o r c e s a f t e r c r a c k i n g was i m p o s s i -b l e . The s h e a r s t i f f n e s s modulus, denoted by G i n t h e u n c r a c k e d s t a t e , was reduced t o z e r o

i m m e d i a t e l y a f t e r c r a c k i n g ( F i g . 1.12). I

T

F i g . 1.12 Shear modulus o f c r a c k e d c o n c r e t e a c c o r d i n g t o | 13, 4 6 ]

M a i n t a i n a n c e of t h e f u l l s h e a r r e s i s t a n c e a f t e r c r a c k i n g was sup-posed i n t h e n o n - l i n e a r f i n i t e element programs d e v e l o p e d by

F r a n k l i n [_ 21 J , Isenberg/Adham Q 31 J , Zienkiewics/Phillips/Owen C ?2] , Swoboda [_ 74 ] and M u l l e r [_ 52 ] ( F i g . 1.13). As was

men-t i o n e d a l r e a d y i n s e c men-t i o n 1.2, t h i s assumption was also used i n the a n a l y t i c a l methods b a s e d on the t h e o r y of p l a s t i c i t y ( B r a e s t r u p / N i e l s e n [ s ] , T h i i r l i m a n n [ 7 8 ] , CEB-Model Code Q 101) •

F i g . 1.13 Shear modulus of cracked concrete according to Q 5 , 10,

31 , 52, 53, 74, 78, 92 ]

A l s o Ebbinghaus Q l 5 ] took a f u l l y maintained G-value a f t e r cracki n g cracki n t o a c c o u n t , b u t added the c o n d cracki t cracki o n t h a t t h cracki s v a l u e cracki s r e -duced t o z e r o , i f the s h e a r s t r e s s i n the c r a c k s exceeds a c e r t a i n l i m i t . T h i s l i m i t was d e f i n e d t o d e c r e a s e f o r i n c r e a s i n g s t r a i n normal t o the c r a c k d i r e c t i o n ( o r , f o r m u l a t e d d i f f e r e n t l y , f o r i n c r e a s i n g c r a c k w i d t h s ) ( F i g . 1.14).

F i g . 1.14 Shear modulus of cracked concrete according to f '5

A f o r m u l a t i o n w h i c h i s s t i l l g e n e r a l l y u s e d , i m p l i e s t h a t the s h e a r s t i f f n e s s of an element a f t e r c r a c k i n g i s reduced to a l o w e r v a l u e ,

b u t r e m a i n s g r e a t e r than z e r o , a c c o r d i n g t o the e q u a t i o n : T = a.G.y i n w h i c h a i s a p r e d e t e r m i n e d c o n s t a n t , w i t h 0 < a < 1 ( F i g . 1.15). X F i g . 1.15 Shear modulus of c r a c k e d c o n c r e t e a c c o r d i n g t o |~ 11, 28, 37, 43, 72, 91 ] T h i s t y p e of f o r m u l a t i o n was used by H a n d / P e c k n o l d / S c h n o b r i c h [_ 28 J (a = 0 . 4 ) , Y u z u g u l l u / S c h n o b r i c h Q 9 l 3 (a = 0 . 2 ) , S u i d a n / S c h n o b r i c h |^72^] (a = 0 . 5 ) , L i n / S c o r d e l i s [_ 43 ] , C e d o l i n / D e i P o l i £ 1 1 J and K r i s n a m o o r t h y / P a n e e r s e l v a m \~ 37 ] • I n a l a t e r p u b l i c a t i o n C e d o l i n and D e i P o l i improved t h i s f o r m u l a t i o n , r e l a t i n g a t o the s t r a i n normal to the d i r e c t i o n of the c r a c k s ( F i g . 1.16) £ 12 ^ .

X

F i g . 1.16 Shear modulus of c r a c k e d c o n c r e t e a c c o r d i n g t o £l2 J

-The s u g g e s t e d e x p r e s s i o n f o r e G /G = C (1 - — ) G /G = 0 c r i n w h i c h C i s a n u m e r i c a l c o n s t a n t ( s u g g e s t e d v a l u e 0.1 E ) , e i s c n t h e f i c t i t i o u s s t r a i n ( t h e c o n t r i b u t i o n of the c o n c r e t e t o t h e s t r a i n was n e g l e c t e d ) i n the d i r e c t i o n normal t o the c r a c k and i s a l i m i t v a l u e beyond w h i c h no s h e a r t r a n s m i s s i o n a c r o s s the c r a c k s i s supposed to be p o s s i b l e (0.0035 < z < 0.0045).

S c h i m m e l p f e n n i g £ 67 J used a reduction f a c t o r a which was defined to d e c r e a s e as a f u n c t i o n of i n c r e a s i n g s t r a i n n o r m a l t o the c r a c k d i r e c t i o n , b u t was a l s o r e l a t e d t o the l e v e l of the s h e a r s t r e s s i n the c r a c k s ( F i g . 1.17).

