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Ill’cl».¢uul€.lllsa£ll )IICR‘II.I0tI..Ql|ItV.IlI.-r‘ tin?! .u‘.. 1". n. a I L! .\‘IQ ’n v “(0’!‘ tail: ,u".lv"llcflbovln . ‘ ‘. .I ’3‘“! {if It’- .u\l(.¢'.n vtl ‘ A v . I -7 v I. ..--J:0II. ul 41“.-.- 9‘: : . I (- .t. .1 ...!»o.. v :1 .l.!ic.nllnr05|1 r V. .Iia...’ . 1A 5 . L -. Ji r6"? ___—_-__ r-o - . - .— IINIHHIIIHII‘ I’ ll ||H||"|| || 3 1293 00574 4523 This is to certify that the dissertation entitled "Constitutive Modeling And Flexural Analysis of Steel Fiber Reinforced Concrete for Structural Applications." presented by Mr. Cha-Don Lee has been accepted towards fulfillment of the requirements for Ph-O- degreein i {“‘l Engimufaj "’ .D - _. Major Professor \ Date Jan. 2,1770 MSU i: an Affirmative Action/Equal Opportunity Institution 0- 12771 PLACE N RETURN BOX to remove this checkout from your record. TO AVOID FINES return on at bdore due due. DATE DUE DATE DUE DATE DUE APR 2 5 8‘32“ ' CONSTITUTIVE MODELING AND FLEXURAL ANALYSIS OF STEEL FIBER REINFORCED CONCRETE FOR STRUCTURAL APPLICATIONS By Cha-Don Lee A DISSERTATION Submitted to Michigan State University in partial fulfillment. of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Environmental Engineering 1990 LO ABSTRACT CONSTITUTIVE MODELING AND FLEXURAL ANALYSIS OF STEEL FIBER REINFORCED CONCRETE FOR STRUCTURAL APPLICATIONS by Cha-Don Lee The objective of this study was to develop techniques for predicting the efbcts of steel fibers on the tensile, compressive and fiexural behavior of concrete. A refined concept ("interaction concept") was proposed for predicting the tensile strength of SFRC. This concept accounts for the contributions of fibers through their pull-out action and also by arresting the growth of microcracks in cementitious matrices. A constitutive model was also developed for predicting the complete tensile stress-strain relationship of SFRC. The modeling of pOSt- pcak performance takes into account the contributions of fibers crossing the critical section through their pull-out action as well as that of matrix in its post-peak softening range of behavior. A simple and practical model was also developed for predicting the compres- sive constitutive behavior of steel fiber reinforced concrete. The model accounts for the efbcts of fiber volume fraction, aspect ratio and type (straight vs. hooked), and the matrix compressive strength, on the compressive behavior of SFRC. A flexural analysis procedure, with some simplifying assumptions made to simulate the fiexural behavior in the vicinity of the cracked section, was developed which gives due consideration to the behavior at and near the critical (cracked) section. The tensile and compressive constitutive models of SFRC developed in this study were used in this fiexural analysis procedure. Analytical studies were conducted using the developed fiexural analysis procedure in order to derive relationships between the tensile constitutive behavior and the fiexural strength of SFRC. The complexity and inStability associated with testing cementitious materi- als in direct tension have led to extensive use of flexural testing for assessing the tensile behavior of SFRC. It is thus important to analyze fiexural test results in order to derive information regarding tensile performance of SFRC. For this purpose, a "System Identification" approach was ad0pted in this investigation. The "System Identification" technique was used together with the developed fiexural analysis procedure in. order to derive information on the tensile behavior of SFRC using flexural test results. This technique was successful in obtaining optimum sets of parameters which provide satisfactory matches between the measured and predicted flexural load-deflection relationships. The tensile characteristics of SFRC obtained from analysis of fiexural test results by "Sys- tem Identification" were superior to those obtained from direct tension test results. This phenomenon was attributed to the positive effects of strain gra- dient existing in SFRC under flexural loads. To Y oung-Soon and Uihwan iii ACKNOWLEDGMENT I wish to express my sincere appreciation to those who have contributed to the development of this dissertation by their academic guidance and by encouragement. In particular, I would like to thank my academic advisor Dr. Parviz Soroushian, Associate Professor of Civil and Environmental Engineering, for his constant help throughout this research. Thanks are also extended to the other members of the thesis committee; Dr. W. Bradley, Dr. N. Altiero, and Dr. G. Ludden for their interest and their comments during this research. The financial support for this project was provided by the Research Excel- lence Fund of the State of Michigan, the DEED Research Program of the Ameri- can Public Power Association, and the GB Faculty Development Program in Computer Aided Engineering. These contributions are gratefully acknowledged. iv TABLE OF CONTENTS LIST OF TABLES ................................................................................................................. LIST OF FIGURES ................................................................................................................ Chapter 1: INTRODUCTION ............................................................................................... Chapter 2: MECHANICAL PROPERTIES OF STEEL FIBER REINFORCED Chapter 3: CONCRETE: A REVIEW OF THE LITERATURE ................. 2.1 INTRODUCTION ........................................................................................... 2.2 STEEL FIBER REINFORCED CONCRETE UNDER TENSION ................... 2.2.1 Microcrackin g in Mortar and Concrete under Tension ............................ 2.2.2 Microcrackin g in Steel Fiber Reinforced Concrete under Tension ............ 2.2.3 Pull-Out Mechanism of Steel Fibers in SFRC under Tension .................. 2.2.4 Tensile Behavior of Steel Fiber Reinforced Concrete ............................... 2.2.5 Constitutive Models ............................................................................... 2.3 STEEL FIBER REINFORCED CONCRETES UNDER COMPRESSION ............................................................................................. 2.3.1 Microcracking in Mortar and Concrete under Compression ..................... 2.3.2 Compressive Behavior of Steel Fiber Reinforced Concrete ....................... 2.3.3 Analytical Modeling ............................................................................... 2. 4 STEEL FIBER REINFORCED CONCRETE UNDER FLEXURE .................. 2.4.1 Flexural Properties of SFRC ................................................................... 2. 4. 2 Analysis of Steel Fiber Reinforced Concrete Beams under Flexure ........... THE CONSTITUTIVE MODEL FOR STEEL FIBER REINFORCED CONCRETE UNDER TENSION ...................................................................... 3.1 INTRODUCTION ........................................................................................... 3.2 EFFECTS OF FIBER REINFORCEMENT VARIABLES ON TENSILE BEHAVIOR OF STEEL FIBER REINFORCED CONCRETE ...................... 3.3 DEVELOPMENT OF EXPRESSIONS FOR THE NUMBER OF STEEL FIBERS CROSSING UNIT CROSS-SECTIONAL AREA IN SFRC ............... 3.3.1 Development of Theoretical Expressions ................................................. 3.3.2 Experimental Assessment of the Orientation Factor ................................ 3.3.3 Theoretical Values vs. Experimental Measurements ................................ 3.4 PREDICTION OF TENSILE STRENGTH: "INTERACTION CONCEPT" .................................................................................................... 3.4.1 "Interaction Concept" ................................. _. ........................................... 3.4.2 Empirical Coeffcicnts ............................................................................. viii ix qub 10 l3 l6 19 27 27 3O 33 35 36 38 4o 40 41 44 45 55 6O Chapter 4: Chapter 5: Chapter 6: 3.4.3 "Interaction Concept" vs. Composite Material and Spacing Concept ................................................... 3. 5 PREoPEAK CONSTITUTIVE MODELING .................................................... 3.6 POST-PEAK CONSTITUTIVE MODELING ................................................. 3.7 COMPARISON WITH TEST RESULTS ........................................................ 3.8 SUMMARY AND CONCLUSION - - ........................................ THE CONSTITUTIVE MODEL FOR STEEL FIBER REINFORCED CONCRETE UNDER COMPRESSION ............................................................. 4.1 INTRODUCTION ................................................ 4. 2 EFFECTS OF VARIABLES ON COMPRESSIVE BEHAVIOR ...................... 4.3 EXPERIMENTAL RESULTS ......................................................................... 4.4 THE CONSTITUTIVE MODELS FOR STEEL FIBER REINFORCED CONCRETE .................................................................................................. 4.5 COMPARISON WITH TEST RESULTS AND PARAMETRIC STUDIES ....................................................................................................... 4.6 SUMMARY AND CONCLUSION ................................................................... FLEXURAL ANALYSIS OF STEEL FIBER REINFORCED CONCRETE ........................................................................................................ 5.1 INTRODUCTION ........................................................................................... 5.2 FLEXURAL ANALYSIS ................................................................................. 5.2.1 Curvature Distributions .......................................................................... 5.2.2 Analysis of the Critical Section ............................................................... 5.2.3 Nonlinear F lexural Analysis of the Critical (Cracked) Section ................. 5.2.4 Comparison with Test Results ................................................................ 5.3 FLEXURAL BEHAVIOR OF CRITICAL (CRACKED) SFRC SECTION ....................................................................................................... 5.4 A PARAMETRIC STUDY OF SFRC BEAM FLEXURAL BEHAVIOR .................................................................................................... 5.5 SUMMARY AND CONCLUSION ................................................................... INTERPRETATION OF FLEXURAL TEST RESULTS USING "SYSTEM IDENTIFICATION " ............................................................................ 6.1 INTRODUCTION ........................................................................................... 6.2 "SYSTEM IDENTIFICATION" ...................................................................... 6.3 DEFINITION OF ERROR FUNCTION .......................................................... 6.4 METHOD OF OPTIMIZATION ..................................................................... 6.5 SELECTION OF PARAMETERS IN "SYSTEM IDENTIFICATION" ............ 6.6 RESULTS OF "SYSTEM IDENTIFICATION" ............................................... 6.7 SUMMARY AND CONCLUSION ................................................................... vi 72 74 86 89 92 92 93 96 100 106 113 115 115 117 117 121 126 129 132 136 153 157 157 158 161 162 173 174 184 Chapter 7: SUMMARY AND CONCLUSION ........................................................................ 188 LIST OF REFERENCES ....................................................................................................... 199 vii LIST OF TAB LES 2.1 Application of Steel Fiber Reinforced Concrete ................................................................. 5 3.1 Mean Values of Fiber Orientation at Different Location on Cross Section and for Diffirent Fiber Types ............................................................................... 58 3.2 Direct Tensile Test Specimen and Results ......................................................................... 67 3.3 Average, Standard Deviations and Sum Total of Squares of Normalized Errors for Difiarent Concepts ........................................................ 71 3.4 Pull-Out Load-Deflection Relationship for Straight-Round Steel Fibers ............................................................................................. 77 3.5 Conditions of Some Direct Tension Test Results of SFRC ................................................ 86 4.1 Fiber Reinforced Concrete Mixtures .................................................................................. 98 5.1 Flexural Test Conditions .................................................................................................. 130 5.2 Effects of Material-Related Factors ................................................................................... 140 5.3 EfEcts of Constitutive Behavior-Related Factors .............................................................. 140 5.4 15st of Material-Related Factors (2* Factorial Design) .................................................. 152 5.5 13st of Material-Related Factors in Order ..................................................................... 152 6.1 Test Conditions and Optimized Values from System Identification .................................................................................................................... 175 6.2 Comparison of the Tension Test Results with the Optimum Values of Parameters in Analysis of Flexural Test Results Using System Identificat ion ........................................................................................................ 176 6.3 Derived Tensile Strength of Matrices and Composites in Direct Tension and System Identification ..................................................................... 183 viii LIST OF FIGURES 2.1 Propagation of Cracks in Cementitous Matrix - .. .......................................... 8 2.2 Tortuosity of Crack Path ................................... 9 2.3 Branching of Microcracks inside the Cement Paste and Shattering of Grains by Crossing Microcracks - _ ............................................. 9 2. 4 Interaction of Microcracks with Steel Fibers - -- ................................................ 10 2. 5 Continuous Nature of Shifted Microcracks around a Steel Fiber ....................................... 11 2.6 Schematic Description of the Microstructure of Steel Fiber-Cement Interface .................... 12 2.7 Debonding and Pseudo-Debonding of Steel Fibers in Cementitious Materials .................... 12 2.8 Interaction of a Crack with a Number of Randomly Oriented Steel Fibers ........................ 13 2.9 A Typical Relationship between Average Bond Stress and Pull-Out Deflection in Pull-Out Tests on Straight-Round Steel Fibers ............................................ 15 2.10 Effect of Fiber Types on Pull-Out Behavior .................................................................... 15 2.11 Deviation from Linear Behavior in SFRC. Resulting from Gradual Microcrack Propagation .................................................................................................................... 16 2.12 Typical Direct Tensile Stress-Strain Relationships for Concrete and Mortar Matrices ......................................................................................................................... 17 2.13 Two Theoretical Types of SFRC Direa Tensile Behavior in the Post-Peak Region ......... 18 2.14 Fiber Reinforced Composite ............................................................................................ 20 2.15 Verification of the Composite Material Concept .............................................................. 24 2.16 Verification of the Spacing Concept ................................................................................ 26 2.17 Localization of Microcracks ............................................................................................. 28 2.18 Microcracks in the Post-Peak Region .............................................................................. 29 2.19 Typical Compressive Stress-Strain Curves for Plain Concrete and Reinforced with 1% Volume Fraction of Steel Fibers ....................................................................... 30 2.20 Efbcts of Fiber Reinforced and Confinement by Transverse Steel on Compressive Behavior of Concrete .................................................................................... 31 2.21 Typical Comparison between Compressive Behavior of Fibrous and Equivalent Confined Concretes .......................................................................................................... 32 2.22 The Compressive Constitutive Model of Reference 5 ....................................................... 33 2.23 Analytical Model of Reference 5 vs. Test ........................................................................ 34 2.24 Flexural Behavior of Steel Fiber Reinforced Concrete Beam under Flexure ...................... 37 2.25 One Major Crack at Critical Seetion in Steel Fiber Reinforced Concrete Beam ................ 37 2.26 Hypothetical Stress-Strain Distributions ......................................................................... 38 3.1 15st of Fiber Volume Fraction on Tensile Behavior of Steel Fiber Reinforced Concrete ........................................................................................ 41 3.2 Influence of Fiber Aspect Ratio on Direct Tensile Behavior of Steel Fiber Reinforced Concrete .................................................................................... 42 3.3 Efiects of Fiber Diameter on Direct Tensile Strength of SFRC .......................................... 43 ix 3.4 Efficts of Steel Fiber Deformation on Direct Tensile Behavior of SFRC ............................................................................................................. 44 3.5 Number of Fibers per Unit Area ....................................................................................... 46 3.6 Orientation of Steel Fibers in Concrete ............................................................................ 47 3.7 Three Dimensional Fiber Orientation ................................................................................ 48 3.8 Two Dimensional Orientation of Fibers ............................................................................ 49 3.9 Diffirent Conditions with Two Boundaries ....................................................................... 51 3.10 Efbct of Height on Orientation Factor in Cases with Two Boudaries ....................................................................................................................... 51 3.11 Effects of Cross-Sectional Dimensions on Orientation Factor in Cases with Four Boundaries ........................................................................................ 53 3.12 Diffirent Conditions with Four Boundaries ..................................................................... 54 3.13 Comparisons between Exact and Approximate Expressions of Orientation Factor in Different Conditions ..................................................................... 55 3.14 Measurement of the Number of Fibers per Unit Area ...................................................... 57 3.15 Mmsured vs. Normal Distribution .................................................................................. 59 3.16 2-D and 3-D Orientation Factors for Diffirent Geometric Conditions ..................................................................................................................... 62 3.17 Influences of Fibers on Craking Characteristics and Tensile Behavior of Concrete .......................................................................................... 63 3.18 Comparisons of the Proposed "Interaction Concept" with ' Tensile Strength Tea Results .......................................................................................... 68 3.19 Comparisons of the Composites Material and the Spacing Concept Predictions with Tensile Strength Test Results .................................................. 70 3.20 Simulation of the Steel Fiber Reinforced Concrete Tensile Behavior upon Cracking at Peak Load ................................................................ 73 3.21 Increase in Strain at Peak-Tensile Stres in the Presence of Steel Fibers as a Function of Some Fiber Reinforcement Properties (N ,-d, -l[) ...................................................................... 73 3.22 Comparisons of the Average Stress vs. Average Crack Width Relationship in the Post-Peak Region with Tea Results ................................................. 76 3.23 Typical Experimental Plot of Fiber Pull-Out vs. Displacement (Slippage) Relationship .................................................................................................... 76 3.24 Model of Pull-Out Load-Deflection Relationship for Straight-Round Steel Fibers ............................................................................................ 77 3.25 Pull-Out Test Procedure ................................................................................................. 78 3.26 Simulation of Pre-Peak Pull-Out Behavior of Straight- Round Steel Fibers .......................................................................................................... 79 3.27 Comparison of the Experimental Pull-Out Load-Deflection Relationships in the Post-Peak Region with the Empirically Derived Model for This Study ...................................................................... 81 3.28 Effias of Fiber Concentration, Volume FraCtion and Orientation on Pull-Out Strength .................................................................................... 82 3.29 Comparisons of Experimental Tensile Stress-Deformation Relationships with Predictions of the Model Developed in This Study ................................................................................................. 87 4.1 Efbcts of DifErent Fibers on Compressive Behavior of SFRC ........................................... 94 4.2 Experimental Compressive Stress-Strain Relationship ....................................................... 99 4.3 General Form of the Compressive Constitutive Model of SFRC ........................................ 101 4.4 Comparison between Empirical Expresions on Test Results for Diffirent Variables of Compressive Constitutive Model .................................................................................. 104 4.5 Analytical Model vs. Tea (1 Mpa =- 145 psi) ................................................................... 106 4.6 thcts of Fiber Reinforcement Index and Concrete Compressive Strength on Compressive Behavior of SFRC as Predicted by the Proposed Constitutive Model (1 Mpa = 145 psi) ................................................................................................ 