FIBER REINFORCED CEMENT AND CONCRETE COMPOSITES

Transcription

1 FIBER REINFORCED CEMENT AND CONCRETE COMPOSITES by Antoine E. Naaman, Ph.D. Fellow and Honorary Member ACI; Fellow ASCE; Fellow PCI; Fellow IFS Professor Emeritus, Department of Civil Engineering and Environmental Engineering University of Michigan, Ann Arbor Hardcover; 765 pages; 7.25x9.5 in. ISBN LCCN This book is written for advanced students, engineers, researchers, educators, and practicing professionals. It is intended as a basic reference, a comprehensive source of information, and a thorough teaching text. It stresses the fundamental understanding of FRC behavior and modeling, and provides the readers with a sufficiently strong basis to design an FRC composite, design structures with an FRC composite, and use FRC composites with confidence. The book is organized into 15 chapters (see below), which can be assembled according to their intended function. Chapter 1 presents a general background and overview of historical evolution of fibers and FRC composites. The following three chapters address the question: What do fibers do to concrete as a material? They offer extensive information on the behavior and properties of the resulting composite (tension, compression, bending, shear, toughness, fracture energy, ductility, time-dependent properties, fatigue, fire resistance, etc). Chapter 5 describes different classes of applications, their purpose, and numerous examples to answer the question: What do fibers do to structural concrete? Chapter 6 deals with fabrication methods and mix design of FRC composites; potential challenges in manufacturing the composite, including proper fiber dispersion and mix workability, are addressed with suggested solutions. Chapter 7 develops the fundamental relations related to fiber and matrix volumes or weights and their fractions, fiber count, fiber spacing, specific surface, equivalence relations, and the like.

2 Chapters 8 to 14 focus specifically on analytical modeling (tension, compression, bending, shear, elastic modulus, bond, stress-strain relationships, etc), and Chapter 15 offers a perspective looking at the challenges ahead. This book will surely help the reader find rational answers and solutions to the many questions that arise when dealing with fiber reinforced concrete. Whenever possible, a consistent notation and widely accepted symbols are adopted, and all symbols used in the text are defined and summarized in Appendix A. The book utilizes primarily the International System of Units (SI); however, all important tables, figures, and design information, are given in dual units (SI and US). Conversion factors from US to SI units and equivalence relations for some dimensionally inconsistent equations are given in Appendix B. An extensive list of references is given at the end of each chapter. Chapter 1 Introduction Overview - History OBSERVED BEHAVIOR BASICS MODELING What Do Fibers Do to Concrete? Chapter 2: Key Mechanical Properties (tension, compression, bending, shear) Chapter 3: Temperature, and Cyclic Dependent Properties (shrinkage, creep, fire, corrosion, durability, fatigue) Chapter 4: Toughness, Ductility, Fracture Energy (surface energy, fracture toughness, indices) What do Fibers do to Structural Concrete? Chapter 5: Applications: Classification, Purpose, Examples Chapter 6 Fabrication and Mix Design Workability Practical V f Apparent V f Typical mixtures SCC UHPC Packing density Chapter 7 Equivalence Relations Volume fraction Weight fraction Specific gravity Specific surface Fiber spacing Fiber number Denier to mm Tex and Dtex Probability Cost Mix proportioning example Chapter 8 Modeling Unit Cell Chapter 9 Modeling Tension Chapter 10 Modeling Compression Chapter 11 Modeling Bending Chapter 12 Elastic Modulus Chapter 13 Bond Chapter 14 Strain-Hardening FRC: Chronological Evolution Chapter 15 Looking Ahead

