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Effect of fibre reinforcement on the crack propagation in concrete 1981

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EFFECT OF FIBRE REINFORCEMENT ON THE CRACK PROPAGATION IN CONCRETE by ANTHONY SZE-TONG (YAM B . S c , The U n i v e r s i t y of Saskatchewan, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1981 (g) Anthony Sze-Tong Yam, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Ci\T^L ^A^MIA* ' Kf\\ The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Pl a c e Vancouver, Canada V6T 1W5 DE-6 ( 2 / 7 9 ) ABSTRACT The i n f l u e n c e of f i b r e reinforcement on crack propagation i n concrete was s t u d i e d . T h i r t y - f i v e double t o r s i o n specimens, made wit h three types of f i b r e s ( f i b r e g l a s s , s t r a i g h t s t e e l f i b r e s and deformed s t e e l f i b r e s ) were t e s t e d . . The v a r i a b l e s were the f i b r e volume and s i z e of the f i b r e s . The t e s t r e s u l t s i n d i c a t e d t h a t the r e s i s t a n c e t o r a p i d crack growth increased somewhat wi t h i n c r e a s i n g f i b r e content up t o about 1.25% - 1.5% by volume. The degree of compaction had an enormous e f f e c t on the f r a c t u r e p r o p e r t i e s . The f r a c t u r e toughness increased w i t h f i b r e content up t o about 1.25% by volume, and then decreased, due t o incomplete compaction. I t was found t h a t i n t h i s t e s t geometry, f i b r e s d i d not s i g n i f i c a n t l y r e s t r a i n crack growth. I t was a l s o observed t h a t once the crack had propagated down the f u l l l ength of the specimen, the system changed from a continuous system t o a discontinuous system, c o n s i s t i n g of two separate p l a t e s held together by the f i b r e reinforcement. D i f f e r e n t types of f i b r e s d i d not s i g n i f i c a n t l y a f f e c t the f r a c t u r e toughness. ACKNOWLEDGEMENT S The author expresses h i s indebtedness t o P r o f e s s o r s S. Mindess and J.S. Nadeau f o r t h e i r v a l u a b l e guidance i n p l a n n i n g and c a r r y i n g out the i n v e s t i g a t i o n . The author i s g r a t e f u l t o P r o f e s s o r R.J. Gray f o r h i s ad v i c e . The author a l s o wishes t o thank the C i v i l E n g i n e e r i n g Department t e c h n i c i a n s and e s p e c i a l l y Mr. B. M e r k l i e f o r h i s a s s i s t a n c e i n making the t e s t equipment and c a r r y i n g out the t e s t . T h i s r e s e a r c h was made p o s s i b l e by grants from the N a t i o n a l Sciences and E n g i n e e r i n g Research C o u n c i l Canada, and the N a t u r a l , A p p l i e d and Health Sciences Grants Committee, U.B.C. TABLE OF CONTENTS Page A b s t r a c t i i Acknowledgements i i i Table of Contents i v L i s t of F i g u r e s v L i s t of Tables v i i L i s t of Symbols i x 1. I n t r o d u c t i o n 1 2. F r a c t u r e Mechanics: General Background 3 2.1 H i s t o r i c a l Background 3 2.2 The S t r e s s I n t e n s i t y Approach 7 2.2.1 S t r e s s I n t e n s i t y F a c t o r 7 2.2.2 E f f e c t i v e Crack Length 10 ' 2.3 R e l a t i o n s h i p Between G and K 11 2.4 F r a c t u r e Mechanics A p p l i e d t o F i b r e R e i n f o r c e d Concrete " 12 3. Measurement of F r a c t u r e Parameters and S t a b l e Crack Growth 15 3.1 T e s t Specimens 15 3.2 Double T o r s i o n Technique 16 4. Experimental Procedure 20 4.1 M a t e r i a l 20 4.2 Design of Specimen and Mold 22 4.2.1 C a s t i n g of Specimens 22 4.2.2 P r e p a r a t i o n of Specimens Before T e s t i n g 24 4.3 T e s t Program 27 4.3.1 Compliance T e s t 27 4.3.2 Double T o r s i o n T e s t 31 5. Experimental R e s u l t s 37 5.1 I n t r o d u c t i o n 37 5.2 Cement Paste Specimens 37 5.3 F r a c t u r e Toughness 44 5.4 R e s i d u a l Strength 54 5.5 Compliance 56 5.6 V-K P l o t 56 6. General D i s c u s s i o n 67 7. C o n c l u s i o n 70 B i b l i o g r a p h y 71 Appendix A . 7 4 " Appendix B 75 - i v - LIST OF FIGURES F i g u r e s Page 2.1 Crack f r o n t c o o r d i n a t e s 5 2.2 The th r e e d i f f e r e n t modes of f a i l u r e 9 3.1 The double t o r s i o n specimen 17 4.1 C a s t i n g mold 23 4.2 Loading j i g 25 4.3 Te s t setup 26 4.4 E x t e r n a l l o a d c e l l 28 4.5 Front view of the t e s t i n g setup 29 4.6 Sideview of the t e s t i n g setup w i t h a specimen i n p l a c e 30 4.7 I n i t i a l stage of crack propagation 33 4.8 Specimen j u s t before f a i l u r e 34 4.9 Specimen a f t e r f a i l u r e 35 5.1 Load r e l a x a t i o n curves 41 5.2 T y p i c a l l o a d r e l a x a t i o n curves 43 5.3 V-K-j. p l o t f o r the average of the two cement specimens 45 5.4 R e l a t i o n s h i p s between f r a c t u r e toughness, weight d e n s i t y , r e s i d u a l s t r e n g t h and f i b r e volume f o r GF 102 s e r i e s 47 5.5 R e l a t i o n s h i p s between f r a c t u r e toughness, weight d e n s i t y , r e s i d u a l s t r e n g t h and f i b r e volume f o r GF 204 s e r i e s 48 5.6 R e l a t i o n s h i p s between f r a c t u r e toughness, weight d e n s i t y , r e s i d u a l s t r e n g t h and f i b r e volume f o r h" SSF s e r i e s 49 5.7 R e l a t i o n s h i p s between f r a c t u r e toughness, weight d e n s i t y , r e s i d u a l s t r e n g t h and f i b r e volume f o r 1" SSF s e r i e s 50 5.8 R e l a t i o n s h i p s between f r a c t u r e toughness, weight d e n s i t y , r e s i d u a l s t r e n g t h and f i b r e volume f o r BSF s e r i e s 51 5.9 R e l a t i o n s h i p between r e s i d u a l s t r e n g t h and f i b r e volume 55 - v - F i g u r e s Page 5.10 R e l a t i o n s h i p between system compliance and crack l e n g t h 59 5.11 V-Kj p l o t s f o r GF 102 s e r i e s 60 5.12 V-Kj p l o t s f o r GF 204 s e r i e s 61 5.13 V-Kj p l o t s f o r h" SSF s e r i e s 62 5.14 V-Kj p l o t s f o r 1" SSF s e r i e s 63 5.15 V-K p l o t s f o r BSF s e r i e s 64 - v i - LIST OF TABLES Table Page 4.1 Mix d e s i g n 21 5.1 Load r e l a x a t i o n data f o r hardened cement paste 38 5.2 Load r e l a x a t i o n data f o r hardened cement paste 39 5.3 F r a c t u r e toughness and r e s i d u a l s t r e n g t h of specimens 46 5.4 Weight d e n s i t y of specimens 53 5.5 R e s u l t s of compliance study f o r specimens h" SSF 1.0 57 5.6 R e s u l t s of compliance study f o r specimens BSF 1.0 58 5.7 Summary of r e s u l t s from the V-K T curves 65 BI Load r e l a x a t i o n data f o r GF - 0 76. B2 Load r e l a x a t i o n data f o r GF 102 - 0.25 77 B3 Load r e l a x a t i o n data f o r GF 102 - 0.5 7:8 B4 Load r e l a x a t i o n data f o r GF 102 - 0.75 79 B5 Load r e l a x a t i o n data f o r GF 102 - 1.0 80 B6 Load r e l a x a t i o n data f o r GF 102 - 1.25 81 B7 Load r e l a x a t i o n data f o r GF 102 - 1.5 82 B8 Load r e l a x a t i o n data f o r GF 102 - 2.0 83 B9 Load r e l a x a t i o n data f o r GF 204 - 0.25 84 BIO Load r e l a x a t i o n data f o r GF 204 - 0.75 85 B l l Load r e l a x a t i o n data f o r GF 204 - 1.25 86 B12 Load r e l a x a t i o n data f o r GF 204 - 1.5 87 B13 Load r e l a x a t i o n data f o r GF 204 - 2.0' 88 B14 Load r e l a x a t i o n data f o r SSF - 0 89 B15 Load r e l a x a t i o n data f o r h" SSF - 0.25 90 B16 Load r e l a x a t i o n data f o r h" SSF - 0.5 B17 Load r e l a x a t i o n data f o r vn SSF - 0.75 92 B18 Load r e l a x a t i o n data f o r I'" SSF - 1.25 93 B19 Load r e l a x a t i o n data f o r L.II *2 SSF - 1.5 94 B20 Load r e l a x a t i o n data f o r L- 11 *2 SSF - 2.0 95 - v i i - Table B21 Load r e l a x a t i o n data B22 Load r e l a x a t i o n data B23 Load r e l a x a t i o n data B24 Load r e l a x a t i o n data B25 Load r e l a x a t i o n data B26 Load r e l a x a t i o n data B27 Load r e l a x a t i o n data B28 Load r e l a x a t i o n data B29 Load r e l a x a t i o n data B30 Load r e l a x a t i o n data B31 Load r e l a x a t i o n data Page f o r 1" SSF - 0. 25 96" f o r 1" SSF - 0. 5 97 f o r 1" SSF - 1. 0 98' f o r 1" SSF - 1. 25 99 f o r 1" SSF - 1. 5 100 f o r BSF - 0 101 f o r BSF - 0 .25 102 f o r BSF - 0 .5 103 f o r BSF - 0 .75 104 f o r BSF - 1 .25 105 f o r BSF - 2 .0 106 - v i i i : — LIST OF SYMBOLS le n g t h of the semi-major a x i s , or one h a l f of the crack l e n g t h e f f e c t i v e crack l e n g t h s l o p e of the V-K^ curve i n t e r a t o m i c e q u i l i b r i u m bond spacing ( l a t t i c e spacing) system compliance the y - i n t e r c e p t of the V-K-j. curve modulus of e l a s t i c i t y energy r e l e a s e r a t e c r i t i c a l s t r a i n energy r e l e a s e r a t e s t r e s s i n t e n s i t y f a c t o r , where s u b s c r i p t I r e f e r s t o Mode I f a i l u r e c r i t i c a l s t r e s s i n t e n s i t y f a c t o r ( a l s o known as f r a c t u r e toughness) pound f o r c e a p p l i e d l o a d c o rresponding p o l a r c o o r d i n a t e s t h i c k n e s s of the double t o r s i o n specimen p l a t e t h i c k n e s s i n the plane of the crack crack v e l o c i t y width of the double t o r s i o n specimen c a r t e s i a n c o o r d i n a t e s with the o r i g i n a t the crack t i p d e f l e c t i o n s u r f a c e t e n s i o n Poisson's r a t i o r a d i u s of c u r v a t u r e at the t i p of the e l l i p s e a p p l i e d s t r e s s r e s u l t a n t s t r e s s i n a-a d i r e c t i o n , s u b s c r i p t aa r e p r e s e n t s the d i r e c t i o n i d e a l f r a c t u r e s t r e n g t h y i e l d s t r e n g t h moment arm Chapter I INTRODUCTION Concrete and r e l a t e d c e m e n t i t i o u s m a t e r i a l s are heterogeneous and composite i n nature. M i c r o c r a c k s are an in h e r e n t c h a r a c t e r i s t i c of such m a t e r i a l s , due to volume changes of the cement paste d u r i n g h y d r a t i o n and due to shrinkage of the hardened cement paste upon d r y i n g . When under l o a d , these microcracks w i l l extend, forming an ex t e n s i v e crack network which e v e n t u a l l y leads to one or more l a r g e cracks and subsequent f a i l u r e . C o n t r o l of c r a c k i n g i s p a r t i c u l a r l y important f o r the s e r v i c e a b i l i t y of r e i n f o r c e d concrete s t r u c t u r e s . Adequate crack c o n t r o l o f t e n can be achieved u s i n g s m a l l e r reinforcement bars, more c l o s e l y spaced. R e s u l t s have shown t h a t concrete s t r u c t u r e s w i t h f i b r e r einforcement develop f i n e r cracks under l o a d i n g . Yet, how do f i b r e s work? Do they a c t as crack a r r e s t o r s or do they simply h o l d the cracked s t r u c t u r e together? I f the f i b r e s do a r r e s t c r a c k s , then how do they a f f e c t the crack growth r a t e ? One means of g a i n i n g an understanding of these phenomena i s through f r a c t u r e mechanics. Using t h i s approach, the f r a c t u r e s t r e n g t h , o , i s i n v e r s e l y p r o p o r t i o n a l to the square r o o t of the s i z e of the c r i t i c a l flaw. When a s t r e s s l e s s thana^ i s applied,, the s t r u c t u r e w i l l support t h a t s t r e s s o n l y as long as the flaw does not grow to the c r i t i c a l s i z e f o r t h a t s t r e s s . Research on b r i t t l e m a t e r i a l s has shown - 1 - t h a t flaws w i l l grow under s u s t a i n e d l o a d i n g - a phenomenon known as s u b c r i t i c a l crack growth. S u b c r i t i c a l crack propagation i s caused by l o c a l i z e d crack t i p s t r e s s e s , which are d i r e c t l y r e l a t e d to the s t r e s s i n t e n s i t y f a c t o r . Thus crack growth can be expressed as a f u n c t i o n of the s t r e s s i n t e n s i t y f a c t o r . The r e l a t i o n s h i p between cr a c k growth and the s t r e s s i n t e n s i t y f a c t o r can b e s t be d e s c r i b e d by a V-K^ p l o t , i . e . a p l o t of crack v e l o c i t y vs s t r e s s i n t e n s i t y . Once the crack growth has been c h a r a c t e r i z e d i n t h i s way, d e s i g n c r i t e r i a can be developed f o r such a m a t e r i a l , and a b e t t e r understanding of the mechanisms governing crack growth can o f t e n be o b t a i n e d . In a d d i t i o n , the l i f e expectancy of s t r u c t u r a l elements made wit h such m a t e r i a l s can then be p r e d i c t e d . The o b j e c t i v e s of the r e s e a r c h r e p o r t e d here.were: 1. To i n v e s t i g a t e the e f f e c t of f i b r e r einforcement on crack growth i n c o n c r e t e . 