111 5.1 Crack Patterns and Possible Curvaure Distributions ........................................................ 116 5.2 Moment and Curvature Distributions .............................................................................. 118 5.3 Actural and Assumed Crack Shapes .................................................................................. 122 5.4 Deformation Compatibility after Cracking ....................................................................... 123 5.5 Estimation of the Initial Strain Configuration .................................................................. 127 5.6 Improvements of Regula-Falsi Method .............................................................................. 128 5.7 Comparisons between Experimental and Analytical Deflection Curves ............................................................................................................. 131 5.8 Typical Load-Deflection Curves (VI: 0.5% and 1.2%) ...................................................... 133 5.9 Strain and Stress Distributions at the Critical Section ...................................................... 134 5.10 Material-Related and Constitutive Behavior-Related Factors .......................................... 137 5.11 Definitions of Four Difiarent Criteria .............................................................................. 141 5.12 Influence of Matrix Strength on Load-Deflection Curve ................................................... 144 5.13 Influence of Matrix Softening in Tension on Load- Deflection Curve .............................................................................................................. 145 5.14 Influences of Fiber Dimensions and Fiber Volume Fraction ............................................. 147 5.15 Influence of Fiber Pull-Out Behavior ............................................................................... 149 6.1 Error Surface in Parameter Space (Case of N = 2) ........................................................... 159 6.2 The Iterative Minimization Procedure ............................................................................... 163 6.3 Main Theorems Used in Powell’s Algorithm ..................................................................... 168 6.4 Main Theorems in Powell‘s Algorithm for N = 2 ............................................................. 170 6.5 Quadratic Line Search ...................................................................................................... 172 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Flexural Load- Deflection Curves ............................................................................................................. 177 xi CHAPTER 1 INTRODUCTION Reinforcement of concrete with short, randomly distributed steel fibers leads to improvements in the tensile strength, and tensile and compressive ductility of the material [1-6].* Improvements in the stiffiiess, cracking characteristics, strength and toughness of reinforced concrete structural elements in the presence of steel fibers under fiexural, shear, torsional and axial forces are direct conse- quences of the improvements in the tensile and compressive performance of the material [7-9]. The advantages associated with the use of steel fibers in load-bearing struc- tural elements can be realized in large scales only if structural engineers are pro- vided with Structural design equations and guidelines for optimizing the use of Steel fibers in conjunCtion with conventional reinforcing bars in structural ele- ments. The very basic tools required for analytical studies on fiber reinforced concrete structural elements are reliable constitutive models of fibrous concrete which have been verified using comprehensive sets of experimental results. In many applications Steel fiber reinforced concrete is subjected to fiexural forces. It is thus important to develop analytical techniques to predicting the fiexural behavior of steel fiber reinforced concrete which is marked by nonlinear stress distribution and dominance of a cracked seetion in deciding the post-peak perfor- 11131106. " Numbers in square brackets refer to the list of references. 2 The objective of this research was to: (1) develop tensile and compressive constitutive models for steel fiber reinforced concrete which reflect the current level of our understanding of the physics of the material behavior under dif’tErent stress systems and are refined using the available experimental data; and (2) develop analytical techniques for flexural analysis of steel fiber reinforced con- crete, and for deriving information on the direct tensile and compressive perfor- mance characteristics of steel fiber reinforced concrete based on the flexural test data. A comprehensive review of the literature on mechanical properties and con- stitutive modeling of steel fiber reinforced concrete under compression and ten- sion is presented in Chapter 2. This chapter also critically reviews some popu- lar concepts used in predicting the tensile strength of steel fiber reinforced con- crete, and presents the background and analysis procedures for steel fiber rein- forced concrete behavior under flexure. The development of an experimental model for predicting the constitutive behavior of steel fiber reinforced concrete under tension is described in Chapter 3. In Chapter 4, an empirical compressive constitutive model is presented for compressive behavior of steel fiber reinforced concrete. In both Chapters 3 and 4, parametric Studies are conduaed using the developed constitutive models in order to assess the performance characteristics of steel fiber reinforced concrete. Chapter 5 describes the process of incorporating the developed tensile and compressive constitutive models into an approximate nonlinear flexural analysis procedure which takes into account the formation of one major crack at the flexural section and the nonlinear distributions of stresses and curvatures in the vicinity of this crack. The system identification technique is used in Chapter 6 together with the developed fiexural analysis procedure and tensile/ compressive constitutive models in order to identify characteristic tensile and compressive values of the stress- strain characteristics of steel fiber reinforced concrete using the flexural load- deflection relationship. Finally, Chapter 7 summarizes the research program and presents the. conclustions; suggestions are also made for future research in this area. CHAPTER 2 MECHANICAL PROPERTIES OF STEEL FIBER REINFORCED CONCRETE: A REVIEW OF THE LITERATURE 2.1 INTRODUCTION Stress system produced in concrete by external loads (compression, tension, fiexure or multi-axial) lead to a tendency towards the propagation and intercon- nection of microcracks in cementitious materials [10-13]. The ease with which microcracks can propagate in concrete results in a brittle failure which is gen- erally considered to be a major shortcoming of cementitious materials. Reinforcement of concrete with short, randomly distributed steel fibers results in improvements in tensile strength and tensile and compressive tough- ness of the material. This is due to the fact that propagating microcracks in cementitious matrices tend to be arrested or deflected [1,5,10,11,14] by fibers. Debonding and pull-out actions of fibers under tension and the confinement of cementitious matrices by steel fibers under compression are also important mechanisms through which steel fibers improve the mechanical behavior of con- crete [1,5,10,11,14]. Improvements in the fiexural performance of concrete in the presence of steel fibers are direct consequences of the corresponding improve— ments in the tensile and compressive performance of the material. The desirable mechanical properties of steel fiber reinforced concrete have encouraged the use of steel fiber reinforced concrete in wide ranges of non- structural and structural applications [15] (see Table 2.1 ). Table 2.1 Application of Steel Fiber Reinforced Concrete EXPERIMENTAL APPLICATIONS APPLICATIONS EXPERIMENTS IHIGHWAY CONSTR. a REPAIRJ [AIRPORT RUNWAYS, TAXIWAYS, APRONS] L [LNQUSTRIAL FLOORS] f , BAN CONST. a REPAIRJ § [REFRACTORY CASTAIBLES I 3" [BRIOGE OECRI CONSTR. a REPAIR] f .. FLUID CONTAINMENT 3 fiNAL. RESERVOIR eiNINp] [ENE a TUNNEL LININ_G] ~ [STRUCTURAL BLDG. ELEMENTS] SECURITY VAULTS [EAiSSON a PILE REPAIR] [, RAILROAO TIES J r PIPE J r [S—UBTERRANMN VAULTSJ I; L MODULAR ,lPKNELS j E FREFRACTORY PRECIFASTJ E [STRUCTURAL PRECAST j FBREAKwATERs’ F CRIB BLOCK WACNINE BASES a PRANESJ l I 6 In order to design or analyze the reinforced concrete structural elements incorporating steel fibers, it is important to understand and to be able to predict the Stress-Strain and load-deformation properties of the material under tensile, compressive and flexural loads. The remainder Of this chapter presents a comprehensive review of the literature on the failure mechanisms of plain and steel fiber reinforced concrete under compression, tension and flcxure. The avail- able tensile and compressive constitutive models and flexural analysis procedures developed for steel fiber reinforced concrete are also critically reviewed. 2.2 STEEL FIBER REINFORCED CONCRETE UNDER TENSION The pre—peak tensile behavior of SFRC may be characterized by the process of microcracking of the matrix prior to the formation of a continuous crack sys- tem across the section which marks the peak load and the appearance of a mac- rocrack. These propagating microcracks tend to be arrested by fibers leading to increased stiffness and peak tensile strength of steel fiber reinforced concrete. The peak tensile load is typically marked by the appearance of one major crack at the critical section, after which the pull-out of fibers bridging the criti- cal crack tends to dominate the behavior in the post-peak region. The fiber pull-out process generally provides the composite material with important post- peak ductility and toughness. In the following discussion, first some observations regarding the nature of microcracking in mortar and concrete are presented and the interaction mechan- isms Of fibers with microcracks are described. The fiber pull-out process and the tensile constitutive behavior of steel fiber reinforwd concrete are also discussed. 2.2.1 Microcracking in Mortar and Concrete under Tension For air dried mortar and concrete, shrinkage-induced bond cracks around large aggregate particles appear prior to any loading (Figure. 2.1(a)). In con- crete, multiple cracking around sand grains is frequently observed (Figure. 2.1(b)), and this phenomenon seems to be more pronounced between adjacent sand grains than around isolated ones. In mortar, under tensile loads, microcracks tend to propagate along the seg- ments of cement-sand grain interface and also around the air voids (Figure 2.1(c)). It can be shown that cracks under tension change orientation when encountering aggregates under tensile stress in order to pass around aggregates without crossing (see the microcrack in Figure 2.1(d) encountering aggregate "A"). This suggests that aggregate surfaces may act as crack arrestors, causing microcracks to stOp prior to reaching aggregate surfaces. In the presence of aggregates, the crack path is thus never straight (see Figure 2.2 for mortar). The overalltortuosity of the crack pattern in concrete is higher than that in mor- tar, because concrete cracks must propagate around the densely spaced aggregate pieces as well as sand grains. Other phenomena sometimes Observed in the microcracking process include the branching of microcracks inside the paste, and shattering of aggregate parti- cles crossed by microcracks. Branching (Figure 2.3(a)) occurs at the crack tip, and usually only one of these branches is activated and increases in width with further loading. In some cases a crack is observed to run through, rather than around, an aggregate grain (Figure 2.3(b)). This might cause the aggregate grain to shatter. The reorientation, branching and multiple cracking associated with the interaction of microcracks with aggregate particles lead to the dissipation of large amounts of energy, which is benefitial to tensile behavior of the material. (a) Shrinkage Induced Bond Cracks (b) Multiple Cracking (c) Crack Propagation (d) Reorientation of Cracks Figure 2.1 Propagation of Cracks in Cementitious Matrix Figure 2.2 Tortuosity of Crack Path A ggregate 7/ (a) Microcrack Branching (b) Aggregate Shattering Figure 2.3 Branching of Microcracks inside the Cement Paste and Shattering of Grains by Crossing Microcracks 10 2.2.2 Microcracking in Steel Fiber Reinforced Concrete Under Ten- sron. . The propagation mechanism of microcracks tends to be influnced by the presence of fibers. Cracks approaching fibers in a direction almost parallel to them tend to run parallel to such fibers for at least some distance along the length (Figure 2.4(a)) and those cracks advancing in a direction inclined with respect to steel fibers are either shifted (Figure 2.4(b), which occurs in 30% of events in steel fibers) or branched out into multiple post-fiber cracks (Figure 2.4(c), observed in 50% of events in steel fiber). The microcrack encountering a fiber stays continous, making the lateral shifts around the fibers, as can be clearly seen in the picture of the groove under a steel fiber which intersected microcracks in Figure 2.5. (a) Parallel Running (b) Shifting (e) Branching Figure 2.4 Interaction of Microcracks with Steel Fibers. ll $ . Groove -;. ‘. 3:“ '0‘ 3(- Hu, ”('5‘ 0..., WW” .' .:. ‘X‘. . '59:... W“ e ”Mme! .A Figure 2.5 Continuous Nature of Shifted Microcracks around a Steel Fiber. The intersection of microcracks with steel fibers is strongly influenced by, the nature of fiber-matrix interfacial zone [11]. This zone in steel fiber rein- forced concrete consists Of 3 regions (Figure 2.6): (l) a thin duplex film in actual contact with steel fibers; (2) outside this, 10 to 30 micrometer-thick porous region incorporating massive calcium hydroxide crystals; and (3) outside this, a highly porous layer parallel to the interface; pseudo-debonding may occur in this very porous region due to the tensile stresses in a direction parallel to the crack generated near the crack tip. ‘ The microcrack propagation in the vicinity of fibers (at the fiber-matrix interface) might take place at the interface itself leading to the separation of the matrix from the fiber (debonding, Figure 2.7(a)), or it might occur (as discussed earlier, see Figure 2.4(c)) at a small distance (about 20 micro-meter) from the fiber and parallel to it (pseudo-debonding, Figure 2.7(b)) [11]. The nature of psuedo-debonding (Figure 2.7(b)) leads to branching and lateral shifting of the advancing crack, and sometimes causes some true debonding between places 12 6 O ‘0 o ~0‘0..Oe . o.‘po Do .0 Porous Layer Bulk Paste Figure 2.6 Schematic Description of the Microstructure of Steel Fiber-Cement Interface [11] Steel Fiber Steel Fiber (a) Debonding Debondin g and Pseudo-debonding Of Steel Fibers in Cementitious Materials [1 l l. (b) P seudo—debondin g Figure 2.7 l3 Figure 2.8 Interaction of a Crack with a Number of Randomly Oriented Steel Fibers [11]. where parallel secondary cracks run past the fiber. The propagation of a micro. crack when encountering a number of randomly oriented fibers might take place with a mixture of fiber-matrix interaction types (see Figure 2.4 for these types), as shown in Figure 2.8. The branching, shifting and parallel running of microcracks involve dissipa- tion of extra energy from the stressed system and illustrate an important mechanism through which fibers enhance the pre-peak stiffness and the ultimate tensile Strength of fiber reinforced concrete. 2.2.3 Pull-Out Mechanism of Steel Fibers in SFRC Under Tension The pre-peak behavior and maximum tensile strength of the composite material depends on local bond characteristics at the fiber-matrix interface, while the post-peak behavior is dominated by an average bond behavior in pull-out action of fibers bridging the critical crack. Thus, pull-out tests on individual 14 fibers embedded in. concrete matrix seem to provide information which are more relevant to average bond behavior in the post-peak tensile behavior of steel fiber reinforced concrete [16]. Fiber-matrix interfacial bond strength is provided by a combination of adhesion, friction and mechanical interlocking [17]. Fiber debonding from the matrix at early stages of loading in the pre-peak region is resisted by the adhe- sion of matrix to fibers. Following debonding, the frictional stress transfer between fiber and matrix and mechanical bonding tend to dominate the pull-out performance and the corresponding energy dissipation which characterize the post-peak behavior of the composite. In the post-peak region under tensile stresses, following the appearance of macrocracks, the resistance to pull-out is provided in fibers aligned in the tensile stress direction primarily by shear stresses along the interfaces. For inclined fibers, the progressive bending of suc- cessive sections Of the fiber will require an additional efbrt which depends on the rigidity and yielding properties of fibers. Inclined fibers may also produce a normal stress component on part of the sliding surface of embedded fibers thus slightly increasing the frictional resistance [17]. It is worth mentioning that an excessive increase in interfacial bond Strength may actually cause fiber rupture (instead Of fiber pull-out) to dominate the failure of the compoiste material, leading to a drop in toughness (which benefits from the energy dissipated by the fiber pull-out process). A typical relationship between average bond stress and pull-out deforma- tion Obtained from pull-out tests on straight-round steel fibers is presented in Figure 2.9 [18-21]. The pull-out behavior is observed to be linear before the peak pull-out load is reached. In the post-peak region the fibers are observed to gradually slip out at decreasing pull-out loads until the completion of fiber pull- out. Mechanical deformations of steel fibers can modify the pull-out 15 2.0 ' 5.0 10.0 15.0 Figure 2.9 A Typical Relationship between Average Bond Stress and Pull-Out 1231elflection in Pull-Out Tests on Straight-Round Steel Fibers [l8- Pull-Out Lmd (N) 150 1" Crimped Fiber 100 '- Hoobd Fiber Straight-Round Fiber Slip (mm) 1 4 8 12 16 Figure 2.10 Effect of Fiber Types on Pull-Out Behavior 16 load—deformation relationships (Figure 2.10). 2.2.4 Tensile Behavior Of Steel Fiber Reinforced Concrete Fibers, with their microcrack arresting action, tend to increase the fracture energy and consequently the tensile strength of concrete. With the progress of microcracking process in SFRC, deviation from the linear behavior occurs in ten- sile stress-strain relationship of the material (see Figure 2.11) [22,23]. The deviation from linear behavior under tension in SFRC takes place at about 80% of the tensile strength [24]. The increase in the tensile strength of the matrix resulting largely from the microcrack-arrest action of steel fibers at a typical volume fraction of 1 to 2% is usually about 20 to 50% [24,25]. The peak load under direct tensile stress in SFRC seems to be reached when a catastrophic microcrack propagation takes place and a continuous syStem of microcracks forms at a critical cross section. Reference 24, based on microscOpic Tensile Stress Tensrle' Strain J Figure 2.11 Deviation from Linear Behavior in FRC, Resulting from Gradual Microcrack Propagation. 17 4 Mpa 1 l—- , a ‘ I I ’ I, ’I ’ ’ average strain over gage length 83 mm I, I. L/ I l l 2.. .’ t .r HHHHT , . l ‘\ O \ ’l . A . I . , average strain computed from gage A I . _, F” i I I I . . . . N A: strain gage outsrde the critical region ‘ ‘ H I ‘ ‘ I; I I l I I I 200 400 600 Figure 2.12 Typical Direct Tensile Stress-Strain Relationships for Concrete and Mortar Matrices. measurements, has concluded that immediately after peak load a single crack becomes visible in steel fiber reinforwd concrete specimens under direct tension. Thereafter, tensile deformations tend to be localized in the cracked location, and unloading takes place outside the cracked region (Figure 2.11). Increasing defor- mations at this stage result in gradual pull-out or rupture of fibers crossing the crack. The crack opening under tension is resisted at this stage by fibers bridging the crack and also by the remainder of the tensile resistance of the matrix at the crack in its softening zone of behavior. Fiber pull-out mechanism in post-peak stage provides steel fiber reinforced concrete with greatly increased ductility com- pared to plain concrete under tension. A typical direct tensile stress- deformation relationship for concrete matrix, which demonstrates the post- cracking tensile resistance and softening behavior Of cementitious matrices is 18 shown in Figure 2.12 [10]. In spite of matrix contributions to tensile resistance at cracks, either fiber pull-out or rupture Of fibers tends to dominate the post-cracking failure mechan- ism Of SFRC under direct tension, depending on fiber length and fiber-matrix interfacial bond characteristics. The longer fibers with better bond to cementi- tious matrices tend to have higher pull-out forces, and thus they rupture before pulling out. Theoretically Speaking, following the appearance of macrocrack and the activization of the pull-out performance of fibers, two types of behavior might be Observed: the tensile resistance might continue to increase with increas- ing tensile deformations, or it might progressively drop following a sudden drop at the peak load (see Figure 2.13). The first case (increasing resistance after VI > Critical Value Tensile Stress VI < Critical Value Average Strain (a) V, 2 VG", (b)V; < Verit. Figure 2.13 Two Theoretical Types Of SFRC Direct Tensile Behavior in the Post-Peak Region [26]. - 19 peak) can take place if the fiber volume fraction is above a critical volume needed for maintaining the tensile resistance of the composite material after cracking bythe pull out action of fibers. This critical fiber volume fracion depends on the geometry, aspect ratio, orientation, and tensile strength of fibers in addition to the fiber-matrix interfacial bond characteristics. For concretcs reinforced with steel fibers in a 3-D random manner and con- structed with conventional mixing techniques, the volume fractionof fibers that can be incorporated into concrete within constraints of sufficient workability and fiber dispersability is normally less than the critical volume fraction. Hence, a sudden dropping of tensile resistance at peak tensile stress followed by gradual softening in the post peak region tends to dominate the direct tensile behavior of SFRC [26]. The tensile resistance is expected to reach zero when the last fiber is completely pulled out of the matrix, and this takes place at relatively large crack widths comparable to half the fiber length. Direct tensile behavior of steel fiber reinforced concrete has been Observed to be influenced by fiber volume fraction, aspect ratio (length over diameter) and fiber shape (mechanical deformation) among other factors. A detailed discus- sion on the efbcts of these factors on the tensile behavior of steel fiber reinforced concrete is presented in Chapter 3. 