F i g . 1.17 Shear modulus of c r a c k e d c o n c r e t e a c c o r d i n g to Q 67 ]

However, a l t h o u g h r e l a t i n g t h e v a l u e a t o the development o f the s t r a i n n o r m a l t o the c r a c k s r e s u l t s i n an improved e x p r e s s i o n f o r a, t h i s f o r m u l a t i o n , t o o , has to be r e g a r d e d as r a t h e r p r o v i s i o n a l s i n c e i t i s o b v i o u s t h a t o t h e r f a c t o r s may i n f l u e n c e the s h e a r s t i f f n e s s as w e l l , such as the d i s t r i b u t i o n of c r a c k w i d t h s and d i s t a n c e s , the c o n c r e t e q u a l i t y , the v a l u e of the s h e a r d i s p l a c e -ment and the l o a d h i s t o r y . These a s p e c t s were t a k e n i n t o a c c o u n t by S c h a f e r £ 65 ] , who assumed t h a t the i n i t i a l r e s i s t a n c e of two

G was :

i f E < E < e c r n p

i f E > e n p

c r a c k f a c e s a g a i n s t s h e a r d i s p l a c e m e n t can by n e g l e c t e d o v e r a c e r t a i n d i s t a n c e , w h i l e a f t e r t h i s " f r e e s l i d i n g r a n g e " a l i n e a r r e -l a t i o n between s h e a r s t r e s s and s h e a r d i s p -l a c e m e n t c a n be assumed ( F i g . 1.18). The l e n g t h o f t h e f r e e s l i d i n g range was t h e o r e t i c a l l y d e r i v e d , w h i l e t h e l i n e a r r e l a t i o n was s u b s e q u e n t l y based on t e s t r e s u l t s ( F 30 ] , see a l s o s e c t i o n 2.2). I n c o m b i n a t i o n w i t h formu-l a s f o r t h e average c r a c k w i d t h and d i s t a n c e , a r e formu-l a t i o n was der i v e d w h i c h der e l a t e d a b o t h t o t h e s t der a i n nodermal t o t h e c der a c k d i der e c -t i o n and -t o -the s h e a r d e f o r m a -t i o n ( F i g . 1.19). f minimi/ .«ff1^ m T rr r v viiniinn CO 2 1 0.5 Gcr< 05 1 CO Y F i g . 1.18 F r e e s l i d i n g range a c c o r d i n g t o |_65 j F i g . 1.19 Shear modulus of c r a c k e d c o n c r e t e , a c c o r d i n g t o |_ 65 ] G e i s t e f e l d |_ 24 ] d e a l t , i n h i s t w o - d i m e n s i o n a l t e n s i o n s t i f f e n i n g m o d e l , w i t h the s h e a r s t i f f n e s s i n a s i m i l a r way, b u t d i d n o t t a k e i n t o a c c o u n t t h e f r e e s l i d i n g r a n g e . I t i s o b v i o u s t h a t the f o r m u l a t i o n o f c h a r a c t e r i s t i c s r e p r e s e n t i n g the t r a n s f e r o f s t r e s s e s a c r o s s c r a c k s i s g e n e r a l l y r a t h e r p r o v i -s i o n a l . A g r e a t number o f v a r i a b l e -s w h i c h may r e a -s o n a b l y be ex-p e c t e d t o i n f l u e n c e t h e b e h a v i o u r i s n o t t a k e n i n t o a c c o u n t . I n none o f t h e programs was t h e p o s s i b i l i t y o f s h e a r d i l a t a n c y due t o wedging a c t i o n o f the c r a c k s implemented. S i n c e up t o now the n u m e r i c a l programs have p r e d o m i n a n t l y been used t o i n v e s t i g a t e r e l a t i v e l y u n c o m p l i c a t e d s t r u c t u r e s s u b j e c t e d t o s i m p l e l o a d i n g c o n d i t i o n s i n w h i c h s h e a r g e n e r a l l y p l a y e d a s u b o r d i n a t e r o l e , the r e l e

-vance o f t r a n s f e r of s t r e s s e s i n c r a c k s has o n l y p a r t i a l l y been r e c o g n i z e d . O n l y i n |_ 12 J , i n which beams w i t h o u t shear r e i n f o r c e -ment s u b j e c t e d t o s h e a r f o r c e s were n u m e r i c a l l y i n v e s t i g a t e d , i t was p o i n t e d o u t t h a t t h e r e was a s t r i c t r e l a t i o n s h i p between t h e

i n s e r t e d c h a r a c t e r i s t i c s r e g a r d i n g a g g r e g a t e i n t e r l o c k and t h e r e -s u l t -s o f t h e c a l c u l a t i o n -s . S i n c e c a l c u l a t i o n program-s a r e bound t o be a p p l i e d t o s t r u c t u r e s and l o a d i n g t y p e s o f g r e a t e r c o m p l e x i t y , a good f o r m u l a t i o n of t h e s e m a t e r i a l p r o p e r t i e s i s a l s o becoming more i m p o r t a n t . 1.4 Aim o f t h e r e s e a r c h program

/The a i m o f t h e r e s e a r c h program i s t o e s t a b l i s h t h e r e l a t i o n between s t r e s s e s and d i s p l a c e m e n t s between c r a c k f a c e s . ) I n t h e f i r s t i n -s t a n c e t h e b e h a v i o u r under m o n o t o n i c i n c r e a -s e c i l o a d i n g w i l l be s t u d i e d . I n t h e p r e v i o u s p a r t i t has a l r e a d y been p o i n t e d o u t t h a t the r e l a t i o n w h i c h i s g e n e r a l l y employed up t o now and w h i c h i s r e p r e s e n t e d by dx = Ct(..dA (1.1) i s n o t t h e b a s i c e q u a t i o n d e s c r i b i n g t h e t r a n s f e r o f s t r e s s e s a c r o s s c r a c k s , b u t t h a t t h i s r e l a t i o n has t o be r e p r e s e n t e d by ( s e e a l s o F i g . 1.20) da • = C nn Cn t dw « dx C t n C t t dA ( 1.2)

The c o e f f i c i e n t s C , C , C and C depend on t h e i n t e r a c t i o n nn n t t n t t

between t h e mechanisms o f a g g r e a t e i n t e r l o c k , dowel a c t i o n and a x i a l r e s t r a i n t f o r c e s i n t h e r e i n f o r c i n g s t e e l . F i r s t l y , i t has to be a s s e s s e d how t h e s e i n d i v i d u a l components a r e i n f l u e n c e d by t e c h n o l o g i c a l f a c t o r s . S e c o n d l y , i t has t o be e s t a b l i s h e d how t h e i n t e r a c t i o n s between t h e components a r e a c h i e v e d .