2. To determine q u a n t i t a t i v e l y the e f f e c t of f i b r e content on the f r a c t u r e toughness of c o n c r e t e . - 2 - Chapter 2 FRACTURE MECHANICS: GENERAL BACKGROUND 2.1 H i s t o r i c a l Background Stresses around cracks have been stud i e d i n d e t a i l l a by many people. In 1913, I n g l i s showed th a t s t r e s s e s around an e l l i p t i c a l hole (an e l l i p s e i s o f t e n used t o c h a r a c t e r i z e the general geometry of a crack) i n a uniformly s t r e s s e d p l a t e could be'expressed as a = ail + 24-) (1) aa p where a = r e s u l t a n t s t r e s s i n a-a d i r e c t i o n aa a = a p p l i e d s t r e s s a = length of the semi-major a x i s , or one-half of the crack length p = ra d i u s ' o f curvature at the t i p of the e l l i p s e 2 Westergaard showed t h a t s t r e s s e s near a sharp crack could be expressed as a x x =  CTJS" c o s | ( 1 " s i n | s i n T ) + ' * ' ( 2 ) a = a l-Jr- c o s ^ ( l - s i n | sinrJ^) + ••• (3) yy J 2r 2 2 2 /a .0 0 3 0 , t A \ a = a hr- S i n s COSs COS-s- + ••• (4) xy J 2r 2 2 2 numbers r e f e r t o b i b l i o g r a p h y at the end - 3 - where: s u b s c r i p t s xx, yy and xy r e p r e s e n t the c o o r d i n a t e d i r e c t i o n s xx and yy = C a r t e s i a n c o o r d i n a t e s w i t h the o r i g i n a t the crack t i p r , 8 = corresponding p o l a r c o o r d i n a t e s and the other symbols are as d e f i n e d above. These crack t i p c o o r d i n a t e s are shown i n F i g u r e 2.1. With the h e l p of these s o l u t i o n s , the t h e o r y of the mechanics of f r a c t u r e can be developed. The t e n s i l e s t r e n g t h of an i d e a l c r y s t a l l i n e body i s the s t r e s s which must be a p p l i e d t o cause i t t o f r a c t u r e across a p a r t i c u l a r c r y s t a l l o g r a p h i c p l a n e . T h i s i d e a l s t r e n g t h can be expressed as ( 3 ) m v b * o where a = i d e a l f r a c t u r e s t r e n g t h m E = modulus of e l a s t i c i t y y = s u r f a c e t e n s i o n b Q = i n t e r a t o m i c e q u i l i b r i u m bond spacing ( l a t t i c e spacing) I f we modify the I n g l i s s o l u t i o n (Eq. 1) by equating p = b Q , and a l s o note t h a t 2 / j ~ >> 1, we o b t a i n " o °aa " 2°JF <6» ' o - 4 - F i g u r e 2.1 Crack F r o n t Coordinates - 5 - By l e t t i n g the aa d i r e c t i o n be the xx d i r e c t i o n , then a = a = 2oHf (7) aa xx ,/b V o Using t h i s m o d i f i e d form and equating a = a as a f r a c t u r e 3 xx m c r i t e r i o n , at f r a c t u r e a = a p . Hence Thus, the f r a c t u r e .strength i s i n v e r s e l y r e l a t e d t o the square r o o t of the l e n g t h of the c r a c k . In 1920, based on t e s t s on precracked g l a s s specimens, 4 G r i f f i t h concluded t h a t " f o r an i n f i n i t e s i m a l l y small amount of crack e x t e n s i o n , the decrease i n s t o r e d e l a s t i c energy of a cracked body under f i x e d g r i p c o n d i t i o n s i s i d e n t i c a l t o the decrease i n p o t e n t i a l energy under c o n d i t i o n s of constant l o a d i n g " . G r i f f i t h showed t h a t the d r i v i n g f o r c e f o r crack e x t e n s i o n was the d i f f e r e n c e between the energy which c o u l d be r e l e a s e d i f the crack was extended and t h a t needed t o c r e a t e new s u r f a c e s . Using an energy-rate balance approach, he showed t h a t a F = v / l i E ? (plane s t r e s s ) (9) which i s very s i m i l a r t o Eq.8, even though they were d e r i v e d from d i f f e r e n t c o n s i d e r a t i o n s . By d e f i n i n g the energy r e l e a s e r a t e (crack d r i v i n g f o r c e ) as G, and n o t i n g t h a t t h i s crack d r i v i n g f o r c e equals the s u r f a c e energy of the newly formed s u r f a c e , 2y, (two new s u r f a c e s are c r e a t e d due t o c r a c k i n g ) , - 6 - i t can be concluded t h a t G = 2y (10) and (11) G r i f f i t h ' s approach p r o v i d e d the b a s i s f o r the concept of t r e a t i n g f r a c t u r e i n terms of the change i n energy remote from the immediate atomic environment of the crack t i p . One drawback of h i s theory was t h a t i t was based on an i d e a l l y e l a s t i c b r i t t l e m a t e r i a l and d i d not i n c l u d e the l o c a l i z e d p l a s t i c deformation near the crack t i p t h a t occurs i n most 5 6 m a t e r i a l s . In 1952, Irwin and K i e s and Orowan modxfied G r i f f i t h ' s t h e o r y by i n t r o d u c i n g a new parameter, y , which r e p r e s e n t s the l o c a l i z e d p l a s t i c deformation energy d i s s i p a t e d at the crack t i p . Thus G = 2( y + Yp ) (12) From e x t e n s i v e experimental and t h e o r e t i c a l approaches to c a l c u l a t i n g t h i s parameter G, numerous data f o r d i f f e r e n t 7 crack geometries and m a t e r i a l s have been made a v a i l a b l e . U n f o r t u n a t e l y , the modulus of e l a s t i c i t y of some m a t e r i a l s , such as c o n c r e t e , i s d i f f i c u l t t o e v a l u a t e . T h e r e f o r e , i t i s d e s i r a b l e t o combine G and E i n t o a s i n g l e parameter. 2.2 The S t r e s s I n t e n s i t y Approach 2.2.1 S t r e s s I n t e n s i t y F a c t o r 2 Using Westergaard's s o l u t i o n , f a i l u r e r e s u l t i n g from the s t r e s s f i e l d which i s a s s o c i a t e d with the crack t i p can be - 7 - d i v i d e d i n t o t h r e e c a t e g o r i e s . These c a t e g o r i e s are g e n e r a l l y r e f e r r e d t o as: Mode I t e n s i o n f a i l u r e (crack opening) Mode I I i n p l a n e shear f a i l u r e Mode I I I a n t i p l a n e shear f a i l u r e ( t w i s t i n g ) These f a i l u r e modes are shown i n F i g u r e 2.2 In f r a c t u r e a n a l y s i s , Mode I f a i l u r e i s the most important mode, and w i l l be the o n l y one d i s c u s s e d here. . 9 In 1959, by r e a r r a n g i n g Westergaard*s s o l u t i o n , Irwin obtained a term, K, which depended o n l y on the a p p l i e d s t r e s s and crack l e n g t h , = cr/ira" (plane s t r e s s ) (13) where the s u b s c r i p t I r e f e r s t o Mode I f a i l u r e . The advantage of t h i s parameter K, the s t r e s s i n t e n s i t y f a c t o r , i s t h a t i t f u l l y d e s c r i b e s the combined e f f e c t of the a p p l i e d s t r e s s and the crack l e n g t h . In the gene r a l case of mixed mode f r a c t u r e , a p p l i e d s t r e s s e s due t o t e n s i o n , t o r s i o n , p o i n t l o a d i n g , e t c . , each make t h e i r own s p e c i f i c c o n t r i b u t i o n s t o a, and the r e s u l t a n t may be c a l c u l a t e d simply by adding the i n d i v i d u a l s t r e s s i n t e n s i t i e s . The e f f e c t s of specimen shape, body c o n f i g u r a t i o n , and boundary c o n d i t i o n s on can be i n c o r p o r a t e d i n t o a g e o m e t r i c a l f u n c t i o n f ( g ) , so t h a t Eq. 13 becomes K = o/Wa f ( g ) (14) - 8 - Figure 2 . 2 The Three D i f f e r e n t Modes of F a i l u r e (After Knott) y = a Mode I opening (displacement u in x direction.) ° " x y - T r _ j _ 0 IV M o d e l sheor (displacement v in y direction.) Mode HI ont ip lone shear vt ModeDH antiplane shear ( displacement win z direction.) - 9 - • 2.2.2 E f f e c t i v e Crack Length In m e t a l l i c m a t e r i a l s , p l a s t i c deformation of the m a t e r i a l near the crack t i p c r e a t e s a p l a s t i c zone. The s i z e of t h i s zone, r , can be estimated by (10) 1 K I 2 * Y where a = y i e l d s t r e n g t h The e f f e c t i v e crack l e n g t h , a e , must i n c l u d e the e f f e c t of t h i s zone, hence a e = (a + r ) (16) In the case of a plane s t r a i n specimen, p l a s t i c deformation i s more d i f f i c u l t a t the c e n t e r of t h i c k specimens, which are more l i k e l y t o c l e a v e than t o p l a s t i c a l l y deform. Irwin"'""'" estimated t h a t the p l a s t i c zone f o r t h i c k specimens i s reduced by a f a c t o r of 3. Thus, f o r m e t a l l i c m a t e r i a l s , 1 K I 2 7 y Concrete and other c e m e n t i t i o u s m a t e r i a l s do not deform and develop a " p l a s t i c " zone as do m e t a l l i c m a t e r i a l s . However, they do form a l a r g e zone of f i n e cracks around the crack t i p . T h i s zone of f i n e c r a c k s w i l l move with the crack t i p as the 12 crack extends. The zone of fxne cracks has been c a l l e d the 13 "pseudo-plastxc zone" , which can be c o n s i d e r e d t o be e q u i - v a l e n t t o the p l a s t i c zone i n metals. - 10 - 2.3 Relationship Between G and K Using Westergaard 1s solution and the energy p r i n c i p l e , Irwin showed that the e l a s t i c s t r a i n energy release rate G and the stress i n t e n s i t y factor K can be related by and where These relationships enable the s t r a i n energy release rate to be expressed d i r e c t l y i n terms of values of the stress i n t e n s i t y . Formulation of the r e l a t i o n s h i p involves only the energy p r i n c i p l e ; no mention of fracture has been made. As the crack moves forward by an amount da, an amount of K 2 energy per unit thickness equal to Gda or (-) * da would hi be released. Whether t h i s i s s u f f i c i e n t to cause catas- trophic crack propagation depends on whether the energy released reaches a c r i t i c a l magnitude. This r e s u l t demon- strates that when the energy release rate i s below some c r i t i c a l value, the system may undergo stable crack growth before catastrophic crack propagation occurs. The r e s u l t also provides a means of understanding stable crack growth. Stable crack growth i s important because i n a system under load, i t may cause a crack which i s i n i t i a l l y smaller than G = — (plane stress) (.18). G = (1 - v 2) (plane strain) (.19) v = Poisson's r a t i o - 11 - the c r i t i c a l s i z e f o r the a p p l i e d s t r e s s to extend to the c r i t i c a l s i z e , at which p o i n t f r a c t u r e would occur. 2.4 Fracture Mechanics A p p l i e d to F i b r e Reinforced Concrete I t i s now w e l l e s t a b l i s h e d t h a t concrete f a i l u r e i s due to progr e s s i v e i n t e r n a l c r a c k i n g . F a i l u r e i s the r e s u l t of an e s s e n t i a l l y continuous m a t e r i a l changing to an e s s e n t i a l l y 14 discontinuous one. R i c h a r t , Brandtzaeg and Brown f i r s t found that the volume of concrete under u n i a x i a l compressive loading i n i t i a l l y decreased, as would be expected from e l a s t i c theory. However, when the a p p l i e d load reached about two-thirds of the u l t i m a t e l o a d , the volume of the concrete s t a r t e d to inc r e a s e . At u l t i m a t e l o a d , they found that the apparent volume of the concrete specimen was l a r g e r than the i n i t i a l volume of the specimen. From t h i s , they concluded that the bulging and eventual f a i l u r e of the m a t e r i a l r e s u l t e d from the gradual development of i n t e r n a l tension-induced microcracking throughout the specimen, and t h i s has subsequently been confirmed by many other i n v e s t i g a t o r s . F a i l u r e takes place when the cracks develop continuous p a t t e r n s . In 1963, by i n t r o d u c i n g a l i g n e d s t e e l f i b r e s p a r a l l e l to the t e n s i l e s t r e s s i n a concrete system, Romualdi and Batson"^ found t h a t the t e n s i l e c r a c k i n g strength of the system increased i n p r o p o r t i o n to the inverse square root of the wire spacing. They reasoned t h a t as an i n t e r n a l t e n s i l e crack propagates i n a given m a t e r i a l , displacements perpendi- c u l a r to the plane of the crack develop i n the v i c i n i t y of the crack t i p as a r e s u l t of the s t r e s s s i n g u l a r i t y i n tha t - 12 - r e g i o n . The presence of a s t i f f e n i n g element i n the v i c i n i t y of the crack opposes these displacements by means of adhesive c o u p l i n g between the s t i f f e n i n g element and the m a t r i x . The r e s u l t i n g bond f o r c e s are d i r e c t e d toward the crack plane and reduce the magnitude of the e x t e n s i o n a l s t r e s s e s i n the v i c i n i t y of the crack t i p . F r a c t u r e mechanics p r i n c i p l e s were used i n t h e i r work t o account f o r the i n f l u e n c e of f i b r e r e i n f o r c e m e n t on the crack r e s i s t i n g mechanism. Since then, numerous s t u d i e s have been c a r r i e d out to i n v e s t i g a t e the crack a r r e s t mechanism. I n i t i a l l y , these s t u d i e s i n v o l v e d o n l y the a p p l i c a t i o n of l i n e a r - e l a s t i c f r a c t u r e mechanics. However, as an i n c r e a s i n g amount of experimental data became a v a i l a b l e , i n c o n s i s t e n c i e s i n the measured f r a c t u r e parameters such as the c r i t i c a l s t r a i n energy r e l e a s e r a t e , G I C or the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r ( a l s o known as f r a c t u r e toughness), K.j.