2.2.5 Constitutive Models There are two popular concepts for analytical simulation of the failure mechanism in fiber reinforced concrete under direct tension: composite material concept and spacing concept. The composite material concept attributes the increase in tensile strength of concrete resulting from steel fiber reinforcement to the mobilization of the fiber- to—matrix interfacial bond resistance through the pull-out action of steel fibers at 20 Failure PI... Fall" Pl“. 1 . l _ _ _ _ .L —. ‘— ~ 1 _____ a j: -9 o— 1“.-- ’ \‘K / : — — — — I ‘\ II _____ i ‘ _. ‘— I “ I, | \\ -. _. _ _ _ _L j —9 g.— t ‘ "’"' l —D d—- ...... ..:\ / -. i I " (a) Continous, Aligned (b) Uniformly Dispersed, Randomly Orineted Figure 2.14 Fiber Reinforced Composite peak tensile stress [1,3,27-29]. The composite material concept was originally developed for matrices reinforced with aligned, continuous fibers with perfect bond to the matrix (Figure 2.14(a)), assuming that the Poisson’s ratios of fibers and matrix are similar [28,30-33]: cc = om'-(1- v,)+o, ovf (2.1) where : om ' = matrix tensile Strength ; - fiber tensile Stress at composite failure ;and L9 I fiber volume fraction. < K... II 21 In the case of- matrices reinformd with short randomly oriented fibers (Fig- ure 2.l4(b)), the composite material concept Should be refined to account for: (1) the randomness Of the orientation and location of fibers with respect to the failure plane which tends to reduce their efficiency in providing resistance against the applied tensile stresses; (2) the failure plane crossing short fibers at random locations along the length, leaving less than half of the fiber length in one side of the crack to resist pull-out forces; and (3) partial cffictiveness of matrix in con- Stributing to the tensile resistance at cracks. These factors result in the follow- in g expression for predicting the tensile strength of steel fiber reinforced concrete: 0. = nt-om't 1 - V, )+ Turbo/v, (2.2) where : 111 = the fraction of matrix tensile strength efl’ective at the composite peak tensile stress = 1.0 [3]; 112 = orientation efficiency factor 0.41 for fibers randomly oriented in space [9,12,16]; T13 = fiber location factor = 0.5 [3]; of' = smaller of the fiber fracture and pull —out strengths = Z'Tu'If/df S of“ ; ‘tu = average f iber—to-matrix interfacial bond stress at peak pull-out resistance ; - fiber tensile strength ; a9 fl I 22 fiber length ; and If d, fiber diameter . In the above expression (Equation (2.2)), the values of 111 and I are found empirically through normalizing expression of Equation (2.2) with respect to 0‘t: n Om'°( 1 " VI ) — = 10 + 2112-1131,, (2.3) Figure 2.15(a) shows how the above expression fits the test results presented by Mangat (1976) [3] with m equal to 1.1 and 1:, equal to l Mpa (145 psi) for concrete reinforced with straight-round steel fibers. A closer analysis of Figure 2.15(a) indicates that this figure represents the strong dependence of the composite material tensile Strength on the matrix ten- sile strength rather than fiber pull-out strength. Contrary to the assumption of the composite material concept that a relatively large fraction of fiber pull-out strength should be mobilized at the composite material peak stress, the measured values of strain and crack width at peak tensile stress in steel fiber reinforced concrete are not sufficiently large to mobilize the pull-out action of fibers [28]. The discrepancy becomes clear when the increase in tensile strength resulting from the presence of fibers is directly related to fiber reinforcement properties: cc - nl-Gm'(1-V,)= an-rb-rqu-lf/d, (2.4) Assuming a value of 1.1 for Th. 1 Mpa (145 psi) for I“ and 0.34 for ( 2412-113 ) ( derived from Figure 2.15(a) for the same set of data ), considerable 23 discrepancies between test results and analytical predictions can be observed in Figure 2.15(b) compared to Figure 2.15(a). This leads to the conclustion that composite material concept (Equation (2.2)) can not satisfactorily describe the trends observed in the tensile strength of cement composites'reinforced with steel fibers. The spacing concept for predicting the tensile strength of steel fiber rein- forced concrete is, on the other hand, based on the assumption that the dom- inant factor deciding the efbctiveness of fibers in contributing to the tensile strength of concrete is the number of fibers available in unit volume of the com- posite to disrupt the propagation of microcracks. Fiber count has been represented in the literature [34,35] by different measures of the average spacing of fibers, or by the number of fibers at unit cross sectional area. Once the parameter representing fiber spacing is defined, test results are used in this so called "fiber spacing" approach to derive the empirical relationship between the tensile strength of fiber concrete and this parameter. Romualdi and Mandel (1964) [34] have derived an expression for average fiber spacing based on the assumption that the projectiles of ramdomly distri- buted fibers in the direction parallel to that of tensile stress decide the effictiveness of fibers in increasing the tensile strength of concrete. The expres- sion for fiber spacing derived in this reference is an average of the spacings of projectiles in a plane normal to the tensile stress direction : s = 13.8-d,/\j fo100 (2.5) An alternative approach based on the spacing concept has been introduced by Soroushian and Lee (1989) [36]. This approach suggests that the fibers in any orientation with respect to tensile stress can play the microcrack-arresting ”c/(Vftlf/df) a.-1.1w..-(1—Vt) 24 4 o -teet ( Ref. 2) q -- 0./(Vfolf/df)v~1.loa. .(1-vr/(Vt-tt/dt))+o..34o145. o r - . r. 0 556 r rtobo ' Y r Frisco «run-(1 -Vf)/(Vf*|f/df) (a) Conventional Verification [3] 150. 3 O tos't(Ref.2) o 125.: -- ”c-lJeavnOU-‘/l)-O.34¢145¢erlf/df 1004 753 3 504 25-; . 1 01 1 o o '25? o -50‘ r r r , , . ‘ . . I . ° 25 5° 75 100 0.34:145thlf/df (b) New Verification of the Composite Material Concept Figure 2.15 Verification of the Composite Material Concept 25 role, and the number of fibers ( N 1 ) per unit cross-sectional area in the compo- site (irrespective of their orientation) is the factor representing the efflectiveness of fibers in increasing the tensile strength of concrete : N1 = aton/Af (2.6) where: a = orientation factor (depends on the section geometry and fiber length as described in the next Chapter) and fiber cross -sectional area b s, ll 2 Figure 2.16(a) and 2.16(b) present typical relationships between tensile strength test results (the same ones used in Figure 2.15) and different measures related to fiber spacing. The correlations between tensile strength test results and diffirent measures related to fiber spacing in Figures 2.16(a) and 2.16(b) show that the spacing concept has deficiencies in representing the failure condi- tions of steel fiber reinforced concrete at peak tensile stress. One factor leading to these deficiencies is the disregard of the spacing concept for the effects of fiber-to—matrix interfacial bond characteristics on the composite material perfor- mance. It can be concluded from the above discussion that a new approach to the prediction of SFRC tensile strength is nwded. This approach should take into account contribution of fibers to tensile strength of the composite through both microcrack arrest and pull-out action. Steps taken towards this goal in this investigation are described in the next Chapter. ac/am 26 1.6 .4 0 test ( Ref. 2) a “" ”c/U... -l.+0.0112080e(—1,45) 1.4- . 1 .1 1.2-» 1 1 1.0- .l 1 008 T T I r‘ I I r I r l’ I I— r I 1 0.00 0.10 0.20 0.30 0.40 S (a) Fiber Spacing (Equation (2.5)) 1.6 j 0 test ( Ref. 2 ) —- ac/am =1.+0.00627¢N1u(0.7922) L; N1 (b) Fiber Spacing (Equation (2.6)) Figure 2.16 Verification of the Spacing Concept. 27 2.3 STEEL FIBER REINFORCED CONCRETE UNDER COMPRES- SION In compression, fibers improve the post-peak ductility, energy absorption capacity and, to some extent, the strength of concrete [37]. These improve- ments result from the arrest of microcrack propagation by fibers as well as the confinement efbcts of fibers in the cementitious matrix. This section presents the nature of microcracking in plain concrete, the role of steel fibers in confining concrete matrices, and the experimentally observed performance characteristics of steel fiber reinforced concrete under compression. No literature is available on the microcracking behavior of steel fiber reinforced concrete under compression. 2.3.1 Microcracking in Mortar and Concrete under Compression The internal stresses at aggregate-matrix interfaces resulting from external compressive loading generally consist of components normal to the aggregate (compressive or tensile) and those acting parallel to the aggregate (i.e. in shear) [13,38]. Bond cracks at aggregate-matrix interfaces and their propagation, which lead to the nonlinear behavior of concrete in compression, are thus caused by either tensile or shear stresses [38]. Under increasing uniaxial compressive Stresses, the extent of microcracking at stresses below 85% of peak stress in the pre-peak region is limited primarily to cracks at the interface between the coarse aggregates and mortar matrix [13.38]. At about 85% of ultimate compressive load, the bond microcracks begin to increase substantially. This marks the increase in Poisson’s ratio and the devia- tion of stress-volumetric strain relationship from linearity. Mortar cracks tend 28 to bridge between. the nearby bond cracks and the appearance of continuous microcracks indicates that the so-called critical load is reached, where significant nonlinearities tend to occur and the volume of compressed concrete starts to increase rather than continue to decrease. As some load paths become inopera- tive due to microcracking under compressive stress, alternative load paths (either entirely through mortar or partly through mortar and partly through aggre~ gates) continue to be available for carrying increasing compressive loads [38]. The decrease in the number of load paths available would eventually bring con- crete to the post-peak region of behavior when compressive resistance decreases with extensive continuous microcracking [38]. Just prior to peak and immediately after it, a localimtion of microcracks seems to take place (Figure 2.17) [39]. This stage is also distinguished by for- mation of continuous microcracks (in which mortar cracks interconnect the bond Figure 2.17 Localization of Microcracks Figure 2.18 Microcracks in the Post-Peak Region microcracks). This involves sharp increase in mortar cracks at or near the peak stress. In the post peak region, microcracks are continuous, uniform and exten- sive in all directions (Figure 2.18) [39]. The continuous microcracks tend to be roughly in the loading direction, particularly if the transverse friction forces at the ends (where external compressive stresses are applied) are completely removed [39]. A contribution to the ultimate strength might be provided also by mechani- cal interlocking of the coarse aggregates after cracking. Factors like this might influence the trends in the effects of coarse aggregates on concrete compressive behavior. The idea that microcracks in concrete under compression are the major cause of nonlinearity has been challenged lately [40]. The nonlinearity of concrete appears to be highly dependent on the nonlinear softening reponse 30 characteristics of cement paste and mortar (due to submicrocracking) rather than bond cracks at the coarse aggregate-matrix interface which penetrate into mortar. More studies are needed to fully understand the nature of failure in concrete materials under compression. 2.3.2 Compressive Behavior of Steel Fiber Reinforced Concrete Short, randomly distributed fibers, when added to concrete, confine the material and delay the crack propagation, thus producing increases in the peak strength, strain at the peak stress, ductility and energy absorption capacity (Fig- ure 2.19) [2.5.14]. The strain at peak compressive stress tends to increase in the the presence of steel fibers [5]. The confinement effect of steel fibers in concrete is shown in Figure 2.20 [41] which compares the effects of increasing the fiber reinforcement index (Figure 2.20(a)) with the efficts of increasing the confinement of non-fibrous concrete by transverse reinforcement (Figure 2.20(b)). Soroushian and Lee (1987) [41], using Coup. Stres 30 (Slang) F V‘ ' 12 Platn Concrete Steel Ether 60 n 60 20 6 8 10 12 16 Conpreeetve Strata (‘10-3) Figure 2.19 Typical Compressive Stress-Strain Curves for Plain Concrete and Reinforced with 1% Volume Fraction of Steel Fibers[33] 31 2.0 ,- 2.0 Strength Into ,3)» {Strength lute V; - 2: tits ’1.” 7 1.5.1. [11(ij 1.5 h- p. o 0.021 1.0 l- h.’. . 7 l. ' 0.016 h;/l - 9.3 _ p. - 0.017 0.5 .- h'ls -‘ 3.5 Plate 1 o l l 1 1 1 6 l 12 3 16 20 6 8 12 16 20 Cmpreestve Ssretn (X10- ) Coepreutve Strata (810-3) (a) Fiber Reinforcement (b) Transverse Reinforcement Figure 2.20 Efficts of Fiber Reinforced and Confinement by Transverse Steel on Compressive Behavior of Concrete the empirical constitutive models presented in Reference 5 for fibrous concrete and in Reference 42 for confined concrete, showed that the improvements in duc- tility and energy absorption capacity resulting from the increase in fiber rein- forcement index are comparable to those resulting from the increased confinement of non-fibrous concrete by transverse reinforcement. Figure 2.21 [41] shows typical comparison between the compressive constitutive relationships of the fibrous and equivalent confined concretes. The behavior of steel fiber reinforced concrete is dependent on the volume fraction and aspect ratio of steel fibers, mechanical deformation of fibers, matrix mix proportions and maximum aggregate size, specimen geometry, and loading versus casting direction [3,5]. The efbcts of these variables on the compressive behavior of steel fiber reinforced concrete will be discussed in detail in 32 50.01 A o a. 2 V m in .13 U) 1 1 -- Confined(Re-0.005. h',’e-0.75) Strain — FibroueCJbI/d-SO.) 0.0 I r I f v r I I t F] 0.0 00 0.004 0.008 0.012 0.01 6 0.020 (a) Steel Fiber 60.0- . A o O. 2 V m e) .93 U) 1 -- Confined(Re-0.01. h'/e-3.5) . Strain -— fibrousgnyI/d-ZOO.) 0.0 f r m r v r f f r fl 0.000 0.004 0.008 0.012 0.01 6 0.020 (b) Steel Fiber Figure 2.21 Typical Comparison between Compressive Behavior of Fibrous and Equivalent Confined Concretes. 33 conjunction with discussions on compressive constitutive modeling of steel fiber reinforced concrete in Chapter 4. 2.3.3 Analytical Modeling Very few analytical Studies on the compressive constitutive behavior of steel fiber reinforced concrete have been reported in the literature. Reference 5 has presented a compressive stress-Strain diagram for Steel fiber reinforced mortar, the details of which are Shown in Figure 2.22. This model consists of two curvi- linear portions, one for the pre-peak and the other for the post-peak regions. The constant coefficients in the curvilinear equations have been derived using some characteristic stress and strain values as the boundary conditions. These characteristic values have been expressed, using experimental stress-Strain rela- tionships, as functions of the fiber reinforcement index and the compressive strength of plain mortar. ‘— m f=f'.- “(MP/”Blimp” f =1. - A2'W)+Bz(fl$l)z H C2'("‘r/)+ 021915,)” All Characteristic Parameters are Functlona of VfIf/df. ‘r/ 5 Figure 2.22 The Compressive Constitutive Model of Reference 5 34 The model of Reference 5 has been based on an experimental strain meas- urement technique which has possibly led to Strain values (e.g., at peak Stress) greater than the typical values reported in the literature. Hence, although this model compares reasonably well with the test results based on which it has been developed (Figure 2.23(a), it can not successquy predict the experimental compressive Stress-Strain relationships reported in the literature (Figure 2.23(b)). It should also be emphasized that the model of Reference 5 has been developed for Steel fiber reinforced mortar, which behaves diffirently from Steel fiber rein- ‘ forced concrete 7° .. — Experimental ‘_ - -—— Analytical h- ’ \ / x \ ‘\ ‘ ‘\ - \ “‘ A \ s‘ '1 ' 3‘ g .. \ ‘~-‘ um - as o \ ‘s I \ ‘s l ' ‘ ‘ \ x I \x 5 ~ ~- : '- \‘ If,“ D ‘7 i s 1.. Vf - 0‘ y- 0 l l l l I 1 l l l 1 l l l l L 5 10 15 melee man (30.001) (a) Test Data from Reference 5 Figure 2.23 Analytical Model of Reference 5 vs. Test 35 Comp. Stress (Mpa) 50 L Experimental Ji- “ ‘ " " Analytical ;' PC I w Mpa V, '11/41 3 0.94 llltllllLllllLJ 0 s 10 15 Compressive Strain (x 10") (b) Test Data from Reference 70 Figure 2.23 Analytical Model of Reference 5 vs. Test(cont’d) 2.4 STEEL FIBER REINFORCED CONCRETE UNDER FLEXURE An important advantage of using Steel fibers in concrete is related to the improvements in fiexural behavior, which result directly from the improvements in the tensile and compressive behavior of Steel fiber reinforced concrete. Mechanisms determining the improved fiexural behavior of steel fiber reinforced concrete are, however, more complex than those responsible for improvements in the tensile and compressive behavior of the material. This partly illustrates why the improvements in flexural behavior of concrete resulting from steel fiber reinforcement are more pronounced than those in tensile and compressive 36 behavior. The experimentally observed behavior of Steel fiber reinforced concrete and flexural analysis procedures applied to the material will be discussed in the fol- lowing sections. The main advantage of the use of Steel fiber reinforced concrete is its high performance related to its fiexural behavior. This benefit is the direct out- come of improvements in its mechanical properties of tension and compression. Mechanisms behind the fiexural behavior of Steel fiber reinforced concrete, how- ever, are quite diffirent from that of plain concrete or conventionally reinforced concrete. This section will describe experimentally observed behavior of steel fiber reinforced concrete beam and present reviews on some of the approaches to analyze it. 2.4.1 Flexural Properties of SFRC The improvements of SFRC performance in compression and in tension result in significantly higher improvements in the flexural strength and ductility of SFRC [43-45]. A typical comparison between the fiexural load-deflection relationships of plain and Steel fiber reinforced concrete is presented in Figure 2.24. SFRC exhibits an obvious deviation from linear load—deflection behavior prior to the peak fiexural load. This point of deviation from linearity has been called the first-crack load (PC, in Figure 2.24). Beyond the first cracking, the flexural load continues to increase at a lower Stiffiiess due to the formation and propagation of a macrocrack at the critical section in matrix until the ultimate load (Pu) is reached. In plain concrete once the deflection corresponding to the ultimate fiexural load of plain concrete is exceeded, failure is brittle and the post-peak load-deflection curve shows a Sharp descending behavior. Steel fiber reinforced concrete, on the Other hand, is able to sustain a considerable fraction 37 Po LOGO I I I l I :Ploin concrete I I L Deformation Figure 2.24 Flexural Behavior of Steel Fiber Reinforced Concrete Beam under Flexure P/2 pf; Figure 2.25 One Major Crack at Critical Section in Steel Fiber Reinforced Con- crete Beam 38 of its flexural resistance ever at deflections considerably larger than those corresponding to the peak load. In the post-peak region, only one major crack occurs at a critical section in SFRC (see Figure 2.25). This can imply that curvature tends to become con- centrated at this critical section. The critical section may be subject to severe distortions and plane sections may no longer remain plane after bending at this location. 2.4.2 Analysis of Steel Fiber Reinforced Concrete Beams under Flexure Limited analytical Studies have been reported in the literature on predicting the flexural behavior of Steel fiber reinforced concrete. Some investigators [46-50] have assumed hypothetical stress-Strain and Strain distributions across the criti- cal section at the ultimate condition (see Figure 2.26) in orde' to compute -. f. 2”t"t'u"sls/ds Figure 2.26 Hypothetical Stress-Strain Distributions [46-50] 39 flexural Strength by considering equilibrium conditions at the critical section. Very limited number of attempts have been made to simulate complete flexural load-deformation relationship of SFRC [45,51]. Studies in this area have been typically based on conventional beam theory, which assumes that plane sections normal to the beam axis remain plane after bending. An overall flexural analysis of SFRC beams, however, Should account for the opening of a crack at the critical section, whee fiexural deformation tends to be concentrated. The nonlinearities occuring in the vicinity of the crack Should also be taken into 300011111. CHAPTER 3 THE CONSTITUTIVE MODEL FOR STEEL FIBER REINFORCED CONCRETE UNDER TENSION 3.1 INTRODUCTION Direct tensile tests on SFRC have Shown that reinforcement of concrete with short, randomly distributed steel fibers leads to improvements in tensile strength and tensile ductility of the cementitious material. These improvements can be attributed to the microcrack-arresting and pull-out actions of fibers [1,5,10,11,14]. 30th of these actions of Steel fibers in concrete tend to be depen— dent on the number of fibers crossing unit cross-sectional area in concrete [35]. Consequently, accurate expressions for computing the number of fibers in unit area are required for the modeling of SFRC under tension. In spite of the Significances of the tensile behavior in structural applications of SFRC, very few tensile constitutive models have been developed [24,26,52]. In this chapter, an empirical constitutive model for SFRC under tension is pro- posed, which reflects our understnding of the physics of SFRC tensile behavior and also takes advantage of the tensile test results reported by different investi- gators. The developed model accounts for both of the microcrack-arresting and pull-out actions of steel fibers in concrete. 