3 CONTENTS. Preface xxiii Acknowledgments xxv Chapter 1 Introduction: Overview - History Composites and Fiber Reinforced Composites Fiber Reinforced Cement and Concrete (FRC) Composites FRC Composites: Historical Brief Development of Fibers for Concrete Pioneering Developments Modern Developments Why the Initial Lack of Success Classification and Characterization of Fibers for Cement and 12 Concrete Matrices Suggested Characterization for Evaluation of 13 Mechanical Performance Classification by Material Type Typical Properties of fibers for Use in Concrete Macro and Micro-Fibers Fiber Mats Common Ranges of Fiber Geometric Parameters Typical Fiber Reinforcing Parameters and 23 Characteristics Important for Design 1.6 Typical Range of V f and Mixing Limitations Fiber-Matrix Reinforcing Effectiveness: Basic Approach Simplified Mechanics for Initial Design Pros and Cons of Fibers Used in Concrete Applications: Types and Level of Usage Cost Considerations: Fiber Volume Fraction versus Weight 33 Fraction Background Density Ratio and Effect on Cost FRC versus other Fiber Reinforced Composites: 35 Fundamental Difference 1.13 FRC Composites: Overall Design and Challenges The Continuous Path to Higher Performance Whiskers and Nano-Fibers Introduction to Other Chapters and Conclusion References 41 Books, Symposia, Web Sites, US Patents, References Cited Chapter 2 What Do Fibers Do To Concrete: Part I Key 47 Mechanical Properties 2.1 Introduction 47

4 2.2 Understanding the Benefits, Drawbacks and Limitations of 49 Fiber Reinforcement Changes at the Material Level: Fresh State Changes at the Material Level: Hardening State Changes at the Material Level: Hardened State Transition Zone Changes at the Structural Level Matrix Compatibility and Optimization Trade-Offs, Limitations and Caution What Do Fibers Do to Concrete in Tension? Brittle and Ductile Materials Typical Failure Modes in Tension General Classification: Strain-Softening and Strain- 57 Hardening Composites Stress at First Cracking or First Percolation Cracking Direct Tension Testing Typical Stress-Strain or Stress-Elongation Curves in 65 Direct Tension Typical Stress versus Crack Opening Curves Remarks: Reliability, Variability, Size Effect, Rate 73 Effect 2.4 What Do Fibers Do to Concrete In Compression? Reference Behavior of Plain Concrete in Compression How Do Fibers Affect the Failure Mode of Concrete in 77 Compression? Stress-Strain Curve of FRC in Compression Summary: Influence of Fibers on the Stress-Strain 88 Response of Concrete in Compression 2.5 What Do Fibers Do to the Response of Concrete in Bending? Typical Bending Response Examples of Load-Deflection Curves as Influenced by 91 Various Parameters Correlation Between Bending and Tensile Response What Do Fibers Do to the Response of Concrete in Shear? Shear Tests Shear Stress versus Strain Behavior Behavior of SIFCON under Monotonic Shear Loading Behavior of SIFCON under Cyclic Shear Loading Analytical Predictions of Shear Strength of FRC ACI Minimum Shear Reinforcement Comparison SIFCON: Properties in a Nutshell Special Applications of SIFCON Early Age Compression Properties Summary: Definitions and Terminology Concluding Remarks References 120 Chapter 3 What Do Fibers Do to Concrete? Part II: Time, 127 Temperature, and Cyclic Dependent Properties 3.1 Introduction Plastic Shrinkage Cracking Control Shrinkage Strain General Observations 134

5 3.3.2 Free Shrinkage Restrained Shrinkage Creep Creep Under Compression Loading Creep Under Tensile Loading Fire Resistance General Observations Material Level Structural Level Special Case of PP Fiber and Similar Fiber Materials Use of Temperature for Property Enhancement and 146 Modification 3.6 Corrosion Resistance, Deterioration, Degradation Corrosion: Background and Consequences Corrosion of Fiber Reinforced Concrete Experimental Variables Mechanical Tests Main Conclusions from Corrosion Testing 151 of FRC Tensile Post-Cracking Response as 155 Affected by Corrosion or Other Degradation 3.7 Durability Issues and Freeze-Thaw Resistance Background on Freeze-Thaw Testing Relevant Observations for Plain Concrete Freeze-Thaw Resistance of Fiber Reinforced Concrete The Need for New Test Standards Suitable for FRC 159 Composites 3.8 Fatigue Reference Fatigue for Plain Concrete in Compression Fatigue of Fiber Reinforced Concrete: Special Care and Precautions Fatigue of FRC in Bending with 2% Hooked Steel Fibers by Volume Key Experimental Observations S-N Relation High Amplitude Cyclic Loading Concluding Remarks References 172 Chapter 4 What Do Fibers Do To Concrete? Part III: 179 Toughness, Fracture Energy, Ductility 4.1 Introduction Load Test versus Deformation Test Special Notation for this Chapter Toughness and Toughness Index ASTM C1018 Standard Test for Flexural Toughness Conceptual Basis Toughness Index Computation Remarks on First Cracking Point Some Issues with Toughness Indices ASTM C1609M Standard Test for Flexural Performance Remarks