I

I

-i

I

-x a

F i g . 1.20 S t r e s s e s and d i s p l a c e m e n t s i n a c r a c k I t i s i m p o r t a n t t h a t n o t m e r e l y e m p i r i c a l r e l a t i o n s f o r t h e c o e f -f i c i e n t s o -f e q u a t i o n (1.2) a r e p r o v i d e d , b u t a l s o t h a t an i n s i g h t i n t o the f u n d a m e n t a l b e h a v i o u r of the mechanism i s o b t a i n e d . T h i s would n o t o n l y g i v e i n f o r m a t i o n o v e r a w i d e r range o f v a l u e s t h a n c o v e r e d by the e x p e r i m e n t a l r e s u l t s a l o n e , b u t would a l s o g i v e some i d e a o f the b e h a v i o u r under o t h e r t y p e s o f l o a d i n g ( c y c l i c l o a d i n g o r l o n g t e r m c o n s t a n t l o a d i n g ) . I n e x t e n d i n g t h e i n v e s t i -g a t i o n , a t a l a t e r s t a -g e , t o the e f f e c t o f such t y p e s o f l o a d i n -g c o n d i t i o n s , a f u n d a m e n t a l i n s i g h t c o u l d c o n t r i b u t e c o n s i d e r a b l y to an e f f i c i e n t d e s i g n o f new t e s t s e r i e s . I t w i l l be i n v e s t i g a t e d what i n f o r m a t i o n i s a l r e a d y a v a i l a b l e i n t h e l i t e r a t u r e and where new r e s e a r c h i s n e c e s s a r y .

-2 STATE OF THE ART

The mechanisms i n v o l v e d i n t h e t r a n s m i s s i o n o f f o r c e s a c r o s s c r a c k s a r e , as m e n t i o n e d b e f o r e , a g g r e g a t e i n t e r l o c k , dowel a c t i o n and a x i a l r e s t r a i n t s t i f f n e s s o f r e i n f o r c i n g b a r s ( s e e a l s o F i g . 1.3). These components w i l l f i r s t be d e a l t w i t h s e p a r a t e l y . Then t h e a v a i l a b l e i n f o r m a t i o n on t h e i n t e r a c t i o n o f t h e s e components i n c r a c k s c r o s s e d by r e i n f o r c e m e n t i s r e v i e w e d . T o p i c s a r e d i s c u s s e d o n l y as f a r as they s u p p l y i n f o r m a t i o n w h i c h i s d i r e c t l y concerned w i t h t h e s u b j e c t t o be s t u d i e d . An e x t e n d e d r e v i e w o f t h e whole f i e l d of a v a i l a b l e i n f o r m a t i o n , f o c u s i n g a l s o on c y c l i c and i m p a c t l o a d i n g , was p u b l i s h e d e a r l i e r ( W a l r a v e n £ 83 ] ) . 2. 1 A g g r e g a t e i n t e r l o c k I n o r d e r t o a v o i d c o n f u s i o n when t a l k i n g about c r a c k w i d t h , i t i s e m p h a s i z e d , b e f o r e d e a l i n g w i t h t h e phenomenon o f a g g r e g a t e i n t e r -l o c k , t h a t t h e c r a c k w i d t h i s d e f i n e d as t h e d i s p -l a c e m e n t t h a t t h e c r a c k f a c e s have undergone i n t h e normal d i r e c t i o n i n r e l a t i o n t o each o t h e r . T h i s d e f i n i t i o n i s n e c e s s a r y because on t h e m i c r o s c a l e the l o c a l c r a c k w i d t h s v a r y c o n s i d e r a b l y when a s h e a r d i s p l a c e m e n t has o c c u r r e d ( F i g . 2 . 1 ) . As d e f i n e d h e r e , t h e c r a c k w i d t h i s i n d e -pendent o f t h e s h e a r d i s p l a c e m e n t .

F i g . 2.1 L o c a l v a r i a t i o n o f " c r a c k w i d t h " on t h e m i c r o s c a l e

F e n w i c k I 18 J conducted tests on specimens as represented i n F i g . 2.2. The specimens were c r a c k e d p r i o r t o t e s t i n g by an e x t e r -n a l t e -n s i l e f o r c e . The c r a c k p l a -n e was p r e d e t e r m i -n e d by a g r o o v e a l o n g t h e o u t l i n e of t h e specimen. A f t e r c r a c k i n g , t h e c r a c k w i d t h

was k e p t c o n s t a n t a t v a l u e s r a n g i n g from 0.06 t o 0. 38 mm. To obt a i n a c o n s obt a n obt c r a c k w i d obt h d u r i n g a obt e s obt , a f obt e r e v e r y l o a d i n c r e -ment the c r a c k w i d t h was a d j u s t e d by means o f an e x t e r n a l f o r c e , n o r m a l t o t h e c r a c k p l a n e . V a l u e s f o r t h i s f o r c e were n o t g i v e n . A l l t h e specimens f a i l e d as a r e s u l t o f f l e x u r a l t e n s i l e c r a c k i n g a t an e a r l y s t a g e o f l o a d i n g ( F i g . 2.2.a). preformed crack -secondary cracks 125mm4JOO-T(N/mm2) 2.0 1.5 1.0 0.5

18

.100,1 125 : 33 N /mm/ b.