41 3.2 EFFECTS OF FIBER REINFORCEMENT VARIABLES ON TEN- SILE BEHAVIOR OF STEEL FIBER REINFORCED CONCRETE Direct tensile behavior of steel fiber reinforced concrete has been observed to depend on the fiber volume fraction, aspect ratio and deformation type. The increase in fiber volume fraction up to a certain limit (beyond which problems with workability and fiber dispersability Start to dominate the behavior) tends to increase the direct tensile strength (Figure 3.1(a)), Strain at peak tensile stress (Figure 3.1(b)) and post-peak energy absorption capacity (Figure 3.1(c)) of steel fiber reinforced concrete. At a specified fiber volume fraCtion, the increase in fiber aspect ratio (defined as the ratio of the fiber length to its diameter) also increases the direct tensile strength (Figure 3.2(a)). strain at peak stress (Figure 3.2(b)) and also enegy absorption capacity of steel fiber reinforced concrete [2,24,28,52,53]. Tensile Strength Increase (‘72) Peak Strain Increase (913) 40 40 circular, straight average of all types of fiber in shape df=0.4 mm ' [f/d{=53 l/d=69 _c:s:w=1:3:0.5 '- _/ 1 L l l 2.0 (95) - 2.0 (96) (a) Tensile Strength (b) Strain at Peak Tensile Stress Figure 3.1 Efficts of Fiber Volume Fraction on Tensile Behavior of Steel Fiber Reinforced Concrete [52] 42 Toughuss Increase (9'6) 80 circular, straight df-OA mu: 0 ~10 - O O O . I J J 2 (95) (c) Energy Absorption Capacity Figure 3.1 Effects of Fiber Volume Fraction on Tensile Behavior of Steel Fiber Reinforced Concrete [52] (cont’d) Strength Increase (95) 5W” [New (95) 40 ~10 vf=l.2% vf=l.2% df=0.25 mm (if-20.25 mm ”c:s:w=l:3:0.5 - c:s:w=l:3:0.5 2° ' 20 " _ —- e ‘1 N! 1 L I o . 0 50 100 so 100 (a) Direct Tensile Strength (b) Strain at Pmk Stress Figure 3.2. Influence of Fiber Aspect Ratio on Direct Tensile Behavior of Steel Fiber Reinforced Concrete [52]. 43 Fibers with high aspect ratios are, however, more difficult to disperse in concrete and have more pronounwd adverse efficts on fresh mix workability. Hence, at each fiber volume fraction, there is a limit on aspect ratio beyond which the problems with fresh mix workability and fiber disperability tend to decide the tensile behavior of SFRC. Some limited test results [25] have also indicated that the direct tensile strength of steel fiber reinforwd concrete increases with decreasing fiber diameter at a constant aspect ratio (Figure 3.3). Steel fibers are generally mechanically deformed for achieving a better mechanical bonding to the matrix. Reference 4 has reported experimental ten- sile Stress-strain relationships for steel fiber reinforced concrete incorporating straight-round, hooked and paddled fibers. The results presented in Figure 3.4 are indicative of some differences in the overall tensile stress-Strain relationship Direct Tensile Strength (Mpa) 3.0 Vt=r.8% ---- Vt=r.2% —°“' Vt=o.6% Fiber Diameter (mm) 1.0 0.2 0.35 0.5 Figure 3.3 Effect of Fiber Diameter on Direct Tensile Strength of SFRC [25]. 1.5 .__ straight, round: dI=0.38 mm, ”=25 mm, Vf=l.73% ----- hooked: df=0.40 mm, ”=30 mm. Vf=1-73% - - paddled: df=0.75 mm. ”=50 mm. W=1-7395 c:s:w=l:2.5:0.35 or 0.45 1.0 f n /-"'\._ ,. .° 0! l Tensile Stress divided by Tensile Strength Average Strain (x0001) I .1 . . 4 8 A 12 Figure 3.4 Effects of Steel Fiber Deformations On Direct Tensile Behavior of SFRC [4]. of steel fiber reinforced concrete, incorporating fibers with diffirent mechanical deformations. 3.3 DEVELOPMENT OF EXPRESSIONS FOR THE NUMBER OF STEEL FIBERS CROSSING UNIT CROSS-SECTIONAL AREA IN SFRC Both microcrack-arresting and pull-out action of fibers in concrete which contribute to the improvements in tensile strength and tensile ductility of Steel fiber reinforced concrete are dependent on the number of fibers crossing unit cross-sectional area in concrete [1,5,10,14,35]. In the following discussion, 45 theoretical expressions are derived for the numbe of fibers per unit cross sec- tional area in fiberireinforced concrete, with due consideration given to the efbcts of the surrounding boundaries. Measurements are made on the number of fibers per unit cross-sectional area in steel fiber reinforced concrete specimens incor- porating various volume fractions of fibers of diffirent types. Based on Statisti- cal evaluation of the measured values, the differences in fibe' concentration at different locations on the cross section are assessed. The effects of vibration on reorientation of steel fibers in concrete are investigated through comparisons between the computed and measured values of number of fibers per unit cross- sectional area. 3.3.1 Development of Theoretical Expressions The average number of fibers per unit area may be considered as the total number of fibers times the possibility of one randomly located fiber crossing the unit area. This probability can be computed using a so-called fiber orientation factor (a), which is basically the average ratio, for all possible fiber orientations, of the projected fiber length in the tensile Stress direction (for cross-sections nor- mal to the tensile Stress direction) to the fiber length itself. Given the orientation factor as a, the average number of fibers per unit area can be obtained as follows: N 1 = p N (3.1) V _ a _f_ A/ where : p = possibility of one fiber crossing unit cross sectional area 46 _alf _lbf in Figure 3.5 ; and N = total number of fibers V’ b h 1 ' F' 3 5 = In Igure . . Ar If It can be Shown from Equation (3.1) that finding the orientation factor with reasonable accuracy under different geometric conditions is important in the development of tensile constitutive models for SFRC. The value of orientation factor is afficted by diffirent factors: (a) boun- daries of the Specimen restricting the orientation of fibers (see Figure 3.6); (b) vibration during concrete construction which may cause reorientation of fibers in Unit Area / Random Plane 4. T I Figure 3.5 Number of Fibers per Unit Area 47 Y I H’— \ / \ / x __\_ :/\“\ / / / z \ \ \ (a) Three-Dimensional (b) Two Boundaries RAB z >$z> . 454/ L (c) Four Boundaries Figure 3.6 Orientation of Steel Fibers in Concrete 48 horizontal planes; (c) fiber types; and (d) location in cross section (top vs. bot- tom) with respect to the casting and vibration directions. Results of analytical and experimental investigations of these factors are presented in this section. Steel fibe's when uniformly dispersed in an infinitely large volume of con- crete, are expected to be randomly orieited, with equal probabilities of being oriented in difirent directions in Space. The orientation factor in this condition (a representing a in Equation (3.1)) can thus be expressed as follows (see Figure 3.7, where projectile is taken along the z-direction); 1:12 1:12 In in If cos0cos¢d0d¢ a0 = (3.2) (m 2ft, = 0.405 Projectile Figure 3.7 Three Dimensional Fiber Orientation 49 Application of steel fibe' reinforced concrete to thin product (e.g., panels) may practically restrict fibers to a two-dimensional distribution. In a pure 2-D distribution (Figure 3.8), with two boundaries restricting the orientation of fibers in the plane, the following equation can be used to derive the orientation factor (arm ): rib/213d): $3.1. forbs If b/2 041) =* l/2 ' I33 I forb> 1 LL -.L f “b lf/Z + (l b)0'64 where ; Lilf cosGd 0 B3 = 7 ykde and 7- = sin“(3) ’f \ / Figure 3.8 Two Dimensional Orientation of Fibers (3.3) 50 The above expressions give a lower limit for 2-D orientation factor equal to 0.64 when the width (b in Equation (3.3)) becomes infinity, and an upper limit equal to 1.0 when the width becomes close to zero. Where two boundaries are present to restrict the fiber orientation (Figure 3.6(a)). the orientation factor (a; representing an in this condition) in the z- direcrion, which is a typical direction of tensile stresses, can be obtained by con- siderin g the efbcts of these two parallel boundaries: ' h/2 dy Lsf/zl— for h< lf( Figure 3.9(a)) “,2 = 4 I’l2 (3'4) 1 Z dy 2 ' ' .9 % L; [521 + (0.405)(1-lf/h) for h lf( figure 3 (b) ) t f where : [on] 1:1, cosecosod 0d 9 Bl " I,(n/2)y 70 = sin"(df/lf) y = sin"(2y/lf) The result is shown in Figure 3.10 and it indicates that at thicknesses smaller than the fiber length, the obtained fiber orientation factor ((12) is very close to a 2-D condition, and for thickness greater than two times the fiber length, there is a gradual approach to 3-D fiber orientation. 51 (a)h$lf (b)h>lf Figure 3.9 Different Conditions with Two Boundaries 0.7 L. 0.5-i 0 4 4.1 o . o q— -I .5. o 5- H '1 o d-J .4 c .9. “ L «II 0 0.4— 1 h/lf O 3 T— U 0 T I I I 1 V I I t r I t I I I I I I I r U—r‘fi W T i I f! I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 3.10 Effect of Height on Orientation Factor in Cases with Two Boun- daries 52 In the condition where four boundaries are present (Figure 3.6(c)), the fol- lowing expressions can be used to derive the orientation factor (a) in the z- direction: iflbfi) for both sides (b,h) < If J,h/2B dy l f d/ 1 . -b-f(lf,h)+ (l-lf/b)—J—hz;-2—— for only one Side (h) s I, a: < 2 ‘ I,/2 9-” IL“, I )+ kWh-25L) [wzfiidy + (0.405)(b-If)(h-lf) bh f ’f bh lf/Z bh for both sides (b ,h)2 I, where: m/2 n/2 Bzdxdy flm’")- LI’ZLI/Z (m/2)(n/2) ’ 7 5 I I If cos0cos¢d0d¢ 62: 40-?“ 7 3 lijojyodedo yo: sin-1(dfxzf) ; y = sin’1(2y/If) ; 5: sin'1(2x/lf) ;and [31 = given in illustration of Equation (3.4). 53 Figure 3.11 presents the effects of cross sectional dimensions on the fiber orientation factor (a) for typical cases representing conditions having four boun- daries as shown in Figure 3.12. Three cases with width-to—height (b/ h) ratios equal to l, 3 and 6 are considered. It may be concluded from Figure 3.11 that the width-to—height ratio plays an important role in deciding the fiber orienta- tion factor at cross sectional dimensions close to or smaller than fiber length. The fiber orientation seems to gradualy approach a 3-D conditions as the cross sectional dimensions exceed two times the fiber length. Approximate equations for 2-D and 3-D orientation factors (which are rather convenient to use) are presented below. These expressions were derived using Taylor Expansions of exact expressions. Comparisons between exact and approximate expressions are given in Figure 3.13. d o d j — b-h J -- b-ah 1.0-4 b-ah 'l 3 0.9-3 H cl 0 .1 3 . 0.8— : q .9 ‘ *6 0.74 '4" 4 c . .92 . 3 0.6: 0.5-1 l h/lf 004 V'UUVWYY—YIYI'UIIV'rYIUYTUI11" IIIVVI’IUTIUVIUYUIY—r 0.0 1.0 2.0 3.0 4.0 5.0 Figure 3.11 Effects of Cross-Scetional Dimensions on Orientation Factor in Cases with Four Boundaries 54 bl’ 9/2-I -15 «IN III I; b'. I, (I /2 I, /2 j_ .\\.\,/ J L .L .L J l l l l I, I2 b-I, I, /2 Figure 3.12 Different Conditions with Four Boundaries r If2 1 6-—-T - r _ bh 0" (his-1,) 0" «Is-,1 I If for both b,hSl, a=1-’€--Ta ngl-R, )-..-(156+0766—) foronlthlf (3.6) 2 6’ for both 12,1: > I, 0.098i—+o.21 {—b-i-h—Honos bh f bh r I wig—L-Ta '1 _ 4 b " (Is-1,) forbSl, “20 = I, for b > I, (3'7) 0.31“;- + 0.64' 55 orientation factor 1.1 :1 —— Exact Expression . . - - - Simplified Expression 1.0-1 2 ’ II I Oeg- \ L __| I r '1 4 b . 0.8:: I \ h/ b 0.7: 0 0...; <24» 3 0.5 0.5:: 1.0 0.4 ‘ .................................................... b/If Figure 3.13 Comparisons between Exact and Approximate Expressions of Orientation Factor in Diffirent Conditions 3.3.2 Experimental Assessment of the Orientation Factor Orientation of Steel fibers in concrete and consequently the number of fibers per unit area are influenced not only by the boundaries restricting the random orientation of fibers, but also by the fact that Steel fibers tend to settle down and reorient in horizontal planes when fibrous concrete is vibrated during place- ment. Hence, as a result of vibration, the orientation of Steel fibers in concrete moves further away from a 3-D condition and tends to approach a 2—D condi- tion. In order to assess the degree of fiber reorientation during vibration, a comparison was made between the values for the number of fibers per unit area obtained theoretically from Equation (3.5) and measured experimentally. This section presents the results of measurements made in this Study on the number of fibers per unit area. These results are compared with the theoretical values 56 in order to derive more representative expressions for the actual fiber orientation conditions in concrete. The measurements of the number of fibers per unit area were performed on fractured cross-sectional surfaces of 152 mm by 152 mm by 457 mm (6 in. by 6 in. by 18 in.) steel fiber reinforced concrete beams tested in fiexure. The concrete matrices had a water/ binder ratio of 0.40, fly ash (type F)/ binder ratio of 0.3, aggregate! binder ratio of 4.0, fine-to-coarse aggregate ratio of 1.0 , and supe- plasticizer (solid content)/ binder ratio of 0.01 by weight. The fiber volume frac- tions were 0.5, 1.0, 1.5 and 2.0%, and the fibers were either straight (51 mm = 2 in. length and 0.5 mm = 0.02 in. diameter ) or hooked (51 mm = 2 in. length and 0.5 mm = 0.02 in. diameter ). The specimens were vibrated externally, and wee tested in flexure after 28 days of air curing (at 40% Relative Humidity and 22°C, 72°F). The fiexural loading was continued until complete separation occurred. A total of 19 flexural Specimens wee tested in this investigation. For each specimen, the number of fibers per unit area was measured using a 51 mm (2 in.) square frame (Figure 3.14(a)), noting that the number of fibers per unit area at a certain location is the sum total of the number of fibers appearing on one side and the number of pulled out fibers on the corresponding opposite Side. Measurements were made at Six locations on each specimen (Figure 3.14(b)). These locations were categorized as top, middle, and bottom with respect to the casting direction, as shown in Figure 3.14(b). The measured values of the number of fibers per unit area were then normalized as follows : 57 (a) Measurement Technique Castlng Direction T<='P Top 1 52 m m Mlddle Middle Bottom Bottom [n— 152 mm —~l (b) Measurement Location Figure 3.14 Measurement of the Number of Fibers per Unit Area 58 N1 (1: —— v,/A, (3.8) where a, the normalized value, is actually the orientation factor in Equation (3.1). The theoretical values for a (a0, a2 and a in Equations (3.2),(3.4) and (3.5), respectively) were influenced only by the boundary conditions, but not the vibration of concrete. The differences between the measured and theoretical “values of a will thus mainly represent the consequences of fiber reorientation in fresh mix under vibration. Table 3.1 summarizes the measurements made in this study for the number of fibers per unit cross sectional area in a total of 19 specimens. The means and Standard deviations of the orientation factors (obtained by normalizing the number of fibers per unit area following Equation (3.1)) are given in this table for diffirent locations on cross section and for diffirent fiber types. In order to verify if [there is any statistically significant diffirence between the fiber orientation factors, and consequently the number of fibers per unit area, Table 3.1 Mean Values of Fiber Orientation at Different Location on Cross Section and for Different Fiber Types Fiber Type Mean and Standard Deviation of alghaLEqB) . Top Middle Bottom All (No. of Specrmen) VI d M can Std. Dev. Mean §td. Dev. Meag Std. Dev. 1 can gt .Eev. . . Straight ' (16 Specimens) 0.609 0.231 0.587 0.235 0.654 0.240 0.617 0.235 Hooked (3 Specimens) 0.605 0.211 0.476 0.0773 0.776 0.214 0.619 0.179 All [19 Specimens) 0.608 0.225 0.569 0.221 0.673 0.237 0.617 0.228 59 at different locations on cross section or for diffirent fiber types, the hypOthesis that the'e are no efficts of location and fiber type was tested statistically using the measured values of fiber orientation factor. T-tests [54] indicated that at a significance level of 0.05, given the measurements made on the available speci- mens, there are no statistically significant effects of location on cross section (top vs. bottom) or fiber type (straight vs. hooked) on orientation factor and conse- quently on the number of fibers per unit area. Hence, the mean and standard deviation of fiber orientation factor can be derived using all the measurements made in this Study, irrespective of the location or fiber type. The resulting values of mean and standard deviation (given in the last two columns of the last row in Table 3.1) are 0.617 and 0.228, respectively. Figures 3.15(a) and 3.15(b) compare the frequency and cumulative frequency distributions of the measured orientation factors with the corresponding normal distribution curves. Some degree of Similarity between the measured values and normal distribution curves can be observed. 0.12 7 .— ‘ a date -—- normal curve(s.d.-0.2279,ovg.-0.617) > 0.09- E3 'A 83 31%; I}; 0 0.06-1 /F a? S: 27.3 Q III/I 2v 1 ’V a) ”b F 0.03- V /M //i a» // 0.00 ....... I ....... , ..... . . ......................... 0 00 0 20 0 40 0.150 0 80 l 00 1 20 1 40 Nt/(W/M) (a) Frequency Distribution Figure 3.15 Measured vs. Normal Distribution 1.20 . E —— from data 3 from normal curve 1.00-3 3‘ : i=- : -0 E g 0.801 a E L. . 0. 0 60 5 g " a: E 2 E 3 0'40? :I 3 o : 0.205 0000 I"rijrr‘r‘t'fititrfirtIfV'UI'U'II'I UUUUUUUUUUUU I IIIIIII 0.0 0.2 0.4 0.6 0.0 1:0 1.2 1.4 Nl/(Vf/Af) (b) Cumulative Frequency Distribution Figure 3.15 Measured vs. Normal Distribution (cont’d) F-tests [54] on measurements made at different locations on cross section and for diffeent fiber types also showed no statistically significant efficts of location and fiber type (at 0.05 Significant level) on the Standard deviation of the measured values of orientation factor. 3.3.3 Theoretical Values vs. Experimental Measurements The theoretical value of orientation factor obtained from Equation (3.5) for the parameters chosen in this study ( fiber length of 51 mm = 2 in., and cross sectional dimensions of 152 mm by 152 mm = 6 in. by 6 in.) is 0.537. The difference between the measured mean value of fiber orientation (0.617) and the calculated value of 0.537 is about 4 times the Standard error of the measured 61 mean value. This significant difbrence can not be Simply attributed to the vari- ation of steel fiber concentration inside concrete. The relatively large value of orientation factor in actual measurements may result from the modification of fiber orientation during the vibration of fresh mix [55]. Vibration efficts cause a reorientation of steel fibers inside concrete and encourage a tendency toward 2-D distribution of fibers in horizontal planes. This efbct of vibration may be used to illustrate the difference between the measured and theoretical values of number of fibers per unit area (and orientation factor). It Should be noted that, for the conditions of test Specimens in this study (152 mm = 6 in. square section, 51 mm = 2 in. fibe' length), a pure 2-D distri- bution in horizontal planes (considering the boundary effects), would lead to a horizontal orientation factor of 0.74 (see section 3.3.1 for theoretical expressions for 2-D fiber orientation) which is larger than the measured value of 0.617. Noting that this measured value (0.617) is at the same time larger than the corresponding 3-D fiber orientation factor of 0.537 obtained theoretically (consid- ering the boundary effects), it may be concluded that the actual fiber orientation factor after vibration of concrete is in between the corresponding 2-D and 3-D orientation factors (calculated considering the boundary efbcts). Approximate values of orientation factors in Specimens with different geometries and fiber lengths can be derived from Figure 3.16. This figure shows values for orienta- tion factors in 2-D conditions with two boundaries (calculated using Equation (3.3)) and 3-D conditions (with different height-to-width ratios calculated using Equation (3.5)), in terms of the ratio of specimen width to fiber length. For each geometric condition, given the specimen width to fiber length ratio and also the ratio of specimen height to its width, Figure 3.16 can be used to derive theoretical values of orientation factor for 2-D and 3-D conditions. The experi- mental results of this study indicate that the actual orientation factor would fall 62 1.0-l : ‘f I ..I. 25- 0.9- 7; 1 m .. .I I v T‘ O «I ‘2 0.8-} O I ““‘s 3 1 \~‘ é“:‘ h/b B 0.7.: \ “‘ ‘==-~:.—~ ‘. ~--::-~_--- oi E “\\‘ ~“‘. ~~~~‘~~.-- 0(2-0) ‘6 ° 5'. ...... ‘0.2s ‘ ..... 0.5 0 51 “'“-------....o.7§ : 1.0 004 TV 1T1 U—V YTlfi r? U rrf' I—I' '''''''' 1 IIIIIIIII r UUUUUUUUU rU—rT—rrrrl Y 0.0 1.0 2.0 3.0 4.0 5.0 6.0 b/lf ' Figure 3.16 2-D and 3-D Orientation Factors for Different Geometric Condi- tions between the 2-D and 3-D values (an average of the two may be used as a rough approximation). 3.4 PREDICTION OF TENSILE STRENGTH: "INTERACTION CONCEPT" As mentioned earlier, there are two dominant analytical simulations of fiber reinforced concrete failure mechanism under direct tension. One considers the pull-out action of fibers as the key mechanism through which fibers contribute to the tensile strength of material (generally referred to as the composite material concept or the law of mixture) [l,3,27-29,56]. The second approach (usually referred to as the spacing concept) suggests that the spacing of fibers,.not their 63 bonding to matrix and pull-out behavior, is the key factor which decides the effictiveness of fibers in concrete [34,35]. One may consider that the spacing concept is based on the assumption that at the composite peak Stress fibers act mainly to arrest microcracks (Figure 3.17) rather than to bridge the macro- cracks, while the opposite applies to the composite material concept. Figures 2.18(b) and 2.19 presented in Chapter 2 are indicative of shortcom- ings of the composite material and spacing concepts, respectively, in describing the performance of steel fiber reinforced concrete at peak tensile Stress. The deficiencies of the composite material concept may have been caused by disregard for the microcrack-arresting action of steel fibers (Figure 3.17(a)) and the inheent assumption at the composite material approach that the pulloout Microcrack Aggregate Fiber (b) Bridging of Macrocracks Figure 3.17 Influences of Fibers on Cracking Characteristics and Tensile Behavior of Concrete 64 resistance of fibers is almost fully mobilized at peak tensile stress (where the strains and crack openings are insufficient to do so) [28]. The spacing concept, on the othe' hand, disregards any partial mobilization of bond stresses (Figure 3.17(b)) at peak tensile stress, which could be the key reason for its discrepancies when compared to test results. 