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Effect of fibre reinforcement on the crack propagation in concrete Yam, Anthony Sze-Tong 1981

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EFFECT OF FIBRE REINFORCEMENT ON THE CRACK PROPAGATION IN CONCRETE by ANTHONY SZE-TONG (YAM B.Sc, The University of Saskatchewan, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1981 (g) Anthony Sze-Tong Yam, 1981 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Ci\T^L ^A^MIA*' Kf\\ The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 DE-6 (2/79) ABSTRACT The influence of fibre reinforcement on crack propagation in concrete was studied. Thirty-five double torsion specimens, made with three types of fibres (fibreglass, straight steel fibres and deformed steel fibres) were tested.. The variables were the fibre volume and size of the fibres. The test results indicated that the resistance to rapid crack growth increased somewhat with increasing fibre content up to about 1.25% - 1.5% by volume. The degree of compaction had an enormous effect on the fracture properties. The fracture toughness increased with fibre content up to about 1.25% by volume, and then decreased, due to incomplete compaction. It was found that in this test geometry, fibres did not significantly restrain crack growth. It was also observed that once the crack had propagated down the full length of the specimen, the system changed from a continuous system to a discontinuous system, consisting of two separate plates held together by the fibre reinforcement. Different types of fibres did not significantly affect the fracture toughness. ACKNOWLEDGEMENT S The author expresses his indebtedness to Professors S. Mindess and J.S. Nadeau for their valuable guidance in planning and carrying out the investigation. The author is grateful to Professor R.J. Gray for his advice. The author also wishes to thank the Civil Engineering Department technicians and especially Mr. B. Merklie for his assistance in making the test equipment and carrying out the test. This research was made possible by grants from the National Sciences and Engineering Research Council Canada, and the Natural, Applied and Health Sciences Grants Committee, U.B.C. TABLE OF CONTENTS Page Abstract ii Acknowledgements iiTable of Contents iv List of Figures v List of Tables viList of Symbols x 1. Introduction 1 2. Fracture Mechanics: General Background 3 2.1 Historical Background 3 2.2 The Stress Intensity Approach 7 2.2.1 Stress Intensity Factor2.2.2 Effective Crack Length 10 ' 2.3 Relationship Between G and K 1 2.4 Fracture Mechanics Applied to Fibre Reinforced Concrete " 12 3. Measurement of Fracture Parameters and Stable Crack Growth 15 3.1 Test Specimens 13.2 Double Torsion Technique 16 4. Experimental Procedure 20 4.1 Material 24.2 Design of Specimen and Mold 22 4.2.1 Casting of Specimens4.2.2 Preparation of Specimens Before Testing 24 4.3 Test Program 27 4.3.1 Compliance Test 24.3.2 Double Torsion Test 31 5. Experimental Results 37 5.1 Introduction5.2 Cement Paste Specimens 35.3 Fracture Toughness 44 5.4 Residual Strength 55.5 Compliance 56 5.6 V-K Plot6. General Discussion 67 7. Conclusion 70 Bibliography 1 Appendix A .74" Appendix B 75 - iv -LIST OF FIGURES Figures Page 2.1 Crack front coordinates 5 2.2 The three different modes of failure 9 3.1 The double torsion specimen 17 4.1 Casting mold 23 4.2 Loading jig 5 4.3 Test setup 6 4.4 External load cell 28 4.5 Front view of the testing setup 29 4.6 Sideview of the testing setup with a specimen inplace 30 4.7 Initial stage of crack propagation 33 4.8 Specimen just before failure 34 4.9 Specimen after failure 35 5.1 Load relaxation curves 41 5.2 Typical load relaxation curves 43 5.3 V-K-j. plot for the average of the two cement specimens 45 5.4 Relationships between fracture toughness, weight density, residual strength and fibre volume for GF 102 series 47 5.5 Relationships between fracture toughness, weight density, residual strength and fibre volume for GF 204 series 48 5.6 Relationships between fracture toughness, weight density, residual strength and fibre volume for h" SSF series 49 5.7 Relationships between fracture toughness, weight density, residual strength and fibre volume for 1" SSF series 50 5.8 Relationships between fracture toughness, weight density, residual strength and fibre volume for BSF series 51 5.9 Relationship between residual strength and fibre volume 55 - v -Figures Page 5.10 Relationship between system compliance and crack length 59 5.11 V-Kj plots for GF 102 series 60 5.12 V-Kj plots for GF 204 series 61 5.13 V-Kj plots for h" SSF series 62 5.14 V-Kj plots for 1" SSF series 63 5.15 V-K plots for BSF series 4 - vi -LIST OF TABLES Table Page 4.1 Mix design 21 5.1 Load relaxation data for hardened cement paste 38 5.2 Load relaxation data for hardened cement paste 9 5.3 Fracture toughness and residual strength of specimens 46 5.4 Weight density of specimens 53 5.5 Results of compliance study for specimens h" SSF 1.0 57 5.6 Results of compliance study for specimens BSF 1.0 8 5.7 Summary of results from the V-KT curves 65 BI Load relaxation data for GF - 0 76. B2 Load relaxation data for GF 102 - 0.25 77 B3 Load relaxation data for GF 102 - 0.5 7:8 B4 Load relaxation data for GF 102 - 0.75 79 B5 Load relaxation data for GF 102 - 1.0 80 B6 Load relaxation data for GF 102 - 1.25 81 B7 Load relaxation data for GF 102 - 1.5 82 B8 Load relaxation data for GF 102 - 2.0 83 B9 Load relaxation data for GF 204 - 0.25 84 BIO Load relaxation data for GF 204 - 0.75 85 Bll Load relaxation data for GF 204 - 1.25 86 B12 Load relaxation data for GF 204 - 1.5 87 B13 Load relaxation data for GF 204 - 2.0' 88 B14 Load relaxation data for SSF - 0 89 B15 Load relaxation data for h" SSF - 0.25 90 B16 Load relaxation data for h" SSF - 0.5 B17 Load relaxation data for vn SSF - 0.75 92 B18 Load relaxation data for I'" SSF - 1.25 93 B19 Load relaxation data for L.II *2 SSF - 1.5 94 B20 Load relaxation data for L- 11 *2 SSF - 2.0 95 - vii -Table B21 Load relaxation data B22 Load relaxation data B23 Load relaxation data B24 Load relaxation data B25 Load relaxation data B26 Load relaxation data B27 Load relaxation data B28 Load relaxation data B29 Load relaxation data B30 Load relaxation data B31 Load relaxation data Page for 1" SSF - 0. 25 96" for 1" SSF - 0. 5 97 for 1" SSF - 1. 0 98' for 1" SSF - 1. 25 99 for 1" SSF - 1. 5 100 for BSF - 0 101 for BSF - 0 .25 102 for BSF - 0 .5 103 for BSF - 0 .75 104 for BSF - 1 .25 105 for BSF - 2 .0 106 - viii: — LIST OF SYMBOLS length of the semi-major axis, or one half of the crack length effective crack length slope of the V-K^ curve interatomic equilibrium bond spacing (lattice spacing) system compliance the y-intercept of the V-K-j. curve modulus of elasticity energy release rate critical strain energy release rate stress intensity factor, where subscript I refers to Mode I failure critical stress intensity factor (also known as fracture toughness) pound force applied load corresponding polar coordinates thickness of the double torsion specimen plate thickness in the plane of the crack crack velocity width of the double torsion specimen cartesian coordinates with the origin at the crack tip deflection surface tension Poisson's ratio radius of curvature at the tip of the ellipse applied stress resultant stress in a-a direction, subscript aa represents the direction ideal fracture strength yield strength moment arm Chapter I INTRODUCTION Concrete and related cementitious materials are heterogeneous and composite in nature. Microcracks are an inherent characteristic of such materials, due to volume changes of the cement paste during hydration and due to shrinkage of the hardened cement paste upon drying. When under load, these microcracks will extend, forming an extensive crack network which eventually leads to one or more large cracks and subsequent failure. Control of cracking is particularly important for the serviceability of reinforced concrete structures. Adequate crack control often can be achieved using smaller reinforcement bars, more closely spaced. Results have shown that concrete structures with fibre reinforcement develop finer cracks under loading. Yet, how do fibres work? Do they act as crack arrestors or do they simply hold the cracked structure together? If the fibres do arrest cracks, then how do they affect the crack