0

^=0.06 rr m 0.13 X'*> - —• -0.32 —538 crac test dng of alocks -mm) A( dng of alocks -mm) 0.05 0.10 0.15 Q20 0.25 T(N/mm2) 2.0 0.05 0.10 0.15 0.20 F i g . 2.2 T e s t equipment and r e s u l t s o f F e n w i c k / P a u l a y |_ 18 J F i g . 2.2.b shows t h e development o f t h e s h e a r s t r e s s a c t i n g on t h e c r a c k p l a n e as a f u n c t i o n o f t h e shear d i s p l a c e m e n t f o r v a r i o u s c o n s t a n t c r a c k w i d t h s , t h e c o n c r e t e s t r e n g t h b e i n g c o n s t a n t . F i g , 2.2.c shows t h e i n f l u e n c e o f t h e c o n c r e t e s t r e n g t h on t h i s r e l a t i o n f o r a c o n s t a n t c r a c k w i d t h : b e f o r e t e n s i l e c r a c k i n g t h e c r a c k s t i f f n e s s w i t h r e g a r d t o shear l o a d i n g appeared t o be an i n -c r e a s i n g 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 . F i g u r e s l i k e 2.2.b and 2.2.c r e p r e s e n t a v e r a g e shear s t r e s s e s . I t must be r e a l i z e d t h a t t h e s h e a r s t r e s s e s a r e g e n e r a l l y n o t u n i f o r m -l y d i s t r i b u t e d , b u t v a r y a -l o n g the -l e n g t h o f t h e c r a c k p -l a n e . T h i s v a r i a t i o n i s a f u n c t i o n o f t h e geometry o f t h e specimens (as

-s t r a t e d by Schwing |_ 69 J) and of the method of i n t r o d u c i n g the l o a d s . T h i s e f f e c t has always t o be t a k e n i n t o a c c o u n t when i n t e r -p r e t i n g t e s t r e s u l t s .

An i n v e s t i g a t i o n , based on t h e same t e s t i n g c o n c e p t as used i n [ l 8 ] , was c a r r i e d o u t by Houde and M i r z a \_ 30 ] . The r e s u l t s o f t h i s i n v e s t i g a t i o n d i d n o t d i f f e r v e r y much from t h o s e o b t a i n e d i n [ 1 8 ] .

P a u l a y and L o e b e r F 60 ] conducted other constant crack width t e s t s , u s i n g a d i f f e r e n t t e s t s e t - u p and l a r g e r s p e c i m e n s , i n o r d e r t o a v o i d e a r l y f l e x u r a l t e n s i l e c r a c k i n g . The t y p e o f specimen used i n t h e i r t e s t s i s r e p r e s e n t e d i n F i g . 2.3. 300 mm stresses to adjust crack width 1.114 J 152 F i g . 2.3 T e s t arrangement a c c o r d i n g t o |_ 60 ]

The lower p a r t o f t h e specimen was c o m p l e t e l y f i x e d , w h i l e t h e upper p a r t c o u l d move f r e e l y . The c r a c k w i d t h c o u l d be a d j u s t e d w i t h an a c c u r a c y o f 2%. C r a c k w i d t h and s h e a r d i s p l a c e m e n t were measured on b o t h s i d e s o f t h e specimen. The t e s t r e s u l t s were n o t i n f l u e n c e d by t h e development of secondary c r a c k s i n t h e s p e c i m e n s . The t e s t s were c a r r i e d out w i t h a c o n s t a n t c o n c r e t e s t r e n g t h e q u a l t o f , = 37 N/mm2. O b i e c t s o f i n v e s t i g a t i o n were t h e i n f l u e n c e

c c y l

of t h e c r a c k w i d t h , t h e e f f e c t o f a g g r e g a t e s i z e and shape, and the e f f e c t o f l o a d h i s t o r y . The maximum v a l u e s o f t h e s h e a r s t r e s s e s

were much h i g h e r than o b t a i n e d i n the t e s t s of |_18, 30 J . The upper l i m i t of s h e a r t r a n s f e r a t T = 7 N/mm2 was n o t r e a c h e d as

a r e s u l t o f a g g r e g a t e i n t e r l o c k i n the c r a c k . E i t h e r f a i l u r e o c c u r r e d as a r e s u l t of l o c a l c r u s h i n g o f the c o n c r e t e i n the top or b o t t o m s e c t i o n o f the s p e c i m e n s , or f u r t h e r i n c r e a s e o f l o a d was n o t p o s s i b l e w i t h the l o a d i n g arrangement u s e d . I t was o b s e r v e d a l -so by P a u l a y and L o e b e r |_ 60 J that the shear stress-displacement r e l a t i o n i s e s s e n t i a l l y dependent on the c r a c k w i d t h . Shape and s i z e of the a g g r e g a t e p a r t i c l e s had no n o t i c e a b l e i n f l u e n c e i n the range t e s t e d (D = 9.5- 19 mm, round and c r u s h e d ) . The r e l a t i o n

max

between s h e a r s t r e s s e s and d i s p l a c e m e n t s i s r e p r e s e n t e d i n F i g . 2.4, and the s t r e s s e s n o r m a l to t h e c r a c k p l a n e , n e c e s s a r y t o keep the c r a c k w i d t h c o n s t a n t , a r e r e p r e s e n t e d as a f u n c t i o n of the shear s t r e s s i n F i g . 2.5. X(N/mm2) 8 ₈ X (N/mm ) 0.1 0.2 0.3