£ became apparent. The v a l u e s of G^^ or appeared t o be s t r o n g l y dependent on the specimen geometry and the method of measurement. Rec e n t l y , a number of i n v e s t i g a t o r s have begun a p p l y i n g e l a s t i c - p l a s t i c f r a c t u r e mechanics t o f i b r e r e i n f o r c e d c o n c r e t e . Two of the reasons f o r extending the l i n e a r - e l a s t i c c r i t e r i a i n t o the e l a s t i c - p l a s t i c r e g i o n are: (1) c o n c r e t e i t s e l f i s not a p e r f e c t l y b r i t t l e m a t e r i a l ; and (2) f i b r e r e i n - forcement g i v e s the concrete more apparent d u c t i l i t y . The most common techniques of e l a s t i c - p l a s t i c f r a c t u r e mechanics which a where s u b s c r i p t s I and C r e f e r t o the Mode I f a i l u r e and the c r i t i c a l v a l u e , r e s p e c t i v e l y . - 13 - have been a p p l i e d to f i b r e r e i n f o r c e d c o n c r e t e are: (1) the c r i t i c a l c rack opening displacement method (COD), (2) the J - i n t e g r a l technique, (3) R-curve techniques and (4) the f i c t i t i o u s c rack model. A b r i e f d e s c r i p t i o n of these 16 17 techniques has been g i v e n by Mindess ' . A number of i n v e s t i g a t i o n s have been c a r r i e d out u s i n g these methods; however, the experimental data i n d i c a t e some u n c e r t a i n t y i n t h e i r a p p l i c a t i o n as w e l l . While N i s h i o k a , Yamakawa, Hirakawa 18 19 and Akihama and Brandt c l a i m t h a t COD and J can be a p p l i e d c c 20 to f i b r e r e i n f o r c e d c o n c r e t e , Halvorsen and V e l a z c o , 21 V i s a l v a n i c h and Shah have found t h a t J and COD depend on c c " the specimen geometry. Although a l i m i t e d number of experimental r e s u l t s support the R-curve technique, more r e s e a r c h i s r e q u i r e d to c l a r i f y i t s a p p l i c a b i l i t y to f i b r e r e i n f o r c e d c o n c r e t e . -• 14 - Chapter 3 MEASUREMENT OF FRACTURE PARAMETERS AND STABLE CRACK GROWTH 3.1 T e s t Specimens A number of specimen geometries have been developed to measure f r a c t u r e parameters and crack p r o p a g a t i o n . The most common are 1. Edge cracked t e n s i l e specimen 2. Centre cracked t e n s i l e specimen 3. Double c a n t i l e v e r beam specimen 4. Double t o r s i o n specimen Edge cracked t e n s i l e specimens and c e n t r e cracked t e n s i l e specimens such as the compact t e n s i o n speciman and the notched beam specimen have been adopted i n ASTM Standard E561-80 f o r f r a c t u r e t e s t i n g of m e t a l l i c m a t e r i a l s . However, the double t o r s i o n and double c a n t i l e v e r beam specimens are more f r e q u e n t l y used on ceramic m a t e r i a l s to measure slow crack growth and f r a c t u r e toughness. One of the reasons f o r t h e i r p o p u l a r i t y i s t h a t w i t h these specimens, the f r a c t u r e toughness i s independent of the crack l e n g t h over a s u b s t a n t i a l range of crack growth. These specimens a l s o a l l o w s e v e r a l determinations of f r a c t u r e toughness on a s i n g l e specimen. Both double c a n t i l e v e r beam and double t o r s i o n specimens have the same i n i t i a l geometry. B a s i c a l l y they are r e c t a n g u l a r p l a t e s w i t h a c e n t r e groove running the f u l l l e n g t h of the p l a t e . However, i n the double t o r s i o n technique, t o r s i o n a l l o a d i n g i s used to propagate the crack, r a t h e r than the t e n s i l e - 15 - l o a d i n g of the double c a n t i l e v e r beam technique. In the double c a n t i l e v e r beam technique, crack v e l o c i t y s t u d i e s are u s u a l l y performed u s i n g t h e f i x e d l o a d i n g technique. The crack v e l o c i t y i s c a l c u l a t e d by measuring the l e n g t h of the crack increment and the time r e q u i r e d f o r such an increment. The p o s i t i o n of the crack i s monitored o p t i c a l l y . An e q u i v a l e n t method can a l s o be used i n crack v e l o c i t y s t u d i e s with the double t o r s i o n technique. Slow crack growth data are a l s o o b tained under a constant l o a d , and the crack growth r a t e can be monitored o p t i c a l l y . Under these c o n d i t i o n s , both techniques should be e q u i v a l e n t . As an a l t e r n a t i v e , u s i n g a constant displacement or a c o n s t a n t displacement 22 r a t e to propagate the crack, Evans showed t h a t by u s i n g compliance methods, the crack growth r a t e can be c a l c u l a t e d d i r e c t l y from the a p p l i e d l o a d , P, or the l o a d r e l a x a t i o n r a t e dp/dt. Both the c r a c k v e l o c i t y and the s t r e s s i n t e n s i t y can be determined from the l o a d . T h i s method t h e r e f o r e p r o v i d e s a simple way of m o n i t o r i n g c r a c k growth when d i r e c t v i s u a l o b s e r v a t i o n of the crack t i p i s i m p o s s i b l e . 3.2 Double T o r s i o n Technique By c o n s i d e r i n g the double t o r s i o n specimen shown i n F i g 3.1 as two r e c t a n g u l a r e l a s t i c s e c t i o n s , W i l l i a m s and 23 Evans showed t h a t the s t r e s s i n t e n s i t y i s a f u n c t i o n only of the specimen dimensions, the a p p l i e d l o a d and Poisson's r a t i o . . The s t r e s s i n t e n s i t y can be expressed as - 16 - F i g u r e 3.1 The Double T o r s i o n Specimen K I = P"mt 3 U 3 + V ) ] H ( 2 0 )  A Wt t n where = s t r e s s i n t e n s i t y f a c t o r P = a p p l i e d l o a d co = moment arm m v = Poisson's r a t i o W = width of the specimen t = t h i c k n e s s of the specimen t = p l a t e t h i c k n e s s i n the plane of the crack They confirmed t h a t the compliance C and the crack l e n g t h "a" are l i n e a r l y r e l a t e d . The system compliance can be expressed as C = | = (Ba + c) (21) where y = d e f l e c t i o n B = s l o p e of the V-Kj curve c = the i n t e r c e p t of the V-K^ curve 23 With the compliance e x p r e s s i o n , Willxams and Evans a l s o showed t h a t f o r constant displacement or constant displacement r a t e , the crack growth r a t e can be r e l a t e d t o the instantaneous l o a d and the corresponding l o a d r e l a x a t i o n r a t e dp/dt as V =--Z- = ~ ( P i , f ) ( a i , f ) (dp_) ( 2 2 ) v  B p 2 [at' p2 {at' - 18 - where s u b s c r i p t s i and f r e p r e s e n t the i n i t i a l and f i n a l s t a t e s , r e s p e c t i v e l y . Hence, the v e l o c i t y of the crack can be measured over a range of K y values from a s i n g l e experiment. - 19 - Chapter 4 EXPERIMENTAL PROCEDURE 4.1 M a t e r i a l s CSA Type 10 (ASTM Type 1) normal P o r t l a n d cement was used t o prepare the c o n c r e t e . The f i n e aggregate was commercially a v a i l a b l e c o n c r e t e sand, and the coarse aggregate was 3/8"(9.5 mm) pea g r a v e l . A l l aggregate were s t o r e d a t ambient l a b o r a t o r y moisture c o n d i t i o n s . To improve the w o r k a b i l i t y of the mixes, two types of admixtures were a b used, an a i r e n t r a i n i n g agent and a water r e d u c i n g agent ,. Three types of f i b r e s were used. These were a l k a l i - c d r e s i s t a n t f i b r e g l a s s , s t r a i g h t s t e e l f i b r e s and deformed s t e e l f i b r e s e . Two types of f i b r e g l a s s were used — 102 f i l a m e n t s per f i b r e bundle and 204 f i l a m e n t s per f i b r e bundle. They were chopped s t r a n d i n 1.0 in.(25.4 mm) l e n g t h s . The s t r a i g h t s t e e l f i b r e s c o n s i s t e d of 0.50 in.(12.7 mm) and 1.0 in.(25.4 mm) long f i b r e s w i t h c r o s s - s e c t i o n a l dimensions 0.01 x 0.022 in.(0.254 x 0.559 mm). The deformed s t e e l f i b r e s were 0.022 in.(0.559 mm) i n diameter w i t h hooked ends, as shown i n Table 4.1. MBVR, s u p p l i e d by Master B u i l d e r s Co. L i q u i d p o z z o l i t h , type 300N, s u p p l i e d by Master B u i l d e r Co. S u p p l i e d by Owens-Corning ^ S u p p l i e d by S t e l c o , Hamilton, O n t a r i o S u p p l i e d by Bekaert S t e e l Wire Corp. - 20 - TABLE 4.1 MIX DESIGNS Mix S e r i e s F i b r e Volume Weight (lb) Dosage ( m l . ) % by Volume Cement Water Sand 3/8" Gr a v e l F i b r e P o z z o l i t h AEA 0 85 42.5 173 53 0 100 12 0.25 173 53 1.14 " " G l a s s 0.5 171.5 56.7 2.27 " f i b r e 0.75 170.7 56.5 3.41 " (GF) 1.0 1.25 1.5 2.0 „ „ 16 9.8 169.0 168.0 166 .4 56.2 55.9 55.6 55 4.54 5.7 6.8 9.1 E l II „ 0 67.5 24 148 148 0 58 23 0.25 25 146 .5 146.5 3.3 " S t r a i g h t 0.5 26.5 144.5 144.5 6.7 " " s t e e l 0.75 27 142.5 142.5 10 " f i b r e 1.0 28 141 141 13 .5 " (SSF) 1.25 28.5 140 140 17 " 1.5 29 13 9. 139 20 " 2.0 " 30 136 136 27 " 0 67.5 33 .8 148 148 0 58 23 0.25 147.4 147.4 3.27 " " Deformed 0.5 146.8 146.8 6.53 " " S t e e l 0.75 146 .3 146.3 9.8 " " F i b r e 1.0 145.7 145.7 13.1 (BSF). 1.25 145.1 145.1 . 16.3 ... y" </" ^ _ - - T r-F 1.5 2.0 » » 144 .5 143.3 144.5 143 .3 19.6 26.1 " » - 21 - 4.2 Design of Specimen and Mold The dimensions of the double t o r s i o n specimens were p r o p o r t i o n e d from s m a l l e r specimens p r e v i o u s l y used by 24 Nadeau, Mindess and Hay . A l l specimens were c a s t xn s t e e l molds, each 48 x 16 x 2.0 inches (1219 x 406.4 x 50.8 mm). Each mold co n t a i n e d a 48 i n . (1219 mm) long bar w i t h a tapered c r o s s s e c t i o n , 1.0 i n . (25.4 mm) i n depth x 0.50 i n . (12.7 mm) at base x 0.25 i n . (6.35 mm) a t the top f o r molding the p r e c a s t groove, as shown i n F i g u r e 4.1. A c o a t i n g of o i l was a p p l i e d t o the mold be f o r e c a s t i n g so t h a t the concrete would not adhere t o the mold. Specimens were removed from the mold by simply d i s a s s e m b l i n g the mold. 4.2.1 C a s t i n g of Specimens A l l mixes were prepared i n a pan-type mixer, w i t h the i n g r e d i e n t s weighed on a balance a c c u r a t e t o 1.0 l b . The L a i r e n t r a i n i n g agent was d i l u t e d w i t h the mixing water w h i l e the w o r k a b i l i t y agent was f i r s t poured i n t o the f i n e aggregate and allowed t o be absorbed. T h i s procedure was intended t o prevent d i r e c t chemical r e a c t i o n between the w o r k a b i l i t y a i d and the a i r e n t r a i n i n g agent. The pan was f i r s t dampened, and the coarse and f i n e aggregates p l u s two t h i r d s of the mixing water were p l a c e d i n the mixer and mixed thor o u g h l y . The cement was then added and mixed i n u n t i l i t was u n i f o r m l y d i s t r i b u t e d throughout the batch. The remaining water was then added. F i b r e r e i n f o r c e m e n t was added by shaking the - 22 - F i g u r e 4.1 C a s t i n g Mold ELEVATION VIEW - 23 - f i b r e s through a s i e v e to prevent " b a l l i n g " , and the concrete was mixed f o r a f u r t h e r f i v e minutes. A f t e r mixing, the concrete was p l a c e d i n t o the o i l e d mold with a shovel and compacted u s i n g an immersion v i b r a t o r . F i n a l l y , the specimens were, f i n i s h e d with a t r o w e l and covered w i t h p l a s t i c to prevent d r y i n g . Specimens were removed from the mold a f t e r 24 hours and kept i n a moist room u n t i l t e s t i n g . The age of the specimens was about 3 years a t the time of t e s t i n g . 