growth rate? One means of gaining an understanding of these phenomena is through fracture mechanics. Using this approach, the fracture strength, o , is inversely proportional to the square root of the size of the critical flaw. When a stress less thana^ is applied,, the structure will support that stress only as long as the flaw does not grow to the critical size for that stress. Research on brittle materials has shown - 1 -that flaws will grow under sustained loading - a phenomenon known as subcritical crack growth. Subcritical crack propagation is caused by localized crack tip stresses, which are directly related to the stress intensity factor . Thus crack growth can be expressed as a function of the stress intensity factor. The relationship between crack growth and the stress intensity factor can best be described by a V-K^ plot, i.e. a plot of crack velocity vs stress intensity. Once the crack growth has been characterized in this way, design criteria can be developed for such a material, and a better understanding of the mechanisms governing crack growth can often be obtained. In addition, the life expectancy of structural elements made with such materials can then be predicted. The objectives of the research reported here.were: 1. To investigate the effect of fibre reinforcement on crack growth in concrete. 2. To determine quantitatively the effect of fibre content on the fracture toughness of concrete. - 2 -Chapter 2 FRACTURE MECHANICS: GENERAL BACKGROUND 2.1 Historical Background Stresses around cracks have been studied in detail la by many people. In 1913, Inglis showed that stresses around an elliptical hole (an ellipse is often used to characterize the general geometry of a crack) in a uniformly stressed plate could be'expressed as a = ail + 24-) (1) aa p where a = resultant stress in a-a direction aa a = applied stress a = length of the semi-major axis, or one-half of the crack length p = radius'of curvature at the tip of the ellipse 2 Westergaard showed that stresses near a sharp crack could be expressed as axx = CTJS" cos|(1 " sin| sinT) + '*' (2) a = a l-Jr- cos^(l - sin| sinrJ^) + ••• (3) yy J 2r 2 2 2 /a .0 0 30, t A\ a = a hr- Sins COSs COS-s- + ••• (4) xy J 2r 2 2 2 numbers refer to bibliography at the end - 3 -where: subscripts xx, yy and xy represent the coordinate directions xx and yy = Cartesian coordinates with the origin at the crack tip r,8 = corresponding polar coordinates and the other symbols are as defined above. These crack tip coordinates are shown in Figure 2.1. With the help of these solutions, the theory of the mechanics of fracture can be developed. The tensile strength of an ideal crystalline body is the stress which must be applied to cause it to fracture across a particular crystallographic plane. This ideal strength can be expressed as (3) m v b * o where a = ideal fracture strength m E = modulus of elasticity y = surface tension bQ = interatomic equilibrium bond spacing (lattice spacing) If we modify the Inglis solution (Eq. 1) by equating p = bQ, and also note that 2/j~ >> 1, we obtain " o °aa " 2°JF <6» ' o - 4 -Figure 2.1 Crack Front Coordinates - 5 -By letting the aa direction be the xx direction, then a = a = 2oHf (7) aa xx ,/b V o Using this modified form and equating a = a as a fracture 3 xx m criterion, at fracture a = ap. Hence Thus, the fracture .strength is inversely related to the square root of the length of the crack. In 1920, based on tests on precracked glass specimens, 4 Griffith concluded that "for an infinitesimally small amount of crack extension, the decrease in stored elastic energy of a cracked body under fixed grip conditions is identical to the decrease in potential energy under conditions of constant loading". Griffith showed that the driving force for crack extension was the difference between the energy which could be released if the crack was extended and that needed to create new surfaces. Using an energy-rate balance approach, he showed that aF = v/liE? (plane stress) (9) which is very similar to Eq.8, even though they were derived from different considerations. By defining the energy release rate (crack driving force) as G, and noting that this crack driving force equals the surface energy of the newly formed surface, 2y, (two new surfaces are created due to cracking), - 6 -it can be concluded that G = 2y (10) and (11) Griffith's approach provided the basis for the concept of treating fracture in terms of the change in energy remote from the immediate atomic environment of the crack tip. One drawback of his theory was that it was based on an ideally elastic brittle material and did not include the localized plastic deformation near the crack tip that occurs in most 5 6 materials. In 1952, Irwin and Kies and Orowan modxfied Griffith's theory by introducing a new parameter, y , which represents the localized plastic deformation energy dissipated at the crack tip. Thus G = 2( y + Yp ) (12) From extensive experimental and theoretical approaches to calculating this parameter G, numerous data for different 7 crack geometries and materials have been made available. Unfortunately, the modulus of elasticity of some materials, such as concrete, is difficult to evaluate. Therefore, it is desirable to combine G and E into a single parameter. 2.2 The Stress Intensity Approach 2.2.1 Stress Intensity Factor 2 Using Westergaard's solution, failure resulting from the stress field which is associated with the crack tip can be - 7 -divided into three categories. These categories are generally referred to as: Mode I tension failure (crack opening) Mode II inplane shear failure Mode III antiplane shear failure (twisting) These failure modes are shown in Figure 2.2 In fracture analysis, Mode I failure is the most important mode, and will be the only one discussed here. . 9 In 1959, by rearranging Westergaard*s solution, Irwin obtained a term, K, which depended only on the applied stress and crack length, = cr/ira" (plane stress) (13) where the subscript I refers to Mode I failure. The advantage of this parameter K, the stress intensity factor, is that it fully describes the combined effect of the applied stress and the crack length. In the general case of mixed mode fracture, applied stresses due to tension, torsion, point loading, etc., each make their own specific contributions to a, and the resultant may be calculated simply by adding the individual stress intensities. The effects of specimen shape, body configuration, and boundary conditions on can be incorporated into a geometrical function f(g), so that Eq. 13 becomes K = o/Wa f(g) (14) - 8 -Figure 2.2 The Three Different Modes of Failure (After Knott) y = a Mode I opening (displacement u in x direction.) °"xy-T r_j_ 0 IV Model sheor (displacement v in y direction.) Mode HI ontiplone shear vt ModeDH antiplane shear ( displacement win z direction.) - 9 - • 2.2.2 Effective Crack Length In metallic materials, plastic deformation of the material near the crack tip creates a plastic zone. The size of this zone, r , can be estimated by (10) 1 KI 2 * Y where a = yield strength The effective crack length, ae, must include the effect of this zone, hence ae = (a + r ) (16) In the case of a plane strain specimen, plastic deformation is more difficult at the center of thick specimens, which are more likely to cleave than to plastically deform. Irwin"'""'" estimated that the plastic zone for thick specimens is reduced by a factor of 3. Thus, for metallic materials, 1 KI 2 7 y Concrete and other cementitious materials do not deform and develop a "plastic" zone as do metallic materials. However, they do form a large zone of fine cracks around the crack tip. This zone of fine cracks will move with the crack tip as the 12 crack extends. The zone of fxne cracks has been called the 13 "pseudo-plastxc zone" , which can be considered to be equi valent to the plastic zone in metals. - 10 -2.3 Relationship Between G and K Using Westergaard1s solution and the energy principle, Irwin showed that the elastic strain energy release rate G and the stress intensity factor K can be related by and where These relationships enable the strain energy release rate to be expressed directly in terms of values of the stress intensity. Formulation of the relationship involves only the energy principle; 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 this is sufficient to cause catas trophic crack propagation depends on whether the energy released reaches a critical magnitude. This result demon strates that when the energy release rate is below some critical value, the system may undergo stable crack growth before catastrophic crack propagation occurs. The result also provides a means of understanding stable crack growth. Stable crack growth is important because in a system under load, it may cause a crack which is initially smaller than G = — (plane stress) (.18). G = (1 - v2) (plane strain) (.19) v = Poisson's ratio - 11 -the critical size for the applied stress to extend to the critical size, at which point fracture would occur. 