OA

0.5 0.6 07 0.8 A (mm) 2 3 A 5 6 _OVitN/mm2) F i g . 2.4 Shear s t r e s s s h e a r d i s F i g . 2.5 Average s h e a r s t r e s s -p l a c e m e n t r e l a t i o n f o r r e s t r a i n i n g normal c o n s t a n t c r a c k w i d t h s , s t r e s s r e l a t i o n s f o r a c c o r d i n g t o Q 60 ] crack widths 0.25-0.51 mm For t h e s e l a s t m e n t i o n e d c u r v e s no s i g n i f i c a n t i n f l u e n c e of a g g r e

-g a t e type o r c r a c k w i d t h was o b s e r v e d . I t i s s e e n i n F i -g . 2.4 t h a t the s h e a r s t r e s s - s h e a r d i s p l a c e m e n t r e l a t i o n s have a " h a r d e n i n g " c h a r a c t e r . T h i s can be e x p l a i n e d by the p h y s i c a l n a t u r e o f t h e mechanism, s i n c e an i n c r e a s e of s h e a r d i s p l a c e m e n t must r e s u l t i n an i n c r e a s e of the t o t a l c o n t a c t a r e a between the c r a c k f a c e s (due t o c r u s h i n g of hardened cement p a s t e ) , w h i c h r e s u l t s i n i n c r e a s i n g r e s i s t a n c e t o s h e a r d i s p l a c e m e n t . The i n f l u e n c e of l o a d h i s t o r y i s shown i n F i g . 2.6. I n t h i s d i a g r a m the mean e x p e r i m e n t a l c u r v e ob-t a i n e d i n s e p a r a ob-t e ob-t e s ob-t s , c a r r i e d ouob-t w i ob-t h a c o n s ob-t a n ob-t r a ob-t i o beob-tween s h e a r s t r e s s and c r a c k w i d t h , i s g i v e n , w i t h on b o t h s i d e s t h e i n t e r v a l of s c a t t e r of 13% (shaded a r e a ) . Upon t h i s c u r v e the r e -s u l t -s from the p r e v i o u -s " c o n -s t a n t c r a c k w i d t h t e -s t -s " ( F i g . 2.4) have been superimposed t o e n a b l e a c o m p a r i s o n t o be made. The d o t t e d l i n e c o n n e c t s the a p p r o p r i a t e s t r e s s v a l u e s f o r the t h r e e d i s -t i n c -t c r a c k w i d -t h s used i n -the -t e s -t s . I -t r e v e a l s -the same f o r m as the r e l a t i o n s h i p o b t a i n e d from the " v a r i a b l e c r a c k w i d t h t e s t s " .

0.1 0.2 0.3 OA 0.5 0.6 0.7 0.8 A (mm) F i g . 2.6 Mean e x p e r i m e n t a l c u r v e f o r s h e a r s t r e s s - d i s p l a c e m e n t r e l a t i o n s h i p w i t h c o n s t a n t s h e a r s t r e s s t o c r a c k w i d t h r a t i o , a c c o r d i n g t o |_ 60 ] T a y l o r [ 7 5 ] c a r r i e d out an i n v e s t i g a t i o n i n t o the f u n d a m e n t a l

b e h a v i o u r o f beams w i t h o u t s h e a r r e i n f o r c e m e n t , i n w h i c h i t was demonstrated how t h e b e h a v i o u r o f t h e s t r u c t u r e as a whole can be e x p l a i n e d as an i n t e r a c t i o n o f a number o f b a s i c mechanisms. One of t h e s e mechanisms was a g g r e g a t e i n t e r l o c k . Measurements on beams showed t h a t c r a c k s do n o t open t o t h e i r f i n a l w i d t h and s h e a r t h e n , but open and s h e a r s i m u l t a n e o u s l y . T h e r e f o r e i t was doubted whether the r e s u l t s o b t a i n e d i n Q l 8 , 30, 60 ] are immediately a p p l i c a b l e to t h e a n a l y s i s o f a c t u a l beam b e h a v i o u r . O b s e r v a t i o n s on t h e beams w i t h o u t s h e a r r e i n f o r c e m e n t appeared t o show a c o n s t a n t r a t i o o f

c r a c k w i d t h t o s h e a r d i s p l a c e m e n t d u r i n g c r a c k o p e n i n g . T h e r e f o r e a g g r e g a t e i n t e r l o c k t e s t s were c a r r i e d o u t , n o t w i t h c o n s t a n t c r a c k w i d t h s as i n [ 18, 30, 60 ] , but w i t h constant crack width to shear d i s p l a c e m e n t r a t i o s ( F i g . 2 . 7 ) .