4.2.2 P r e p a r a t i o n of Specimen b e f o r e t e s t i n g A day before the t e s t i n g of any specimen, a c o a t i n g of p l a s t e r of P a r i s was a p p l i e d on the non-grooved s i d e of the specimen. T h i s procedure was intended to a i d i n o b s e r v i n g crack growth on the t e n s i o n s i d e of the specimen d u r i n g t e s t i n g . J u s t p r i o r to t e s t i n g , the specimen was removed from the moist room and a coat of a commercial c u r i n g agent was a p p l i e d on the specimen to prevent l o s s of moisture due to e v a p o r a t i o n d u r i n g t e s t i n g . An i n i t i a l c rack, 4.0 i n . (101.6mm) long, was cut i n the c e n t r e of one end of the specimen along the groove, u s i n g a diamond saw. The crack was c u t i n such a way t h a t the l e a d i n g edge was on the grooved s u r f a c e . A l o a d i n g j i g , with f o u r clamps and f o u r l o a d i n g p o i n t s , was f i x e d i n the t e s t i n g machine. D e t a i l s of t h i s j i g are shown i n F i g u r e 4.2. The prepared specimen was secured to the l o a d i n g j i g with the precracked edge over the l o a d p o i n t s and the t e n s i o n s i d e up (Figure 4.3). The edges of the specimens were a l i g n e d p a r a l l e l w i t h the s i d e s of the l o a d i n g j i g . The d i s t a n c e s between the edges of the - 24 - F i g u r e 4.2 Loading J i g 48 i d 2 i T El 12 25 135 ,n3"3" '4V ftJM . • ! TV' 11 r l l -T T I I I I J_L ! —14 11 9 * I 9 | Flexible clamps Hous ing- for attaching the loading jig to the loading machine P L A N VIEW ELEVATION V I E W - 25 - F i g u r e 4.3 T e s t S e t u p specimen and the loading points were kept equal. An external load c e l l of 2000 l b f capacity, with two ball-bearings as loading points, was placed under the precracked side of the specimen (Figure 4.4). The loading configuration was so arranged that when the cross-head of the t e s t i n g machine descended the load c e l l acted as a r i g i d support. The two ball-bearings, one on either side of the crack, thus applied the required force to the specimen. Figures 4.5 and 4.6 show the t e s t i n g setup. There are two main advantages of t h i s loading con- f i g u r a t i o n . F i r s t , the ball-bearings were placed under the specimen instead of hanging above i t . This eliminated the necessity of a f i x t u r e to hold the ball-bearings i n t h e i r housing. Second, the tension surface was located on the top of the specimen. This provided the p o s s i b i l i t y of observing crack propagation with ease. A s t r i p chart recorder, c a l i - brated with the external load c e l l , was connected to the load c e l l to record the applied load and elapsed time. The chart speed was 2.0 in/min (50.8 mm/min). 4.3 Test Program Two types of tests were performed. 4.3.1 Compliance Test The material compliance of the specimen was obtained pi Tinius Olsen, 200000 l b capacity mechanical loading machine - 27 - Figure 4.4 External Load C e l l  - 30 - by measuring the a p p l i e d load and the corresponding d e f l e c t i o n as f o l l o w s . The specimen was f i r s t prepared and set on the loading j i g as described i n Sec t i o n 4.2.2. Two d i a l gauges, one placed next to the load c e l l and one placed at the outer edge of the specimen, were used to measure d e f l e c t i o n s . The load was a p p l i e d to the specimen i n 200 l b increments. Readings of the d i a l gauges and the corresponding a p p l i e d load were recorded. The d e f l e c t i o n of the p l a t e was obtained by s u b t r a c t i n g the reading of the d i a l gauge placed on the outer edge of the specimen from the reading of the one near the load c e l l . A f t e r a set of compliance measurements was made, the specimen was removed from the loading apparatus. A longer crack was cut using a p o r t a b l e ceramic e l e c t r i c power saw. These procedures were then repeated w i t h d i f f e r e n t crack lengths. 4.3.2 Double Tor s i o n Test 2 5 26 The load r e l a x a t i o n method ' was used on double t o r s i o n specimens to determine the r e l a t i o n s h i p between crack v e l o c i t y and s t r e s s i n t e n s i t y . Readings of a p p l i e d load and corresponding elapsed time were recorded. During t e s t i n g , the cross-head of the t e s t i n g machine was f i r s t lowered so that the specimen was j u s t touching the two loading p o i n t s of the load c e l l . Once contact was made, the two clamps on the loading side were re l e a s e d . Load.was a p p l i e d by lowering the cross-head w i t h a constant speed of 0.0373 in/min (0.947 - 31 - mm/min). Load r e l a x a t i o n s were performed a t every two hundred pound increment, by simply s t o p p i n g the cross-head movement. When the crack s t a r t e d t o propagate, t h i s u s u a l l y r e s u l t e d i n the l o a d - e l a p s e d time curve d e v i a t i n g from l i n e a r i t y , and the cross-head was stopped immediately. The n o n l i n e a r i t y of the curve s i g n a l l e d a change of compliance i n the specimen. With the cross-head f i x e d , the l o a d continued t o f a l l s l o w l y as the crack grew a t a d e c r e a s i n g r a t e . Each p o i n t on the r e s u l t i n g load-time curve corresponded to a d i f f e r e n t crack v e l o c i t y and s t r e s s i n t e n s i t y . Thus, a V-K-j. p l o t c o u l d be ob t a i n e d from a s i n g l e l o a d - r e l a x a t i o n curve. Background r e l a x a t i o n was measured as the l o a d - r e l a x a t i o n curve f o r the l a s t r e l a x a t i o n o btained f o r the specimen be f o r e the l o a d dropped, and t h i s was s u b t r a c t e d from the apparent l o a d r e l a x a t i o n curve. The l o a d i n g procedure was resumed a f t e r measuring the f i r s t crack p r o p a g a t i o n and stopped when the cra c k f i n a l l y propagated down the p l a t e . F i g u r e s 4.7 t o 4.9 show the crack appearance a t v a r i o u s stages. The new crack t i p was l o c a t e d with the a i d of a lOx magnifying g l a s s and marked on the specimen. However, t h i s apparent crack t i p was found t o be m i s l e a d i n g , due t o the nature of the observed c r a c k . Before a v i s i b l e crack c o u l d be observed on the s u r f a c e , m i c r o c r a c k s v i s i b l e o n l y u s i n g v e r y h i g h m a g n i f i c a t i o n have probably a l r e a d y propagated - 32 - Figure 4.7 I n i t i a l Stage of Crack Propagation - 34 -  beyond the v i s i b l e c r a c k . T h e r e f o r e , the measured value would tend to underestimate the t r u e crack l e n g t h . Thus t h i s procedure was d i s c o n t i n u e d . Other e f f o r t s were made to measure the t r u e crack l e n g t h by u s i n g a p e n e t r a t i n g dye. U n f o r t u n a t e l y , the r e s u l t s were l e s s than s a t i s f a c t o r y due to the roughness of the crack s u r f a c e . The p e n e t r a t i n g dye tended to d i s p e r s e and became b l u r r e d ; thus a d e f i n i t e crack f r o n t c o u l d not be o b t a i n e d . - 36 - Chapter 5 EXPERIMENTAL RESULTS 5.1 I n t r o d u c t i o n The main object of t h i s research was to i n v e s t i g a t e the e f f e c t of f i b r e reinforcement on crack growth i n concrete. Crack propagation data were obtained using the load r e l a x a t i o n method on a double t o r s i o n specimen as described above. In a l l , t h i r t y - o n e specimens were t e s t e d and the r e s u l t s are shown i n Tables B l to B31 (Appendix B). In the g l a s s and deformed s t e e l f i b r e specimens, the f i b r e content was the only v a r i a b l e . The w/c r a t i o f o r these specimens was 0.5. Both the f i b r e content and the w/c r a t i o were v a r i e d i n the s t r a i g h t s t e e l f i b r e s e r i e s (see Table 4.1). 5.2 Cement Paste Specimens To o b t a i n some i n d i c a t i o n of the v a l i d i t y of the t e s t r e s u l t s , two specimens of cement paste (w/c = 0.4) were made and t e s t e d . The r e s u l t s were then compared wi t h a v a i l a b l e data from the l i t e r a t u r e t o see whether the values obtained were of the r i g h t order of magnitude. The age of the specimens at t e s t i n g was.90 days. The t e s t r e s u l t s are tab u l a t e d i n t a b l e s 5.1 and 5.2. The data i n Tables 5.1 and 5.2, and Tables B l to B31 can be described as f o l l o w s : - 37 - TABLE 5.1 Load R e l a x a t i o n Data f o r Cement Paste No. 1 F i b e r volume: 0 Load a t f a i l u r e : 600 l b . Load a f t e r f a i l u r e : 0 Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (bl True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b p l o a d l b dp paper (in) dm time (sec) d t slope ldt/b lo a d l b dp paper (in) dm time (sec) d t slope . ldt\a ,dp, _ .dp. l d t J l d t * a b -2 VxlO i n / s e c K h k s i — i n 300 280 0.6815 20.6 13.57 75 2 60 1.25 12.32 11.4 0.364 280 200 1.038 58.125 3.44 50 2 60 0.833 2.607 1.8 0.34 265 62 4.5 135 0.459 48 6.75 202.5 0.237 0.222 0.18 0.325 260 10 2.5 75 0.133 10 6.75 202.5 0.049 0.084 0.067 0.316 I n i t i a l Load 330 l b . - 38 - TABLE 5.2 Load R e l a x a t i o n Data f o r Cement Paste No. 2 F i b e r volume: 0 Load a t f a i l u r e : 540 l b . Load a f t e r f a i l u r e : 0 Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) . True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b p. l o a d l b dp. paper ( i n l . . dm . . time (sec) d t slope >dt J l o a d l b dp paper (in) . dm time (sec) d t s l o p e /dp. AdtV ldt ;- a b -2 VxlO i n / s e c K '2 k s i - i n 280 150 0.5 15 10 120 4.69 140 0.853 9.147 16.7 0.332 260 160 1.44 43 .13 3 .71 4Q 2.56 76.88 0.52 3.19 6.1 0.313 245 64 2.16 64.69 0.989 18 4 120 0.15 0.839 1.8 0.294 240 40 4.5 135 0.296 8 9 27Q Q.029 0.267 0.34 0.291 235 8 7.5 225 Q.Q36 2 6 18Q Q.011 0.Q25 0.015 0.285 I n i t i a l Load 318 l b . Column 1 a p p l i e d l o a d Columns 2-5 d e s c r i b e the sl o p e of the r e l a x a t i o n curve a t the a p p l i e d l o a d In d e s c r i b i n g the slope of the r e l a x a t i o n curve a t the a p p l i e d l o a d , a h o r i z o n t a l l i n e was drawn which i n t e r c e p t e d the apparent r e l a x a t i o n curve a t the a p p l i e d l o a d (Figure 5.1). The slope of the curve was obtained, by c o n s t r u c t i n g .a tangent to the curve a t the i n t e r c e p t e d p o i n t . Column 2 d i f f e r e n c e i n the v e r t i c a l a x i s (y-axis) i . e . d i f f e r e n c e i n l o a d dp Column 3 d i f f e r e n c e i n the h o r i z o n t a l a x i s (x-axis) i . e . d i f f e r e n c e i n c h a r t l e n g t h dm Column 4 d i f f e r e n c e i n time, dt where d t = -T-—§2 = 3 0 dm c h a r t speed Column 5 slope of the apparent r e l a x a t i o n curve a t the a p p l i e d l o a d . The corresponding background r e l a x a t i o n was obt a i n e d by reproducing the background r e l a x a t i o n curve of the specimen below the apparent r e l a x a t i o n curve (see F i g u r e 5.1) with the i n i t i a l l o a d drop of the curves a t the same x-value. A v e r t i c a l l i n e was drawn through the i n t e r c e p t e d p o i n t of the a p p l i e d l o a d on the apparent r e l a x a t i o n curve which cut a p o i n t on the background r e l a x a t i o n curve. A t a n g e n t i a l l i n e was drawn to the background r e l a x a t i o n curve a t t h i s p o i n t . The slope of t h i s l i n e was the corresponding background -40- - 4 1 - relaxation. The slope of the background curve i s described i n Columns 6-9, which are s i m i l a r to Columns 2-5. Column 10 true relaxation = apparent relaxation - corresponding background relaxation Column 11 crack v e l o c i t y at the applied load Column 12 stress i n t e n s i t y at the applied load Typical load relaxation curves are shown i n Figure 5.2. These curves can divided into two parts; i ) p o s i t i v e slope region, and i i ) negative slope region. In the f i r s t part, as the load increases, the slope of the curve w i l l remain constant as long as the compliance of the specimen remains constant. When the crack starts to propagate, the slope of the curve f i r s t decreases d r a s t i c a l l y , and then decreases at a much more gradual rate. This suggests that most of the crack propagation occurs during the early part of the relaxation curve. The background relaxation curve i s obtained shortly before the crack s t a r t s to propagate. This background re- laxation i s due to the relaxation of the loading machine, and perhaps also due to creep i n the specimens. The background relaxation curve i s subtracted from the load relaxation curve i n order to get the true load relaxation of the specimen. T y p i c a l l y , the curves are s u f f i c i e n t l y d i f f e r e n t only for the f i r s t 90 seconds. A sample ca l c u l a t i o n of the crack v e l o c i t y i s shown i n Appendix A. - 42 - 1400 r Figure 5.2 T y p i c a l Load Relaxation Curve 1 2 0 0 1000 8 0 0 6 0 0 4 0 0 2 0 0 Region I > the difference between the two curves is significant Region 2 1 the difference between the two curves is insignificant Load relaxation curve Background relaxation curve J L 30 60 90 120 150 Time (sec. ) 180 210 240 - 43 - A V-Kj p l o t f o r the r e s u l t s of the two cement paste specimens, and the average V-K^ p l o t , are shown i n F i g u r e 5.3. The slope of the average V-K^ curves i s 37.6. With the same geometry but a s m a l l e r specimen (9.0 x 3.0 x 0.5 i n . or 24 228 x 76.2 x 12.7 mm), Nadeau, Mindess and Hay found t h a t the slope of the V-K^ curve was approximately 35. The val u e of the f r a c t u r e toughness of the c o n t r o l specimen was 0.69 k s i - i n 2 (0.75 MN _ 3 / / 2) compared t o the v a l u e of 0.293 k s i - i n 3 5 (0.32 MN~ 3 / / 2 24 27 obtained by Nadeau, Mindess and Hay . Wecharatana and Shah , us i n g 32 x 6.0 x 1.5 i n . (812 x 152 x 38.1 mm) double t o r s i o n specimens, found t h a t the f r a c t u r e toughness was 1.2 k s i - i n 2 (1.31 MN - 3/ 2). In both cases (24,27), the w/c r a t i o was 0.5. The v a l u e o b t a i n e d seems t h e r e f o r e t o be w i t h i n the range of va l u e s r e p o r t e d i n the l i t e r a t u r e . 