2.4 Fracture Mechanics Applied to Fibre Reinforced Concrete It is now well established that concrete failure is due to progressive internal cracking. Failure is the result of an essentially continuous material changing to an essentially 14 discontinuous one. Richart, Brandtzaeg and Brown first found that the volume of concrete under uniaxial compressive loading initially decreased, as would be expected from elastic theory. However, when the applied load reached about two-thirds of the ultimate load, the volume of the concrete started to increase. At ultimate load, they found that the apparent volume of the concrete specimen was larger than the initial volume of the specimen. From this, they concluded that the bulging and eventual failure of the material resulted from the gradual development of internal tension-induced microcracking throughout the specimen, and this has subsequently been confirmed by many other investigators. Failure takes place when the cracks develop continuous patterns. In 1963, by introducing aligned steel fibres parallel to the tensile stress in a concrete system, Romualdi and Batson"^ found that the tensile cracking strength of the system increased in proportion to the inverse square root of the wire spacing. They reasoned that as an internal tensile crack propagates in a given material, displacements perpendi cular to the plane of the crack develop in the vicinity of the crack tip as a result of the stress singularity in that - 12 -region. The presence of a stiffening element in the vicinity of the crack opposes these displacements by means of adhesive coupling between the stiffening element and the matrix. The resulting bond forces are directed toward the crack plane and reduce the magnitude of the extensional stresses in the vicinity of the crack tip. Fracture mechanics principles were used in their work to account for the influence of fibre reinforcement on the crack resisting mechanism. Since then, numerous studies have been carried out to investigate the crack arrest mechanism. Initially, these studies involved only the application of linear-elastic fracture mechanics. However, as an increasing amount of experimental data became available, inconsistencies in the measured fracture parameters such as the critical strain energy release rate, GIC or the critical stress intensity factor (also known as fracture toughness), K.j.£ became apparent. The values of G^^ or appeared to be strongly dependent on the specimen geometry and the method of measurement. Recently, a number of investigators have begun applying elastic-plastic fracture mechanics to fibre reinforced concrete. Two of the reasons for extending the linear-elastic criteria into the elastic-plastic region are: (1) concrete itself is not a perfectly brittle material; and (2) fibre rein forcement gives the concrete more apparent ductility. The most common techniques of elastic-plastic fracture mechanics which a where subscripts I and C refer to the Mode I failure and the critical value, respectively. - 13 -have been applied to fibre reinforced concrete are: (1) the critical crack opening displacement method (COD), (2) the J-integral technique, (3) R-curve techniques and (4) the fictitious crack model. A brief description of these 16 17 techniques has been given by Mindess ' . A number of investigations have been carried out using these methods; however, the experimental data indicate some uncertainty in their application as well. While Nishioka, Yamakawa, Hirakawa 18 19 and Akihama and Brandt claim that COD and J can be applied c c 20 to fibre reinforced concrete, Halvorsen and Velazco, 21 Visalvanich and Shah have found that J and COD depend on c c " the specimen geometry. Although a limited number of experimental results support the R-curve technique, more research is required to clarify its applicability to fibre reinforced concrete. -• 14 -Chapter 3 MEASUREMENT OF FRACTURE PARAMETERS AND STABLE CRACK GROWTH 3.1 Test Specimens A number of specimen geometries have been developed to measure fracture parameters and crack propagation. The most common are 1. Edge cracked tensile specimen 2. Centre cracked tensile specimen 3. Double cantilever beam specimen 4. Double torsion specimen Edge cracked tensile specimens and centre cracked tensile specimens such as the compact tension speciman and the notched beam specimen have been adopted in ASTM Standard E561-80 for fracture testing of metallic materials. However, the double torsion and double cantilever beam specimens are more frequently used on ceramic materials to measure slow crack growth and fracture toughness. One of the reasons for their popularity is that with these specimens, the fracture toughness is independent of the crack length over a substantial range of crack growth. These specimens also allow several determinations of fracture toughness on a single specimen. Both double cantilever beam and double torsion specimens have the same initial geometry. Basically they are rectangular plates with a centre groove running the full length of the plate. However, in the double torsion technique, torsional loading is used to propagate the crack, rather than the tensile - 15 -loading of the double cantilever beam technique. In the double cantilever beam technique, crack velocity studies are usually performed using the fixed loading technique. The crack velocity is calculated by measuring the length of the crack increment and the time required for such an increment. The position of the crack is monitored optically. An equivalent method can also be used in crack velocity studies with the double torsion technique. Slow crack growth data are also obtained under a constant load, and the crack growth rate can be monitored optically. Under these conditions, both techniques should be equivalent. As an alternative, using a constant displacement or a constant displacement 22 rate to propagate the crack, Evans showed that by using compliance methods, the crack growth rate can be calculated directly from the applied load, P, or the load relaxation rate dp/dt. Both the crack velocity and the stress intensity can be determined from the load. This method therefore provides a simple way of monitoring crack growth when direct visual observation of the crack tip is impossible. 3.2 Double Torsion Technique By considering the double torsion specimen shown in Fig 3.1 as two rectangular elastic sections, Williams and 23 Evans showed that the stress intensity is a function only of the specimen dimensions, the applied load and Poisson's ratio.. The stress intensity can be expressed as - 16 -Figure 3.1 The Double Torsion Specimen KI = P"mt3 U3+ V)]H (20)  A Wt t n where = stress intensity factor P = applied load co = moment arm m v = Poisson's ratio W = width of the specimen t = thickness of the specimen t = plate thickness in the plane of the crack They confirmed that the compliance C and the crack length "a" are linearly related. The system compliance can be expressed as C = | = (Ba + c) (21) where y = deflection B = slope of the V-Kj curve c = the intercept of the V-K^ curve 23 With the compliance expression, Willxams and Evans also showed that for constant displacement or constant displacement rate, the crack growth rate can be related to the instantaneous load and the corresponding load relaxation rate dp/dt as V =--Z- = ~(Pi,f) (ai,f) (dp_) (22) v Bp2 [at' p2 {at' - 18 -where subscripts i and f represent the initial and final states, respectively. Hence, the velocity of the crack can be measured over a range of Ky values from a single experiment. - 19 -Chapter 4 EXPERIMENTAL PROCEDURE 4.1 Materials CSA Type 10 (ASTM Type 1) normal Portland cement was used to prepare the concrete. The fine aggregate was commercially available concrete sand, and the coarse aggregate was 3/8"(9.5 mm) pea gravel. All aggregate were stored at ambient laboratory moisture conditions. To improve the workability of the mixes, two types of admixtures were a b used, an air entraining agent and a water reducing agent ,. Three types of fibres were used. These were alkali-c d resistant fibreglass , straight steel fibres and deformed steel fibrese. Two types of fibreglass were used — 102 filaments per fibre bundle and 204 filaments per fibre bundle. They were chopped strand in 1.0 in.