3.4.1 "Interaction Concept" The formulation presented below for the prediction of SFRC tensile strength can potentially account for the contributions of both the microcrack- arresting and partial pull-out actions of steel fibers at the peak tensile stress of the composite material: a, = A ~o,,,' + 0.251t-d, -l,-1:-N1 (3.9) where : A = 1+ a-N11’3; 1: = interfacial bond stress at the composite peak tensile stress t'O’m ; and a , t = coefli cients to be derived empirically. The contribution of matrix at peak tensile stress in the above equation is represented by A -o,,,', where A is dependant on the number of fibers per unit cross-seetional area (N 1). This reflects the fact that a higher number of fibers (with a smaller fiber Spacing) is more efbctive in arresting microcracks (Figure 3.17(a)) and thus in increasing the contribution of the matrix to the composite material tensile Strength. Equation (3.8) can be used to derive N 1, with the 65 orientation factor (or) being a function of the fiber reinforcement properties (geometry and volume fraction) and the cross-sectional dimensions of test speci- men as described in section 3.3 in this Chapter. The contribution of fiber pull-out at peak tensile stress is represented in Equation (3.9) by the multiplication of the average fiber interfacial area reSiSting pull-out (0.251tid, -l,) times the average fraction of bond stress mobilized at the composite peak tensile stress (1:) times the number of fiber pe' unit area (N 1). The proportionality of 1: and matrix tensile Strength (om') reflects the fact that strange matrices may be capable of activating a larger fraction of fiber pull-out force at peak tensile stress. It is assumed that the inclination of fibers with respect to the tensile stress direction has a negligible effect on the pull-out action of fibers. It is worth mentioning that the decisions on the dependence of the matrix contribution to tensile Strength of the composite on the number of fibers per unit cross-sectional area (N 1), and also the dependence of the fiber pull-out contribution on the matrix tensile Strength (6",) were made based on the physics of the composite material behavior at peak tensile Stress, and also based on an extensive trial and adjustment verification of different concepts fordescribing the composite material performance at peak tensile Stress. The proposed approach to the prediCtion of SFRC tensile Strength accounts for the physical interactions that exist between fibers and matrix at peak tensile Stress, and it may thus be referred to as the "interaction concept" for predicting the tensile Strength of fiber reinforced concrete. Coefficients a and t of the "interaction concept" in Equation (3.9) have to be decided empirically using tensile Stress test results. A comprehensive set of test data was used for this purpose. 66 3.4.2 Empirical Coefficients A total of 50 SFRC tensile strength test results were used to derive the empirical coefficients of the proposed "interaction concept" equation for the prediction of SFRC tensile strength (Equation (3.9)). The direct tension test results used in this study were obtained for mortars reinforced with straight (round or rectangular) steel fibers [24,25,29,52,57]. These tests were performed on Specimens with rectangular cross sections of diffirent dimension. Table 3.2 summarizes the following properties of the tension test speciments : (a) cross sectional dimensions ; (b) fiber reinforcement properties ; and (c) matrix mix pro- portion and tensile Strength test results. Least square fitting of the "interaction concept" expression (Equation (3.9)) to the test data. presented in Table 3.2 provided the basis for calculating coefficients a .and t of Equation (3.9), which were found to be equal to 0.138 and 0.2, respectively. Hence, the proposed "interaction concept" leads to the follow- ing equation for calculating the tensile Strength of Steel fiber reinforced concrete a, = om'-( 1+ 0.133-N,1/3 + 0.051t-df-l,-N1) (3.10) Figure 3.18 Shows the desirable comparisons between predictions of the pro- posed "interaction concept" (Equation (3.10)) and SFRC tensile strength test results. The bond stress mobilized at the composite tensile strength is represented in Equation (3.9) by t ~o,,, '. An empirical value of 0.20 for t indicates that, for a typical matrix tensile Strength of 2.41 Mpa (350 psi), the bond stress developed at the composite material tensile strength is typically 0.48 Mpa (70 psi), which is Table 3.2 Direct Tensile Test Specimen and Results 67 matrix fiber . en size mposste Ref. tensile strain diameter length Vf widthxdepth tensile strain czszw strength at peak strength at peak (Mpa) (110.0001) (mm) (mm) (‘5) (“m " mm) (Mpa) (110.0001) 24 1:2:0.5 2.8 1.74 0.41 25.4 0.5-1.5 76x19 3.0-3.6 1.9-2.2 1:25:06 1.68 ‘ 0.2.5 18.8 1-3 38x51 2.0-2.5 ‘ 5 " " " 0.25 12.7 1-3 " 1.8-2.1 ’ " " ‘ 0.25 3.4 1-3 " ' 2.2-2.7 ‘ 13:05 3.38 1.43 0.25 19.1 0.6-1.7 102x102 3.7-4.0 1.6-1.8 " " " " 38.1 03-1.7 " 3.5-3.8 1.5-1.9 " " " " 25.4 0.3-1.7 " 3.0-4.3 1.6-2.2 " " " 0.41 13.8 0.6-1.7 " 3.4-3.5 15-l.6 29 " " " " 25.4 06-1.? " 3.6-4.0 1.5-1.8 " " " " 38.1 06-1.? " 3.8-4.5 1.7-2.0 " " " 0.43 15.2 1.7 " 3.7 1.6 " " " '° 30.5 l.2-1.7 " 3.5-4.1 1.7 " " " " 45.7 0.6-1.7 " 3.7-4.6 1.6-1.8 1:20.45 1.74 0.85 0.5 50 0.6-1.8 16x100 1.9-2.6 1.2-2.1 34 " " " 0.35 35 0.6-1.8 " 1.9-2.4 1.1-2.2 " " " 0.25 7.5 0.6-1.8 " 1.6-2.0 0.9-1.7 2.8 0.43 12.7 1-3 127106 3.6-5.3 1.64.4 35 " " 19.1 1-3 " 2.7-4.6 1.2-6.6 " " 25.4 1-3 " 3.5-5.3 1.3-6.0 ‘ Not Reported 68 1.8 A m 4 O'- “J . v a. . ° 8' EL) {c 1.4- o 90 0 50 on o o C ‘ ° 0 .9 ° 0 0 +1 0 o o 0 . ° oo o 0 ° 0 o e 0 ° ° L. o e o 3 0° 0“ o O .5 ‘g . 1.0 f r U I U V I 1.0 1.4 1.8 ac/am Figure 3.18 Comparisons of the Proposed "Interaction Concept" with Tensile Strength Test Results only about 25% of a typical fiber-to-matrix bond strength of about 2 Mpa (reported in Reference 28). This result is compatible with the discussion made earlier on strains and crack openings at the peak tensile stress of the composite indicating that they are not large enough to fully mobilize the pull-out action and interfacial bond strength of steel fibers in concrete. The matrix contribution to tensile strength is represented in Equation (3.9) by 0,,"( 1 + cerl’3 ). With the empirical value of 0.138 for a, at a typical value of 0.047 for N 1 (corresponding to a volume fraction of 1% in a direct ten- sion test specimen with typical cross-sectional dimensions), the contribution of matrix to the composite material tensile strength is 1.05 times the matrix tensile strength. This increase in the tensile strength of matrix may be attributed to the microcrack-arresting aCt ion of fibers inside the matrix which tends to Strengthen the matrix under the action of tensile stresses. 69 3.4.3 "Interation Concept" vs. Composite Material and Spacing Con- cepts This section uses tensile strength test results presented in Table 3.2 to per- form a comparative study on the accuracy of the proposed "interaction concept" versus thoses of the composite material concept (Equation 2.4 [3])) and the spac- ing concept (Equation 2.5 [34]) and modified spacing concept (Equation 2.6 [35]) in predicting the tensile strength of SFRC. The comparison between predictions of the new "interaction concept" and test results is presented in Figure 3.18. Figures 3.19(a), (b), and (c) compare the same test results with the predictions of the composite material concept (Equation 2.4), spacing concept (Equation 2.5), and modified spacing concept (Equation 2.6), respectively. Relatively large scatters between test results and predictions based on the composite material and spacing concepts are observed in Figure 3.19. The sum total of the squares of normalized errors ( the normalimd error represents the diffirence between theoretical and experimental tensile strength values normalized with respect to the experimental strength ) for each of the four approaches introduced in Figures 3.18 and 3.19 are as follows : 0.429 for the "interaction concept" (Equation (3.10)), 0.553 for the composite material concept (Equation (2.4)), 0.567 for the spacing concept (Equation (2.5)), and 0.551 for the modified spacing concept (Equation (2.6)). This confirms the favorable comparison of the "interaction concept" prediction with test results. The average and standard deviation of errors (differences between normalized theoretical and experimental values) are presented in Table 3.3. The average error of the "interaction concept" prediction is observed to be closer to zero and the standard deviation of its errors is also 7O A g, 1.3 13 1 V O 4.; J a. 8 0 g o bEQ ac \ B 1.4“ 50': 0 3 T O o O O 0 g 9 o o 9 o 00 0 Q) 0 0° 0° 0° 0 O O O O :3; o O O {a g 0 o o a E 1 o e r T— I I T I r V 0 1.0 1.4 1.8 .4... (a) Composite Material Concept (Equation (2.4)) 1.8 A 1 in o'- 14.1 v e ‘5. a 8 e" g 0 as .E o 8- o (n 1.0 [— fi r V T " 1.0 1.4 1.8 ac /am (b) Spacing Concept (Equation (2.5)) Figure 3.19 Comparisons of the Composites Material and the Spacing Concept Predictions with Tensile Strength Test Results 71 '00 1.4- (Ia/0m Modified Spacing Concept ( Eq. 6 ) T . 1.4 1.8 ac/am (0) Modified Spacing Concept (Equation (2.6)) Figure 3.19 Comparisons of the Composites Material and the Spacing Concept Predictions with Tensile Strength Test Results (cont ’d) ..A O ' 1 J 1 1 Table 3.3 AVG-{380. Standard Deviations. and Sum Total of Squares of Nor- mahud Errors for Diffirent Concepts Concept Avg. Std. Dev. Sum Total of Sqrs. " Interaction Concept " (Ermation 3.10) -0.0014 0.0925 0.429 Composite Material (Equation 2.4) 0.0456 0.0945 0.553 S acing (Equation 2.5) 0.0248 0.104 0.567 Modified Spacingjquuation 2.6) 0.0234 0.102 0.551 72 seen to be the lowest, when compared with those obtained for the other con- cepts. Hence, Equation (3.10) based on the "interation concept" seems to predict the direct tensile strength of steel fiber reinforced concretes with a more reasonable accracy. 3.5 PRES-PEAK CONSTITUTIVE MODELING The pre-peak tensile behavior of steel fiber reinforwd concrete deviates from linearity when microcrack propagarion has already occured (see Figure 2.14). Thus, as shown in Figure 3.20 , the behavior was assumed ( based on the reported tension test results) to be linear, with a slope equal'to the elastic modulus of the matrix, up to the matrix tensile strength (6”,). At this point, due to major microcrack propagation in the matrix, the stiffiiess was assumed to be reduced, and the stress-Strain relationship was continued linearly up to the peak tensile stress. Strain at peak tensile stress was derived empiricaly, using the test data summarized in Table 3.2, as a function of some fiber reinforcement properties: a, = Em-(l + 0.3s-Nl-df-1f) (3.11) where: 8c = composite tensile strain at peak tensile stress ,' and 8,, = matrix tensile Strain at peak tensile Stress. 73 Tensile Stress 5. v. ------------ P! I It.) Tensile Strain( t ) R 0 Tensile Strain Figure 3.20 Simulation of the Steel Fiber Reinforced Concrete Tensile Behavior upon Cracking at Peak Load 1.. ‘ a 1"an — (¢ - (.(l + 0735N‘dl ‘1’) 1.2-1 4 . O I $ 4 w. u. I 4 0.4- 1 O J O O f T f V f v r “0.0 Y 03 ' V 15) 13 2.0 N 1 Odftlf Figure 3.21 Increase in Strain at Peak-Tensile Stress in the Presence of Steel Fibers as a Function of Some Fiber Reinforcement Properties (N 1'51] '0‘) 74 Figure 3.21 presents the relatively desirable comparison between predictions of the above equation and the reported test results for strain at peak tensile stress in steel fiber reinforced mortar. 3.6 POST-PEAK CONSTITUTIVE MODELING In the pre-peak region the matrix and fibers interact and both contribute to the tensile resistance of fibrous concrete. Crack opening at the peak load, how- ever, sharply reduces the contribution of matrix and tends to transfer tensile loads mainly to the fibers bridging the crack. The matrix contributes to the post-peak tensile resistance of the composite through its softening behavior. The tensile behavior of the composite in the post-peak region can thus be simu- lated by superimposing the pull-out performance of fibers with the matrix softening behavior. Due to difficulties in direct tensile testing of concrete only limited experi- mental data are available in this area. In the interpretation of the direct ten- sion test results it should be considered that the post-peak deformations in fiber reinforced concrete tend to localize in one major crack at the critical section. The tensile behavior of fiber reinforced concrete can thus be represented by a stress-strain relationship in the pre-peak region, and an average stress vs. aver- age crack width (deformation) relationship in the softening (post-peak) region [10,24,58]. An empirical model presented in Reference 10 for the softening (post-peak) behavior of the matrix was simplified (by a bilinear presentation of the curvilinear model) to represent the contribution of the matrix to the post- peak behavior of fiber reinforced concrete: 75 1 s, 0 S s Ssc, 3 12 0’”- .s-s sSsSs (') 04' . CO or CO . Om scO - scr L where: am = tensile stress (in post— peak region); Om ' = peak tensile Stress; s = crack opening; sc, = crack width at 0),, equal to 0.4-om' = 0.015 mm ,' and 3c0 = crack opening at cm equal to zero. The above equation is compared with test results in Figure 3.22. It should be noted that a more elaborate modeling may require the consideration of fiber efbms on the matrix post -peak tensile behavior. Upon the cracking of matrix at the peak tensile strength of the composites a crack strats to open and the pull-out mechanism of fibers tends to be mobilized. Typical experimental Now of fiber pull-out versus displacement (slippage) rela- tionships for straight round steel fibers are given in Figure 3.23. Based on 36 experimental pull-out load-slip relationships reported in References 16,18,19,59- 61, an empirical expression for pull-out behavior was developed in this investiga- tion (Figure 3.24). The model consists of three straight lines: a linear pre-peak ascending portion, and a bilinear post-peak descending branch. This trilinear model includes two characreristic bond values (1,, and t, in Figure 3.24) and three characteristic pull-out slip values (st , s, and so). 76 0' 0m ' Normalized Stress . Concrete 0 Mortar 0 Paste Figure 3.22 Comparisons of the Average Stress vs. Average Crack Width Rela- tionship in the Post-Peak Region with Test Results 1:“ ...... 1 l l 1 a, 1 t : U) I 3 1 S . '3 1 9" 1 03-1,, ----- : .......... . ' 1 ' 1 ' 1 : 1 0 L 1 Figure 3.23 Typical Experimental Plot of Fiber pull-out vs. Displacement (Slip- page) Relationship 77 Pull-Out Stress (Mpa) 2.62 0.79 ‘* 0.025 2.8 Figure 3.24 Model of Pull-Out Load-Deflection Relationship for Straight- Round Steel Fibers Table 3.4 Pull-Out Test Conditions and Results [16,18,19,59,60,61] matrix fiber avg. bond peak Ref. czszw diameter em; $1.”! strength displacement (mm) (mm) (Mpa) (mm) 16 l:4:0.5 0.38 50.8 1.8 ‘ l:2.5:0.55 0.4 12.7 2.62 "' 18 1:25:06 0.25 12.7 2.62 0.51 1:2.5:0.55 0.15 12.7 1.02 "' " 0.4 12.7 2.25 " 1:3:0.3l 0.4 20. l 2.4 0.04 or 0.4 30. 2.4 ‘ 19 0.4 20. 2.4 " l:3:0.65 0.3 30. 2.4 ‘ 0.3 20. 2.3 ‘ 59 1:2:0.4 0.64 31 2. l ' 1:0:0.31 0.64 50.8 0.64 " 1:0:0.55 0.38 30.3 " 0.203 60 " 0.51 " "' 0.45 " 0.41 " ‘ 0.23 61 1:25:04 0.4 12.5 0.194 0.2 l:2.5:0.6 0.4 12.5 0.42 0.2 78 (0.) Ref. 18 0)) Ref. 19 (c) Ref. 21 ((1) Ref. 22 M x “\‘\ \\\‘1L\\\‘ h P i i (i) Ref. 24 (c) Ref. 23 Figure 3.25 Pull-Out Test Procedure Several pull-out test results reported in the literature were used to derive the characteristic bond stress and slip values of the proposed model. Table 3.4 summarizes some fiber and matrix properties and experimental procedures (see Figure 3.25) as well as test results for the pull-out tests used in this study. The average characteriStic bond stress values derived from pull-out test results are as follows: 2.62 Mpa (380psi) ; and (3.13) :d II t, = 0.3'1?“ While the stability of the pull-out test is vital to proper monitoring of the 79 fracture behavior and pull-out deformations, most pull-out tests have not been conducted under stable test conditions [62]. It was pointed out in Reference 62 that with the method of crosshead or overall displacement-controlled tests, very low initial sitiffi'tesses, and thus large slip values at the peak pull-out load, are generally recorded. To avoid the problems addressed above, slip-controlled tests which ensure greater stability during specimen softening were perfomed and the typical results are shown in Figure 3.26(a) for the pull-out behavior of straight- round steel fibers. It has been reported in this reference that considerable scatter observed in the measured peak slip values possibly result from the flat nature of the load-slip characteristics in the vicinity of the peak-load. Pull-out test results of Reference 62 are distinguished from others [16.18.19.59-61] by a much larger initial pull-out stiffness. Based on the test results reported in Reference 62, the fiber pull-out slip at peak pull-out load (spk) was selected to Pull-Out Stress (Mpa) 3 r 2 I- 1 1- Slip (mm) 0 1 1 l l t 1 0.05 0.1 0.15 (a) Pull-Out Test Results in Reference 62 Figure 3.26 Simulation of Pre-Peak Pull-Out Behavior of Straight-Round Steel Fibers 80 Mpa —-— test analytical . r. 2.62 -- - '/ I I I I l l | l l I I l 1 0.025 mm (b) Pre-Peak Pull-Out Model vs. Test Results from Reference 62 Figure 3.26 Simulation of Pre-Peak Pull-Out Behavior of Straight-Round Steel Fibers (cont’d) be 0.025 mm (0.001 in.) for use in the model of this investigation.. The model is shown in Figure 3.26(b) to closely simulate the initial pull-out stiffness in the pull-out test results of Reference 62 (see Figure 3.26(b)). It is worth mentioning that this value of slip at peak pull-out load is roughly five times the maximum crack opening at the peak tensile stress of SFRC under direct tension as given in Reference 28. With the limited fiber pull-out data available, the other characteristic pull- out slip value at residual strength (3, in Figure 3.24 corresponding to a bond strength of 1:, ) was obtained as the average of test results reported in the litera- ture (s, = 2.8mm) [18.19.6061]. The slip at zero pull-out load (so in Figure 3.24) was assumed to occur when complete pull-out has been made (i.e., when 81 the slip value equals half of the fiber length). A summary of the selected values for characteristic bond stress and slip values is given below: sp,‘ = 0.025mm (0.001 in. ); (3.14) s, = 2.8mm (0.11 in. ); and So: lf/2. Typical comparisons beween the post-peak branch of the proposed fiber pull-out model with the above empirical characteristic values of bond stress and slip and experimental post-peak results are observed in Figure 3.27 to be reason- ble. In the pull-out tests used in this investigation, except for Reference 62 (where multiple fibers were pulled out simultaneously), a single straight-round ' Pull-Out. Load, N 1 4 8 12 Pull-Out. Distance (mm) Figure 3.27 Comparison of the Experimental Pull-Out Load-DefieCtion Rela- tionships in the Post-Peak Region with the Empirically Derived Model of This Study 82 steel fiber (which was aligned in the loading direction) was pulled out of the matrix. In the actual conditions of the composite material, however, fibers are closely spaced and also randomly oriented. An increase in the number of fibers per unit cross sectional area has been shown to reduce the pull-out strength of fibers (Figure 3.28(a)). Another observation has been made by Reference 60 that, as the volume fraction in matrix (and thus the number of fibers per unit cross sectional area) increases, the pull-out strength of fibres tends to increase (see Figure 3.28(b)). More comprehensive test results are needed if the e&cts of the number of fibers per unit cross-sectional area on the pull-out performance of fibers are to be considered. Peak Load per Fiber/ Peak Load of One Fiber 0 straight, parallel fibers 1,5 J. . fibers at. 60' 1.0 ' 0.5 '- 0 l l 0.03 0.06/mm2 (a) Fiber Concentration (No. of Fibers per Area) Figure 3.28 Effect of Fiber Concentration, Volume Fraction and Orientation on Pull-Out Strength 83 Relative Increase 0 Single Pull-Out ,. C Group Pull-Out l 2 Vi (95) (b) Fiber Volume Fraction Peak Load, N O df=0.~t mm 0 df=0.25 mm 50" . df=0.15 mm 40%/\0\O 1 1 1 l 1 20 40 50 Degrees Angle of Orientation (c) Fiber Orientation Figure 3.28 Effect of Fiber Concentration, Volume Fraction and Orienation on Pull-Out Strength (cont’d) 84 As far as the fiber orientation efbcts on pull-out strength are concerned, as shown in Figure 3.28(c), the increase in fiber inclination (with respect to the pull-out load direction) first increases the pull-out strength and then starts to reduce the pull-out resistance. Based on this observation it was assumed that the pull-out performance of fibers aligned in the direction of pull-out load roughly represents an average performance for randomly oriented fibers. The composite material post-peak behavior was simulated assuming that the fiber slippage in the pre-peak region is negligible. This assumption was made based on the discussions presented in Reference 28, where it is stated that the crack opening at peak load is too small to significantly mobilize pull-out action of fibers. The composite material post-peak behavior may thus be assumed to depend on the pull-out behavior of fibers crossing the critical section (with fiber pull-out starting near the peak load) and the matrix softening behavior in the post -peak region: 0 = of + cm (3.15) where: O' = total resistance after peak tensile strength; and of = average tensile stress provided by the pull-out resistance of fibers across the critical section. In the use of fiber pull-out behavior for simulating the post-peak behavior of composite materials, it was assumed that the fiber embedment length is equal to the statistically derived average value of l,/4. The tensile resistance pro- vided by fibers can thus be derived through multiplying the average value of bond stress by the interfacial area of all fibers crossing the cracked section 85 assuming an average embedment length of If! 4 : 1 a, = err-d, --£—-N1 (3.16) where: t = average in tetfacial bond stress. The value of 1: in the above expression can be obtained, using the proposed fiber pull-out constitutive model, as a fraction of the crack opening (5) in the post -peak region: f til '3 forOSsSs,‘ spk P T T T 'S "T'Sk 17:4 ' “ s “ ’ ’ P forspk< sS 3,, (3.17) Sr-spk Sr_spk 1 1'3 - ' 3+ '0 fors,'f’0 for 8) Epf . where: f = concrete compressive Stress ; e = concrete compressive Strain ; f ’cf = compressive strength of steel fiber reinforced concrete ; e” = strain at peak stress; and 102 z,f ’0 = coefficients derived empirically in terms of the compressive Strength and fiber reinforcement index. Coefficients z and f ’0, and the stress and strain at peak compressive stress (f '6] and epf) in this model were derived empirically for different fiber types as functions of the matix compressive strength and the fiber reinforcement index. The empirical expressions for different variables of the proposed model are given below (see Figure 4.3). These expressions have been derived through least square fitting of curves to experimental results. Figure 4.4 presents comparisons between the empirical expressions given below and test results. Compressive Strength, f ’cf (see Figure 4.4(a)) : f'cf =f’c + aHVfIf/df (4.2) where ; 3.6 Mpa (515 psi) for straight fibers or = 6.0 Mpa (872 psi) for hooked fibers Residual Compressive Strength, f ’0 (see Figure 4.4(b)) : where ; B = 11.8 Mpa (1700 psi) for both straight and hooked fibers 103 Slope of the Descending Branch, 2 (see Figure 4.4(c)) : where; 0.66 for straight fibers 7 = 0.70 for hooked fibers Strain at Peak Stress, Er! (see Figure 4.4(d)) : 2,, = 5p + Wilt/‘1! where ; 5 {0.0007 for straight fibers "" 0.0017 for hooked fibers ep 0.0021 strain at peak stress for plain concrete (4.4) (4.5) It should be mentioned that variations in specimen geometry,-loading versus casting direction, rate of loading, and maximum aggregate size (e.g. mortar versus concrete) will modify the compressive constitutive behavior of fibrous con- crete. More test results are needed in order to refine the developed model for considering the effects of these factors. 