5.3 F r a c t u r e Toughness One of the q u e s t i o n s about f i b r e r einforcement i s i t s e f f e c t i v e n e s s i n i n c r e a s i n g the f r a c t u r e toughness of co n c r e t e . The f r a c t u r e toughness, K j C , of the specimens i s t a b u l a t e d i n Table 5.3, and i s p l o t t e d a g a i n s t the f i b r e volume i n F i g u r e s 5.4 t o 5.8. (A sample c a l c u l a t i o n of the f r a c t u r e toughness i s shown i n Appendix A ) . In g e n e r a l , the f r a c t u r e toughness i n c r e a s e s w i t h f i b r e volume up t o about 1.25%. The s t r e s s i n t e n s i t y f a c t o r , K^, a t the f i r s t c rack (the f i r s t observed v i s i b l e crack) i s t a b u l a t e d i n Table 5.3. The va l u e of t h i s i s approximately equal t o 70% of the corresponding K j C - F i g u r e 5.3 V-K P l o t For The Average Of The Two Cement Specimens 10 10 -I 10 - 2 10 - 3 10 -41 0 .2 2 3 I 1 0 . 3 1 O SPECIMEN # I SLOPE = 37.0 2 • SPECIMEN # 2 S L O P E = 38 .0 3 A V E R A G E S L 0 P E = 37.6 0.4 0 . 5 0 . 6 0.7 0.8 0.9 KjUsi -in.z) - 45 - TABLE 5.3 FRACTURE TOUGHNESS AND RESIDUAL STRENGTH OF SPECIMEN SERIES FIBRE CONTENT O, K I C(KSI-IN J 5) CRITICAL K (KSI-IN^) AT FIRST RESIDUAL STRENGTH o CRACK (lb) 0 1.39 1.11 100 0.25 1.25 1.15 120 0.50 1.44 0.96 80 GF102 0.75 1.62 1.21 400 1.0 1.94 1.21 1050 1.25 2.05 1.32 580 1.50 2.42 1.94 1420 2.0 1.81 1.69 1040 0.25 1.39 1.21 240 0.5 1.25 1.21 240 GF204 0.75 1.69 1.45 244 1.25 2.32 2.05 800 1.5 1.99 0.97 960 2.0 1.30 0.97 780 0 2.36 1.93 240 0.25 0.81 0.73 300 0.5 1.79 0.97 740 V S S F 0.75 1.86 1.21 680 1.25 0.99 0.91 700 1.5 2.05 1.69 960 2.0 2.40 1.93 1290 0.25 0.87 1.12 580 0.5 1.88 1.58 540 1"SSF 1.0 2.42 1.45 1100 1.25 2.29 1.38 1180 1.5 1.25 0.97 660 0 1.41 1.33 120 0.25 1.69 1.44 600 BSF 0.5 1.42 1.21 820 1.25 2.00 1.69 1300 1.5 2.04 1.69 1390 2.0 2.36 1.45 1680 Cement 1 0 0.728 1.485 0 Cement 2 0 0.655 0.448 0 - 46 - W E I G H T D E N S I T Y ( lb. / f t . 3 ) - 48 - F i g u r e 5.6 R e l a t i o n s h i p s Between F r a c t u r e T o u g h n e s s , W e i g h t D e n s i t y , R e s i d u a l S t r e n g t h and F i b r e V o l u m e f o r h" SSF S e r i e s F I B R E V O L U M E (%) - 4 9 - W E I G H T D E N S I T Y ( lb . / f t . 3 ) cn O 1 o o Z k i c ( k s i - i n . " 2 ) r o cn H- c l-S (6 (t> (0 Ul h-1 p. 0) W f t l-i fD 0 i d r t V D> 3 O 3 Ul 3 " H-•O Ul CO fD r t S, fl> CD 13 cr H fD < o M c 3 0) Hi o 1-1 l-i B> o r t C M CO >-3 O c < 0 3 " 3 ft) u i «i c n CO c n ro n r -ro u i ro cr r t o ro 3 u i r -r t •< cn O O O O O R E S I D U A L S T R E N G T H ( lb.) - M 1 o o \ F i g u r e 5 .8 R e l a t i o n s h i p s Be tween F r a c t u r e T o u g h n e s s , VJe igh t D e n s i t y , F I B R E V O L U M E (%) - 51 - F i b r e r e i n f o r c e d concrete i s d i f f i c u l t to compact f u l l y , and a poorly compacted specimen w i l l leave voids and pores i n the f i n i s h e d product. At the higher f i b r e contents, weight d e n s i t i e s (Table 5.4) decreased due to incomplete compaction. The weight d e n s i t i e s of the specimens are p l o t t e d against t h e i r f i b r e volume i n Figures 5.4 to 5.8, and the degree of compaction i s r e l a t e d d i r e c t l y to the weight d e n s i t y . The weight d e n s i t y curves obtained can be c h a r a c t e r i z e d by an i n v e r t e d V. The weight d e n s i t y of the specimens normally increased w i t h i n c r e a s i n g f i b r e content and reached i t s highest value at about 1 to 1.25 per cent f i b r e by volume, then s t a r t e d to d e c l i n e . In genera l , the shape of the f r a c t u r e toughness vs f i b r e volume curves f o l l o w s the same p a t t e r n as the weight d e n s i t y vs f i b r e volume curves. This i n d i c a t e s that the f r a c t u r e toughness i s a f f e c t e d by the degree of compaction of the concrete. At higher f i b r e contents, the trend of the f r a c t u r e toughness vs f i b r e volume f o r the BSF s e r i e s does not f o l l o w the same p a t t e r n as the weight d e n s i t y vs f i b r e volume curve. This d i s p a r i t y may be accounted f o r by the f a c t t h a t BSF i s a more e f f i c i e n t f i b r e , and the increase i n f i b r e volume compensates f o r the adverse e f f e c t of the poor compaction. Thus the e f f e c t of poor compaction i s l e s s severe on the BSF s e r i e s . - 52 - TABLE 5.4 WEIGHT DENSITY OF SPECIMENS S e r i e s F i b r e Content Weight Weight D e n s i t y % (lb) ( l b / f t 3 ) 0. 129.1 147.0 0.25 125.3 142.6 GP-1Q2 0.5 121.7 138.5 0.75 125.4 142.8 1.0 131.5 149.7 1.25 126 .0 143.4 2.0 123 .1 140.1 0.25 125.8 143.2 0.5 122.5 139.4 GF-204 0.75 128.4 146.2 1.0 127.0 144.5 1.25 133 .5 152.0, 1.5 126.0 143.4 2.0 114.4 130.2 0 125.5 142.8 0.25 125.9 143.3 0.5 13 6.7 155.6 V S S F 0.75 134.5 153 .1 1.0 13 7.7 156.8 1.25 144.2 164.2 1.5 141.3 160.8 2.0 144.4 164.4 0 134.1 152.7 0.25 150.6 171.4 •0.5 143 .7 163.6 1" SSF 0.75 142.8 162.6 1.0 148.8 169.4 1.25 150.6 171.4 1.5 142.1 161.8 2.0 148.1 168.6 0 129.9 147.9 0.25 130.7 148.8 BSJf 0.5 125.4 142.8 0.75 132.0 150.3 l.Q 133.1 151.5 1.25 128.0 145.7 1.5 127.7 145.4 2.0 125.2 142.5 - 53 - 5.4 R e s i d u a l Strength R e s i d u a l s t r e n g t h i s d e f i n e d as the s t r e n g t h remaining a f t e r the crack has v i s i b l y extended a c r o s s the e n t i r e l e n g t h of the specimen. The r e s i d u a l s t r e n g t h s of the specimens are t a b u l a t e d i n Table 5.3, and are p l o t t e d a g a i n s t f i b r e volume to g e t h e r w i t h the f r a c t u r e toughness i n F i g u r e s 5.4 t o 5.8. The r e s i d u a l s t r e n g t h of the specimens i s the combined e f f e c t of the i n t e r l o c k i n g f o r c e between aggregates and the p u l l o u t r e s i s t a n c e of the f i b r e r e i n f o r c e m e n t . Zero r e s i d u a l s t r e n g t h was o b t a i n e d f o r the two cement paste specimens. The r e s i d u a l s t r e n g t h s of the p l a i n c o n c r e t e specimens are t h e r e f o r e due to the i n t e r l o c k i n g of the aggregates. The p a t t e r n of these curves c o i n c i d e s w i t h t h a t of the f r a c t u r e toughness curves, and the r e s u l t s are s i m i l a r l y a f f e c t e d by the compaction of the specimens. T h i s i s i l l u s t r a t e d by the s i m i l a r s l o p e s of the c u r v e s . In g e n e r a l , as the weight d e n s i t y curve goes up, the f r a c t u r e toughness and the r e s i d u a l s t r e n g t h curves a l s o go up. When the weight d e n s i t y curves go down, so do the other two curves. The r e l a t i o n s h i p between r e s i d u a l s t r e n g t h and f i b r e volume i s shown i n F i g u r e 5.9. The d i f f e r e n c e i n s l o p e s of the r e s i d u a l s t r e n g t h vs f i b r e volume curves i s r e l a t e d t o the d i f f e r e n c e i n p u l l o u t r e s i s t a n c e f o r d i f f e r e n t types of f i b r e s . Higher s l o p e s g e n e r a l l y i n d i c a t e a h i g h e r p u l l o u t r e s i s t a n c e . However, because of the s c a t t e r shown i n F i g u r e 5.9, i t i s d i f f i c u l t t o assess the p u l l o u t r e s i s t a n c e of the f i b r e s used i n t h i s study from the a v a i l a b l e data. Figure 5.9 Relationship Between Residual Strength and F i b r e 5.5 Compliance Two specimens were used to measure the system compliance. The t e s t r e s u l t s are tabulated i n Tables 5.5 and 5.6, and plotted i n Figure 5.10. The values of the slope and y-intercept of the compliance vs crack length curve of specimen BSF 1.0 are higher than those of the specimen V'SSF 1.0. This implies that specimen BSF 1.0 i s more compliant than specimen h"SSF 1.0, as might be expected from the fact that the BSF 1.0 specimen also had a lower density. 5.6 V-K T Plot The load relaxation data are tabulated i n Tables Bl to B31 (see Appendix B), and V-Kj plots on a log-log scale are shown i n Figures 5.11 to 5.15. Values of the fracture tough- ness and the crack v e l o c i t y were calculated using Equations 20 and 22 respectively. Values of the i n i t i a l and f i n a l crack lengths are 4 i n . (101.6 mm) and 48 i n . (1219 mm). (At f a i l u r e , the crack always ran r i g h t to the end of the specimen, therefore, the f i n a l crack length i s always equal to 48 i n . (1219 mm)). The relationships between the crack v e l o c i t y and stress i n t e n s i t y of the specimens are summarized i n Table 5.7. They were analysed using l i n e a r regression analysis. The co r r e l a t i o n c o e f f i c i e n t s of these regression analyses ranged from 0.80 to 0.99, and they were s i g n i f i c a n t at the 5% l e v e l . Therefore, a good c o r r e l a t i o n between the crack v e l o c i t y and the stress i n t e n s i t y factor e x i s t s . - 56 - Table 5.5' R e s u l t s of Compliance Study f o r Specimen h" SSF Crack l e n g t h (in) l o a d ( k i p ) x l O - 1 Gauge Reading ( i n ) x l 0 ~ J D e f l e c t i o n ( i n ) x l O - 3 Compliance ( i n / k i p ) x l 0 ~ 3 Average Compliance ( i n / k i p ) x l O a P i n out y y/p 2 3 5 2 10 4 6 9 3 7.5 Ll 4 6 10 13 .2 3.2 5.3 6.96 8 14 18 4 5 10 21 28 7 7 2 3 5 2 10 4 6 9.2 3.2 8 7 6 9 14.5 5.5 9.16 9.03 8 12 18.8 6.8 8.5 10 15 24.5 9.5 9.5 2 3.4 5.2 1.8 9 4 6.1 9 2.9 7.25 10 6 8.8 13 4.2 7 7.73 8 11 17 6 7.5 10 13 .1 21 7.9 7.9 2 3 6.5 3.5 17.5 4 5 10.5 5.5 13.75 16 6 7 14 .8 7.8 13 13 .95 8 8.5 18.5 10 12.5 10 10 23 13 13 2 5 9 4 20 4 8 14.4 6.4 16 19 6 11 19.2 8.2 13 .6 15.70 8 13 .1 24.8 11.7 14.6 10 15.8 30 14.2 14.2 - 57 - Table 5.6 R e s u l t s of Compliance Study f o r Specimen BSF 1.0 Crack l e n g t h (in) l o a d ( k i p ) x l O - 1 Gauge Reading , ( i n ) x l O J D e f l e c t i o n ( i n ) x l O - 3 Compliance ^ ( i n / k i p ) x l 0 ~ J Average Compliance ( i n / k i p ) x l O - 3 a P i n out y y/p 2 1.5 3 1.5 7.5 7 4 3 7 4 10 9.375 6 4 10 6 10 8 5 13 8 10 2 2.5 5.5 3 15 4 4.5 10 5.5 13.75 10 6 6.5 14 7.5 12.5 13 .35 8; 9 19 10 12.5 10 11 24 13 13 2 2.5 6 3.5 17.5 4 4.5 11 6.5 16.25 13 6 6.5 15 8.5 14.17 15.48 8 8 20 12 15 10 9.5 24 14.5 14.5 2 2 5.5 3.5 17.5 4 4.5 11.5 7 17.5 16 6 6.5 16.5 10 16.7 17.11 8 8 21.5 13 .5 16.88 10 9.5 26.5 17 17 2 3.5 7 3.5 17.5 4 6 14 8 20 19 6 8 19 11 18.3 18.92 8 10 25 15 18.75 10 11 31 20 20 2 2 6 4 20 22 4 4.5 12 7.5 18.75 19.84 6 6 18 12 20 8 7.5 24 16.5 20.6 - 58 - F i g u r e 5.10. R e l a t i o n s h i p Between System Compliance and Crack Length F i g u r e 5.11 V - K PLOTS FOR GF102 SERIES - 6 0 - F i g u r e 5.12 v-K i PLOTS FOR GF204 SERIES - 61 - F i g u r e 5.13 V-K PLOTS FOR SSF SERIES - 62 - F i g u r e 5.14 V-Kj PLOTS FOR 1" SSF SERIES 10 -I 10 - 2 u 10 - 3 10 - 4 1 1 I L 0 .5 0 . 