(25.4 mm) lengths. The straight steel fibres consisted of 0.50 in.(12.7 mm) and 1.0 in.(25.4 mm) long fibres with cross-sectional dimensions 0.01 x 0.022 in.(0.254 x 0.559 mm). The deformed steel fibres were 0.022 in.(0.559 mm) in diameter with hooked ends, as shown in Table 4.1. MBVR, supplied by Master Builders Co. Liquid pozzolith, type 300N, supplied by Master Builder Co. Supplied by Owens-Corning ^Supplied by Stelco, Hamilton, Ontario Supplied by Bekaert Steel Wire Corp. - 20 -TABLE 4.1 MIX DESIGNS Mix Series Fibre Volume Weight (lb) Dosage (ml.) % by Volume Cement Water Sand 3/8" Gravel Fibre Pozzolith AEA 0 85 42.5 173 53 0 100 12 0.25 173 53 1.14 " " Glass 0.5 171.5 56.7 2.27 " fibre 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 El II „ 0 67.5 24 148 148 0 58 23 0.25 25 146 .5 146.5 3.3 " Straight 0.5 26.5 144.5 144.5 6.7 " " steel 0.75 27 142.5 142.5 10 " fibre 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 " " Steel 0.75 146 .3 146.3 9.8 " " Fibre 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 torsion specimens were proportioned from smaller specimens previously used by 24 Nadeau, Mindess and Hay . All specimens were cast xn steel molds, each 48 x 16 x 2.0 inches (1219 x 406.4 x 50.8 mm). Each mold contained a 48 in. (1219 mm) long bar with a tapered cross section, 1.0 in. (25.4 mm) in depth x 0.50 in. (12.7 mm) at base x 0.25 in. (6.35 mm) at the top for molding the precast groove, as shown in Figure 4.1. A coating of oil was applied to the mold before casting so that the concrete would not adhere to the mold. Specimens were removed from the mold by simply disassembling the mold. 4.2.1 Casting of Specimens All mixes were prepared in a pan-type mixer, with the ingredients weighed on a balance accurate to 1.0 lb. The L air entraining agent was diluted with the mixing water while the workability agent was first poured into the fine aggregate and allowed to be absorbed. This procedure was intended to prevent direct chemical reaction between the workability aid and the air entraining agent. The pan was first dampened, and the coarse and fine aggregates plus two thirds of the mixing water were placed in the mixer and mixed thoroughly. The cement was then added and mixed in until it was uniformly distributed throughout the batch. The remaining water was then added. Fibre reinforcement was added by shaking the - 22 -Figure 4.1 Casting Mold ELEVATION VIEW - 23 -fibres through a sieve to prevent "balling", and the concrete was mixed for a further five minutes. After mixing, the concrete was placed into the oiled mold with a shovel and compacted using an immersion vibrator. Finally, the specimens were, finished with a trowel and covered with plastic to prevent drying. Specimens were removed from the mold after 24 hours and kept in a moist room until testing. The age of the specimens was about 3 years at the time of testing. 4.2.2 Preparation of Specimen before testing A day before the testing of any specimen, a coating of plaster of Paris was applied on the non-grooved side of the specimen. This procedure was intended to aid in observing crack growth on the tension side of the specimen during testing. Just prior to testing, the specimen was removed from the moist room and a coat of a commercial curing agent was applied on the specimen to prevent loss of moisture due to evaporation during testing. An initial crack, 4.0 in. (101.6mm) long, was cut in the centre of one end of the specimen along the groove, using a diamond saw. The crack was cut in such a way that the leading edge was on the grooved surface. A loading jig, with four clamps and four loading points, was fixed in the testing machine. Details of this jig are shown in Figure 4.2. The prepared specimen was secured to the loading jig with the precracked edge over the load points and the tension side up (Figure 4.3). The edges of the specimens were aligned parallel with the sides of the loading jig. The distances between the edges of the - 24 -Figure 4.2 Loading Jig 48 id 2i T El 12 25 135 ,n3"3" '4V ftJM . • ! TV' 11 rll-TT I I I I J_L ! —14 11 9* I9| Flexible clamps Housing-for attaching the loading jig to the loading machine PLAN VIEW ELEVATION VIEW - 25 -Figure 4.3 Test Setup specimen and the loading points were kept equal. An external load cell of 2000 lbf 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 testing machine descended the load cell acted as a rigid 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 testing setup. There are two main advantages of this loading con figuration. First, the ball-bearings were placed under the specimen instead of hanging above it. This eliminated the necessity of a fixture to hold the ball-bearings in their housing. Second, the tension surface was located on the top of the specimen. This provided the possibility of observing crack propagation with ease. A strip chart recorder, cali brated with the external load cell, was connected to the load cell 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 lb capacity mechanical loading machine - 27 -Figure 4.4 External Load Cell - 30 -by measuring the applied load and the corresponding deflection as follows. The specimen was first prepared and set on the loading jig as described in Section 4.2.2. Two dial gauges, one placed next to the load cell and one placed at the outer edge of the specimen, were used to measure deflections. The load was applied to the specimen in 200 lb increments. Readings of the dial gauges and the corresponding applied load were recorded. The deflection of the plate was obtained by subtracting the reading of the dial gauge placed on the outer edge of the specimen from the reading of the one near the load cell. After a set of compliance measurements was made, the specimen was removed from the loading apparatus. A longer crack was cut using a portable ceramic electric power saw. These procedures were then repeated with different crack lengths. 4.3.2 Double Torsion Test 2 5 26 The load relaxation method ' was used on double torsion specimens to determine the relationship between crack velocity and stress intensity. Readings of applied load and corresponding elapsed time were recorded. During testing, the cross-head of the testing machine was first lowered so that the specimen was just touching the two loading points of the load cell. Once contact was made, the two clamps on the loading side were released. Load.was applied by lowering the cross-head with a constant speed of 0.0373 in/min (0.947 - 31 -mm/min). Load relaxations were performed at every two hundred pound increment, by simply stopping the cross-head movement. When the crack started to propagate, this usually resulted in the load-elapsed time curve deviating from linearity, and the cross-head was stopped immediately. The nonlinearity of the curve signalled a change of compliance in the specimen. With the cross-head fixed, the load continued to fall slowly as the crack grew at a decreasing rate. Each point on the resulting load-time curve corresponded to a different crack velocity and stress intensity. Thus, a V-K-j. plot could be obtained from a single load-relaxation curve. Background relaxation was measured as the load-relaxation curve for the last relaxation obtained for the specimen before the load dropped, and this was subtracted from the apparent load relaxation curve. The loading procedure was resumed after measuring the first crack propagation and stopped when the crack finally propagated down the plate. Figures 4.7 to 4.9 show the crack appearance at various stages. The new crack tip was located with the aid of a lOx magnifying glass and marked on the specimen. However, this apparent crack tip was found to be misleading, due to the nature of the observed crack. Before a visible crack could be observed on the surface, microcracks visible only using very high magnification have probably already propagated - 32 -Figure 4.7 Initial Stage of Crack Propagation - 34 -beyond the visible crack. Therefore, the measured value would tend to underestimate the true crack length. Thus this procedure was discontinued. Other efforts were made to measure the true crack length by using a penetrating dye. Unfortunately, the results were less than satisfactory due to the roughness of the crack surface. The penetrating dye tended to disperse and became blurred; thus a definite crack front could not be obtained. - 36 -Chapter 5 EXPERIMENTAL RESULTS 5.1 Introduction The main object of this research was to investigate the effect of fibre reinforcement on crack growth in concrete. Crack propagation data were obtained using the load relaxation method on a double torsion specimen as described above. In all, thirty-one specimens were tested and the results are shown in Tables Bl to B31 (Appendix B). In the glass and deformed steel fibre specimens, the fibre content was the only variable. The w/c ratio for these specimens was 0.5. Both the fibre content and the w/c ratio were varied in the straight steel fibre series (see Table 4.1). 