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Impact behaviour of concrete under multiaxial loading Sukontasukkul, Piti 2002

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IMPACT BEHAVIOUR OF CONCRETE UNDER MULTIAXIAL LOADING by PITI S U K O N T A S U K K U L B . Eng., K i n g Mongkut Institute of Technology-Ladkrabang, Thailand, 1990 M . Eng., Asian Institute of Technology (AIT), Bangkok, Thailand, 1994 THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E S T U D I E S (Department of C i v i l Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A December 2001 © Piti Sukontasukkul, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia Vancouver, Canada Date ^ , 2~0<? \ 11 A B S T R A C T Impact testing of concrete under different types of loading conditions (compression, tension and bending) has been carried out worldwide. However, it has been found that the results are mostly not comparable due to the differences in testing techniques and specimen configurations. In this study, a series of impact tests: compression, 1-dimensional bending (beam tests) and 2-dimensional bending (plate tests) were carried out using the same testing technique (drop weight machine) and similar configuration (hammer weight, height and impact energy). Concrete exhibited more rate sensitivity under 1-D flexure than under compression. However, in 2D flexure, the rate sensitivity of concrete decreased significantly, to a level about the same as that under compression. In addition, the impact behavior of concrete under confining stress was also investigated. The confinement technique adopted here was of the active type using the instrumented confinement apparatus designed and constructed at the University of British Columbia. These experiments were also carried out in compression and flexure ( I D and 2D). In all cases, the measured properties of concrete (failure mode, stress-strain response, strength, inertial load, and rate sensitivity) changed significantly with increasing confining stress. Under confined compression, the failure mode of concrete changed from a shear cone to splitting failure and resulted in higher stress rate sensitivity. However, under confined flexure, the stress rate sensitivity decreased with increasing confinement due to the change of the failure mode from flexure to shear. To complete the confinement study, a further investigation on the effect of the loading platen stiffness (soft or hard) and tup diameter (plate specimens only) was also carried out. Analytically, scalar damage mechanics (SDM) was used to predict the behavior of confined and unconfined concretes under both static and impact loading. The damage measurement technique was selected differently, depending upon the kind of loading (static or impact). Under static loading, the damage was defined in terms of the change or degradation of elastic modulus (E). However, the strain rate variation was used to define Ill the damage of concrete under impact loading. It was found that the use of S D M to predict the response of concrete works quite well ; the predicted responses of concrete under both loading conditions agreed reasonably with the measured responses. iv TABLE OF CONTENT Abstract Ii Table of Content iv List of Tables x i i List of Figures x iv Acknowledgement xx iv Dedication xxv Chapter 1 Introduction and Scope of Study 1 Chapter 2 Literatures Review 4 2.1 Impact testing methods 4 2.2 Concrete under high strain rate 7 2.2.1 Experimental evidence 7 a) Mechanical properties and fracture parameters 7 b) Inertial effects on impact testing 10 c) Fracture process zone, fracture toughness and fracture 11 energy d) Dynamic crack propagation and velocity 13 2.3 Steel fibre reinforced concrete 14 2.3.1 