355 mm 140 mm

loading

^direction

n

F i g . 2.7 S c h e m a t i c i l l u s t r a t i o n o f t e s t equipment, used i n [ 7 5 ]

The r a t i o o f normal t o s h e a r d i s p l a c e m e n t c o u l d be changed between t e s t s b u t was c o n s t a n t d u r i n g a t e s t . T h i s r a t i o was i n t r o d u c e d by means o f a p a r a l l e l r u l e r system. However, a l t h o u g h a good a p p r o x i -m a t i o n o f r e a l c r a c k b e h a v i o u r i n c o n s t r u c t i o n a l s i t u a t i o n s see-med

to be o b t a i n e d , i t i s f e l t t h a t a l s o o b j e c t i o n s c o u l d be r a i s e d , due t o which no g e n e r a l v a l i d i t y c a n be a t t r i b u t e d t o t h e r e s u l t s . A t f i r s t o n l y t h e normal d i s p l a c e m e n t s a t t h e c r a c k were measured:

hence i t i s n o t c e r t a i n t h a t t h e r a t i o o f normal t o s h e a r d i s p l a c e -ment, w h i c h was supposed t o be i n t r o d u c e d by t h e t e s t arrange-ment, was a l s o o b t a i n e d a t t h e l e v e l o f t h e c r a c k . T h i s may o n l y be

con-s i d e r e d t o be t r u e i f i t were c e r t a i n t h a t t h e c r a c k f a c e con-s have no r e s i s t a n c e a g a i n s t t h e imposed d i s p l a c e m e n t s , so t h a t no p r e f e r e n c e f o r any d i r e c t i o n o f c r a c k o p e n i n g would e x i s t . F u r t h e r , t h e o b s e r -v a t i o n s t h a t t h e r e i s a l i n e a r p r o p o r t i o n a l i t y between normal and s h e a r d i s p l a c e m e n t s i n beams w i t h o u t s h e a r r e i n f o r c e m e n t were based on measurements by means o f s t r a i n gauges w h i c h were s t u c k on t h e beam a f t e r t h e c r a c k s had formed, so t h a t a c e r t a i n i n t e r v a l o f d i s p l a c e m e n t was n o t measured. E x p e r i m e n t s i n w h i c h t h e measure-ments on s i m i l a r type o f beams were c a r r i e d o u t f r o m t h e b e g i n n i n g

of l o a d i n g , thus p r o v i d i n g a c o m p l e t e p i c t u r e o f t h e b e h a v i o u r a f t e r c r a c k i n g , r e v e a l e d an i n c r e a s i n g r a t i o between shear and n o r -mal d i s p l a c e m e n t s of t h e c r a c k f a c e s [_ 82 ^ •

I m p o r t a n t i n f o r m a t i o n about t h e way i n w h i c h t h e s t r e s s e s a r e a c t u -a l l y t r -a n s f e r r e d from one c r -a c k f -a c e t o t h e o t h e r c -a n be d e r i v e d f r o m c y c l i c l o a d i n g t e s t s . T e s t s by L a i b l e e t a l . \_ 40 ] , w h i c h were aimed a t s t u d y i n g t h e e f f e c t o f s e i s m i c l o a d i n g on t h e behav-i o u r o f c r a c k s behav-i n r e behav-i n f o r c e d c o n c r e t e , t h e r e f o r e c l a behav-i m e d a t t e n t behav-i o n . These t e s t s were c a r r i e d o u t on specimens as r e p r e s e n t e d i n F i g .

restraint r 7 'ods r

h.

r OOU lit Ml Hi rf h

1

Tspecimen J - -1 " i i .1 I I J restraint rods Tspecimen J - -1 " i i .1 I

o o

_ J J restraint rods 1 crack plane •h r I - -1 _{" i} i .1 I f 1

|

L

_i

₄

_f

V jyt F i g . 2.8 T e s t specimen used i n |_ 40 J

b a r s had a n e g l i g i b l e s h e a r s t i f f n e s s b u t a c t e d to h o l d the s p e c i men h a l v e s t o g e t h e r when s h e a r i n g and o v e r r i d i n g o c c u r r e d . The s p e c

imens were c r a c k e d a t m i d d e p t h , p r i o r t o t e s t i n g , by f o r c i n g c r a c k -i n g wedges -i n t o the s -i d e s o f the spec-imen. The d e s -i r e d -i n -i t -i a l c r a c k w i d t h was then s e t by p o s i t i o n i n g the upper h a l f of the specimen w i t h r e s p e c t t o the l o w e r h a l f by a d j u s t m e n t o f the n u t s on the r e s t r a i n t rods t h a t p a s s e d t h r o u g h the upper r e s t r a i n t beams. The h o r i z o n t a l s h e a r i n g s u r f a c e had a n e t c r o s s - s e c t i o n a l a r e a o f 194000 mm2. F u l l y r e v e r s i n g c y c l i c s h e a r s t r e s s e s o f about 1.24 N/mm2 were a p p l i e d a c r o s s i n i t i a l c r a c k w i d t h s of 0.25, 0.51 and 0.76 mm by h y d r a u l i c rams. A r e s u l t o b t a i n e d on a specimen w i t h an i n i t i a l c r a c k w i d t h o f 0.76 mm and a r e s t r a i n t s t i f f n e s s o f 600 kN/mm i s r e p r e s e n t e d i n F i g . 2.9. T(N/mm2) cycle 1 15 - -1.5 F i g . 2.9 Shear s t r e s s - s h e a r d i s p l a c e m e n t c u r v e f o r c y c l i c l o a d i n g , a c c o r d i n g to |_ 40 J (£' . = 21 N/mm2, D = 3 8 mm, w = 0.75 mm, r e s t r a i n t c c y l max o s t i f f n e s s Ao = 0.3 N/mm2 f o r Aw = 0.1 mm)