5.2 Cement Paste Specimens To obtain some indication of the validity of the test results, two specimens of cement paste (w/c = 0.4) were made and tested. The results were then compared with available data from the literature to see whether the values obtained were of the right order of magnitude. The age of the specimens at testing was.90 days. The test results are tabulated in tables 5.1 and 5.2. The data in Tables 5.1 and 5.2, and Tables Bl to B31 can be described as follows: - 37 -TABLE 5.1 Load Relaxation Data for Cement Paste No. 1 Fiber volume: 0 Load at failure: 600 lb. Load after failure: 0 Apparent Relaxation (a) Corresponding Background Relaxation (bl True Relaxation Velocity Stress Intensity load lb p load lb dp paper (in) dm time (sec) dt slope ldt/b load lb dp paper (in) dm time (sec) dt slope .ldt\a ,dp, _ .dp. ldtJ ldt* a b -2 VxlO in/sec K h ksi—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 Initial Load 330 lb. - 38 -TABLE 5.2 Load Relaxation Data for Cement Paste No. 2 Fiber volume: 0 Load at failure: 540 lb. Load after failure: 0 Apparent Relaxation (a) Corresponding Background Relaxation (b) . True Relaxation Velocity Stress Intensity load lb p. load lb dp. paper (inl . . dm . . time (sec) dt slope >dtJ load lb dp paper (in) . dm time (sec) dt slope /dp. AdtV ldt;-a b -2 VxlO in/sec K '2 ksi-in 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 Initial Load 318 lb. Column 1 applied load Columns 2-5 describe the slope of the relaxation curve at the applied load In describing the slope of the relaxation curve at the applied load, a horizontal line was drawn which intercepted the apparent relaxation curve at the applied load (Figure 5.1). The slope of the curve was obtained, by constructing .a tangent to the curve at the intercepted point. Column 2 difference in the vertical axis (y-axis) i.e. difference in load dp Column 3 difference in the horizontal axis (x-axis) i.e. difference in chart length dm Column 4 difference in time, dt where dt = -T-—§2 = 30 dm chart speed Column 5 slope of the apparent relaxation curve at the applied load. The corresponding background relaxation was obtained by reproducing the background relaxation curve of the specimen below the apparent relaxation curve (see Figure 5.1) with the initial load drop of the curves at the same x-value. A vertical line was drawn through the intercepted point of the applied load on the apparent relaxation curve which cut a point on the background relaxation curve. A tangential line was drawn to the background relaxation curve at this point. The slope of this line was the corresponding background -40-- 41 -relaxation. The slope of the background curve is described in Columns 6-9, which are similar to Columns 2-5. Column 10 true relaxation = apparent relaxation - corresponding background relaxation Column 11 crack velocity at the applied load Column 12 stress intensity at the applied load Typical load relaxation curves are shown in Figure 5.2. These curves can divided into two parts; i) positive slope region, and ii) negative slope region. In the first part, as the load increases, the slope of the curve will remain constant as long as the compliance of the specimen remains constant. When the crack starts to propagate, the slope of the curve first decreases drastically, 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 is obtained shortly before the crack starts to propagate. This background re laxation is due to the relaxation of the loading machine, and perhaps also due to creep in the specimens. The background relaxation curve is subtracted from the load relaxation curve in order to get the true load relaxation of the specimen. Typically, the curves are sufficiently different only for the first 90 seconds. A sample calculation of the crack velocity is shown in Appendix A. - 42 -1400 r Figure 5.2 Typical Load Relaxation Curve 1200 1000 800 600 400 200 Region I > the difference between the two curves is significant Region 21 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 plot for the results of the two cement paste specimens, and the average V-K^ plot, are shown in Figure 5.3. The slope of the average V-K^ curves is 37.6. With the same geometry but a smaller specimen (9.0 x 3.0 x 0.5 in. or 24 228 x 76.2 x 12.7 mm), Nadeau, Mindess and Hay found that the slope of the V-K^ curve was approximately 35. The value of the fracture toughness of the control specimen was 0.69 ksi-in2 (0.75 MN_3//2) compared to the value of 0.293 ksi-in35 (0.32 MN~3//2 24 27 obtained by Nadeau, Mindess and Hay . Wecharatana and Shah , using 32 x 6.0 x 1.5 in. (812 x 152 x 38.1 mm) double torsion specimens, found that the fracture toughness was 1.2 ksi-in2 (1.31 MN-3/2). In both cases (24,27), the w/c ratio was 0.5. The value obtained seems therefore to be within the range of values reported in the literature. 5.3 Fracture Toughness One of the questions about fibre reinforcement is its effectiveness in increasing the fracture toughness of concrete. The fracture toughness, KjC, of the specimens is tabulated in Table 5.3, and is plotted against the fibre volume in Figures 5.4 to 5.8. (A sample calculation of the fracture toughness is shown in Appendix A). In general, the fracture toughness increases with fibre volume up to about 1.25%. The stress intensity factor, K^, at the first crack (the first observed visible crack) is tabulated in Table 5.3. The value of this is approximately equal to 70% of the corresponding KjC-Figure 5.3 V-K Plot 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 SLOPE = 38.0 3 AVERAGE SL0PE= 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, KIC(KSI-INJ5) 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 VSSF 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 -WEIGHT DENSITY ( lb./ft.3) - 48 -Figure 5.6 Relationships Between Fracture Toughness, Weight Density, Residual Strength and Fibre Volume for h" SSF Series FIBRE VOLUME (%) - 49 -WEIGHT DENSITY (lb./ft.3) cn O 1 o o Z kic(ksi -in."2) ro cn H-c l-S (6 (t> (0 Ul h-1 p. 0) W ft l-i fD 0 id rt V D> 3 O 3 Ul 3" H-•O Ul CO fD rt S, fl> CD 13 cr H fD < o M c 3 0) Hi o 1-1 l-i B> o rt C M CO >-3 O c <0 3" 3 ft) ui «i cn CO cn ro n r-ro ui ro cr rt o ro 3 ui r-rt •< cn O O O O O RESIDUAL STRENGTH ( lb.) -M1 o o \ Figure 5.8 Relationships Between Fracture Toughness, VJeight Density, FIBRE VOLUME (%) - 51 -Fibre reinforced concrete is difficult to compact fully, and a poorly compacted specimen will leave voids and pores in the finished product. At the higher fibre contents, weight densities (Table 5.4) decreased due to incomplete compaction. The weight densities of the specimens are plotted against their fibre volume in Figures 5.4 to 5.8, and the degree of compaction is related directly to the weight density. The weight density curves obtained can be characterized by an inverted V. The weight density of the specimens normally increased with increasing fibre content and reached its highest value at about 1 to 1.25 per cent fibre by volume, then started to decline. In general, the shape of the fracture toughness vs fibre volume curves follows the same pattern as the weight density vs fibre volume curves. This indicates that the