6 0.7 0 .8 0.9 I 3 2 4 y i K x( k s i - i n . " 2 ) L E G E N D 2 3 4 5 O I" SSF =0.25 + » = 0.5 V »' * 1.0 A « =1.25 x " =1.5 - 6 3 - F i g u r e 5.15 v - ^ PLOTS FOR BSF SERIES - 6 4 - TABLE 5.7 Summary of R e s u l t s f o r the V-K T Curves Mix F i b r e Content Slope Y - i n t e r c e p t C o r r e l a t i o n S e r i e s % n A C o e f f i c i e n t 0 33.3 1.35 x 10" 3 0.8 0.25 22.4 8.39 x 10" 4 0.847 0.5 26.5 3.16 x 10" 4 0.95 GF102 0.75 31.6 1.3 x 10~ 4 0.98 1.0 33 .8 1.02 x I O - 4 0.96 1.25 50 5.0x 10" 8 0.98 1.5 33.8 -12 3.0 x 10 ± z 0.94 2.0 41.8 2.25 x 1 0 - 1 1 0.96 0.25 28.0 2.79 x 10" 4 0.99 0.5 29.6 4.11 x 10" 4 0.96 GF204 0.75 46.3 1.54 x 10~ 9 0.96 1.25 46 1.59 x 1 0 ~ 1 6 0.92 1.5 30.4 1.35 x 1 0 _ 1 0.94 2.0 26.6 6.77 x 10~ 2 0.96 0 29.0 2.34 x 1 0 ~ 1 0 0.97 0.25 16.0 2.22 0.98 0.5 11.3 3.93 x 10~ 4 0.9 h" SSF 0.75 32.0 6.30 x 10~ 5 0.97 1.25 29.3 1.01 0.99 1.5 29.9 4.03 x 10" 9 0.92 2 63.0 7.4 x 1 0 ~ 2 0 0.95 0.25 53.0 1.07 x 10~ 2 0.97 0.5 61.6 5.0 x 1 0 " 1 4 0.96 1"SSF 1.0 58.3 7.6 x 1 0 " 1 1 0.98 1.25 85.8 3.06 x 1 0 " 2 3 0.99 1.5 20.2 2.7 x 10~ 2 0.98 0 37.1 6.62 x 10~ 7 0.90 0.25 41.1 9.7 x 10" 9 0.99 . BSF 0.5 48.6 5.98 x 10~ 6 0.96 0.75 30.9 5.56 x 10~ 2 0.97 1.25 86 4.08 x 1 0 ~ 2 0 0.99 2 56 7.73 x l Q - 1 1 Q.96 - 6 5 - In F i g u r e s 5.11 to 5.15, s h i f t i n g of the V-K curves to the r i g h t occurs as the f i b r e volume i n c r e a s e s up to about 1.25 to 1.5%. No s p e c i a l p a t t e r n i s observed a f t e r the f i b r e content i n c r e a s e s to more than 1.5% of the t o t a l volume - probably due to unequal compaction. The s l o p e of the V-K y p l o t s was g r e a t e s t a t f i b r e contents of 1.25%. - 66 - Chapter 6 GENERAL DISCUSSION The r e s u l t s described i n the previous chapter are an attempt to evaluate the e f f e c t of f i b r e reinforcement on crack v e l o c i t y i n concrete. In the cement paste specimens, the w/c r a t i o was 0.4, which i s lower than the 0.5 used by Nadeau, Mindess and 24 27 Hay and Wecharatana and Shah . The value of the f r a c t u r e \, toughness obtained i n the experiment (0.69 k s i - i n 2 or -3/2 0.75 MN ), was l e s s than that obtained by Wecharatana 27 3/2 and Shah (1.2 k s i - i n 2 or 1.31 MN ), but, i t was twice 21 . h the value obtained by Nadeau, Mindess and Hay (0.293 k s i - i n -3/2 or 0.32 MN ). However, the agreement between the slope of the V-Kj curves obtained i n these t e s t s (37.6) and the 24 r e s u l t s of Nadeau, Mindess and Hay (35) was good. When ev a l u a t i n g the f r a c t u r e toughness, the s i z e of the specimen must be l a r g e enough to accommodate the s u b c r i t i c a l crack growth, and perhaps some amount of crack growth i s needed before a " v a l i d " I< I C can be obtained. These r e s u l t s i n d i c a t e the need to define a minimum specimen s i z e when t e s t i n g cementitious m a t e r i a l s . The weight d e n s i t y of the specimen u s u a l l y s t a r t e d to decrease when the f i b r e content was about 1.25% by volume. This suggests that when the f i b r e volume was more than 1.25%, f u l l compaction was not achieved. Figures 5.4 to 5.8 i n d i c a t e d t h a t the trend of f r a c t u r e toughness curves was — 67 - s i m i l a r t o t h a t of the weight d e n s i t y c u r v e s . The f r a c t u r e toughness i n c r e a s e d w i t h f i b r e content t o about 1.25%, and then decreased, due t o incomplete compaction. Thus, the advantages of p u t t i n g more f i b r e r e i n f o r c e m e n t i n the specimen may be o f f s e t by the high e r number of v o i d s c r e a t e d due t o d i f f i c u l t i e s i n compaction. In the BSF s e r i e s , the e f f e c t of compaction on the f r a c t u r e toughness seemed t o be l e s s acute. The slope and y - i n t e r c e p t are the two major p a r a - meters of the V-K-j. curves shown i n F i g u r e s 5.11 t o 5.15. A s m a l l y - i n t e r c e p t i n d i c a t e s t h a t s u b c r i t i c a l crack growth i s l e s s s i g n i f i c a n t a t low f r a c t u r e toughness v a l u e s . A lower slope i m p l i e s t h a t changes i n f r a c t u r e toughness have a s m a l l e f f e c t on the crack v e l o c i t y . T h e r e f o r e , f o r a m a t e r i a l which i s l e s s s u s c e p t i b l e t o s u b c r i t i c a l crack growth, the v a l u e s of the y - i n t e r c e p t and s l o p e of the V-Kj p l o t should be s m a l l . F i g u r e s 5.11 t o 5.15 i n d i c a t e t h a t by i n c r e a s i n g f i b r e content up t o about 1.25 t o 1.5% of the t o t a l volume, the V-K^ curves g e n e r a l l y s h i f t e d t o the r i g h t , g i v i n g a s m a l l e r y - i n t e r c e p t . At h i g h e r f i b r e c o n t e n t s , no p a t t e r n i n the p o s i t i o n of the V-K^ curves was observed. Large s l o p e s were g e n e r a l l y a s s o c i a t e d w i t h s m a l l y - i n t e r c e p t v a l u e s . T h i s suggests t h a t by adding about 1.25% to 1.5% by volume of f i b r e r e i n f o r c e m e n t , c o n c r e t e can be made l e s s s u s c e p t i b l e t o s u b c r i t i c a l c r a c k growth. However, - 68 - the crack v e l o c i t y i s q u i t e s e n s i t i v e t o changes i n the f r a c t u r e toughness. The f a c t t h a t high f i b r e a d d i t i o n s ( g r e a t e r than about 1.25%) do not improve the r e s i s t a n c e t o crack growth i s b e l i e v e d t o be caused by the d i f f i c u l t i e s i n f u l l y compacting the specimens. T h i s f i n d i n g i s supported by the weight d e n s i t y r e s u l t s i n Table 5.4. Table 5.3 showed t h a t the r e s i d u a l s t r e n g t h of the specimens i n c r e a s e d as the f i b r e content of the specimens i n c r e a s e d . T h i s i s probably a s s o c i a t e d w i t h the p u l l o u t r e s i s t a n c e of the f i b r e r e i n f o r c e m e n t . " F a i l u r e " of the specimen o c c u r r e d when the crack propagated down the f u l l l e n g t h of the specimen. Once f a i l u r e o c c u r r e d , the system changed from a continuous system t o a d i s c o n t i n u o u s system, c o n s i s t i n g of two separate p l a t e s h e l d together by f i b r e s . Due t o the l o a d i n g c o n f i g u r a t i o n , the crack w i l l open up at f a i l u r e . T h i s crack opening can o n l y be accommodated i f the f i b r e s a t the opening s u r f a c e elongate or s l i p w i t h i n the m a t r i x . The f i b r e s thus h o l d the specimen to g e t h e r a f t e r f a i l u r e has oc c u r r e d . T h e r e f o r e , by i n c r e a s i n g the f i b r e content, the r e s i d u a l s t r e n g t h of the specimens can a l s o be i n c r e a s e d . S e v e r a l l o a d r e l a x a t i o n s were performed on each specimen. Attempts were made t o measure the crack p o s i t i o n a t the end of each r e l a x a t i o n . These i n c l u d e d dye pene- t r a t i o n methods and d i r e c t measurement u s i n g a magnifying g l a s s . Both methods were found t o be inadequate i n measuring the t r u e crack p o s i t i o n . Chapter 7 CONCLUSIONS From the a n a l y s i s of the t e s t r e s u l t s , the f o l l o w i n g c o n c l u s i o n s can be drawn: 1. S u b c r i t i c a l c r a c k growth should be c o n s i d e r e d when measuring the f r a c t u r e parameters of cem e n t i t i o u s m a t e r i a l s . 2. A minimum specimen s i z e should be determined i n order t o get v a l i d r e s u l t s . 3. D i f f e r e n t types of f i b r e do not s i g n i f i c a n t l y a f f e c t the s l o p e and i n t e r c e p t of the V-Kj c u r v e s . 4. The degree of compaction a f f e c t s the f r a c t u r e p r o p e r t i e s of the specimens. Unless s p e c i a l a t t e n t i o n i s g i v e n t o the compaction procedure, f i b r e contents g r e a t e r than 1.5% of the t o t a l volume are not recommended. 5. The f r a c t u r e toughness i n c r e a s e s with f i b r e content t o about 1.25%. 6. R e s i d u a l s t r e n g t h of the specimen i n c r e a s e s with i n c r e a s - i n g f i b r e c ontent. T h i s s t r e n g t h seems a l s o t o depend on the p u l l o u t r e s i s t a n c e of the f i b r e r e i n f o r c e m e n t . 7. In t h i s t e s t geometry, f i b r e s do not s i g n i f i c a n t l y r e s t r a i n c rack growth. - 70 - BIBLIOGRAPHY 1. C E . I n g l i s , "Stress i n a Plate due to the Presence of Cracks and Sharp Corners," Trans. I n s t i t u t i o n of Naval Architect, Vol. LV, pp. 219-230 (1913). 2. H.M. Westergaard, "Bearing Pressure on Cracks," Journal of Applied Mechanics, Vol. 61, pp. A49-A53 (1939). 3. M.M. Eisenstadt, "Introduction to Mechanical Properties of Materials," The Macmillan Company, pp. 187-210 (1971). 4. A.A. G r i f f i t h , "The Phenomena of Rupture and Flow i n Solids," Philosophical Transactions of Royal Society of London 226, pp. 163-198 (1920). 5. G.R. Irwin and J . Kies, "Fracturing and Fracture Dynamics," Welding Journal Research Supplement, pp. 95-100 (1952). 6. E. Orowan, "Fundamentals of B r i t t l e Behaviour of Metals," Fatigue and Fracture of Materials, John Wiley and Sons, pp. 136-167 (1952). 7. G.C. Sih, "Handbook of Stress - Intensity Factors for Researchers and Engineers," In s t i t u t e of Fracture and S o l i d Mechanics, Lehigh University, Bethlehem, Philadelphia (1973). 8. J.F. Knott, "Fundamentals of Fracture Mechanics," John Wiley and Sons (1973). 9. G.R. Irwin, "Linear Fracture Mechanics, Fracture Transition and Fracture Control," Engineering Fracture Mechanics, Vol. 1, No. 2, pp. 241-257 (1968) 10. D.S. Dugdale, "Experimental Study of V-Notch Fatigue Test," Journal of Mechanics and Physics of Solids, Vol. 8, NO. 2, pp. 100-104 (1960). 11. G.R. Irwin, " P l a s t i c Zone Near a Crack and Fracture Toughness," 1960 Sagamore Ordnance Materials Conference, Syracuse University (1961). 12. W.A. Patterson and H.C. Chan, "Fracture Toughness of Glass Fibre-Reinforced Cement," Composites, Vol. 6, No. 3, pp. 102-104 (1975). - 71 - BIBLIOGRAPHY 13. A.E. Naaman, A.S. Argon and F. Moavenzaden, "A Fracture Model f o r Fiber Reinforced Cementitious Materials," Cement and Concrete Research, Vol. 3, No. 4, pp. 397-411 (1973). 14. F.E. Richart, A. Brandtzaeg and R.L. Brown, "A Study of the F a i l u r e of Concrete Under Coinbined Compressive Stresses," B u l l e t i n No. 185, Engineering Experiment Station, University of I l l i n o i s (1928). 15. J.P. Romualdi and G.B. Batson, "Mechanics of Crack Arrest i n Concrete," Journal of the Engineering Mechanics D i v i s i o n , American Society of C i v i l Engineers, Vol. 89, No. EM3, pp. 143-168 (1963). 16. S. Mindess, "The Fracture of Fibre-reinforced and Polymer Impregnated Concretes," International Journal of Cementitious Composites, Vol. 2, No. 1, pp. 3-11 (1980) 17. S. Mindess, "The Cracking and Fracture of Concrete: An Annotated Bibliography, 1928-1980," Materials Research Series Report No. 2, Department of C i v i l Engineering, University of B r i t i s h Columbia (1981). 18. K. Nishioka, S. Yamakawa, K, Hirakama and S. Akihama, "Test Method f o r the Evaluation of the Fracture Toughness of Steel Fibre Reinforced Concrete," RILEM Symp. 1978, Testing and Test Methods of Fibre Cement Composites, The Construction Press Ltd., Lancaster, England, pp. 87-98 (1978). 19. A.M. Brandt, "Crack Propagation Energy i n Steel Fibre Reinforced Concrete," International Journal of Cement Composites, Vol. 2, No. 1, pp. 35-42 (1980).. 20. G.T. Halvorsen, "J-Integral Study of Steel Fibre Reinforced Concrete," International Journal of Cement Composites, Vol. 2, No. 1, pp. 13-22 (1980). 21. G. Velazco, K. Visalvanich and S.P. Shah, "Fracture Behavior and Analysis of Fibre Reinforced Concrete Beam," Cement and Concrete Research, Vol. 10, No. 1, pp. 41-51 (1980). 22. A.G. Evans, "A Method for Evaluating the Time-dependent F a i l u r e C h a r a c t e r i s t i c s of B r i t t l e Materials — and Its Application to P o l y c r y s t a l l i n e Allumina," Journal of Material Science, Vol. 7, pp. 1137-1146. (1972). - 72 - BIBLIOGRAPHY 23. D.P. Williams and A.C Evans, "A Simple Method for Studying Slow Crack Growth," Journal of Testing and Evaluation, V o l . 1, No. 4, pp. 264-270' (1973) . 24. J.S. Nadeau, S. Mindess and J.M. Hay, "Slow Crack Growth i n Cement Paste," Journal of the American Ceramic Society, Vol. 57, No, 2, pp. 51-54 (1974)., 25. E.R. F u l l e r , J r . , "An Evaluation of Double Torsion Testing - Analysis," Fracture Mechanics Applied to B r i t t l e Materials, ASTM STP 678, S.W.. Freiman, Ed,, American Society f o r Testing and Materials, Philadelphia, pp. 3-18 (1978). 26 28 S.M. Wiederhorn, " S u b c r i t i c a l Crack Growth i n Ceramics," In s t i t u t e for Material Research, National Bureau of Standards, Washington (1974). 