104 24 ‘ 0 feet can (W M) — Eq. 2 (W Flam) . 0 root Date (Hm Flinn) . -- £4. 2 (Hooked more) . '6‘ ‘ “,o““ A 0 """"" o ..... Q "g I _,- V '0 .- 1 a. 0 fl -. V ' ' V r v r— w v T . V r ' 0.0 1.0 2.0 3.0 Vfclf/df (a) Compressive Strength eo . a footnote (strewn...) 4 . 7.8“. M“ M) . - £4!- 3 60... A 1 a . Q 2 V ’23 ‘o' 3 o’ I . h J -20 ' r r l v v r 0-0 1.0 2f0 r f 3.0 VMf/df (b) Residual Compressive Strength Figure 4.4 Comparison between Empirical Expressions on Test Results for Different Variables of Compressive Constitutive Model 105 Z / (-343¢fc‘) 2.0 sqt‘thcIf/df) (c) Slope of the Descending Branch VMf/df (d) Strain at Peak Stress Figure 4.4 Comparison between Empirical Expressions on Test Results for Diffirent Variables of Compressive Constitutive Model (cont’d) 106 4.5 COMPARISON WITH TEST RESULTS AND PARAMETRIC STUDIES Typical comparisons between the predictions of the constitutive model developed in this study and the experimental compressive stress-strain relation- ships for 150 mm by 300 mm (6 in. by 12 in.) concrete cylindrical specimens reinforced with straight fibers are given in Figures 4.5(a) through 4.5(e). The experimental results presented in this figure cover wide ranges of fiber reinforce- ment index (V f 'If / (1,) and concrete compressive strength. The proposed model is observed to predict experimental results with a reasonable accuracy. Limited compression test results on crimped fibers have been reported in Reference 64. Figure 4.5(0 shows that the model developed for straight fibers satisfactorily predicts the compressive performance of concretes reinforwd with crimped fibers. fl - rat out. -- M A J 3. . 2 . V 40'] C " 1 3 m C .2 i"- 8 o o , a a 0 10 an Compressive Strain ( x 0.001 ) f'e - 42 M Vfclf/df - (£37 (a) Tesr Data from Reference 64 (Straight Fibers) Figure 4.5 Analytical Model vs. Test (1 Mpa = 145 psi) Compreeelve Streee ( Mpa ) Oompreeelve Streee ( Mpa ) 107 6 v - - . 1'0 ' f . ' m Campreeefi've Strain ( x 0.001 ) Vftclf/df - 1.71 (b) TeSt Data from Reference 64 (Straight Fibers) I ' j 10 comprerift'tive:58 Strain 5x6 x.0 001 ) Wolf/tif- 50. 9 (0) Tea Data from Reference 4 (Straight Fibers) Figure 4.5 Analytical Model vs. Test (1 Mpa = 145 psi) (cont’d) 108 Compreeelve Streee (' Mpa ) 0 u ' ' ' ' 1'0 ' ' - ' 20 Compreeeive Strain 6 x 0.001 ) Pa - 62.1 pa Welt/d1 - 2.0 ((1) Test Data from Reference 64 (Straight Fibers) l A A _A A k L Compreeelve Streee ( Mpa ) 8 0 6 ' ' ' ' 1‘0 ' T j - 20 Compreeeive Strain 515 x 0. 001 ) “i .83 VfCOIf/df - 1 (e) Tesr Data from Reference 64 (Straight Fibers) Figure 4.5 Analytical Model vs. Test (1 Mpa = 145 psi) (cont’d) 109 ea . ] -- rune-(ahead mar) ‘ -- MMM 3 ea- 2 1 E in 40~ O .1 .2 3 J g J E 20~ 8 - 1 1 ' 0 Com Strai 0.001 we: 11:15,: ’ Welt/at - 1 4 (1') Test Data from Reference 64 (Crimped Fibers) fl 4 — rater-nearer) ‘ -- MMM A 4 8. 00- z 1 g in .5 i 5 o 0 ‘ r - - I 10 an Compreefeéve Strain’spa x 0.001 ) (g) Test Data from Reference 64 (Hooked Fibers) Figure 4.5 Analytical Model vs. Test (1 Mpa = 145 psi) (cont’d) 110 U - renown.) 1 -- Mona-urea) 3. ea- : - q V 5”“ o - E f f - 0 1a 20 Compreeeiv' e Strain x 0.001 f'c - 37.2 ISpa ) Vftlf/df - 0.0 (h) Test Data from Reference 64 (Hooked Fibers) Figure 4.5 Analytical Model vs. Test (1 Mpa = 145 psi) (cont’d) Compression test results on concretes incorporating hooked steel fibers are also observed in Figures 4.5(g) and 4.5(h) to be closely predicted by the developed compressive constitutive model. The developed compressive constitutive model of steel fiber reinforced con- crete was also used for a numerical study on the emts of concrete compressive strength and fiber reinforcement index on the fibrous material behavior under compression. Figure 4.6(a) shows the overall constitutive performance of steel fiber reinforced concrete (with straight fibers) as influenced by the variations in steel fiber reinforcement index and compressive strength of concrete. Figure 4.6(b) shows the effects of these factors on the energy absorption capacity of fibrous concrete in compression (represented by the total area underneath the compressive stress-strain curve up to a strain of 0.01). Higher values of fiber 111 reinforcement index and concrete compressive strength are observed to produce significantly higher energy absorption capacities for fiber reinforced concrete. Figure 4.6(c) shows the tendency in strain at peak compressive stress to increase with increasing fiber reinforcement index in steel fiber reinforced concretes incor- porating straight fibers. Finally, Figure 4.6(d) presents typieal improvements in compressive performance of steel fiber reinforced concrete resulting from the use of hooked instead of straight fibers. fl __ re-aelln -- rel-ooh ? 1 g ................... m ”d “I? O .2 4 I ’ “\ g. m: If .wf’df = 2.0 8 117,7 1,... “°' _ aIi r . . 1 ‘ ' j 1 Y I 1 “It! Compreeeive Strain (a) Fiber Reinforcement Index (Straight Fibers) Figure 4.6 Effects of Fiber Reinforcement Index and Concrete Compressive Strength on Compressive Behavior of SFRC as PrediCted by the Proposed Constitutive Model (1 Mpa = 145 psi) 112 -- rc-ee up. -- Hike - H “I ’0‘ 0.0 a I 2 _________ V I - - c / o / z I ----------------------- e M // I” " I 8 , ’ x’ _____,._ I ’o’ e ’ ’ ’0” / a f f '0" / 8' ” .-" / 3' 0.2 , r ’ -"‘ / .1 ‘0‘ m r """ / Lee” M r r r M 1.0 2.0 3.0 Wolf/d! (b) Energy Absorption Capacity (Straight Fibers) 0.000 a I «5" 0.004- V 8 8 a. a O .5 g m. m 0.000, , 0-0 1.0 £0 3.0 Vftif/df (c) Strain at Peak Stress (Straight Fibers) Figure 4.6 Effects of Fiber Reinforcement Index and Concrete Compressive Strength on Compressive Behavior of SFRC as Predicted by the Proposed Constitutive Model (1 Mpa = 145 psi) (cont’d) 113 N g“ =2.0 1.. i. 520 =l.0 2511' ' 'oiaifr'm'fi 'em CampreeiveStrain (d) Hooked vs. Straight Fibers Figure 4.6 Effects of Fiber Reinforcement Index and Concrete Compressive Strength on Compressive Behavior of SFRC as Predicted by the Proposed Constitutive Model (1 Mpa = 145 psi) (cont’d) 4.6 SUMMARY AND CONCLUSIONS Reinforcement of concrete with randomly oriented short steel fibers increases the ultimate strength and especially the post-peak ductility and energy absorp- tion capacity of concrete under compression. The efbctiveness of steel fibers in enhancing concrete behavior under compression depends on the mix proportions of the matrix, the volume fraction, aspect ratio and deformation configurations of fibers, loading versus casrin g direction, specimen geometry, and rate of load- in g. 114 An empirical constitutive model was developed in this study for steel fiber reinforced concretes loaded in compression. This model accounts for the efficts of fiber volume fracrion, aspect ratio and type (straight vs. hooked) as well as the matrix compressive strength on the compressive behavior of steel fiber rein- forced concrete. The model has been developed using results of ninety eight compression tests performed on 150 mm (6 in.) by 300 mm (12 in.) cylindrical concrete specimens with maximum aggregate sizes ranging from 9.5 mm (3/ 8 in.) to 19 mm (3/ 4 in.), incorporating straight or hooked fibers and loaded quasi- statically in the direction of casting. The relatively simple empirical model developed in this study predicts experimental results (for fibrous concretes with relatively wide ranges of fiber and matrix variables) with a reasonable accuracy. More test results are needed for refining the model to consider the effects of maximum aggregate size, speci- men geometry, loading versus casting direction, and the rate of loading. CHAPTER 5 FLEXURAL ANALYSIS OF STEEL FIBER REINFORCED CONCRETE 5.1 INTRODUCTION The improvements in SFRC behavior under compression and tension result in significantly higher improvements in the fiexural strength and ductility of SFRC [24,41,64]. The flexural behavior of SFRC is typically marked by the formation of only one major crack of a critical section, a phenomenon that dis- tinguishes the flexural behavior of SFRC beams from that of beams reinforced with conventional continuous bars (see Figure S.1(a)). This implies that more damage is done to the cracked critical section and a concentration of curvature occurs in the vicinity of this section (Figure 5.1(b)). The critical section tends to suffer severe distortions and, after cracking, plane sections fail to remain plane in the vicinity of the critical section. As loading continues, the crack at the critical section begins to widen and this prompts the use of stress-crack relation- ships rather than stress-strain relationships on the tensile region of bending sec- tion. None of these features was considered in previous investigations [45-51]. Complete flexural load-deflection relationships are analytically constructed in this Study through conducting a flexural analysis of the critical section, and using some assumptions regarding curvature distributions in the vicinity of the critical section. The predicted fiexural load-deflection relationships are com- pared with experimental results, and the model is also used to conduct parametric studies on the effects of different matrix and fiber variables on the 115 116 Conventional Beam Steel Fiber Reinforced Beam Moment Distribution Moment Distribution fl Compressive Curvature Distribution Compressive Curvature Distribution mm Tensile Curvature Distribution Tensile Curvature Distribution (3) Conventional Beam (b) Fiber Reinforced Beam Figure 5.1 Crack Patterns and Possible Curvature Distributions 117 fiexural performance of SFRC. The results of these parametric studies are evaluated using Statistical analysis (by factorial design). 5.2 FLEXURAL AN ALYSIS The flexural analysis procedures developed for steel fiber reinforced concrete beams are described in this section. Attempts are made to consider the effects of cracking at the critical section on the release of tensile strains and the concentra- tion of compressive strains in the vicinity of the critical section. 5.2.1 Curvature Distributions Before the cracking the behavior of fiexural beams can be regarded as elas- tic, and thus conventional beam theories are applieable. With increasing flexural loads the maximum tensile Strain eventually reaches the tensile strain at peak tensile stress of steel fiber reinforced concrete, where the crack starts to open and this marks the first crack strength of the beam (PC, in Figure 2.24). Up to this stage, the moment and curvature diSIributions are similar in shape (Figure 5.2(a)). Once one major crack starts to open at the critical section, ten- sile strains tend to be released near the crack, thus generally preventing the for- mation of another crack near the first crack. The opening of the crack on the tensile side will be accompanied by the concentration of compressive strains near the crack on the compressive side. Hence, upon cracking the distributions of moment, tensile strains and compressive Strains cease to be similar in shape (see Figure 5.2(b)). In the post-cracking region, the pull-out action of fibers gen- erally provides the beam with the capacity to resist increasing loads after crack- ing, and to maintain large fractions of its peak flexural load at large 118 Deformation Moment fl 83:11:11." A A Tensile Curvature (a) Before Cracking Figure 5.2 Moment and Curvature Distributions 119 Deformation P/2 P/2 Tension Relief Zone Moment Distribution Compressive Cuvature ‘10 Distribution Tensile Curvature Distribution 4’: (b) After Cracking Figure 5.2 Moment and Curvature Distributions (cont’d) 120 deformations in the post-peak region (Figure 2.24). Further widening of the crack with increasing deformations further disturbs the beam in the critical region, leading to increased concentration of compressive strains. The fact that only one crack appears in most beams subjected to fiexural loads indicates that at. a distance outside the critical section, where tensile strains drop below the cracking resistance, the curvature and strain distributions would tend to follow conventional elastic beam distributions. A region is defined in this investigation as the critical region (with a length 2°Lc, along the beam span) in which the external moment along the beam is greater than or equal to the one correspond- ing to the first crack moment (Ma). Outside the critical region the elastic beam theory is assumed to be applicable. As the fiexural load increases beyond the first-crack load, the critical region will Spread outward continuously until the external load reaches its ultimate value. The exact distributions of tensile and compressive strains in the critical region are rather complex. Simplifying assumptions have been used in this investigation in order to simulate the complex behavior in this region. Once the beam reaches its ultimate flexural load and resistance starts to decrease with increasing deformations, the critical region is assumed to stabilize (in length), with curvature at the boundary assumed to stay constant at the first-crack value. Increased fiexural deformations in the post-peak region, in spite of the continuous drop in load, lead to further increase in compressive strains in the crtical region. Elastic fiexural deformations outside critical region tend to decrease with drop in load in the post-peak phase of behavior. Hence, hating that at the boundaries of the critical region, curvatures are assumed to be constant, there is a tendency in deformations to increasingly concentrate near the center of the critical region. This concentration takes the form of crack widening and increased strains in tension and compression regions, respectively. 121 5.2.2 Analysis ofthe Critical Section The crack shape at the critical section is assumed to be linear and sym- metric about a plane normal to the beam longitudinal axis (see Figure 5.3). As the crack opens, the tension part of the beam in the critical region can- tinously relieves its Strains while compressive strains continue to increase. The increase in compressive Strains after cracking is needed to satisfy the deformation compatibility requirements which would have been disturbed if, after cracking, the tensile and compressive strains were still assumed to be comparable (see Fig- ures 5.4(a) and (b)). Thus, the crack opening angle (BC, in Figure 5.4(b)) can be obtained by computing the difference in ratations associated with compressive and tensile strains in the critical region (Figure 5.4(b)): 6cr = 9c - et (51) LC? 111 (¢c(x)- ¢. Tensile Strain Figure 5.4 Deformation Compatibility after Cracking 124 ¢c1¢, = compressive and tensile side curvatures , respectively ; and x = distance from center (cracked section). The tensile Side curvature is assumed to vary linearly from zero at the crack to a value equal to the compressive side curvature at the boundary of the critical region. The assumed post-cracking distributions of compressive Side and tensile side curvatures in the critical region are thus as follows: 0.0:) = fl ¢. comvcrm-z >- NON-CONJUGATES (b) Theorem 2 Figure 6.3 Main Theorems Used in Powell’s Algorithm 169 The k"I iteration of this method starts with a current point xk and n direc- tions, dlw’ , j= 1,2,...,n. At the beginning, x1 and (In are assumed to be given. 1. LEI y“): Xk. 2. Find 7t? 1 which minimizes f (yk.j_1 + lei-d”) and let ykJ = yk_j-1+ 11’2ko fOI' j= 1,2,...,II. 3. LCtfizyk'n—Xk. 4. Find 1,: which minimizesf (ykm + Rafi) and let ka = y,” + KJ-Sk. 5. LC! dk+IJ = dk,j+l’ j: 1,2,...,H'1, and dk+l,n = 5". The dII'CCtIOII ko is discarded in favor of a new direction 8,,. 6. Go to step 1 and restart for (k + I)” step. The k‘h cycle which contains (n+1) subcycles for finding minimum along the given direction is schematically shown in Figure 6.4 for n = 2. In this Fig- ure, superscripts and subscripts represent the subcycle number and iteration number in a certain subcycle, respectively. The above Powell’s algorithm produces n mutually conjugate directions in 11 iterations (i.e., k = n ). The method, however, breaks down when the n direc- tions for an iteration become linearly dependent. This happens if (ykfi - yk.0)T °(y,‘.1 - y”) becomes zero. This implies that by discarding d“ for (k + 1)"' step, one of the conjugate directions is lost and thus it never reaches minimum of the funtion. Powell [74] modified his basic method to overcome this type of difficulty by allowing a direction Other than d“ to be dis- carded after the It” iteration. This modification, however, sometimes allows one of the mutually conjugate direcr ions to be discarded, so that more than n itera- tions are required in order to find the minimum of a positive definite quadratic 170 SECOND CYCLE (a) Iterative Procedure “31,,‘01, ..... ...... ..... ...... ..... ..... ..... ...... ....... ...... ..... ..... ..... ..... ....... ~~~~~ ....... ........ ...... Ic- ..... ......... ...... ..... ........ ..... ..... ..... ..... ....... GLOBAL MINIMUM IS GUARANTEED BY THEOREM 2. (b) Conjugacy of Vetors (x3l - x01) and (x32 - x02) Figure 6.4 Main Theorems in Powell’s Algorithm for N = 2 171 function. The direction to be discarded, if any, is chosen using the condition that by discarding one of du’s, j- 1,2,...,n, the determinant of the matrix, [ dk.1,dk3,...,dk.j-1, a, du,,,...,tt,, 1 is made as large as possible. For It“ iteration, it proceeds as follows : 1. Let “.0 = x, and for j= 1,2,...,n, search from yaw--1 in the direction ko for a minimum at y”. 2- Find A = Max.lf (I'm-1)" fUrJ)|= If (In-1) " fU'IuQH Whflcq is the value of j maximizing A in j= 1,2,...,n. 3. Define f, = f (Yea) and f2 = f (3'th ). Then evaluate f3 = f(2-yt,. ' 3m)- . 1 4. If either f32 1 or (f,- 2°f2+f3)-(f,-f2- A)22 E-A-(fl— f3)2 then use old directions, d”, j= 1,2,...,n, for (It + l)”' iteration and put Xtt+1 = y” = “+1.0. Otherwise use rule 5. 5. Determine Xr+1 as in the It“ iteration of Powell’s basic method, but take the directions of {dk.1,dm,...,dk4-1,dm+1,"..ko A} . The supporting theorems and proofs are given in References 73 and 74. In Powell’s method, (n+1) line searches are nwded to generate one conju- gate direction. Therefore, to find the global minimum point (assuming that the given function is quadratic and positive definite) a total of n(n+ 1) line searches are required. Since in the Powell’s method, the error function is being approxi- mated by a quadratic function, it seems to be appropriate to use quadratic line search. In the present study, the method of quadratic line search described by Powell ['73] has been used. This method basically discards one of three points 172 which is farthest from the turning point and then obtains a new current point by taking the specified distance in the given direction such that the function value is decreasing (see Figure 6.5). The newly found point and two existing points are then used to fit the quadratic curve, and the same procedures are repeated until the approximated minimum is within a satisfactory tolerance. The Point To be Discarded Quadratic Curve —\ Current Minimum Point "0 I‘ First Interval 'l V Reduced Inverval 7' Figure 6.5 Quadratic Line Search 173 6.5 SELECTION OF PARAMETERS IN ”SYSTEM IDENTIFICA- TION" The flexural model contains ten material-related and ten constitutive behavior-related factors (see Figure 5.10). The variations in some Of these fac- tors have significant efficts on the behavior of SFRC under flexure, while vara- tions in other factors result in negligible efhcts on the flexural behavior of SFRC. Since it is not economical and practical to Optimize all these factors in the process of "System Identification," factors whose variations result in significant efficts on the flexural behavior Of SFRC need tO be selected as the ”System Identification " parameters. Chapter 5 has examined the influence Of each factor on the flexural peak load (P), flexural ductility (D), flexural toughness (A) and overall flexural behavior of steel fiber reinforced concrete (V). It was Observed that in the case of material-related factors, the fiber peak pull-out strength (1,, ), fiber diameter ((1,), fiber length (I, ), fiber volume fraction (V, ), matrix tensile strength (6“,), and fiber slip at residual pull-out strength (5,) are the most influential factors deciding the flexural behavior Of SFRC. As far as the constitutive behavior- related factors are concerned, it was shown in Chapter 5 that their efbcts are negligible when compared with those of the material-related factors. Among the six influential material-related factors, those representing fiber dimensions (i.e., df and If) as well as the volume fraction of fibers (Vf) should be known inputs while analyzing some flexural test data Obtained for SFRC. This further reduces the number Of "System Identification" parameters and leaves only three material-related factors to be entered as parameters in "System Identificationz" fiber peak pull-out strength (1,, ), fiber slip at residual pull-out strength (5,) and matrix tensile strength (Om'). It is worth mentioning that 174 the tensile strength Of SFRC can be determined once the values of these three factors are Obtained through analysis Of flexural test results using "System Identification.” 