The r e s u l t may be c o n s i d e r e d t o be r e p r e s e n t a t i v e o f the g e n e r a l l y o b s e r v e d b e h a v i o u r . A l t h o u g h the l o a d i n g p o r t i o n o f t h e l o a d - s h e a r d i s p l a c e m e n t c u r v e d u r i n g the f i r s t c y c l e i s n e a r l y l i n e a r , the v e r y n e x t c y c l e o f a l l specimens d e m o n s t r a t e d a marked degree o f n o n - l i n e a r i t y s i m i l a r t o the r e s u l t s f o r c y c l e 15 i n F i g . 2.9. In o n l y a few cases d i d the s h e a r d i s p l a c e m e n t f r e e l y r e t u r n t o

as l i t t l e as 50% of the maximum v a l u e . U s u a l l y the r e t u r n s h e a r d i s p l a c e m e n t was i n the range o f 0 - 2 0 % of t h i s v a l u e . T h i s i s bel i e v e d n o t t o be caused by a type of " bel o c k i n g e f f e c t " between p a r -t i c l e s , s i n c e h a r d l y any s h e a r s -t r e s s was n e c e s s a r y -t o r e -t u r n -t h e specimen t o i t s n e u t r a l p o s i t i o n . I t t h e r e f o r e appears o b v i o u s t h a t t h i s i r r e v e r s i b i l i t y i s due to l o c a l d e f o r m a t i o n o f t h e m a t r i x (hardened cement p a s t e ) . T h i s f e a t u r e i s i m p o r t a n t as a b a s i c p r o p e r t y i n f u r t h e r a n a l y s i s of t h e mechanism. Summarizing the a v a i l a b l e i n f o r m a t i o n on a g g r e g a t e i n t e r l o c k , i t i s o b v i o u s t h a t t h e r e i s a s e r i o u s l a c k o f i n f o r m a t i o n . E x p e r i m e n t s have g e n e r a l l y been c a r r i e d out d i s r e g a r d i n g the e f f e c t o f s h e a r d i l a t a n c y . Only i n [_ 60 J a r a t h e r wide range of v a l u e s has been i n d i c a t e d f o r the s t r e s s normal to the c r a c k p l a n e , n e c e s s a r y t o a d j u s t the c r a c k w i d t h t o i t s o r i g i n a l v a l u e a f t e r e v e r y l o a d i n -crement. However, i t i s n o t o n l y u n c e r t a i n whether a c o n s t a n t c r a c k w i d t h t e s t i s r e p r e s e n t a t i v e o f the b e h a v i o u r o f a c r a c k i n a s t r u c t u r e , b u t a l s o i t i s q u e s t i o n a b l e w h e t h e r t h i s n o r m a l s t r e s s i t s e l f i s a r e a l i s t i c v a l u e , s i n c e i t has t o exceed the i n t e r n a l f r i c t i o n between the c r a c k f a c e s b e f o r e b e i n g a b l e t o a d j u s t the c r a c k w i d t h : hence i t may o v e r e s t i m a t e the s t r e s s t h a t r e a l l y o c c u r s due t o the s h e a r d i l a t a n c y i n a c r a c k (compare a l s o a c t i v e and p a s s i v e s o i l p r e s s u r e ) . T e n d e n c i e s r e v e a l e d by the a v a i l a b l e t e s t d a t a a r e t h a t the s h e a r r e s i s t a n c e i n c r e a s e s w i t h i n c r e a s i n g c o n c r e t e s t r e n g t h and i n c r e a s i n g s h e a r d i s p l a c e m e n t , and d e c r e a s e s w i t h i n c r e a s i n g c r a c k w i d t h . F u r t h e r m o r e i t i s a p p a r e n t t h a t i r r e -v e r s i b l e d e f o r m a t i o n of m a t r i x m a t e r i a l i s d i r e c t l y r e l a t e d t o the o v e r a l l o b s e r v e d b e h a v i o u r . No r e l i a b l e i n f o r m a t i o n was found about the r o l e of the s c a l e of the a g g r e g a t e (D ) , the p a r t i c l e s i z e d i s t r i b u t i o n ( g r a d i n g c u r v e ) and the i n f l u e n c e of t h e c r a c k o p e n i n g p a t h . I t i s o b v i o u s t h a t i n o r d e r t o d e f i n e a p p r o p r i a t e m a t e r i a l p r o p e r t i e s c o n c e r n i n g a g g r e g a t e i n t e r l o c k i t w i l l be n e c e s s a r y t o o b t a i n more i n f o r m a t i o n . 2.2 Dowel a c t i o n Dowel a c t i o n i s d e f i n e d as the c a p a c i t y of r e i n f o r c i n g b a r s t o