fracture toughness is affected by the degree of compaction of the concrete. At higher fibre contents, the trend of the fracture toughness vs fibre volume for the BSF series does not follow the same pattern as the weight density vs fibre volume curve. This disparity may be accounted for by the fact that BSF is a more efficient fibre, and the increase in fibre volume compensates for the adverse effect of the poor compaction. Thus the effect of poor compaction is less severe on the BSF series. - 52 -TABLE 5.4 WEIGHT DENSITY OF SPECIMENS Series Fibre Content Weight Weight Density % (lb) (lb/ft3) 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 VSSF 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 Residual Strength Residual strength is defined as the strength remaining after the crack has visibly extended across the entire length of the specimen. The residual strengths of the specimens are tabulated in Table 5.3, and are plotted against fibre volume together with the fracture toughness in Figures 5.4 to 5.8. The residual strength of the specimens is the combined effect of the interlocking force between aggregates and the pullout resistance of the fibre reinforcement. Zero residual strength was obtained for the two cement paste specimens. The residual strengths of the plain concrete specimens are therefore due to the interlocking of the aggregates. The pattern of these curves coincides with that of the fracture toughness curves, and the results are similarly affected by the compaction of the specimens. This is illustrated by the similar slopes of the curves. In general, as the weight density curve goes up, the fracture toughness and the residual strength curves also go up. When the weight density curves go down, so do the other two curves. The relationship between residual strength and fibre volume is shown in Figure 5.9. The difference in slopes of the residual strength vs fibre volume curves is related to the difference in pullout resistance for different types of fibres. Higher slopes generally indicate a higher pullout resistance. However, because of the scatter shown in Figure 5.9, it is difficult to assess the pullout resistance of the fibres used in this study from the available data. Figure 5.9 Relationship Between Residual Strength and Fibre 5.5 Compliance Two specimens were used to measure the system compliance. The test results are tabulated in Tables 5.5 and 5.6, and plotted in 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 is 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-KT Plot The load relaxation data are tabulated in Tables Bl to B31 (see Appendix B), and V-Kj plots on a log-log scale are shown in Figures 5.11 to 5.15. Values of the fracture tough ness and the crack velocity were calculated using Equations 20 and 22 respectively. Values of the initial and final crack lengths are 4 in. (101.6 mm) and 48 in. (1219 mm). (At failure, the crack always ran right to the end of the specimen, therefore, the final crack length is always equal to 48 in. (1219 mm)). The relationships between the crack velocity and stress intensity of the specimens are summarized in Table 5.7. They were analysed using linear regression analysis. The correlation coefficients of these regression analyses ranged from 0.80 to 0.99, and they were significant at the 5% level. Therefore, a good correlation between the crack velocity and the stress intensity factor exists. - 56 -Table 5.5' Results of Compliance Study for Specimen h" SSF Crack length (in) load (kip)xlO-1 Gauge Reading (in)xl0~J Deflection (in)xlO-3 Compliance (in/kip)xl0~3 Average Compliance (in/kip)xlO a P in 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 Results of Compliance Study for Specimen BSF 1.0 Crack length (in) load (kip)xlO-1 Gauge Reading , (in)xlO J Deflection (in)xlO-3 Compliance ^ (in/kip)xl0~J Average Compliance (in/kip)xlO-3 a P in 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 -Figure 5.10. Relationship Between System Compliance and Crack Length Figure 5.11 V-K PLOTS FOR GF102 SERIES - 60 -Figure 5.12 v-Ki PLOTS FOR GF204 SERIES - 61 -Figure 5.13 V-K PLOTS FOR SSF SERIES - 62 -Figure 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 Kx( ksi-in."2) LEGEND 2 3 4 5 O I" SSF =0.25 + » = 0.5 V »' * 1.0 A « =1.25 x " =1.5 - 63 -Figure 5.15 v-^ PLOTS FOR BSF SERIES - 64 -TABLE 5.7 Summary of Results for the V-KT Curves Mix Fibre Content Slope Y-intercept Correlation Series % n A Coefficient 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 IO-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 10-11 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 10~16 0.92 1.5 30.4 1.35 x 10_1 0.94 2.0 26.6 6.77 x 10~2 0.96 0 29.0 2.34 x 10~10 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 10~20 0.95 0.25 53.0 1.07 x 10~2 0.97 0.5 61.6 5.0 x 10"14 0.96 1"SSF 1.0 58.3 7.6 x 10"11 0.98 1.25 85.8 3.06 x 10"23 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 10~20 0.99 2 56 7.73 x lQ-11 Q.96 -65 -In Figures 5.11 to 5.15, shifting of the V-K curves to the right occurs as the fibre volume increases up to about 1.25 to 1.5%. No special pattern is observed after the fibre content increases to more than 1.5% of the total volume - probably due to unequal compaction. The slope of the V-Ky plots was greatest at fibre contents of 1.25%. - 66 -Chapter 6 GENERAL DISCUSSION The results described in the previous chapter are an attempt to evaluate the effect of fibre reinforcement on crack velocity in concrete. In the cement paste specimens, the w/c ratio was 0.4, which is lower than the 0.5 used by Nadeau, Mindess and 24 27 Hay and Wecharatana and Shah . The value of the fracture \, toughness obtained in the experiment (0.69 ksi-in2 or -3/2 0.75 MN ), was less than that obtained by Wecharatana 27 3/2 and Shah (1.2 ksi-in2 or 1.31 MN ), but, it was twice 21 . h the value obtained by Nadeau, Mindess and Hay (0.293 ksi-in -3/2 or 0.32 MN ). However, the agreement between the slope of the V-Kj curves obtained in these tests (37.6) and the 24 results of Nadeau, Mindess and Hay (35) was good. When evaluating the fracture toughness, the size of the specimen must be large enough to accommodate the subcritical crack growth, and perhaps some amount of crack growth is needed before a "valid" I<IC can be obtained. These results indicate the need to define a minimum specimen size when testing cementitious materials. The weight density of the specimen usually started to decrease when the fibre content was about 1.25% by volume. This suggests that when the fibre volume was more than 1.25%, full compaction was not achieved. Figures 5.4 to 5.8 indicated that the trend of fracture toughness curves was — 67 -similar to that of the weight density curves. The fracture toughness increased with fibre content to about 1.25%, and then decreased, due to incomplete compaction. Thus, the advantages of putting more fibre reinforcement in the specimen may be offset by the higher number of voids created due to difficulties in compaction. In the BSF series, the effect of compaction on the fracture toughness seemed to be less acute. The slope and y-intercept are the two major para meters of the V-K-j. curves shown in Figures 5.11 to 5.15. A small y-intercept indicates that subcritical crack growth is less significant at low fracture toughness values. A lower slope implies that changes in fracture toughness have a small effect on the crack velocity. Therefore, for a material which is less susceptible to subcritical crack growth, the values of the y-intercept and slope of the V-Kj plot should be small. Figures 5.11 to 5.15 indicate that by increasing fibre content up to about 1.25 to 1.5% of the total volume, the V-K^ curves generally shifted to the right, giving a smaller y-intercept. At higher fibre contents, no pattern in the position of the V-K^ curves was observed. Large slopes were generally associated with small y-intercept values. This suggests that by adding about 1.25% to 1.5% by volume of fibre reinforcement, concrete can be made less susceptible to subcritical crack growth. However, - 68 -the crack velocity is quite sensitive to changes in the fracture toughness. The fact that high fibre additions (greater than about 1.25%) do not improve the resistance to crack growth is believed to be caused by the difficulties in fully compacting the specimens. This finding is supported by the weight