6.6 RESULTS OF SYSTEM IDENTIFICATION Table 6.1 summarizes conditions of the SFRC flexural tests considered for "System Identification," and also presents the Optimized values Of the three main parameters Obtained from "System Identification." Figures 6.6(a) through 6.6(k) illustrate the comparisons between the experimentally Obtained and theoretically optimized flexural loadodeflection curves. Satisfactory correlations are Observed in these Figures. From Table 6.1, the optimized values of three parameters are found to be larger than the values obtained from direct tension and material tests (see the comparison presented in Table 6.2). The experimen- tal data presented in Table 6.2 are the averages Obtained from several direct ten- sion and fiber pull-out tests performed on materials comparable to those used in flexural tests. The matrix tensile strength (Om') and performance Of fibers Obtained from the analysis of flexural‘test results may be improved in com- parison with those Obtained from direct tension and pull-out tests due to the strain gradient efficts under flexural loading condition, which generally lead to improved tensile performance of the material [27]. The improvements in pull- out performance in flexural test specimens over those Obtained from single fiber pull-out tests may also be attributed tO the positive efbcts Of fiber reinforcement at the surrounding matrix (noting that single fiber pull-out tests are generally conducted using non-fibrous surrounding matrices) in flexural test specimens. Reference 27, using an analysis Of experimental data, has also reported increase 175 Table 6.1 Test Conditions and Optimized Values from System Ident ificat ion . Refs. Test Specimen Fiber f'c Opt'd Values Error Itr. NO, N°' “WW W 0131' W St 44 1 100 100 300 strt 0.56 30 0.01 (40) 5.032 6.174 2.72 30004000025: 4; 300 strt 0.56 30 0.015 (40) 5.895 5.036 3.441 . 3 1% 18.0 300 strt 0.56 30 0.02 (40) 7.132 4.413 3.447 0.000011 6 100 300 strt 0.56 30 0.01 34.6 3.332 3.831 2.198 0.010869 6 45 ; I% 100 300 strt 0.56 30 0.015 34.6 4.649 3.933 3.121 0.000273 3 6 100 100 300 strt 0.56 30 0.01 48 3.032 5.0 3.0 0.026306 2 7 100 100 300 strt 0.56 30 0.01 24.7 2.564 2.752 2.56 0.000494 100 300 hook 0.5 30 0.01 (40) 3444 9.291 3.085 0.000231 3 75 3 I3 100 300 hook 0.5 30 0.01 (40) 3.831 7.73 6.247 0.000967 4 10 100 100 300 hook 0.5 30 0.01 (40) 3.695 5.371 2.957 0.004180 2 11 100 100 300 hook 0.5 30 0.01 (40) 2.57 6.25 2.887 0.003061 3 Values in parenthesis are assumed ones. 176 Table 6.2 Comparison of the Tension Test Results with the Optimum Values Of Parameters in Analysis of Flexural Test Results Using "System Identification.” Refs. Test Fiber 0; Ratios No. No. me (1f 1r vr 50.33207) 0.0/ 0.: 3°! 3.. St°l st 44 1 strt 0.56 30 0.01 2.1 2.4 2.35 0.97 2 strt 0.56 30 0.015 2.1 2.8 1.92 1.23 3 strt 056 30 0.02 2.1 3.4 1.68 1.23 45 4 strt 0.56 30 0.01 1.95 1.7 1.45 0.97 5 m 056 30 0.015 1.95 2.4 150 0.11 6 strt 0.56 30 0.01 2.30 1.32 1.90 1.07 7 strt 0.56 30 0.01 1.65 1.55 1.05 0.91 75 8 book 0.5 30 0.01 2.1 1.63 2.07 1.10 9 book 0.5 30 0.01 2.1 1.82 1.72 2.23 10 hook 0.5 30 0.01 2.1 1.76 1.20 1.05 11 hook 0.5 30 0.01 2.1 1.22 1.35 1.03 177 40000 2 :34) test matrix fiber (strt) opt’d vort- " No. f'c df If Vf rm to Sr (MIN) (nu) (mu!) 5) Mt») (Mpa) (Inn) 'i 1 40- oss 30 1.0 5.0 at 27 J t Assumed Value- Load(N) Deflection(mm) (a) Test Results from Reference 44 40000 : Niki-2) test matrix fiber (strt) opt'd values '3 No. f'c df If Vf an tn Sr (Min) (mm) (mm) 0‘) (MP!) (WI) (mm) ~ 2 40- 056 30 1.5 5.9 5.0 3.4 i 4 1" ‘ Assumed Vslnu '1 o l I I 4 T r I I T I I s s I I ‘ T I s I ' s I I T 0.0 0.5 1.0 1.5 210 2.5 Deflectionlmmi (b) Test Results from Reference 44 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Flexural Load-Deflection Curves 178 m . : m4 test matrix fiber (strt) . qu’d vshra ‘ No. f'c df If V! an to Sr 11. (MP!) (mm) (mu) (9) 041») (MP9) (min) ', 3 40' I ‘ Assumed Vsha I I I I I I I I— I I I I I I I I I I r r I I T r 0.0 0.5 130 as 2.0 Deflection(mml (c) Test Results from Reference 44 40000 2: m.) test matrix fiber (strt) opr‘d vshes '1 No. f’e dr 1: vr an In Sr (Mpa) (mm) (M) (*1 (Mpa) (Mpa) (mm) ‘ 4 34.6 0.56 30 1.0 33 3.8 22 r I T s I I . I s I I r r I I 1 I . . 010 0:5 1.0 1.5 2.0 Deflection(mm) (d) TeSt Results from Reference 45 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Flexural Load-Deflection Curves (cont’d) 179 4oooo Z... m7) test matrix fiber (strt) Opt’d who ‘ Na f'e at it vr an to St (Mpa) (mm) (M) (‘5) (Min) (Mpa) (mm) 7 5 34.6 0.56 30 1.5 4.7 3.9 3.1 2.5 Deflectionlmml (e) Test Results from Reference 45 ' apthnlnd -- - “be (un) ”I'd vshra _ u“ a I, test matrix 4 No. f‘c dr 1: vr an to 8: (Mar) (mm) (mm) (‘5) (MP!) (Min) (mm) 7 6 48 ass 30 10 3.0 5.0 3.0 O I T 1 . . . j F . . . I . r F r I I . I ‘ 0.0 0': 1.0 1.5 2.0 2.5 Deflection(mm) (1) Test Results from Reference 45 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Flexural Load-Deflection Curves (cont’d) 40000 ; -. m fi'fl—fl — “(3%) test ““"i‘ 555' (strt) Opt’d values ‘ No. r: at it vr an to Sr (MP0 (mm) (nun) (5) (MP9) ("I”) (mm) -1 7 24.7 0.56 30 1.0 2.6 28 2.6 .l A Z ’0’ 20000 8 ..J 0 r r t I IIIIIIIIIIIFYTTT‘TTTT—I" 0.0 0.5 1.0 1;: 2.0 2.5 Deflection(mm) (g) Test Results from Reference 45 40000 . : ”(MO-1) test matrix fiber (hook) opt’d values .. No. PC at 1r vr an to st (MP!) (mm) (min) (‘5) (MP!) Mpa) (mm) a ......... s 40' 05 30 1.0 3.4 9.3 3.1 .’ ~ ~-. ‘ Assumed Values A I 1' ~~~~~~ Z .3 ~~~~~~~ I; 20000— .’ """" 8 t' ..l r' t’ l I I I I l I o ' I I I Y I j I r I I I I I I I I fl I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 Deflectionlmm) (h) Test Results from Reference 75 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically 180 Optimized Flexural Load-Deflection Curves (cont’d) 181 40000 —- mod , __ NR_5_2’ tea matrix fiber (hook) opt’d values NO. f'c df If Vf an tn Sr (MP!) (In!!!) (M) ('1) (MP!) (Mpa) (mm) 9 40' 0.5 30 1.0 3.8 7.7 6.3 ‘ Assumed Value ‘- - -- - --- .- - - - -- l I I I I I l I I I I I r 1 1'0 ‘05 Deflection(mm) (i) Test Results from Reference 75 4oooe : 3% test matrix fiber (hook) opt'd values “ No. f'c df 1f Vf an to Sr (Mpa) (min) (mm) (‘5) (MIN) (MP!) (mm) ~ 10 40' 0.5 30 1.0 3.7 5.4 3.0 .1 ‘ Assumed Valrres I I ’ . i . z I . I 0.0 0.5 1.0 1.5 2.0 2.5 Deflectionlmm) 0) Test Results from Reference 75 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Flexural Load-Deflection Curves (cont’d) 182 : a“ ’ test matrix fiber (hook) Qt’d valrres .. NO. P: df If V! an tu Sr (MP!) (mm) (min) (5) (MP!) (MP!) (mm) - 11 40- 05 30 1.0 26 6.3 29 4 ‘Aaanrrred Vahres A Z V '0 20000-1 -------- g ~~~~~~~~~~~~~ ..1 d ............. t' 44 o 1 a I o I fl I I T T 7 a I " I a I I— I I I e I 0.0 015 1.0 1.5 2.0 2.5 Deflectionlmm) - (k) Test Results from Reference 75 Figure 6.6 Comparisons between Experimentally Obtained and Theoretically Optimized Felxural Load-Deflection Curves (cont’d) in pull-out strength under flexure when compared with pull-out strength under tension. In Table 6.3, the tensile strengths Of SFRC calculated using Equation 3.10 with the Optimized values of parameters Obtained from the analysis Of flexural test data are compared with the corresponding predictions Of Equation 3.10 with average values obtained from direct tension and fiber pull-out tests. The com- posite material tensile strengths Obtained from flexural test results is Observed to be larger that those obtained from direct tension and pull-out test results. The ratios of flexure-based to tension-based composite material tensile strengths are presented in the last column of Table 6.3. It is interesting that the average of these ratios is 1.82, which is close to the value Of 1.87 Obtained by taking the e a q a e e ratro of the matrix modulus of rupture (0.623] [C ) to the matrix direct tens1le 183 Table 6.3 Derived Tensile Strengths of Matrices and Composites in Direct Tension and "System Identification." Ref . Test Matrix Tensile Strength Ratios SFRC Tensile Strength (Eq. 3.10) Ratios No. No. o_° tr {-0.624 f'c) 6,0/ fr o,° a, o,°/ o, 44 1 5.03 3.92 1.28 6.85 2.87 2.39 2 5 .9 3.92 1.505 7.45 2.87 2.59 3 7.13 3.92 1.82 8.42 2.87 2.9 45 4 3.32 3.64 0.91 4.45 2.71 1.64 5 4.65 3.64 1.28 6.4 3.07 2.08 6 3.03 4.29 0.706 4.47 3.08 1.451 7 2.56 3.08 0.83 3.38 2.41 1.407 75 8 3.44 3.92 0.88 6.38 3.54 1.8 9 3.83 3.92 0.98 6.31 3.54 1.782 10 3.7 3.92 0.95 5.47 3.54 1545 1 1 257 3.92 0.65 4.56 3.54 1.288 Averages 1.008 1.82 184 strength (0.3324! .fc'). This suggests that the tensile behavior of steel fiber reinforwd concrete is also influenced by strain gradient in a manner similar to the tensile strength of plain concrete. Large variations in the values of parameters (1,, Om', and 3,) Obtained from "System Identification" in Table 6.1 suggest that the highly variable (and unreliable) measurements Of flexural deflections in the pre—peak region have some influence on the analysis of flexural test data using the "System Identification" approach. These variations may also partly result from the fact that some flexural test results reported in the literature were not accompanied by reliable information on basic material properties and thus some assumptions had tO be made on thses properties through the course of "System Identification." 6.7 SUMMARY AND CONCLUSION The complexity and instability involved in testing cementitious materials under direct tension have led to wide spread use of flexural testing for the assess- ment Of the tensile performance characteristics of cement-based materials. The analytical study presented in this Chapter was aimed at analyzing flexural test results for steel fiber reinforced concrete in order to derive information on the tensile constitutive behavior Of the material. An indirect method, "System Identification", was adopted for the analysis Of flexural teSt results. "System Identification" is a process of selecting the form of the model (with a number Of unknown parameters), and then systematically adjusting the parameters until, based on a predefined criterion, the best correla- tion is achieved between the predicted and measured responses. In this study, flexural test results were used to adjust characteristic parameters in the 185 constitutive model Of SFRC. The flexuaral analysis procedure developed in Chapta' 5, which incorporates tensile and compressive constitutive models of SFRC was employed tO predict the flexural behavior of SFRC in the "System Identification" approach. The characteristic parameters initially considered consisted of ten material-related and ten constitutive behavior-related factors. The number Of these parameters was reduced by selecting factors with significant efficts on the overall flexural behavior of SFRC. These factors (parameters) were optimized through "System Identification." Based on the study performed in Chapter 5, six material-related factors (t,,d,,t,,v,,o,,,' and 5,) were found to dominate the overall flexural behavior of SFRC, and considering the fact that fiber dimensions (d, and If) as well as the volume fraction of fibers (VI) are known inputs, the remaining three material-related factors (ta, Om' and 5,) were entered as parameters in "System Identification." In order to measure the correlation between the predicted and measured flexural load-deflection curves, an error function (E) which is defined in a similar manner as the function (V) measuring the overall flexural behavior Of SFRC in Chapter 5, was established. The error function (E) takes into account difhences between predicted and measured values Of peak flexural strength (P), ductility (D), and toughness (A) of SFRC. In order to find the optimum set Of "System Identification" parameters which minimize the value of error function, a nonlinear programming technique based on Modified Powell’s Method was used together with quadratic line search. The method iteratively searches for the minimum error point in the n- dimensional parameter space by producing 11 mutually conjugate directions in n subcycles and then discarding one Of the mutually conjugate directions to avoid the linear dependency of direction vectors. The minimum points selected along these direction vectors eventually lead to the global minimum of a given error 186 function. The use of Modified Powell’s Method has advantages ova Other methods in application to the specific problem Of this study because it does not require the calculation of gradient vector for which analytical expressions should exist. The method could successftu produce the optimum set of parameters which lead to a satisfactory match between the measured and predicted flexural load-deflection relationships. The values of parametas Obtained from "System Identification" are, in most cases, larger than those daived from direct tension tests paformed on SFRC. An analysis Of the results indicated that: (1) The matrix tensile strength (0") and pull-out performance of fibas Obtained from the analysis of flexural test results wae supaior to those Obtained from direct tension and pull-out tests. This may be attributed to the strain gradient effects unda flexural loads. (2) The improvements in pull—out paformance in flexural tests ova those Obtained from single fiba pull-out tests (whae fibas are genaally pulled out Of non-fibrous matrices) may also be attributed to the positive effects Of fiba rein- forcement of the surrounding matrix in flexural test specimens. (3) Tensile strength Of SFRC Obtained from the analysis of flexural test results is larga than tensile strength obtained from direct tension tests on SFRC. The tensile strength of SFRC unda flexure was, on the avaage, 1.82 times the tensile strength of SFRC Obtained fi'om direct tension tests, which is comparable to the ratio Of the modulus Of rupture to direct tensile strength Of plain concrete. This suggests that the tensile behavior Of SFRC is influenced by strain gradient in a manna similar to the tensile strength of plain concrete. (4) Large variations wae Obsaved in the values Of parametas (I... 0,”, and S,) Obtained from "System Identification." This could result from both unreliable measurements of flexural deflections in the pre-peak region in 187 some test results reported in the litaature, and also from the lack Of information on some basic material properties for flexural tests conducted by otha investiga- lOfS. CHAPTER 7 SUMMARY AND CONCLUSION Reinforcement of concrete with randomly oriented short steel fibers improves the tensile strength, and the tensile and compressive toughness of the mataial. Fibas in cementitious matrices arrest and deflect the propagating microcracks. The debonding and pull-out action of fibas under tension, and the confinement of cementitious matrices by steel fibas under compression, are also important mechanisms leading to improvements in concrete behavior in the presence Of steel fibers. Improvements in the flexural paformance of concrete resulting from steel fiba reinforcement are direct consequences of the correspond- ing improvements in the tensile and compressive paformance Of the the mataial. In orda to develop methods for analysis Of reinforced concrete structures incorporating steel fibas, reliable tensile and compressive constitutive models are needed for fiber reinforced concrete. In many applications, SFRC is subjected to flexural forces and thus it is important to develop analytical techniques for predicting the flexural behavior Of SFRC which account for the nonlinear stress distributions and the dominance Of a cracked seCtion in deciding the post-peak peformance Of SFRC. This investigation dealt with three aspects Of SFRC behavior: (1) compres- sive; (2) tensile; and (3) flexural. In the first phase Of this investigation, sum- marized in Chapters 3 and 4, empirical constitutive models were developed for predicting the complete stress—deformation relationships of SFRC unda tensile and compressive stresses. In Chapter 5, an analytical approach was developed 188 189 for simulating the flexural paformance of SFRC, which incorporated the tensile and compressive constitutive models of SFRC developed in Chaptas 3 and 4. Parametric studies wae conducted using the developed constitutive models and flexural analysis procedures in orda to assess the efbcts of different fiba and matrix propaties on the paformance charactaistics of the composite mataial. Finally, the "System Identification" technique was used togetha with the developed flexural analysis prmdure and constitutive models in orda to derive information on the tensile behavior of SFRC using flexural test data. Diffirent phases of this investigation and the related conclusions are sum- marized below. Tensile Constitutive Modeling Theoretical expressions wae daived for the numba of fibas pa unit cross sectional area in fiba reinforced concrete as functions of fiba volume fraction and length, assuming that cross sectional boundaries are the only factors dis- turbing the 3-D random orientation of fibers. Measurements wae made on frac— tured surfaces Of steel fiber reinforced concrete specimens in orda to assess the actual values for the number Of fibas pa unit area in steel fiber reinforced con- crete. Nineteen steel fiber reinforced concrete specimens incorporating different fiba volume fractions and different fiber types were considered in this investiga- tion. Statistical studies were conducted on the measured values of the number of fibas pa unit area for determining the possible efficts of fiba type and loca- tion on the number of fibers per unit area. Comparisons were also made between the theoretical and measured values Of the number Of fibers per unit area in order to determine the efbcts of reorientation of steel fibers inside con- crete during vibration. Recommendations were made, based on the findings of this research, for approximating the number of fibers per unit area in steel fiber 190 reinforced concrete. The following conclusions wae derived from the results Of this investiga- tion: (1) The type of steel fiba (straight vs. hooked) and the location in cross sec- tion with respect to the casting direction (top vs. bottom) did not have any sta- tistically significant effict on the measured value of numba of fibas pa unit area. (2) Vibration of steel fiba reinforced concrete seems to reorient the fibas, resulting in a tendency towards orienting the fibas in horizontal planes. This phenomenon illustrates the higha values for numba Of fibas pa unit area in actual measurements when compared with theoretical predictions. (3) The numba Of fibas pa unit cross sectional area in steel fiba reinforced concrete afta vibration is between the theoretical values daived for 3—D and 2-D random orientation conditions considering the boundary efbcts. A refined concept ("intaaction concept") was proposed for predicting the tensile strength of SFRC. This concept accounts for the partial mobilization Of the fiba pull-out action (intafacial bond stresses) at the peak tensile strength of composite material, and also considas the microcrack arresting action Of fibers and the consequent strengthening of matrix in the presence Of steel fibers. The proposed "intaaction concept" leads to an expression for predicting the tensile strength of steel fiba reinforwd concrete, which incorporates some unknown coefficients to be determined empirically. These coefficients wae decided in this study using a relatively large number Of SFRC tensile strength test results. The theoretical predictions based on the proposed "interaction concept", when compared with those of the commonly used composite mataial and spac~ ing concepts, show a reasonable correlation with test results. More importantly, the relative matrix and fiber contributions to the composite material tensile 191 strength in the proposed "intaaction concept" are representative of the physics of the composite mataial paformance at peak tensile stress. A constitutive model was also developed for predicting the pre-peak tensile stress-strain relationship as well as the post-peak tensile stress-deformation rela- tionship of steel fiba reinforced concrete. The developed post-peak constitutive model accounts for the constributions of fibers crossing the critical section through their pull-out action as well as that of matrix in its post-peak softening range of behavior. Empirical fiba pull-out load-slip and matrix post-peak con- stitutive models wae combined to daive the composite mataial post-peak ten- sile stress-deformation model. The pre-peak constitutive model of the composite mataial developed in this study was an empirical one based on the tension test results reported in the litaature for steel fiba reinforced concrete. The pro- posed constitutive model is shown to compare reasonably well with tension test results paformed on steel fiba reinforced concrete in both the pre- and post- peak regions. Compressive Constitutive Modeling Reinforcement Of concrete with randomly oriented short steel fibas increases the ultimate strength and especially the post-peak ductility and energy absorp- tion capacity Of concrete under compression. The effectiveness of steel fibas in enhancing concrete behavior under compression depends on the mix proportions Of the matrix, the volume fraction, aspect ratio and deformation configurations Of fibers, loading versus casting direction, specimen geometry, and rate of load- ing. An empirical constitutive model was developed in this study for steel fiber reinforced concretes loaded in compression. This model accounts for the effects Of fiber volume fraction, aspect ratio and type (straight vs. hooked), and the 192 matrix compressive strength, on the compressive behavior Of steel fiber reinforced concrete. The model has been developed using results Of ninety eight compres- sion tests paformed on 150 mm (6 in.) by 300 mm (12 in.) cylindrical concrete specimens with maximum aggregate sizes ranging from 9.5 mm (3/8 in.) to 19 mm (3/4 in.), incorporating straight or hooked fibas and loaded quasi-statically in the direction of casting. The relatively simple empirical model developed in this study predicts experimental results (for fibrous concretes with relatively wide ranges of fiba and matrix variables) with a reasonable accuracy. More test results are nwded for refining the model to consida the efbcts of maximum aggregate size, speci- men geometry, loading vasus casting direction, and the rate of loading. Flexural Analysis of Steel Fiber Reinforced Concrete Unda flexural loads, one major crack genaally forms in steel fiba rein- forced concrete at the critical section, in the vicinity of which a relief of tensile stresses occurs. Afta cracking, the critical section sufbrs sevae distortions and thus plane sections do not remain plane in its vicinity. A flexural analysis pro- cedure was developed which gives due considaation to the behavior at and near the critical (cracked) section. The exact distributions of tensile and compressive strains in the critical region are ratha complex. Some simplifying assumptions wae made in order to simulate the flexural behavior at critical region. Before the crack starts to open, the moment and curvature distributions are similar in shape. As the flexural load increases beyond the first-crack load, the critical region is assumed to spread outward, and it stabilizes when the beam reaches its ultimate load, with curvature at the boundary of the critical region assumed