t r a n s f e r f o r c e s p e r p e n d i c u l a r t o t h e i r a x i s ( F i g . 1.3.b). The e f f e c t of dowel a c t i o n depends e s s e n t i a l l y on t h e c o n s t r u c t i o n a l c i r c u m -s t a n c e -s . T h i -s can be e x p l a i n e d by c o n -s i d e r i n g t h e -s t r e -s -s e -s around a b a r , l o a d e d by a dowel f o r c e , and t h e consequences of a t e n s i l e c r a c k . For a t h i n s l i c e o f an e l a s t i c m a t e r i a l , l o a d e d as i n d i c a t e d i n F i g . 2.10, t h e s t r e s s e s i n t h e m a t e r i a l c a n be d e t e r m i n e d by u s i n g p l a n e s t r e s s e l a s t i c i t y methods Q 70, 90 ] . F i g . 2.10 T h i n s l i c e o f an e l a s t i c m a t e r i a l , l o a d e d by a dowel f o r c e I n t h i s way i t was f o u n d t h a t t h e r a d i a l s t r e s s e s a r e e q u a l t o : F , TlR cos l|l (2.1) V a l u e s o f t h e c i r c u m f e r e n t i a l t e n s i l e s t r e s s e s i n t h e c o n c r e t e a r e : 2F, a, = 0.344 I|I TTR a, = 0.637 ijj irR if) = 0 * = 2 (2.2) (2.3) Thus t h e t e n s i l e s t r e s s i n t h e d i r e c t i o n o f t h e dowel a c t i o n f o r c e i s h i g h e s t and t h e t e n s i l e s t r e s s normal t o t h e dowel f o r c e , w h i c h tends t o produce a wedging s p l i t t i n g a c t i o n i s o n l y 54% o f the maximum. T h i s t r e n d was e x p e r i m e n t a l l y c o n f i r m e d by Weaver and C l a r k f_ 86 ] . Wh en t h e t e n s i l e s t r e n g t h o f t h e c o n c r e t e i s reached

and a c r a c k i s formed, an a d j u s t m e n t i n t h e l o a d c a r r y i n g s y s t e m may be e x p e c t e d . I n an i n - p l a n e l o a d e d p l a n a r element a r e d i s t r i b u t i o n o f s t r e s s e s o c c u r s , r e s u l t i n g i n h i g h e r s t r e s s e s under t h e b a r ( F i g . 2.11.a). When t h e dowel l o a d i s i n c r e a s e d , a p r o g r e s s i v e d e t e r i o r a t i o n o f the c o n c r e t e under t h e b a r o c c u r s , r e s u l t i n g i n a g r a d u a l d e c r e a s e i n s t i f f n e s s t i l l t h e u l t i m a t e l o a d i s r e a c h e d ( c u r v e a i n F i g . 2.11.c). a b c

F i g . 2.11 a,b: Dowel c r a c k i n g i n a p l a n a r s t r u c t u r e and i n a beam c : L o a d - d e f l e c t i o n c u r v e s f o r b o t h cases

I n a beam, however, a f t e r t h e f o r m a t i o n o f a c r a c k g e n e r a l l y no r e -d i s t r i b u t i o n o f s t r e s s e s i s p o s s i b l e , an-d a r i g o r o u s e x t e n s i o n o f the c r a c k a l o n g t h e b a r a x i s , r e s u l t i n g i n f a i l u r e must be e x p e c t e d ; o n l y i f t h e beam i s r e i n f o r c e d w i t h s t i r r u p s , t h e dowel c r a c k may be stopped and a c o m p l e t e l y d i f f e r e n t mechanism i s a c t i v a t e d t o

t r a n s f e r dowel f o r c e s .

T h i s s t u d y f o c u s e s on dowel a c t i o n i n c r a c k s i n p l a n a r e l e m e n t s , i n w h i c h p a r t i c u l a r l y t h e r e l a t i o n between f o r c e s and d i s p l a c e m e n t s between t h e c r a c k f a c e s i s c o n s i d e r e d . An e x t e n d e d r e v i e w o f a v a i l

-a b l e knowledge -about dowel -a c t i o n i n be-ams, b e f o r e -and -a f t e r dowel c r a c k i n g , t h e d e t e r m i n a t i o n o f u l t i m a t e b e a r i n g c a p a c i t i e s a t l a r g e d e f o r m a t i o n s and t h e b e h a v i o u r under c y c l i c l o a d i n g has been r e -p r e s e n t e d i n |_ 83 3 •

between the axes of the undeformed p a r t s o f the b a r s on b o t h s i d e s of the c r a c k ( F i g . 2.12).

F i g . 2.12 D e f l e c t i o n o f a b a r s u b j e c t e d t o a dowel f o r c e

As has been i n d i c a t e d , the t o t a l d e f l e c t i o n i s b o t h a r e s u l t o f the d e f o r m a t i o n o f the p a r t o f the b a r embedded i n the c o n c r e t e and t h e p a r t w h i c h i s f r e e o v e r a c e r t a i n l e n g t h .

F o r the d e f o r m a t i o n o v e r the f r e e l e n g t h , P a u l a y | 59 ] d i s t i n -g u i s h e d t h r e e mechanisms ( F i -g . 2.13):

F i g . 2.13 Mechanisms o f s h e a r t r a n s f e r o v e r the f r e e l e n g t h a c c o r d i n g t o Q 59 "}

Load t r a n s f e r by b e n d i n g : the c a p a c i t y o f t h i s mechanism i s l i m -i t e d by the f o r m a t -i o n o f p l a s t -i c h -i n g e s -i n t h e b a r .

- Load t r a n s f e r by pure s h e a r .

- Load t r a n s f e r by k i n k i n g : i f t h e r e i s a c o n s i d e r a b l e s h i f t be-tween the two main b a r a x e s , f o r i n s t a n c e as a r e s u l t o f p l a s t i c