6 4.2.3 JCI-SF4 Methods of Tests of Flexural Strength and 192 Flexural Toughness of Steel FRC RILEM TC 162-TDF Test and Design Methods for 193 Steel FRC European Standard EN Conceptual Definition of Toughness and Energy Absorption 197 Capacity Total Energy to Failure in Tension: Qualitative 197 Observations 4.4 Basic Fracture Mechanics as Applied to FRC Composites Fracture Toughness and Stress Intensity Factor: Stress 201 Based Criterion Fracture Energy and Toughness: Energy Based 204 Criterion Correlation Between Fracture Energy and Fracture 205 Toughness Non-Linear Materials Typical Experimental Tests to Obtain G Ic and K Ic Notched Tensile Prisms Compliance Method Multiple Criteria for Fracture Fracture Processes in Fiber Reinforced Cement Composites: 211 Experimental Observations Background Crack Model and R-Curve Behavior Examples of Numerical Results for Fracture Energy Fracture Energy for Strain-Hardening FRC Composites 4.6 Surface Energy of FRC Composites All Fibers Fail at Failure of the Matrix at the Cracked 220 Section Some Fibers Fail and All Remaining Fibers Pull Out 221 from the Cracked Section All Fibers Pull-Out from the Cracked Section Surface Energy of Pull-Out Theoretical Prediction from a Tension Test Experimental Results from Tension Tests Experimental Results from Bending Tests 4.7 Ductility and Ductility Index for Deflection-Hardening FRC 227 Beams Conventional Definition of Ductility Ductility Index Applied to Deflection-Hardening FRC 229 Composites Energy Based Ductility Index for Deflection- 230 Hardening FRC Composites Example of Application Remarks on Ductility Index Concluding Remarks References 236 Chapter 5 Applications: Classification, Purpose, 243 Examples 5.1 Introduction: Applications Classification 243

7 5.2 Applications by Fiber Type and Limitations Design Trade-Offs for a Particular Application Using SIFCON in Over-Reinforced Concrete Beams Using Fibers in Concrete Beams Reinforced with Prestressing 255 Strands 5.6 Using Fibers in Bridge Decks with Reduced Reinforcement Effect of Fibers on Punching Shear Behavior of RC Decks, 264 Slabs, Foundations Conclusions from Study on Punching Shear of 271 HPFRCC Slabs 5.8 Moment-Rotation Response of Beams with Fibers, 272 Reinforcing Bars, or Prestressing Strands Conclusions from Study on Moment-Rotation Capacity 275 of HPFRCC Beams 5.9 FRC Seismic Resistant Joints, Connections, Coupling Beams, 276 etc Typical Examples Typical Experimental Tests for Joints and Connections Typical Experimental Tests for Coupling Beams Remarks on Material Design and Joint Design General Conclusion on Energy Absorption Capacity Enhancement of Material Toughness Resilience, Robustness and Damage Tolerance Bond of Deformed Reinforcing Bars Bond Stress-Slip Relationship Pull-Out Test Specimen and Special Testing Fixture Bond of Strain-Hardening FRC and SIFCON Effect of Fiber Type on Bond Stress versus Slip 292 Response 5.11 Bond of Prestressing Strands Conclusions from Tests on Prestressing Strands General Conclusions from Tests on Strands Bond of Reinforcing Bars from Beam Tests Conclusions from Bond of Bars Embedded in 307 SIFCON Beams 5.13 Improving Shear Resistance of Structural Concrete with 311 Fibers 5.14 Summary: What Do Fibers Do to Structural Concrete? Desirable Properties of Repair Materials Concluding Remarks References 314 Chapter 6 Fabrication and Mix Design Introduction Cementitious Matrices Manufacturing Fabrication - Production Premixing Mixers Steps for Pre-Mixing Common Problems with Pre-Mixing Methods to Facilitate Fiber Addition to the Mixer Lay-Up Placement Method Shotcrete Spray-Up Process 328