density results in Table 5.4. Table 5.3 showed that the residual strength of the specimens increased as the fibre content of the specimens increased. This is probably associated with the pullout resistance of the fibre reinforcement. "Failure" of the specimen occurred when the crack propagated down the full length of the specimen. Once failure occurred, the system changed from a continuous system to a discontinuous system, consisting of two separate plates held together by fibres. Due to the loading configuration, the crack will open up at failure. This crack opening can only be accommodated if the fibres at the opening surface elongate or slip within the matrix. The fibres thus hold the specimen together after failure has occurred. Therefore, by increasing the fibre content, the residual strength of the specimens can also be increased. Several load relaxations were performed on each specimen. Attempts were made to measure the crack position at the end of each relaxation. These included dye pene tration methods and direct measurement using a magnifying glass. Both methods were found to be inadequate in measuring the true crack position. Chapter 7 CONCLUSIONS From the analysis of the test results, the following conclusions can be drawn: 1. Subcritical crack growth should be considered when measuring the fracture parameters of cementitious materials. 2. A minimum specimen size should be determined in order to get valid results. 3. Different types of fibre do not significantly affect the slope and intercept of the V-Kj curves. 4. The degree of compaction affects the fracture properties of the specimens. Unless special attention is given to the compaction procedure, fibre contents greater than 1.5% of the total volume are not recommended. 5. The fracture toughness increases with fibre content to about 1.25%. 6. Residual strength of the specimen increases with increas ing fibre content. This strength seems also to depend on the pullout resistance of the fibre reinforcement. 7. In this test geometry, fibres do not significantly restrain crack growth. - 70 -BIBLIOGRAPHY 1. CE. Inglis, "Stress in a Plate due to the Presence of Cracks and Sharp Corners," Trans. Institution 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. Griffith, "The Phenomena of Rupture and Flow in 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 Brittle 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," Institute of Fracture and Solid 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, "Plastic 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 for 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 Failure of Concrete Under Coinbined Compressive Stresses," Bulletin No. 185, Engineering Experiment Station, University of Illinois (1928). 15. J.P. Romualdi and G.B. Batson, "Mechanics of Crack Arrest in Concrete," Journal of the Engineering Mechanics Division, American Society of Civil 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 Civil Engineering, University of British Columbia (1981). 18. K. Nishioka, S. Yamakawa, K, Hirakama and S. Akihama, "Test Method for 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 in 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 Failure Characteristics of Brittle Materials — and Its Application to Polycrystalline 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, Vol. 1, No. 4, pp. 264-270' (1973) . 24. J.S. Nadeau, S. Mindess and J.M. Hay, "Slow Crack Growth in Cement Paste," Journal of the American Ceramic Society, Vol. 57, No, 2, pp. 51-54 (1974)., 25. E.R. Fuller, Jr., "An Evaluation of Double Torsion Testing - Analysis," Fracture Mechanics Applied to Brittle Materials, ASTM STP 678, S.W.. Freiman, Ed,, American Society for Testing and Materials, Philadelphia, pp. 3-18 (1978). 26 28 S.M. Wiederhorn, "Subcritical Crack Growth in Ceramics," Institute for Material Research, National Bureau of Standards, Washington (1974). 27. M. Wecharatana and S.P. Shah, "Double Torsion Tests for 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, "Elastic Moduli of Hardened Portland Cement and Tricalcium Silicate Pastes.: Effect of Porosity," Symposium on Structure of Portland Cement Paste and Concrete, Highway Research Board, SR 90, Washington D.C., pp. 135-144 (1966)., - 73 -APPENDIX A Sample Calculations Using data obtained from specimen GF 102 - 0.25 fracture toughness (KI(-.) ksi-in2 use v = 0.20 after Helmuth & Turk28 K = P co (-3(1 + IC cr m Wt3tn = (1.03) (6.875 )C 3(1 + °*2-0) (15.25)(2)J(1) = 1.216 ksi-in2 where P =1.03 kips measured cr * CO m = 6.875 in W = 15.25 in t = 2 in t n = 1 in V = 0.2 Crack Velocity slope of relaxation curve = 510 lb/1 in corresponding background machine relaxation = 360 lb/1 in paper speed = 2 in/min corresponding applied load =950 lb initial crack length a. = 4 in applied load corresponding to initial crack length P^ = 950 lb P mn#/- o - / • min ->/-n#/- o • / • min - = 510 /in • 2 m/min • TTT - 360 /in • 2 in/mm • rA t ' ' 60 sec ' ' 60 sec = 5#/sec V = 9-50# * 4 in • 5#/sec = 0.021 in/sec (950*r - 74 -APPENDIX B - 75- -TABLE BI Load Relaxation Data for Glass Fiber Fiber volume: 0 Load at failure: 1150 lb Load after failure: 100 lb Ap parent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope vdt'b load lb dp paper (in) dm time (sec) dt slope (dt} a , dp. _ , dp . 1 dt; { cTt ! a b -2 VxlO in/sec K h ksi-in *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 *Initial Load TABLE B2 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 0.25% Load at failure: 1030 lb Load after failure: 120 lb Apparent Relaxation (a) Correspondin Background Relaxation ( g b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper ( in) dm time (sec) dt slope 1 at;b load lb dp paper (in) dm time (sec) dt slope 1 at'a r dP^ _ ' dP\ c aT} v at) a b -2 VxlO in/sec K h ksi-in *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 *Initial Load Initial Crack Length 4 in a Final Load Final Crack Length 48 in TABLE B3 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 0.5% Load at failure: 1040 lb Load after failure: 80 lb A ppareni b Relax ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper ( in) dm time ( sec) dt slope (£iE) Kat'b load lb dp paper (in) dm time (sec) dt slope (SiR) ldt'a Kat' a (dt)b -2 VxlO in/sec K h ksi-in *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 *Initial Load - .78 -TABLE B4 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 0.75% Load at failure: 1340 lb Load after failure: 400 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time ( sec) dt slope /dp, load lb dp paper (in) dm time C sec) dt slope ^dt;a <af>-<ai> a b -2 V x 10 in/sec K h ksi-in *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 *Initial Load TABLE B5 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 1.0% Load at failure: 1600 lb. Load after failure: 1050 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper ( in) dm time (sec) dt slope ldt'b load lb dp paper (in) dm time (sec) dt slope (dt'a ldtJ ldt', a b -2 Vx 10 in/sec K h ksi-in *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 •Initial Load - 80 -TABLE B6 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 1.25% Load at failure: 1680 lb Load after failure: 580 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope ^dt'b load lb dp paper (in) dm time ( sec) dt slope (*£) 1 dt; ^dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 81 -TABLE B7 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 1.5% Load at failure: 2000 lb Load after failure: 1420 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (^P) ldt;b load lb dp paper (in) dm time (sec) dt slope dp ldt'a vdt; ^dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load -82 -TABLE B8 Load Relaxation Data for Glass Fiber size 102 series Fiber volume: 2.01 Load at failure: 1500 lb Load after failure: 1040 lb Apparent-Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope ldtJb load lb dp paper (in) dm time ( sec) dt slope (9£) [dt> a dp_ _ dp ld-t; *dt' a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 83 -TABLE B9 Load Relaxation Data for Glass Fiber size 204 series Fiber volume: 0.25% Load at failure: 1150 lb Load after failure: 240 lb A pparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dP) vdt;b load lb dp paper (in) dm time (sec) dt slope ldt'a (^P) _ (d£) a b -2 VxlO in/sec K h ksi-in *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 •Initial Load TABLE BIO Load Relaxation Data for Glass Fiber size 204 series Fiber volume: 0.75% Load at failure: 1400 lb Load after failure: 244 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (SlE) ydt'b load lb dp paper (in) dm time (sec) dt slope . dp. _ .dp. ldt; Kdt' a b -2 V x 10 in/sec K h ksi-in *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 •Initial Load TABLE Bll Load Relaxation Data for Glass Fiber size 204 series Fiber volume: 1.25% Load at failure: 1920 lb Load after failure: 800 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (^P) ^dt; b load lb dp paper ( in} dm time (sec) dt slope {at> a <g> - c§i>b a b -2 VxlO in/sec K h ksi-in *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 * Initial Load - 86. -TABLE B12 Load Relaxation Data for Glass Fiber size 204 series Fiber volume: 1.5% Load at failure: 1660 lb Load after failure: 960 lb A pparen t Relax ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope {dt'h load lb dp paper (in) dm time (sec) dt slope -dp. (at> K dti a b -2 VxlO in/sec X h ksi-in *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 •Initial Load - 87 -TABLE B13 Load Relaxation Data for Glass Fiber size 204 series Fiber volume: 2.0% Load at failure: 1080 lb Load after failure: 780 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope vdt'b load lb dp paper (in) dm time (sec) dt slope (dp_j 'dP^ _ (dP\ 1 dt; dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load TABLE 'B14 Load Relaxation Data for Straight Steel Fiber Fiber volume: 0 Load at failure: 1860 lb Load after failure: 240 lb Apparent Relaxation (a) Corresponding Background Relaxation ( b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope ' dt;b load lb dp paper ( in) dm time (sec) dt slope, 1 dt;£ rdP} _ /dP^ ldt; ^dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load -89 .-TABLE B15 Load Relaxation Data for 1/2" Straight Steel Fiber Series Fiber volume: 0.25% Load at failure: 670 lb Load after failure: 300 lb p ipparen t Relas :ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (d£) . load lb dp paper (in) dm time (sec) dt slope (d£) <§t> - <st> a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - .90 -TABLE B16 Load Relaxation Data for 1/2" Straight Steel Fiber Series Fiber volume: 0.5% Load at failure: 1490 lb Load after failure: 7.40 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope ( ^) 1 at'b load lb dp paper (in) dm time (sec) dt slope ldtJa (dp} _ (dp} ^dt; dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load TABLE B17 Load Relaxation Data for 1/2" Straight Steel Fiber Series Fiber volume: 0.75% Load at failure: 1540 lb Load after failure: 680 lb A pparen t Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope ( ^) load lb dp paper (in) dm time C sec) dt slope ldt]a ldt; ldt;, a b -2 VxlO in/sec K h ksi-in *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 •initial Load - .92 -TABLE B18 Load Relaxation Data for 1/2" Straight Steel Fiber Series Fiber volume: 1.25% Load at failure: 780 lb Load after failure: 700 lb A pparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dP) load lb dp paper (in) dm time (sec) dt slope Kaz' a <i> - <n> a b -2 VxlO in/sec K H ksi-in *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 •Initial Load •r 93 -TABLE B19 Load Relaxation Data for 1/2" Straight Steel Fiber Series Fiber volume: 1.50% Load at failure: 1700 lb Load after failure: 96 0 lb Apparent Relaxation (a) Corresponding Background True Relaxation Velocity Stress Intensity Relaxation (b) load lb P load lb dp paper (in) dm time (sec) dt slope (£P_) load lb dp paper (in) dm time (sec) dt slope (d£) /dp, _ (dp, Kdt' ^dt; a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 94 -TABLr, B 2 0 Load Relaxation Data for l/2n Straight Steel Fiber Series Fiber volume: 2.0% Load at failure: 1980 lb Load after failure: 1290 lb A pparenl : Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper ( in) dm time ( sec) dt slope ldt;b load lb dp paper (in) dm time (sec) dt slope (dP) (dP) _ (dp (at> Kat> a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 95-TABLE B21 Load Relaxation Data for' 1" Straight Steel Fiber Series Fiber volume: 0.25% Load at failure: 920 lb Load after failure: 580 lb A pparen t Relaxation (a) Corresponding Background Relaxation ( b) True Relaxation Velocity Stress Intensity load lb P ' load lb dp paper (in) dm . time (sec) dt slope vdt;b load lb dp . paper ( in) dm time (sec) dt slope ( ^) ( dt'a ,dp, ,dp. a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - .96 -TABLE B22 Load Relaxation Data for 1" Straight Steel Fiber Series Fiber volume: 0.5% Load at failure: 1560 lb Load after failure: 540 lb Apparent Relaxation (a) [ Corresponding J Background i. Relaxation (b) I-True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope *dt;b load lb dp paper (in) dm time (sec) dt slope (dP) (dtJa <§t>- <s> a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 97 > TABLE B23 Load Relaxation Data for 1" Straight Steel Fiber Series Fiber volume: 1.0% Load at failure: 2000 lb Load after failure: 1100 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper ( in) dm time (sec) dt slope ldt'b load lb dp paper (in) dm time (sec) dt slope (d£} ldt'a (-P.) _ (dP) (d±> dt;, a b -2 VxlO in/sec K h ksi-in *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 •Initial Load TABLE B24 Load Relaxation Data for 1" Straight Steel Fiber Series Fiber volume: 1.25% Load at failure: 1900 lb Load after failure: 1180 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope Mt'b load lb dp paper (in) dm time • (sec) dt slope ( dP^ _/dP\ at; Kdt' a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 99 -TABLE B25 Load Relaxation Data for 1" Straight Steel Fiber Series Fiber volume: 1.5% Load at failure: 1040 lb Load after failure: 660 lb Al ^parent : Relax< ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope .dp Mt'b load lb dp paper (in) dm time (sec) dt slope (dp. ldt;a (dp" dp. a b -2 VxlO in/sec K h ksi-in *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 •Initial Load a Final Load - 100 -TABLE B26 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 0 Load at failure: 1170 lb Load after failure: 120 lb A pparen t Relax ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb "P load lb dp paper (in) dm time (sec) dt slope (d£) load lb dp paper (in) dm time (sec) dt slope (dp) ^dt; *dt', a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 101 -TABLE B27 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 0.25% Load at failure: 1400 lb Load after failure: 600 lb A] pparen t Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dP) ^dt;b load lb dp paper (in) dm time (sec) dt slope ldt'a ldt; ldt' a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - 102 •:-TABLE B28 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 0.5% Load at failure: 1180 lb Load after failure: 820 lb Dparen t Relax ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope /dpv (apb load lb dp paper (in) dm time (sec) dt slope /dp, (dT}a (dP) _ (dp.) a b -2 VxlO in/sec K h ksi-in *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 •Initial Load -.103 -TABLE B29 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 0.75% Load at failure: 1420 lb Load after failure: 1100 lb A pparen t Relax ation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dt'b load lb dp paper (in) dm time (sec) dt slope (dP) [dt'a a b -2 VxlO in/sec K h ksi-in *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 •Initial Load - .104 -TABLE B30 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 1.25% Load at failure: 1660 lb Load after failure: .13 00 lb Apparent Relaxation (a) Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dP) (dt'b load lb dp paper (in) dm time (sec) dt slope (dPy (dt'a (dP) _ (dp} a b -2 VxlO in/sec K h ksi-in *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 ^Initial Load - .105 -TABLE B31 Load Relaxation Data for Bent Steel Fiber Series Fiber volume: 2.0% Load at failure: 2000 lb Load after failure: 16 80 lb Apparent Relaxation (a) " Corresponding Background Relaxation (b) True Relaxation Velocity Stress Intensity load lb P load lb dp paper (in) dm time (sec) dt slope (dP) ^dt;b load lb dp paper (in) dm time (sec) dt slope (dp} ldt'a dp _ (dp} (at' at' a b -2 VxlO in/sec K h ksi-in *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 * Initial Load 

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