Properties of hardened S F R C under static loading 15 2.3.2 Properties of hardened S F R C under impact loading 17 2.4 Analytical models for concrete under high rates of loading 19 2.4.1 Continuum damage mechanics model 19 2.4.2 Linear fracture mechanics model 20 a) Macro level (scale of meter) 20 b) Micro level (scale of centimeter) 22 2.4.3 Thermodynamic model 23 V 2.4.4 Zielinski 's model 24 2.5 Concrete under multiaxial stress 26 2.5.1 Testing technique 26 2.5.2 Static loading 27 2.5.3 Impact loading 30 Chapter 3 Exper imenta l Procedure 33 3.1 Introduction 33 3.2 Specimen preparation 33 3.3 Testing equipment and apparatus 34 3.3.1 The 578 kg impact machine 34 3.3.2 Tup load cell 37 3.3.3 Accelerometer ; 39 3.3.4 A R O M A T laser sensor 40 3.3.5 Instrumented confinement apparatus 42 3.3.6 Data acquisition system 45 3.4 Testing program 46 3.4.1 Static tests 46 3.4.2 Impact tests 46 3.4.2.1 Unconfined tests 46 a) Prism tests 46 b) Beam tests 48 c) Plate tests 48 3.4.2.2 Confined tests 50 a) Prism tests 50 b) Beam tests 53 c) Plate tests 53 Chapter 4 Data Analysis 59 4.1 Introduction 59 4.2 Data filtering 59 4.2.1 Moving average 60 4.2.2 Example: data filtering analysis 61 4.3 Acceleration, velocity and deflection 64 4.4 Energies 67 4.4.1 Energy lost by the hammer 67 4.4.2 Fracture energy or energy absorbed by specimen 67 4.5 True load 69 4.5.1 Inertial load of beam specimens 69 4.5.1.1 Beams 69 a) Unconfined beams 69 b) Confined beams 70 4.5.1.2 Plates 72 a) Unconfined plates 72 b) Biaxially confined plates 73 c) Uniaxially confined plates 74 d) Unfractured plates 75 4.6 Stress (or strain) rate and its relationship with strength (n) 79 Chapter 5 Results and Discussions-Prisms 82 5.1 Introduction 82 5.2 Failure mode 82 5.2.1 Unconfined prisms 82 a) Static loading 82 b) Impact loading 84 5.2.2 Confined prisms 86 5.3 Stress-strain response 87 5.3.1 Unconfined prisms 87 a) Static loading 87 b) Impact loading 89 v i i 5.3.2 Confined prisms 92 5.4 Material properties 95 5.4.1 Compressive strength 95 a) Unconfined prisms 95 b) Confined prisms 97 5.4.2 Ultimate strain 101 a) Unconfined prisms 101 b) Confined prisms 102 5.4.3 Elastic modulus 104 a) Unconfined prisms 104 b) Confined prisms 106 5.4.4 Relationship between stress (or strain) rate with strength... 108 a) Unconfined prisms 110 b) Confined prisms 110 5.4.5 Fracture energy 114 a) Unconfined prisms 114 b) Confined prisms 115 Chapter 6 Results and Discussions-Beams 123 6.1 Introduction 123 6.2 Failure mode 123 6.2.1 Unconfined beams 123 6.2.2 Confined beams 124 6.3 Load-deflection response 127 6.3.1 Unconfined beams 128 6.3.2 Confined beams 131 6.4 True peak load 134 6.4.1 Unconfined beams 134 6.4.2 Confined beams 136 6.5 Inertial load 1 3 7 v i i i 6.6 Relationship between stress and stress rate 141 6.6.1 Unconfined beams 141 6.6.2 Confined beams 142 6.6.3 n value 145 6.7 Fracture energy 149 Chapter 7 Results and Discussions-Plates 152 7.1 Introduction 152 7.2 Failure mode 152 7.2.1 Unconfined plates 152 7.2.2 Confined plates 153 7.3 Load-deflection response 156 7.3.1 Unconfined plates 156 a) Load-deflection curves 156 b) Multiple peaks in F R C plates under impact loading... . 160 c) Effect of loading rate (hammer drop height) 163 d) Effect of fibre content 165 7.3.2 Confined plates 167 a) Effect of confinement 168 - Biaxial confinement 168 - Uniaxial confinement 170 b) Effect of fibre type and content 171 c) Effect of loading rate (hammer drop height) 174 7.4 Peak bending load and inertial load 177 7.4.1 Unconfined plates 177 a) Peak bending load 177 b) Inertial load 181 7.4.2 Confined plates 184 a) Measured peak load 184 b) Inertial load 191 ix 7.5 Energy lost by the hammer (impact loading) and fracture energy 192 7.5.1 Unconfined