27. M. Wecharatana and S.P. Shah, "Double Torsion Tests f o r Studying Slow Crack Growth of Portland Cement Mortar," Cement and Concrete Research, Vol. 10, No. 6, pp. 833-844 (1980). R.A. Helmuth and D.H. Turk, " E l a s t i c Moduli of Hardened Portland Cement and Tricalcium S i l i c a t e Pastes.: E f f e c t of Porosity," Symposium on Structure of Portland Cement Paste and Concrete, Highway Research Board, SR 90, Washington D.C., pp. 135-144 (1966)., - 73 - A P P E N D I X A Sample Calculations Using data obtained from specimen GF 102 - 0.25 fracture toughness (KI(-.) k s i - i n 2 use v = 0.20 af t e r Helmuth & T u r k 2 8 K = P co (-3(1 + I C c r m Wt 3tn = (1.03) (6.875 ) C 3 ( 1 + ° * 2 - 0 ) (15.25)(2) J(1) = 1.216 k s i - i n 2 where P =1.03 kips measured cr * CO m = 6.875 i n W = 15.25 in t = 2 i n t n = 1 i n V = 0.2 Crack V e l o c i t y slope of relaxation curve = 510 lb/1 i n corresponding background machine relaxation = 360 lb/1 i n paper speed = 2 in/min corresponding applied load =950 l b i n i t i a l crack length a. = 4 i n applied load corresponding to i n i t i a l crack length P^ = 950 l b P m n # / - o - / • min ->/-n#/- o • / • min - = 510 / i n • 2 m/min • TTT - 360 / i n • 2 in/mm • r A t ' ' 60 sec ' ' 60 sec = 5 #/sec V = 9 - 5 0 # * 4 i n • 5 #/sec = 0.021 in/sec (950*r - 74 - APPENDIX B - 7 5 - - TABLE BI Load R e l a x a t i o n Data f o r Glass F i b e r F i b e r volume: 0 Load a t f a i l u r e : 1150 l b Load a f t e r f a i l u r e : 100 l b Ap parent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope vdt'b load l b dp paper (in) dm time (sec) dt slope (dt} a , dp. _ , dp . 1 d t ; { cTt ! a b -2 VxlO i n / s e c K h k s i - i n *910 300 0.75 22.5 13.33 330 1.03 30.9 10.667 2.666 1.77 1.075 880 280 1.125 33.75 7.78 240 1.5 45 5.333 2.444 1.15 1.04 860 250 2.5 75 3.333 150 2 60 2.5 0.833 0.41 1.015 840 82 2.25 67.5 1.215 120 3.5 105 1.143 0.;0721 0.37 0.99 820 70 4 120 0.583 40 2.5 75 0.533 0.05 0.027 0.968 800 38 5.5 160 0.23 30 5.5 165 0.182 0.049 0.025 0.94 * I n i t i a l Load TABLE B2 Load R e l a x a t i o n Data f o r G l a s s F i b e r s i z e 102 s e r i e s F i b e r volume: 0.25% Load a t f a i l u r e : 1030 l b Load a f t e r f a i l u r e : 120 l b Apparent R e l a x a t i o n (a) C o r r e s p o n d i n Background R e l a x a t i o n ( g b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper ( i n ) dm time (sec) d t s l o p e 1 a t ;b l o a d l b dp paper (i n ) dm time (sec) d t s l o p e 1 a t ' a r d P ^ _ ' dP\ c aT} v at ) a b -2 VxlO i n / s e c K h k s i - i n *950 510 1 30 17 360 1 30 12 5 2.1 1.15 900 560 2.5 75 7.46 360 2.5 75 4.4 3.067 1.43 1.089 880 240 2.5 75 3.2 160 3.5 105 1.52 168 0.824 1.065 860 100 2 60 1.66 80 4.5 135 0.593 1.07 0.549 1.04 840 100 4 120 0.833 80 4.5 135 0.593 0.24 0.13 1.014 820 40 4.5 135 0.296 20 3.5 105 0.19 0.106 0.06 0.99 800 20 6 180 0.111 10 4.5 135 0.074 0.037 0.022 0.963 a1030 470 1 30 15.6 360 1 30 12 3.66 14.2 1.246 980 300 2.5 75 4 160 3.5 105 1.52 2.44 10.6 1.18 960 120 2 60 2 160 3.5 105 1.52 0.48 2.15 1.15 940 100 3.31 99 1 80 4.5 135 0.593 0.413 1.93 1.14 920 60 3 90 0.67 80 4.5 135 0.593 0.077 0.375 1.11 900 140 2.5 75 1.86 20 3.5 105 0.19 1.67 0.085 1.09 * I n i t i a l Load I n i t i a l Crack Length 4 i n a F i n a l Load F i n a l Crack Length 48 i n TABLE B3 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 102 s e r i e s F i b e r volume: 0.5% Load at f a i l u r e : 1040 l b Load a f t e r f a i l u r e : 80 l b A ppareni b Relax a t i o n (a) Corresponding Background Relaxation (b) True R e l a x a t i o n V e l o c i t y S tress I n t e n s i t y load l b P load l b dp paper ( in) dm time ( sec) dt slope (£iE) Kat'b load l b dp paper (in) dm time (sec) dt slope (SiR) l d t ' a Kat' a ( d t ) b -2 VxlO in/sec K h k s i - i n *1000 380 1 60 12.66 380 1.818 54.38 6.98 5.68 2.2 1 .18 960 300 2 60 5 140 1.688 50.63 2.76 2.24 0.95 1 .15 940 80 0.938 28.1 2.84 70 2 60 1.167 1.67 0.757 1 .11 920 60 1.375 41.25 1.45 50 4.5 135 0.37 1.08 0.51 1 .086 900 60 3 180 0.667 50 4.5 135 0.37 0.296 0.146 1 .063 880 40 6 180 0.222 30 9 270 0.111 0.111 0.057 1 .04 * I n i t i a l Load - .78 - TABLE B4 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 102 s e r i e s F i b e r volume: 0.75% Load at f a i l u r e : 1340 l b Load a f t e r f a i l u r e : 400 l b Apparent Relaxation (a) Corresponding Background Relaxation (b) True Re l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time ( sec) dt slope /dp, load l b dp paper (in) dm time C sec) dt slope ^dt ;a <af>-<ai> a b -2 V x 10 i n / s e c K h k s i - i n *1000 640 1.5 45 14.2 420 1.5 45 9.333 4.88 1.95 1.18 960 340 2 60 5.66 100 1 30 3.33 2.333 1.01 1.13 940 60 1. 30 2 90 2.125 63.75 1.41 0.59 0.267 1.11 920 40 1.5 : 45 0.888 50 3 90 0.555 0.333 0.157 1.08 900 40 3.5 105 0.381 20 3.5 105 0.19 0.191 0.094 1.06 880 20 4 120 0.167 20 7 210 0.095 0.072 0.037 1.04 * I n i t i a l Load TABLE B5 Load R e l a x a t i o n Data f o r G l a s s F i b e r s i z e 102 s e r i e s F i b e r volume: 1.0% Load a t f a i l u r e : 1600 l b . Load a f t e r f a i l u r e : 1050 l b Apparent R e l a x a t i o n (a) C o r r e s p o n d i n g Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper ( i n ) dm time (sec) d t s l o p e l d t ' b l o a d l b dp paper (i n ) dm time (sec) d t s l o p e ( d t ' a l d t J l d t ' , a b -2 Vx 10 i n / s e c K h k s i - i n *1000 970 960 940 910 900 240 210 150 40 50 40 0.56 0.875 1.625 1 3 6 16.8 26 48.8 30 90 180 14.8 8 3.08 1.333 0.555 0.222 380 180 80 40 30 20 1.5 1.75 1 1.75 4 6 45 52.5 30 52.5 120 180 8.44 3 .43 0.688 0.762 0.25 0.111 5.78 4.51 1.497 0.57 0.305 0.111 2.07 1.9 0.65 0.25 0.147 0.0548 1.18 1.146 1.134 1.11 1.07 1.06 • I n i t i a l Load - 80 - TABLE B6 Load Re l a x a t i o n Data f o r Glass F i b e r s i z e 102 s e r i e s F i b e r volume: 1.25% Load at f a i l u r e : 1680 l b Load a f t e r f a i l u r e : 580 l b Apparent Relaxation (a) Corresponding Background Relaxation (b) True R e l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time (sec) dt slope ^dt'b load l b dp paper (in) dm time ( sec) dt slope (*£) 1 d t ; ^dt ;, a b -2 VxlO in/sec K h k s i - i n *1100 205 0.438 13.13 15.62 520 1.5 45 11.55 4.07 1.49 1.292 1080 380 1.5 45 8.44 320 1.94 58.2 5.50 2.94 1.1 1.275 1060 180 1.5 45 4 280 3.25 97.5 2.887 1.113 0.433 1.25 1040 180 3.5 105 1.71 150 3.75 112.5 1.33 0.384 0.155 1.223 1020 60 4.81 144.3 0.416 50 4.75 142.5 0.35 0.0646 0.027 1.204 1000 40 7.75 232.5 0.172 20 6 180 0.111 0.0609 0.026 1.18 • I n i t i a l Load - 81 - TABLE B7 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 102 s e r i e s F i b e r volume: 1.5% Load a t f a i l u r e : 2000 l b Load a f t e r f a i l u r e : 1420 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (^P) l d t ; b load l b dp paper (in) dm time (sec) dt s l o p e dp l d t ' a v d t ; ^dt ;, a b -2 VxlO i n / s e c K h k s i - i n *1600 400 1 •30 13.33 680 2 60 11.33 2 0.5 1.89 1560 480 2.78 83.4 5.75 420 * 3 90 4.67 1.08 0.28 1.84 1540 380 4.688 140.6 270 260 4.44 133 1.95 0.747 0.20 1.82 1520 80 1.75 52.5 1.52 160 5 150 1.067 0.457 0.126 1.79 1500 100 4 120 0.833 60 4 120 0.5 0.33 0.094 1.77 1480 60 4.5 13.5 0.444 30 5.75 172.5 0.174 0.27 0.079 1.75 1460 40 6 180 0.222 30 5.75 172.5 0.174 0.048 0.014 1.72 • I n i t i a l Load - 8 2 - TABLE B8 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 102 s e r i e s F i b e r volume: 2.01 Load at f a i l u r e : 1500 l b Load a f t e r f a i l u r e : 1040 l b Apparent-Relaxation (a) Corresponding Background Relaxation (b) True Re l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time (sec) dt slope l d t J b load l b dp paper ( i n ) dm time ( sec) dt slope (9£) [ d t > a dp_ _ dp ld-t ; *dt' a b -2 VxlO in/sec K h k s i - i n *1400 440 1 30 14.66 500 2 60 8 .333 6.33 1.81 1.65 1380 430 1.5 45 9.555 200 1.5 45 4.44 5.111 1.48 1.63 1340 140 1 30 4.667 140 2.125 63.75 2.196 2.47 0.77 1.58 1320 120 2 60 2 80 2.938 88.13 0.908 1.09 0.35 1.56 1300 60 2.25 67.5 0.885 50 4.5 135 0.37 0.519 0.17 1.535 1280 80 6 180 0.444 30 7 210 0.143 0.301 . 0.102 1.51 1260 40 7 210 0.19 20 9.5 285 0.07.01 0.047 0.016 1.488 • I n i t i a l Load - 83 - TABLE B9 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 204 s e r i e s F i b e r volume: 0.25% Load a t f a i l u r e : 1150 l b Load a f t e r f a i l u r e : 240 l b A pparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (dP) v d t ; b load l b dp paper (in) dm time (sec) dt s l o p e l d t ' a (^P) _ (d£) a b -2 VxlO i n / s e c K h k s i - i n *1000 970 940 920 880 860 440 340 140 120 40 20 1 1.5 1.375 3 2.75 3.75 30 45 41.3 90 82.5 112.5 14.67 7.55 3.39 1.333 0.485 0.178 260 160 100 80 30 20 1 1.06 1.75 3.5 3 6 30 31.8 52.5 105 90 180 8.667 5.02 1.905 0.762 0.33 0.111 6 2.535 1.492 0.571 0.152 0.067 2.4 1.07 0.675 0.27 0.078 0.038 1.18 1.14 1.11 1.08 1.04 1.015 • I n i t i a l Load TABLE BIO Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 204 s e r i e s F i b e r volume: 0.75% Load a t f a i l u r e : 1400 l b Load a f t e r f a i l u r e : 244 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (SlE) ydt'b load l b dp paper (in) dm time (sec) dt slope . dp. _ .dp. l d t ; Kdt' a b -2 V x 10 i n / s e c K h k s i - i n *1200 600 1 30 20 500 1 30 1.667 3.333 1.11 1.417 1180 270 0.813 24.4 11.07 280 1.031 31 9.05 2.02 0.696 1.39 1140 140 2.812 84.4 6 200 1.563 46.9 4.267 1.733 0.64 1.37 1130 90 2 60 1.5 120 3 90 1.33 0.167 0.062 1.33 1110 50 2.75 82.5 0.606 60 4 • 120 0.5 0.106 0.04 1.31 • I n i t i a l Load TABLE B l l Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 204 s e r i e s F i b e r volume: 1.25% Load a t f a i l u r e : 1920 l b Load a f t e r f a i l u r e : 800 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P loa d l b dp paper (in) dm time (sec) dt slope (^P) ^dt ; b load l b dp paper ( in} dm time (sec) dt slope {at> a < g > - c§i>b a b -2 VxlO i n / s e c K h k s i - i n *1700 720 1 30 24 410 1 30 13.67 10.33 2.43 2.0 1680 520 1.5 45 11.55 290 1 30 9.67 1.885 0.454 1.98 1650 270 2 60 4.5 210 2 60 3.5 0.993 0.248 1.95 1640 240 3 90 2.66 308 * 2.75 165 1.866 0.794 0.2 1.94 1580 40 3.25 97.5 0.41 20 4 120 0.167 0.243 0.066 1.867 * I n i t i a l Load - 86. - TABLE B12 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 204 s e r i e s F i b e r volume: 1.5% Load a t f a i l u r e : 1660 l b Load a f t e r f a i l u r e : 960 l b A pparen t Relax a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) d t slope {dt'h l o a d l b dp paper (in) dm time (sec) dt s lope -dp. (at> K d t i a b -2 VxlO i n / s e c X h k s i - i n *800 500 1.5 45 11.11 260 1 30 8.667 2.444 1.22 0.945 780 410 2 60 6.83 280 2 60 4.617 2.167 1.139 0.921 •760 280 2.5 75 3.73 150 2 60 2.5 1.23 0.68 0.897 750 120 2 60 2 90 2.75 82.5 1.09 0.91 0.517 0.886 730 60 3 90 0.667 60 6.25 187.5 0.32 0.346 0.208 0.86 720 40 6 180 0.222 20 14.5 435 0.046 0.176 0.137 0.85 700 20 10 300 0.066 20 14.5 435 0.046 0.02 0.014 0.82 • I n i t i a l Load - 87 - TABLE B13 Load R e l a x a t i o n Data f o r Glass F i b e r s i z e 204 s e r i e s F i b e r volume: 2.0% Load a t f a i l u r e : 1080 l b Load a f t e r f a i l u r e : 780 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P lo a d l b dp paper (in) dm time (sec) dt slope vdt'b load l b dp paper (in) dm time (sec) dt slope (dp_j ' dP^ _ (dP\ 1 d t ; d t ; , a b -2 VxlO i n / s e c K h k s i - i n *800 380 1 30 12.667 140 0.5 15 9.333 3.1333 1.667 0 .945 770 230 1.31 39. 34 5.84 300 1.94 58.1 5.16 0.68 0.367 0.909 740 360 3.5 105 2.476 186 3 90 2.067 0.409 0.239 0.874 720 100 3.5 105 0.952 100 4.25 127.5 0.784 0.168 0.104 0.838 700 20 5.5 165 0.121 10 4 120 0.083 0 .038 0.025 0.826 • I n i t i a l Load TABLE 'B14 Load R e l a x a t i o n Data f o r S t r a i g h t S t e e l F i b e r F i b e r volume: 0 Load a t f a i l u r e : 1860 l b Load a f t e r f a i l u r e : 240 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n ( b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) d t slope ' d t ; b load l b dp paper ( in) dm time (sec) dt slope, 1 d t ; £ rdP} _ / dP^ l d t ; ^dt ;, a b -2 VxlO i n / s e c K h k s i - i n *1600 236 0.5 15 15.7 450 2 60 7.5 8.25 2.05 1.89 1560 680 3 90 7.555 400 4 120 3.333 4.22 1.111 1.84 1520 300 2.5 75 4 140 3.31 99.3 1.408 2.592 0.718 1.79 150U 100 2 60 1.667 80 5.5 165 0 .485 1.182 0.336 1.77 1480 100 4 120 0.833 40 5.5 165 0.242 0.59 0.172 1.74 • I n i t i a l Load - 8 9 .