8 6.2.4 Hatschek and Similar Machines Extrusion Slurry Infiltrated Fiber Concrete - SIFCON Mix Design What Do Fibers Do to Fresh Concrete: Key Parameters and 330 Design Trade-Offs Workability Aggregate to Fiber Relative Size and Matrix Type Practical Ranges of Fiber Content Apparent versus Actual Volume Fraction Understanding the Limitation on Maximum V f or V f L/d Fiber Efficiency Cost Considerations Overall Design Parameters and Trade-offs Examples of Mixtures for Steel Fiber Reinforced Concrete Example Study on Self-Consolidating Strain-Hardening FRC 346 Composites Mixing Procedure for Self Consolidating FRC 348 Mixtures Remarks Ultra-High Performance Fiber Reinforced Concrete: UHP- 350 FRC Background and Definition Mix Design: Example of Experimental Study Materials Selection Mixing Procedure Typical UHPC and UHP-FRC Mixtures Particle Packing Considerations and Increasing Matrix 356 Packing Density 6.9 Optimized Non-Proprietary UHP-FRC Mixtures Precautions for Successful FRC Mixtures Concluding Remarks References 366 Chapter 7 Fiber Content, Fiber Number, Fiber Spacing 371 and Equivalence Relations 7.1 Volume, Mass, Weight and their Fractions Definitions, Notation, and Basic Relationships Examples Typical Examples of Fiber Weight (or Mass) for 378 Normal Weight Concrete 7.2 Number of Fibers Circular and Substantially Circular Fibers Non-Circular Fibers Examples: Number of Fibers Typical Number of Fibers for V f = 1% Average Fiber Spacing Average Spacing of Fiber Intersections of a Plane Circular Fibers Non-Circular Fibers Average Spacing of Fiber Centroids in Space Examples: Fiber Spacing Typical Fiber Spacing for V f = 1% Specific Surface of Fibers 387

9 7.4.1 Circular or Primarily Circular Fiber Cross Section Non-Circular Fibers Examples of Specific Surface of Fiber Reinforcement Example: Specific Surface Comparison between 390 Reinforced Concrete and Fiber Reinforced Concrete 7.5 Equivalence Relations Denier to Diameter Examples: Denier to Diameter Tensile Strength Conversion: Pounds per Denier to 393 Pounds per Square Inch (psi) or to Newton/mm 2 (MPa) Examples: Tensile Strength Equivalence Tex and Dtex Probabilistic Analysis Related to N v and N s Proof: Number of Fibers Crossing a Unit Area of Plane Example: Random Number of Fiber Intersections Influence of Mold Walls on the Value of α Cost Considerations: Volume Fraction, Weight Fraction and 400 Willingness-to-Pay Price 7.8 Example of Mix Proportioning by Weight or Volume Concluding Remark References 405 Chapter 8 Modeling the Unit Cell in Tension Introduction Nomenclature and Definitions Assumptions for Modeling Physical Model of Composite in Tension and Unit Cell Stress Distribution Prior to Matrix Cracking in Unit Cell 414 Model Stresses in Fiber and Matrix within Transfer Distance Distance at Which Equal Strains Occur: Transfer 419 Distance x o Stresses in Matrix and Fiber at and Beyond the 422 Transfer Distance x o Prior to Matrix Cracking Trapezoidal Stress Profile Triangular Stress Profile External Tensile Load Leading to First Matrix 423 Cracking Particular Value of Transfer Distance at Onset of 424 Matrix Cracking Stresses in Matrix and Fiber for T = T cr Just Prior to 425 Matrix Cracking Stresses in Matrix and Fiber Just After Cracking Average Stress in the Fiber at Onset of Cracking Partial Bond Activation at Fiber-Matrix Interface: 430 Trapezoidal Stress Profile in Fiber Full Bond Activation at Fiber-Matrix Interface: 432 Triangular Stress Profile in Fiber 8.7 Examples: Stress Distribution in Unit Cell Model at Onset of 433 Matrix Cracking Steel Fiber Reinforced Concrete: Fiber Type Steel Fiber Reinforced Concrete (UHPC): Fiber Type 2 436