tO stay constant at the first-crack value. In the post-peak region, compressive strains in the critical 193 region furtha increase while elastic flexural deformations outside critical region tend to decrease. This, togetha with the assumed constant values of curvature at the boundaries of the critical region, result in a tendency in deformations to increasingly concentrate near the centa (cracked section) of the aitical region. The crack shape at the critical section is assumed to be linear and sym- metric about a plane normal to the beam longitudinal axis. Assuming linear variations in compressive and tensile curvatures at the critical region (where the tensile curvature is assumed to vary from zero at the crack to a value equal to the compressive side curvature at the boundary of the critical region), the crack Opening angle could be Obtained by computing the difference in rotations associ- ated with compressive and tensile strains in the critical region. Maximum crack opening at the extreme bottom laya of the critical section could be Obtained using this crack Opening angle and the neutral axis position obtained by satisfy- ing the equilibrium of tensile and compressive forces at the critical (cracked) sec- tion. A step-by-step incremental approach was adopted for flexural analysis of SFRC beams. In each step, an inaernent is made in curvature on the compres- sive side of the cracked section, and numerical techniques (based on the Modified Regula-Falsi method) wae used to itaatively decide the neutral axis position which satisfies equilibrium conditions. The tensile and compressive constitutive models of SFRC developed in this study wae used in flexural analysis of the critical section. The assumptions described above were then used to derive the flexural behavior Of complete beam using the aitical section behavior at the end Of each step. Using the proposed analytical approach, the flexural behavior and stress profiles at the critical section were investigated at different loading stages with two diffirent fiber volume fractions (0.5% and 1.2%). The results indicated 194 that the peak flexural load at both fiba volume fractions is attained when the crack at critical section has already Opened. This implies that flexural strength is reached in SFRC beams when the tensile behavior has already reached the post-peak region. The flexural strength of SFRC, thaefore, seems to be depen- dent nOt only on the tensile strength of the mataial, but also on its post-peak tensile behavior. This furtha explains why the increases in flexural strength is typically higha than the corresponding increase in tensile strength for given fiba reinforcement conditions. In addition, calculation of modulus of rupture based on linear-elastic flexural analysis equations does not seem to give a characteristic stress value which directly relates to the peak tensile strength of SFRC. While a major fraction of the peak flexural resistance is maintained in the post-peak region for conditions with 1.2% fiba volume fraction, the load- carrying capacity with 0.5% fiba volume fraction drops suddenly in the post- peak regions. The. developed flexural analysis procedure was also used for a numerical parametric study on the influences Of ten material-related and ten constitutive behavior—related factors on the flexural behavior Of SFRC. The significance of these factors in deciding flexural paformance characteristics was examined by simple Obsavations of flexural load-deflection curves and also through statistical analysis based on 2“ factorial design. The aspects of flexural behavior con- sidaed in this study wae flexural peak load (P), ductility (D), toughness (A) and the ovaall flexural behavior of SFRC (V). The following conclusions could be derived using the results of this parametric study: (1) The flexural Strength of SFRC was most sensitive to the variation in matrix tensile strength. (2) Ductility (D), toughness (A) and overall flexural behavior (V) are most influenced by fiber diameter and fiber pull-out strength. 195 (3) The efficts on flexural behavior of the matrix compressive strength, crack opening at which matrix tensile stress diminishes, and fiba slip at peak pull-out load are negligible. (4) Fiber dimensions (fiba diameta and fiba length) as well as fiba volume fraction have almost equally important efbcts on flexural behavior. (5) While the matrix crack Opening at residual matrix tensile Strength has little efficts on different aspects of flexural behavior, fiba slip at residual pull- out strength has relatively important efficts on flexural ductility and ovaall flexural behavior Of SFRC. (6) Fiba—tO-matrix bond strength (1,, ), fiber dimensions and volume frac- tion (61,, If and Vf)’ matrix tensile strenth (Om') and slip at residual pull-out strength (S,) are the most influential factors deciding the flexural behavior of SFRC. (7) Similar Obsavations were derived through analysis using 2" factorial design and also through simple Observation of flexural load—deflection curves. Interpretation of Flexural Test Results Using "System Identification" The complexity and instability involved in testing cementitious materials under direct tension have led to wide spread use of flexural testing for the assess- ment of the tensile performance characteristics Of cement-based materials. The analytical study presented in this Chapta was aimed at analyzing flexural test results for steel fiber reinforced concrete in order to derive information on the tensile constitutive behavior Of the mataial. An indirect method, "System Identification", was adopted for the analysis of flexural test results. "System Identification" is a process Of selecting the form of the model (with a number of unknown parameters), and then systematically adjusting the parameters until, based on a predefined criterion, the hem 196 correlation is achieved between the predicted and measured responses. In this study, flexural test results wae used to adjust charactaistic parametas in the constitutive model Of SFRC. The flexural analysis procedure developed in Chapta 5, which incorporates tensile and compressive constitutive models Of SFRC was employed to predict the flexural behavior of SFRC in the "System Identification" approach. The charactaistic parametas initially considaed consisted of ten mataial-related and ten constitutive behavior-related factors. The number Of these parametas was reduced by selecting factors with significant efbcts on the overall flexural behavior of SFRC. These factors (parameters) wae optimized through "System Identification." Based on the study paformed in Chapta 5, six material-related factors (1“,df.lf,Vf, Om, and S,) were found to dominate the ovaall flexural behavior Of SFRC, and considering the fact that fiber dimensions (d, and If) as well as the volume fraction of fibas (Vf) are known inputs, the remaining three mataial-related factors (1:, , O‘m ' and S ,) were entered as parameters in "System Identification." In orda to measure the correlation between the predicted and measured flexural load-deflection curves, an error function (E) which is defined in a similar manner as the function (V) measuring the ovaall flexural behavior of SFRC in Chapter 5, was eStablished. The error function (E) takes into account differences between predicted and measured values of peak flexural strength (P), ductility (D), and toughness (A) of SFRC. In order to find the optimum set Of "System Identification" parameters which minimize the value of error function, a nonlinear programming technique based on Modified Powell's Method was used togetha with quadratic line search. The method iteratively searches for the minimum error point in the n- dimensional parameter space by producing 11 mutually conjugate directions in n subcycles and then discarding one of the mutually conjugate directions tO avoid 197 the linear dependency of direction vectors. The minimum points selected along these direction vectors eventually lead to the global minimum of a given aror function. The use of Modified Powell’s Method has advantages over otha methods in application to the specific problem of this study because it does not require the calculation Of gradient vector for which analytical expressions should exist. The method could successquy produce the optimum set of parametas which lead tO a satisfactory match between the measured and predicted flexural load-deflection relationships. The values of parametas obtained from "System Identification” are, in most cases, larga than those daived from direct tension tests paformed on SFRC. An analysis of the results indicated that: (1) The matrix tensile strength (0”,) and pull-out paformance Of fibers Obtained from the analysis of flexural test results wae supaior to those obtained from direct tension and pull-out tests. This may be attributed tO the strain gradient effects unda flexural loads. (2) The improvements in pull-out paformance in flexural tests ova those Obtained from single fiba pull-out tests (whae fibers are genaally pulled out of non-fibrous matrices) may also be attributed to the positive efficts of fiba rein- forcement of the surrounding matrix in flexural test specimens. (3) Tensile strength of SFRC Obtained from the analysis of flexural test results is larga than tensile strength Obtained from direct tension tests on SFRC. The tensile strength Of SFRC unda flexure was, on the avaage, 1.82 times the tensile strength Of SFRC obtained from direct tension tests, which is comparable to the ratio of the modulus Of rupture tO direct tensile strength of plain concrete. This suggests that the tensile behavior of SFRC is influenced by strain gradient in a manna similar tO the tensile strength of plain concrete. 198 (4) Large variations wae Obsaved in the values of parametas (1,, on, and S,) Obtained from "System Identification." This could result from both unreliable measurements of flexural deflections in the pre—peak region in some test results reported in the litaature, and also from the lack Of information on some basic mataial propaties for flexural tests conducted by otha investiga- IOI'S. 10. 11. 12. 13. 14. 15. 16. 17. LIST OF REFERENCE ACI Committee 544, "State-Of-the-Art Report on Fiba Reinforced Con- crete," Report:AC1544, IR-82, Amaican Concrete Institute, Detroit, May, 1982. PP.16 Shah, S.P. and Rangan, V., "Fiba Reinforced Concrete Propaties," Journal Of the American Concrete Institute, Feb. 1971, pp. 126-135. Mangat, P.S., "Tensile Strength of Steel Fiba Reinoforced Concrete," Cement and Concrete Research, 1976, Vol.6, pp. 245-252. Shah, S.P., Stroeven, P., Dalhuisen, D. and van Stekelenburg, P., "Complete Stress-Strain Curves for Steel Fiba Reinforced Concrete in Uniaxial Tension and Compression," RILEM Symposium, 1978. Fanella, D. and Naaman, A., "Stress-Strain Properties Of Fiba Reinforced Mortar in Compression," Journal of the Amaican Concrete Institute, Vol.82, No.4, July-August, 1985, pp475-483. William, G.R., "The Effect Of Steel Fibas on the Compressive Strength of Concrete," Fiba Reinforced Concrete, SP44-11, pp. 195-207. J indal, R.L., "Shear and Moment Capacities of Steel Fiber Reinforced Con— crete Beams," Fiba Reinforced Concrete, SP81, American Concrete InSti- tute, 1983, pp. 1-16. Swamy, R.N. and Sa’ad A. Al-Taan, "Deformation and Ultimate Strength in Flexure of Reinforced Concrete Beams Made with Steel Fiber Concrete," Journal Of the Amaican Concrete Institute, Sept.-Oct. 1981, pp. 395-405. Taketo UomOtO, "Shear Strength of Reinforced Concrete Beams with Fiba- Reinforcement," Journal of the Amaican Concrete Institute, Sept.-0ct. 1981. PP. 395-405. GOpalaratnam, V.S. and Shah, S.P., "Softening Response of Plain Concrete in Direct Tension," Journal Of the Amaican Concrete Institute, May-June 1985. pp. 310-323. Diamond, S. and Bentur, A., "On Cracking in Concrete and Fiber Rein- forced Cements," Application of Fracture Mechanics to Cementitious Com- posites, NATO-ARW, Sept. 4-7, 1984, pp. 87-140, Editted by Shah, S.P. GOpalaratnam, V.S. and Shah, S.P., "Failure Mechanisms and Fracture of Fiba Reinforced Concrete," Proceedings of the Fiber Reinforced Concrete Symposium, ACI Convention, Baltimore, November 1986. Shrive, N.G., "Compression Testing and Craking of Plain Concrete," Maga- zine Of Concrete Research, Vol.135, No.122, March 1983, pp. 27-39. Craig, R., McConnell, J., Gamann, H., Dib, N. and Kashani, F., "Behavior Of Reinforced Fibrous Concrete Columns," American Concrete Institute, SP-81:Fiber Reinforced Concrete, 1981, pp. 69-105 20110, RE, "An Overview of Progress in Applications Of Steel Fiber Rein- forced Concrete," Steel Fiber Concrete US-SWEDEN Joint Seminar (NSF- STU), Stockholm 3-5 June, 1985, Editted by Shah, S.P. and Skarendahl, A. Gray, RJ. and Johnston, CD, "The Measurement of Fibre-Matrix Interfa- cial Bond Strength in Steel Fiber Reinforced Cementitious Composites," RILEM, 1978. PP. 317-328. Bentur, A., "Interfaces in Fibre Reinforced Cements," Materials Research Society Symposium, Proceedings, Vol.114, 1988, pp. 133-261. 199 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 200 Naaman, A. and Shah, S.P., "Pull-Out Mechanism in Steel Fiba Reinforced Concrete," Journal of the Structural Divison, ASCE, Aug. 1976, pp. 1537- 1549. ‘ Narayanan, R. and Kareen-Palanjian, A.S., "Factors Influencing the Strength of Steel Fiba Reinforced Concrete," RILEM Symposium, 1978. Pinchin, D.J ., "Intafacial Contact Pressure and Frictional Stress Transfa in Steel Fibre Cement," RILEM Symposium, 1978, pp. 337-344. Burakiewicz, A., "Testing of Fibre Bond Strength in Cement Matrix," RILEM Symposium, 1978, pp. 355-365. Cancho, A.V. and Galvez V.S., "A Micromechanical Model for the Tensile Stress-Strain Curve of Fiba Reinforced Cements," RILEM, 1986. Hannant, D.J., "Fiba Reinforced Cement and Concrete: Part 1; Theoretical Principles," Concrete Society Current Practice Sheet, No.92 GOpalaratnam, V.S. and Shah, S.P., "Micromechanical Model for the Tensile Fracture of Steel Fiba Reinforced Concrete," RILEM Symposium, Proceed- ings, 1986. Walkus, B.R., Januszkiewicz, A. and J auzal, J ., "Concrete Composites with Cut Steel Fiba Reinforcement Subjected to Uniaxial Tension," ACI Journal, October, 1979, pp. 1079-1093. Lim, T.Y., Paramasivam, P. and Lee, S.L., "Analytical Model for Tensile Behavior Of Steel-Fiba Concrete," Journal Of the Amaican Concrete InSti- tute (Mataial Journal), July-August 1987. Swamy, R.N. and Mangat, P.S., "A Theory for the Flexural Strength Of Steel Fiber Reinforced Concrete," Cement and Concrete Research, Vol.4, 1974. pp. 313-325. Soroushian, P. and Bayasi, 2., "Prediction Of the Tensile Strength Of Fiber Reinforced Concrete: A Critique of the Composite Mataial Concept," Fiber Reinforced Concrete, SP 105, Amaican Concrete Institute, pp. 71-84. Naaman, A.E. and McGarry, F.J., "Probabilistic Analysis of Fiber Rein- forced Concrete," Journal Of the Engineering Mechanics, Vol.100, NO.EM2, April 1984, pp. 397-413. Shah, S.P., "Strength Evaluation and Failure Mechanisms of Fiba Reinoforced Concrete," Proceedings of the International Symposium on Fiba Reinforced Concrete, Dec. 16-19, 1987, Madras, India, pp. l.3-1.9. Pakotiprapha, B. and Lee, 3.1., "Mechanical Properties Of Cement Mortar with Randomly Oriented Steel Wires," Magazine of Concrete Research, Vol.26, No.86, March 1974, pp. 3-14. Hughes, B.P., "Fibre Reinforced Concrete in Direct Tension," Proceedings, ICE Confaence on Fiba Reinforced Materials: Design and Engineaing Applications, London, March 1977. Soroushian, P., Lee, CD. and Bayasi, 2., "Fiber Reinforced Concrete: Theoretical Concepts and Structural Design," Proceedings of the Interna- tional Symposium on Fiber Reinforced Concrete, Michigan State University, Feb. 1987, pp. 81-819. Romualdi, JP. and Mandel, J.A., "Tensile Strength Of Concrete Affected by Uniformly Distributed Closely Spaced Short Lengths Of Wire Reinforce- ment," Journal Of the American Concrete InStitute, Vol.61, June 1964, pp. 657-671. 35. 36. 37. 38. 39. 41. 42. 43. 45. 47. 49. 50. 51. 52. 53. 201 Soroushian, P. and Lee, C.D., "Tensile Strength of Steel Fiba Reinforced Concrete: Correlation with Some Fiba Reinforcement Properties," Journal Of the Amaican Concrete Institute (Accepted for Publication). Soroushian, P. and Lee, C.D., "Distribution and Orientation of Fibers in Steel Fiba Reinforced Concrete," Journal Of the American Concrete Insti- tute (Accepted for Publication). Soroushian, P. and Lee, C.D., "Tensile and Compressive Constitutive Models for Steel Fiba Reinforced Concrete," Proceedings, Intanational Symposium on Steel Fiba Reinforced Concrete, London, U.K., 1989. Slate, F.O., "Stress-Strain Response and Fracture of a Concrete Model in Biaxial Loading," Journal of the Amaican Concrete Institute, Aug. 1971, pp. 590-599. Shah, S.P., "Intanal Cracking and Strain-Softening Response of Concrete Unda Uniaxial Compression," Journal Of the Amaican Concrete Institute (Mataials Journal), May-June 1987, pp. 200-212. Soroushian, P. and Lee, C.D., "A Physical Simulation of the Fiber Rein- forced Concrete Behavior unda Compression," Proceedings Of International Symposium on Fiber Reinforced Concrete, Madras, India, Dec. 16-19, 1987, pp. 1.3-l.9. Parviz Soroushian, Ki-Bong ChOi, and Abdulaziz Alhamad, "Dynamic Con- stitutive Behavior of Concrete," Journal of the American Concrete Institute, March-April 1986, pp. 251-259. Kormeling, H.A., Reinhardt, H.W. and Shah, S.P., "Static and Fatigue Pro- perties Of Concrete Beams Reinforced with Continuous Bars and with Egbegs," Journal Of the American Concrete Institute, J an.-Feb. 1980, pp. 4 . ChO, R. and Kobayashi, K., "Flexural Charactaistics of Steel Fiba and Polyegllj’tylene Fibre Hybrid-Reinforced Concrete," Composites, April 1982, pp.1 -168. Sakai, M. and Nakamura, N., "Analysis Of Flexural Behavior of Steel Fiba Reinforced Concrete," RILEM Symposium, 1986. Parameswaran, V.S. and Rajagopalan, K., "Strength of Concrete Beams with Aligned and Random Steel Fibre Micro-Reinforcement," RILEM Sym- posium, 1977, pp. 95-103. Swift, DO. and Smith, R.B.L, "The Physical Significance Of the Flexure Test for Fibre Cement Composites," RILEM Symposium, 1978, pp. 463-478. Rafagopalan, K., Parameswaran, V.S. and Pamaswamy, G.S., "Strength Of Steel Fiber Reinforced Concrete Beams," Indian Concrete Journal, Jan. 1974, pp. 17-25. Hughes, B.P. and Fattuhi, N.I., "Predicting the Flexural Strength Of Steel and Polypropylene Fiber-Reinforced Cement-Based Beams," Composites, Jan. 1977, pp.57-61. Laws, V. and Walton, P.L., "The Tensile-Bending Relationship for Fibre Reinforced Brittle Matrices," RILEM Symposium, 1978, pp. 429-438. Johnston, CD. and Coleman, R.A., "Strength and Deformation of Steel Fiber Reinforced Mortar in Uniaxial Tension," An International Symposium; Fiba Reinforced Concrete. AC1, SP44, pp. 178-193. Lim, T.Y., Paramasivam, P., Mansur, M.A. and Lee, S.L., "Tensile Behavior Of Steel Fibre Reinforced Cement Composites," RILEM Symposium, 1986. 54. 55. 56. 57. 58. 59. 61. 62. 63. 65. 66. 67. 68. 69. 70. 202 Walpole, R ..e and Mya, R. H. "Probability and Statistics for Engineers and Scientists," McMillan Publishing CO., Reading, 1978. Bonzel, J. and Schmidt, M., "Distribution and Orientation Of Steel Fibas in Concrete and Their Influence on the Charactaistics of Steel Fiba Concrete," Igggeedings, RILEM Symposium on Fiba Reinforced Cement and Concrete, Swamy, R. N. and Mangat, P. S., "Flexural Strength Of Steel Fiba Rein- forced Concrete," Proceedings, Institute of Civil Engineas, Part 2, Dec. 1974, pp. 701-707. Fanella, D. and Krajcinovic, D., "Continuum Damage Mechanics of Fiba Reinforced Concrete," Journal of Engineaing Mechanics, Vol.111, No.8, Aug. 1985, pp. 995-1009. Hillaborg, A., "Numaical Methods to Simulate Softening and Fracture Of Concrete," Fracture Mechanics of Concrete: Structural Application and Numerical Calculation, Editted by Sih, G.C. and Ditommaso, A. Hughes, B.P. and Fattuhi, N.I., "Fibre Bond Strength in Cement and Con- cr6e6te," Magazine of Concrete Research, Vol. 27, No.92, Sept. 1975, pp. 161- 1 . Burakiewicz, A., "Testing Of Fibre Bond Strength in Cement Matrix," RILEM Symposium on Fiba Reinforced Cement and Concrete, 1978 Stroeven, P., de Haan, Y.M. and Bouta, C., "PUll-Out Tests of Steel Fibas," RILEM Symposium, 1978 GOpalaratnam, V. S. and Abu- Mathkour, J., "Investigation Of the Pull-Out Charactaistics Of Steel Fibas from Mortar Matrices," Proceedings Of Inta- national Symposium on Fiba Reinforced Concrete, Dec. 16-19, 1987, Madras, India, pp. 2.-2012.211. Naaman, A..,E Argon, AS. and Moavenzadeh, F., "A Fracture Model for Fiba Reinforced Cementitious Materials," Cement and Concrete Research, Vol.3, 1973, pp.39-47. Bayasi, 2., "Mechanical Propaties and Structural Application of Steel Fiba Reinforced Concrete," Ph.D Thesis, Vol. 11, Michigan State Univasity, 1988. Birkirna, D.L. and Hossley, J.R., "Comparison Of Static and Dynamic Behavior of Plain and Fibrous-Reinforced Concrete Cylinders," Technical Report No.4-69, Dept. of Army, Ohio River Division Laboratories, Corps of Engineers, Dec. 1968. Soroushian, P. and Bayasi, 2., "Optimum Use of Pozzolonic Materials in Steel Fiba Reinforced Concrete," Report No.88, Transportation Research Board 68th Annual Meeting, Jan. 22-26, 1989, Washington, DC. Scott, B.D., Park, R. and Priestley M.J.N., "Stress-Strain Behavior of Con- crete Confined by Overlapping Hoops of Low and High Strain Rates," J our- nal Of the Amaican Concrete Institute, Jan-Feb. 1982, pp. 13-27. Soroushian, P. and Sim, J., "Axial Behavior Of R/ C Element under Dynamic Loads," Journal of the Amaican Concrete Institute, Vol.83, No.6, Nov.-Dec. 1986. PP. 1018-1025. Conte, SD. and deBoor, C., "Elementary Numerical Analysis," McGraw-Hill CO., Reading, 1980. GOpalaratnam, V.S. and Shah, S.P., "Properties of Steel Fiber Reinforced Concrete Subjected to Impact Loading," Journal of the American Concrete Institute, Jan-Feb. 1986, pp. 117-126. 71. 72. 73. 74. 75. 76. 203 Stanton, J .F. and McNiven, H.D., "The Developement of A Mathematical Model to Predict the Flexural Response Of Reinforced Concrete Beams to Cyclic9Loads,‘Using System Identification," Report NO.UCB/ EERC-70/ 02, Jan.1 79. Luenbaga, D.G., "Introduction to Linear and Nonlinear Programming," Addison-Wesley Publishing Company, Reading, Mass, 1973. Powell, M.J.D., "An Efficient Method for Finding the Minimum of a Func- tion of Sevaal Variables Without Calculating Daivatives," Computa Jour- nal, Vol.7, 1964. Walshg, G.R, "Methods Of Optimimtion," John Wiley and Sons Ltd., Read- ing, 1 75. Soroushian, P. and Ateff, K., "Mechanical Performance Of Latex Modified Steel Fiba Reinforced Concrete," Report, Michigan State University, Feb. 1989. Soroushian, P. and Mirza, F., "Efficts Of Different Steel Fibers on the Flex- ural Paformance of SFRC," Report, Michigan State University, Mar. 1989.