- TABLE B15 Load R e l a x a t i o n Data f o r 1/2" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 0.25% Load a t f a i l u r e : 670 l b Load a f t e r f a i l u r e : 300 l b p ipparen t Relas :ation (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (d£) . load l b dp paper (in) dm time (sec) dt slope (d£) <§t> - <st> a b -2 VxlO i n / s e c K h k s i - i n *600 280 1 30 9.333 260 1.06 31.88 8.151 1.18 0.78 0.708 580 240 2.5 75 3.2 110 1.5 45 2.44 0.76 0.54 0.685 560 120 3.5 105 1.143 40 2 60 0.667 0.46 0.364 0.66 540 40 3 90 0.443 30 4 120 0.25 0.194 0.159 0.638 530 20 2.75 82.5 0.223 8 5.25 157.5 0.051 0.172 0.146 0.625 510 8 2 60 0.133 8 5.25 157.5 0.051 0.083 0.073 0.614 • I n i t i a l Load - .90 - TABLE B16 Load R e l a x a t i o n Data f o r 1/2" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 0.5% Load at f a i l u r e : 1490 l b Load a f t e r f a i l u r e : 7.40 l b Apparent Relaxation (a) Corresponding Background Relaxation (b) True R e l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time (sec) dt slope ( ^ ) 1 a t ' b load l b dp paper (in) dm time (sec) dt slope l d t J a (dp } _ (dp } ^dt ; d t ; , a b -2 VxlO in/sec K h k s i - i n *1490 1450 1440 1400 1370 1280 1220 700 250 520 440 330 20U 540 1 0.5 1.5 2.5 3.5 1.5 2.59 30 15 45 75 105 45 77.7 23.3 16.67 11.55 5.86 3.143 4.44 6.95 520 240 280 100 70 70 20 1.5 1 2.5 2.28 2.5 2.5 5 45 30 75 8.44 75 75 150 11.56 8 3.73 1.46 0.93 0.93 0.133 11.77 8.6 7.8 4.4 2.21 3.51 6.0 24.37 18.65 17.3 10.23 5.37 8.4 1.838 1.75 1.71 1.70 1.65 1.62 1.512 1.44 • I n i t i a l Load TABLE B17 Load R e l a x a t i o n Data f o r 1/2" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 0.75% Load a t f a i l u r e : 1540 l b Load a f t e r f a i l u r e : 680 l b A pparen t R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope ( ^ ) load l b dp paper (in) dm time C sec) dt slope l d t ] a l d t ; l d t ; , a b -2 VxlO i n / s e c K h k s i - i n *1000 490 1 30 16.33 150 0.5 15 10 6.33 2.3 1.19 970 360 2 60 6 300 2 60 5 1 0.43 1.145 950 170 2.5 75 2.26 340 6 180 1.88 0.386 0.17 1.122 940 80 2.5 75 1.06 " 80 3.5 105 0.762 0.305 0.14 1.11 920 40 2.5 75 0.533 20 2 60 0.333 0.2 0.095 1.08 910 40 4.5 135 0.296 10 2.5 75 0.133 0.165 0.08 1.075 • i n i t i a l Load - .92 - TABLE B18 Load R e l a x a t i o n Data f o r 1/2" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 1.25% Load at f a i l u r e : 780 l b Load a f t e r f a i l u r e : 700 l b A pparent Relaxation (a) Corresponding Background Relaxation (b) True Re l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time (sec) dt slope (dP) load l b dp paper (in) dm time (sec) dt slope K a z ' a <i> - <n> a b -2 VxlO in / s e c K H k s i - i n *750 490 1.5 45 10.89 200 1.5 45 4.45 6.44 8.44 0.886 720 340 2 60 5.667 200 1.5 45 4.45 1.22 0.706 0.85 700 120 2 60 2 190 4.25 127.5 1.5 0.5 0.306 0.826 670 60 3.5 105 0.857 60 3.25 97.5 0.6 0.258 0.16 0.79 660 40 5 150 0.267 40 8 240 0.167 0.1 0 .069 0.78 650 20 6 180 0.111 10 5.5 164 0 .061 0.05 0.036 0.7G7 • I n i t i a l Load •r 93 - TABLE B19 Load R e l a x a t i o n Data f o r 1/2" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 1.50% Load a t f a i l u r e : 1700 l b Load a f t e r f a i l u r e : 96 0 l b Apparent R e l a x a t i o n (a) Corresponding Background True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y R e l a x a t i o n (b) l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (£P_) load l b dp paper (in) dm time (sec) d t slope (d£) /dp, _ (dp, K d t ' ^dt ; a b -2 VxlO i n / s e c K h k s i - i n *1400 740 1.5 45 16.44 200 0.5 15 13.33 3.11 3.11 1.65 1360 660 3 90 7.33 540 4.31 129.3 8 2.63 2.63 0.82 1.606 1340 460 4.5 135 3.4 180 3.5 105 1.71 1.59 0.527 1.58 1320 2.5 75 1.333 80 4 120 0. 667 0.667 0.214 1.56 1300 80 4.81 144.38 0.554 20 3 90 0.222 0.332 0.11 1.535 • I n i t i a l Load - 94 - TABLr, B 2 0 Load R e l a x a t i o n Data f o r l / 2 n S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 2.0% Load at f a i l u r e : 1980 l b Load a f t e r f a i l u r e : 1290 l b A pparenl : Relaxation (a) Corresponding Background Relaxation (b) True Re l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper ( in) dm time ( sec) dt slope l d t ; b load l b dp paper (in) dm time (sec) dt slope (dP) (dP) _ (dp (at> Kat> a b -2 VxlO in/sec K h k s i - i n *1600 580 1 30 19.3 582 1.25 37.5 15.47 3.866 0.967 1.89 1570 460 1.75 52.5 8.76 340 1.875 56.25 6.04 2.72 0.707 1.85 1550 360 2.25 67.5 5.33 200 2 60 3.33 2 0.532 1.83 1520 200 2.5 75 2.666 100 3.75 112.5 2.05 0.616 0.17 1.8 1500 100 3.5 105 0.952 100 3.75 112.5 0.888 0.064 0.018 1.77 1480 40 5.5 165 0.242 30 5 150 0.2 0.042 0.012 1.75 • I n i t i a l Load - 9 5 - TABLE B21 Load R e l a x a t i o n Data for' 1" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 0.25% Load a t f a i l u r e : 920 l b Load a f t e r f a i l u r e : 580 l b A pparen t R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n ( b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P ' lo a d l b dp paper (in) dm . time (sec) dt slope v d t ; b lo a d l b dp . paper ( in) dm time (sec) dt slope ( ^ ) ( d t ' a ,dp, ,dp. a b -2 VxlO i n / s e c K h k s i - i n *720 380 1.5 45 8.44 340 2 60 5.667 2.777 1.5 0.85 700 240 2.5 75 3.2 140 2 60 2.333 0.867 0.51 0.827 680 60 1.875 56.25 1.06 80 3.5 105 0.762 0.304 0.189 0.803 670 30 ( 3 90 0.333 40 4.5 135 0.296 0.037 0.024 0.791 660 20 6.5 195 0.103 20 9 270 0.074 0.029 0.019 0. 780 • I n i t i a l Load - .96 - TABLE B22 Load R e l a x a t i o n Data f o r 1" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 0.5% Load a t f a i l u r e : 1560 l b Load a f t e r f a i l u r e : 540 l b Apparent R e l a x a t i o n (a) [ Corresponding J Background i. R e l a x a t i o n (b) I- True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope *dt ;b lo a d l b dp paper (in) dm time (sec) dt slope (dP) ( d t J a <§t>- <s> a b -2 VxlO i n / s e c K h k s i - i n *1300 470 1 30 15.667 300 1 30 10 5.667 1.74 1.535 1280 350 1.5 45 7.778 300 1.938 58.1 5.16 2.616 0.83 1.512 1270 190 1.56 46.88 4.05 210 2 60 3.5 0.553 0.172 1.5 .1260 120 1.75 52.5 2.286 180 3.25 97.5 1.846 0.44 0.144 1.488 1240 60 2.375 71.25 0.842 60 2.5 7.5 0.8 0.042 0.014 1.465 . 1200 20 6 180 0.111 20 8 240 0.083 0.028 0.01 1.417 • I n i t i a l Load - 97 > TABLE B23 Load R e l a x a t i o n Data f o r 1" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 1.0% Load a t f a i l u r e : 2000 l b Load a f t e r f a i l u r e : 1100 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper ( i n ) dm time (sec) d t slope l d t ' b lo a d l b dp paper (in) dm time (sec) dt s l o p e ( d £ } l d t ' a (-P.) _ (dP) (d±> d t ; , a b -2 VxlO i n / s e c K h k s i - i n *1200 300 ' 0.5 15 20 358 1.5 45 7.95 12.05 4 1.417 1180 440 1.5 45 9.77 280 2.938 88.1 3.177 6.60 2.27 1.394 1160 290 3 90 3.222 190 4.25 127.5 1.49 1.732 0.618 1.37 1140 120 2.5 75 1.6 60 3.25 97.5 0.615 0.985 0.36 1.346 1120 60 3.25 97.5 0.615 20 3 90 0.222 0.393 0.15 1.322 1110 30 3 90 . 0.333 20 3 90 0.222 0.111 0.043 1.311 1100 20 6 180 0.111 10 5 150 0.056 0.056 0.022 1.299 • I n i t i a l Load TABLE B24 Load R e l a x a t i o n Data f o r 1" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 1.25% Load a t f a i l u r e : 1900 l b Load a f t e r f a i l u r e : 1180 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope Mt'b loa d l b dp paper (in) dm time • (sec) dt slope ( dP^ _/ dP\ a t ; K d t ' a b -2 VxlO i n / s e c K h k s i - i n *1500 500 1 30 16.67 300 1 30 10 6.67 1.8 1.748 1460 200 1 30 6.67 340 3 90 3.778 2.889 0.8 1.724 . 1459 220 2.5 75 2.93 140 2.5 75 1.867 1.06 0.302 1.713 1430 50 1.25 37.5 1.33 20 0.75 22.5 0.888 0.445 0.129 1.689 1420 40 2.5 75 0.533 60 5 150 0.4 0.133 0.039 1.677 1400 42 6 180 0.233 20 4 120 0.168 0.066 0.020 1.654 • I n i t i a l Load - 99 - TABLE B25 Load R e l a x a t i o n Data f o r 1" S t r a i g h t S t e e l F i b e r S e r i e s F i b e r volume: 1.5% Load a t f a i l u r e : 1040 l b Load a f t e r f a i l u r e : 660 l b Al ̂ parent : Relax< ation (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope .dp Mt'b loa d l b dp paper (in) dm time (sec) dt s lope (dp. l d t ; a (dp" dp. a b -2 VxlO i n / s e c K h k s i - i n *800 160 0.625 18.8 8.53 340 1.81 54.3 6.25 2.28 1.114 0.945 780 220 1.325 40 5.33 180 1.56 46.8 3.84 1.49 0.78 0.92 770 130 1 30 4.5 108 2.50 75 1.44 0.69 0.37 0.91 760 140 2.5 75 1.86 100 4.50 135 0.741 0.42 0.236 0.90 740 120 7 120 0.576 30 3.75 112.5 0.266 0.309 0.181 0.87 730 70 5.5 165 0.424 28 6 180 0.155 0.269 0.16 0.86 720 20 5.5 165 0.121 10 6 180 0.055 0.066 0.04 0.85 a1040 660 1 60 22 340 1.81 54.3 6.25 15.75 60 1.22 980 500 1.125 3.38 7.080 180 1.56 46.8 3.84 10.92 50 1.16 940 320 1 60 5.380 108 2.50 75 1.44 6.82 34 1.11 320 320 1.5 45 4.9 100 4.50 13.5 0.741 5.64 29 1.08 • I n i t i a l Load a F i n a l Load - 100 - TABLE B26 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r S e r i e s F i b e r volume: 0 Load a t f a i l u r e : 1170 l b Load a f t e r f a i l u r e : 120 l b A pparen t Relax a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b "P lo a d l b dp paper (in) dm time (sec) d t slope (d£) load l b dp paper (in) dm time (sec) dt s lope ( d p) ^dt ; *dt', a b -2 VxlO i n / s e c K h k s i - i n *1170 370 1 30 12.33 420 1.5 45 9.333 3 10.5 1.38 1140 160 1 30 5.33 180 1.5 45 4 1.333 4.9 1.346 1120 80 1.5 45 1. 77 150 3.189 95.6 1.56 0.217 0.83 1.320 1100 310 0.875 26.25 10.33 420 1.5 45 9.333 2.47 0.898 1.299 1070 270 1.5 45 6 180 1.5 45 . 4 2 0.768 1.270 1050 132 1.81 54.3 2.42 150 . 3.188 95.6 1.56 0.86 0.343 1.240 1040 60 2.5 75 0.80 80 4.5 135 0.59 0.21 0.085 1.228 1030 60 5 150 0.40 30 3 90 0.333 0.067 0.027 1.216 • I n i t i a l Load - 101 - TABLE B27 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r S e r i e s F i b e r volume: 0.25% Load a t f a i l u r e : 1400 l b Load a f t e r f a i l u r e : 600 l b A] pparen t R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (dP) ^dt ;b l o a d l b dp paper (in) dm time (sec) dt s l o p e l d t ' a l d t ; l d t ' a b -2 VxlO i n / s e c K h k s i - i n *1190 610 1.75 52.5 11.6 460 1.2 36 7.66 3.95 1.33 1.41 1180 580 2.688 80.6 7.19 230 1.5 45 5.11 2.087 0.713 1.39 1160 260 3.440 103 2.52 40 1 30 1.333 1.188 0.420 1.37 1140 80 3 90 0.888 40 3 90 0.444 0.444 0.163 1.34 1120 60 5.5 165 0.362 20 5 150 0.133 0.23 0.087 1.32 • I n i t i a l Load - 102 •:- TABLE B28 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r S e r i e s F i b e r volume: 0.5% Load a t f a i l u r e : 1180 l b Load a f t e r f a i l u r e : 820 l b Dparen t Relax a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) d t slope /dpv ( a p b load l b dp paper (in) dm time (sec) dt s l o p e /dp, ( d T } a (dP) _ (dp.) a b -2 VxlO i n / s e c K h k s i - i n *1000 340 1 30 11.33 250 1 30 8.333 3 1.2 1.18 980 200 1 30 6.67 180 1.5 45 4 2.67 1.1 1.157 970 90 0.97 29.1 3.096 70 1.5 45 1.553 1.54 0.65 1.15 960 60 2.125 63.8 0.941 40 2.5 75 0.533 0.408 0.177 1.133 940 40 3 90 0.444 10 3.5 105 0.095 0.349 0.156 1.11 920 20 5 150 0.133 10 3.5 105 0.095 0.038 0.018 1.08 • I n i t i a l Load -.103 - TABLE B29 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r Series F i b e r volume: 0.75% Load at f a i l u r e : 1420 l b Load a f t e r f a i l u r e : 1100 l b A pparen t Relax a t i o n (a) Corresponding Background Relaxation (b) True Re l a x a t i o n V e l o c i t y Stress I n t e n s i t y load l b P load l b dp paper (in) dm time (sec) dt slope ( d t ' b load l b dp paper (in) dm time (sec) dt slope (dP) [ d t ' a a b -2 VxlO in/sec K h k s i - i n *840 440 1.5 45 9.77 200 1.5 45 4.45 5.33 2.5 0.99 800 400 3.5 105 3.81 120 3.15 94.5 1.27 2.54 1.33 0.94 780 140 2.5 75 1.867 60 3.25 97.5 0.61 1.258 0.695 0.92 760 60 2.5 75 0.8 60 3.25 9715 0.6 0.2 0.116 0.89 740 40 3.75 112.5 0.356 40 8 240 0.167 0.189 0.1 0.87 720 20 6 180 0.111 20 6 180 0.061 0.05 0.03 0.85 • I n i t i a l Load - .104 - TABLE B30 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r S e r i e s F i b e r volume: 1.25% Load a t f a i l u r e : 1660 l b Load a f t e r f a i l u r e : .13 00 l b Apparent R e l a x a t i o n (a) Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (dP) (dt'b load l b dp paper (in) dm time (sec) dt slope (dPy ( d t ' a (dP) _ ( d p } a b -2 VxlO i n / s e c K h k s i - i n *1380 300 0.906 27.18 11.03 420 1.656 19.69 8.45 1.86 7.5 1.63 1350 210 .1.250 37.5 5.6 150 1.75 52.50 2.857 2.743 0.84 1.59 1330 130 2 60 2.17 140 3.90 117 1.196 0.97 0.37 1.57 : 1320 80 4 120 0.66 40 3.50 105 0.38 0.28 0.089 1.55 1300 20 1.5 45 0.444 40 3.50 105 0.38 0.064 0.021 1.535 ^ I n i t i a l Load - .105 - TABLE B31 Load R e l a x a t i o n Data f o r Bent S t e e l F i b e r S e r i e s F i b e r volume: 2.0% Load a t f a i l u r e : 2000 l b Load a f t e r f a i l u r e : 16 80 l b Apparent R e l a x a t i o n (a) " Corresponding Background R e l a x a t i o n (b) True R e l a x a t i o n V e l o c i t y S t r e s s I n t e n s i t y l o a d l b P l o a d l b dp paper (in) dm time (sec) dt slope (dP) ^dt ;b load l b dp paper (in) dm time (sec) dt s l o p e (dp } l d t ' a dp _ (dp } (at' at' a b -2 VxlO i n / s e c K h k s i - i n *12Q0 830 2.312 69.38 11.96 400 2.5 75 5.33 6.631 2.2 1.417 1170 620 3.188 95.60 6.48 380 3.81 114 3.322 3.16 1.1 1.382 1150 210 3 90 2.333 160 3.75 112.5 1.422 0.911 0.33 1.358 1140 120 3.75 112.5 1.06 40 4 120 0.75 0.317 0.117 1.346 1120 60 3.75 112.5 0.533 140 9.5 285 0.491 0.042 0.016 1.32 1100 42 7 210 0.2 40 8 240 0.167 0.033 0.013 1.299 1080 20 7 210 0.095 8 4 120 0.0667 0.0285 0.012 1.27 * I n i t i a l Load

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