10 8.7.3 Steel Fiber Reinforced Concrete (UHPC): Fiber Type Slurry Infiltrated Fiber Concrete: SIFCON PVA Fiber Reinforced Cement Composite: Coarse Fiber PVA Fiber Reinforced Cement Composite: Fine Fiber PVA Fiber Reinforced Aerated Concrete Polypropylene Fiber Reinforced Cement Composite Carbon Fiber Reinforced Cement Composite Spectra Fiber Reinforced Cement Composite Asbestos Fiber Reinforced Cement Composite Unit Cell in Compression: Steel Fiber Reinforced 444 Concrete 8.8 Practical Observations Stress Estimate in Overall Composite at Onset of 446 Matrix Cracking 8.9 Concluding Remark References 448 Chapter 9 Modeling Tension Introduction Notation Simplified Stress-Elongation Models and Assumptions Assumptions Design Strategy Tensile Stress and Strain in Composite at First Percolation 457 Cracking ( σ cc, ε cc ) Stress at First Crackingσ cc Strain at First Cracking ε cc Upper Bound Stress in Composite at Cracking of 460 Matrix Proof Leading to Eq. (9.1) Maximum Post-Cracking Stress (Strength) and Strain in 462 Tension ( σ pc, ε pc ) Possible Cases for Maximum Post-Cracking Stress, σ pc Composite Strength Assuming All Fibers 463 Fail Simultaneously Composite Strength Assuming All Fibers 464 Pull- Out Simultaneously Non-Dimensional Form for σ pc Upper Bound Limit for σ pc Derivations Leading to Eq. (9.18) for σ pc Unidirectional Fibers Randomly Oriented Fibers Strain at Maximum Post-Cracking Stress, ε pc General Pull-Out Response After σ pc : Part III of Figure Kosa and Naaman Visalvanich and Naaman Gopalaratnam and Shah 1985, Li Summary: Idealized Tensile Response for Modeling 477

11 9.8 Critical Volume Fraction of Fibers to Achieve Strain- 478 Hardening Behavior in Tension Graphical Illustration Critical Volume Fraction of Fibers to Achieve Deflection- 481 Hardening Behavior in Bending Example: Critical Volume Fraction of Fibers Practical Limitations Numerical Examples of Cracking and Maximum Post- 484 Cracking Tensile Stresses Steel Fiber Reinforced Mortar or Concrete Steel Fiber Reinforced Concrete Fibers with High 485 Bond Strength Slurry Infiltrated Fiber Concrete - SIFCON Ultra-High Performance Fiber Reinforced Concrete, 487 UHP-FRC Carbon Fiber Reinforced Cement Composite PVA Fiber Reinforced Cement Composite, Coarse 489 Fiber PVA Fiber Reinforced Cement Composite, Fine Fiber PVA Fiber Reinforced Aerated Concrete Polypropylene Fiber Reinforced Cement Composite Polypropylene Fiber Reinforced Lightweight 492 Cement Composite Glass Fiber Reinforced Cement Board Fiber Reinforced Cement Composite with Spectra 494 Fiber Steel Fiber Reinforced Mortar with Rectangular 495 Fiber 9.11 Modeling a More Complete Tensile Stress-Elongation Curve Desirable Characteristics of the Tensile Stress-Strain Curve 498 for Structural Design 9.13 Fiber Effectiveness and Efficiency Fiber Reinforcing Index Fiber Intrinsic Efficiency Ratio Generation of Other Fiber Cross-Sectional Shapes Concluding Remark References 504 Chapter 10 Modeling Compression Introduction Parameters Influencing the Stress-Strain Response in 514 Compression Experimental Observations Influence of Testing Variables Stress-Strain Equations for Compression Notation Common Boundary Conditions Analytical Relationships: Primarily for Plain Concrete Recommended Analytical Equation for Plain Concrete Recommended Analytical Equations for Fiber 525 Reinforced Concrete Exponential Equation for Inverted S- Curve: Descending Branch 527

12 10.4 Example: Curve Fit for Fiber Reinforced Concrete Ascending Branch: Parabola Ascending Branch: Fractional Equation Ascending Branch: Exponential Function Descending Branch: Fractional Equation Descending Branch: Exponential Passing by Two 534 Prescribed Points Descending Branch: Exponential with Zero Curvature 535 at Inflection Point Recommendations Example: Curve Fit for Plain Concrete Analytically Generated Stress-Strain Relation for: 536 f c = 69 MPa = 10 ksi Curve Fit of Experimental Curve for Plain Concrete Ascending Branch Descending Branch Implementation in Nonlinear Analysis Example: Resultant Location Using Parabolic Stress 542 Block 10.7 FRC Under Cyclic and Random Amplitude Loading Concluding Remarks References 548 Chapter 11 Modeling Bending Response Introduction - Background Notation Used in This Chapter Assumptions for Modeling Elastic Moment and Plastic Moment Plastic Material with Different Yield Strengths in 555 Tension and compression 11.2 Bending Behavior of FRC Composites Typical Load-Deflection Curves and Correlation with 557 Tensile Response Cracking Moment Modulus of Rupture Section Response at First Cracking: Hinge Behavior Definition of Average Post-Cracking Stress for 560 Design 11.3 Nominal Bending Resistance, M n Compression and Tensile Stress Profiles ACI Rectangular Stress Block and Uniform Stress 564 Distribution in Tension Examples Triangular Stress Block in Compression and Uniform 570 Stress Distribution in Tension Example Parabolic Stress Block in Compression and Uniform 572 Stress Distribution in Tension Example ACI Stress Block in Compression and Triangular 575 Stress Distribution in Tension Example General Observations and Discussion Simplified Results Assuming: ACI Rectangular 578

13 Stress Block and Uniform Tensile Stress Distribution Condition for Deflection Hardening for a Rectangular 581 Section 11.5 General Condition for Deflection-Hardening FRC 582 Composites 11.6 Influence of Crack Opening on Average Tensile Stress 583 Resistance 11.7 Computation of Deflections in FRC Beams Concluding Remark References 589 Chapter 12 Elastic Modulus Background on Modulus of Elasticity Shear and Bulk Modulus Prediction Models for the Elastic Modulus of Composites Fibers with Axis Parallel to Loading Direction: Upper Bound 598 Equal-Strain Solution (Voigt s Model) Proof of Voigt s Model Examples of Upper-Bound Modulus: Unidirectional 600 Fibers Parallel to Loading Direction 12.4 Fibers with Axis Normal to Loading Direction: Lower-Bound 602 Equal-Stress Solution (Reuss s Model) Proof of Reuss s Solution Examples of Lower Bound Modulus: Fibers Normal 605 to Loading Direction Comparison Summary Other Models for Unidirectional Discontinuous Fibers Cox Model (1952) for Unidirectional Short Fibers of 606 Length L Example Using Cox s Prediction Equation Halpin and Tsai smodel (1969) Example Using Halpin and Tsai s Prediction Equation Models for Randomly Oriented Discontinuous Fibers Cox s Solution for a Two-Phase Model Halpin and Tsai s Solution for a Two-Phase Model Najm and Naaman s Solution Alwan and Naaman s Solution for a Pseudo Three- 615 Phase Model 12.7 Practical Recommendation for Predicting the Elastic Modulus 617 of FRC Composites with Randomly Oriented and Distributed Fibers 12.8 Measurements of Elastic Modulus Concluding Remarks References 622 Chapter 13 Bond: Characterization and Modeling Introduction Bond Components Adhesive Bond: Chemical and Physical Frictional Bond Mechanical Bond: Anchorage and Clamping Fiber to Fiber Interlock Basic Pull-Out and Pull-Through Tests 631

14 Preferred Behavior of a Fiber under Pull-Out Typical Pull-Out and Pull-Through Test Set-ups Basic Calculations for Bond or Shear Stress Average Bond Stress and Equivalent Bond Stress Definition of Slip Typical Pull-Out Response and Pull-Out Mechanism of 638 Straight Smooth Steel Fibers 13.5 Typical Pull-Out Response and Pull-Out Mechanism of 641 Hooked-End Steel Fiber 13.6 Typical Pull-Out Response and Pull-Out Mechanism of 644 Indented and Crimped Steel Fibers 13.7 Typical Pull-Out Response and Pull-Out Mechanism of 646 Twisted Polygonal Steel Fiber Observations with Twisted Fibers Isolating the Mechanical Component of Bond Pull-Out with UHPC General Observations on the Pull-Out Response of Steel 657 Fibers and Comparisons Early Age Bond Loading Rate Effect on Pull-Out Pull-Out under Impact Loading Pull-Out of Aligned and Oriented Steel Fiber and Influence of 664 Number of Fibers Pulling Out Analytical Modeling: Pull-Out Response and Bond Shear 669 Stress versus Slip Relationship of Smooth Fibers Simplified Assumptions for Modeling Notation and Basic Equations Solution of the Primal Problem: Prediction of the 673 Entire Pull-Out Load versus Slip Response Solution of the Dual Problem: Bond Shear Stress 678 versus Slip Relationship Slip Hardening Bond Shear Stress versus Slip Response Analytical Modeling of Deformed Steel Fiber Pull-Out 681 Behavior Model for Corrugated and Crimped Steel Fiber Model for Hooked-Ends Steel Fiber Model for Fiber with End Anchorage Model for Twisted Steel Fiber Considerations for Structural Applications Comparison of Typical Slip at Peak Pull-Out Load Correlation with Crack Width in Structural Concrete 687 Members Pull-Out Work and Pull-Out Energy Concluding Remarks References 691 Chapter 14 Strain Hardening FRC Composites: 697 Chronological Evolution 14.1 Introduction Historical Background Evolution of Nomenclature to Describe Strain Hardening 705 Behavior 14.4 Evolution Summary Conditions for Strain-Hardening Behavior: Fracture 707

15 Mechanics Approach versus Composite Mechanics Approach 14.6 Similarities and Differences in the Known Conditions for 712 Strain-Hardening Behavior 14.7 Converging Studies Correlation Between Tensile Stress-Displacement Response 716 and Tensile Stress-Strain Response 14.9 Concluding Remarks References 718 Chapter 15 Looking Ahead Introduction Hybrid Fiber Reinforcement Very Lightweight and Cellular Fiber Reinforced Concrete Main Observations Effect of Fibers on Ferrocement and Thin Cement 730 Composites 15.5 Effect of Conventional Reinforcement on Tensile Response 732 of FRC Observations and Conclusions The Need for Slip-Hardening Bond and Limitations Self-Stressing FRC Composites New and Improved Materials Size Effects, Loading Rate Effects, Tension Stiffening Fiber Consumption Estimates Potential Growth Looking Ahead References 741 Appendix A List of Symbols 745 Appendix B Unit Conversions 752 INDEX INDEX 755 Author 764

16 Typical sections of steel fiber reinforced concrete (top) and mortar (bottom).