plates 192 a) Static loading 192 b) Impact loading 193 7.5.2 Confined plates 196 Chapter 8 Comparison between Prisms, Beams and Plates 201 8.1 Introduction 201 8.2 Physical properties 202 8.3 Strength and fracture energy 202 8.4 Stress rate 205 8.5 Relationship between stress rate and strength 206 8.6 Rate sensitivity 210 Chapter 9 Parameters Affecting Confined Tests 213 9.1 Introduction 213 9.2 Effect of loading platen on confined impact test 214 9.2.1 Prism tests 215 a) Failure patterns 215 b) Load-deflection response and peak load 217 9.2.2 Beam tests 220 a) Fai lure patterns 221 b) Acceleration distribution 223 c) Load-deflection response and peak load 225 9.2.3 Plate tests 226 a) Failure patterns 227 b) Load-deflection response and peak load 228 c) Confining force vs. time response 229 9.3 Effect of the loading head (tup) geometry on confined plates 229 9.3.1 Failure patterns 231 9.3.2 Load-deflection response and peak load 234 Chapter 10 Development of Pre-Peak Stress-Strain Relationship using Scalar Damage Mechanics (SDM) 236 10.1 Introduction 236 10.2 Kachanov's concept 236 10.2.1 Macromechanical or one dimensional damage model 237 10.2.2 Effective stress concept 238 10.2.3 Strain equivalence principle 239 10.2.4 Relationship between strain and damage 239 10.2.5 Damage measurement 240 10.3 Predicting the pre-peak response o f concrete under static loading using Scalar Damage Mechanics (SDM) 241 10.3.1 Pre-peak compressive response 241 a) Proposed model 241 b) Comparison with other model 245 10.4 Predicting the pre-peak response of concrete under impact loading using scalar damage mechanics (SDM) 247 10.4.1 Pre-peak compressive response 247 a) Scalar damage mechanics: strain rate approach 247 b) Proposed model 249 10.4.2 Impact strength prediction 251 Chapter 11 Prediction of the Impact Response of Confined Concrete 253 11.1 Introduction 253 11.2 Determination of the stress-strain response 253 11.2.1 Proposed compressive model 253 11.2.2 Comparison with the actual response 254 Chapter 12 Conclusions 257 Appendix A 261 xi Appendix B 263 Appendix C 266 Bibliography 268 X l l L I S T O F T A B L E S Table 2.1 Reduction of Pre-peak Crack Growth with the Increasing of Strain Rate 12 Table 2.2 Fracture Toughness, Elastic Modulus and Surface Energy of Concrete, 26 Steel and Aggregates Table 3.1 Geometry of Fibres 34 Table 3.2 M i x Proportions, by weight 34 Table 3.3 Static Testing Program (For prisms, beams and plates) 47 Table 3.4 Impact Testing Program of the Unconfined Prisms 47 Table 3.5 Impact Testing Program for the Unconfined Beams 49 Table 3.6 Impact Testing Program of Unconfined Plate 50 Table 3.7 Impact Testing Program for Confined Prisms 51 Table 3.8 Impact Testing Program for Confined Beams 54 Table 3.9 Impact Testing Program of Biaxially-Confined Plates 55 Table 3.10 Impact Testing Program for Uniaxially Confined Plates 56 Table 5.1 Average static compressive strength of plain and F R C prisms 96 Table 5.2 Impact strength of plain and F R C prisms in comparison with static Strength 97 Table 5.3 Strength of confined plain concrete and F R C under impact loading 98 Table 5.4 Confined/Unconfined strength ratios of plain concrete and F R C 98 under impact Table 5.5 Ultimate strains of plain concrete and F R C at different rates of loading 101 Table 5.6 Ultimate strains of confined plain and F R C under impact loading 103 Table 5.7 E s o f plain concrete and F R C subjected to static and impact loading 105 Table 5.8 Slopes and n-values of unconfined and confined concrete prisms 111 Table 6.1 Peak Load of Unconfined Beams Subjected to Static Loading 135 Table 6.2 Peak and Inertial Load of Confined Beams 138 Table 6.3 Ratio between Confined and Unconfined Peak Load 138 Table 6.4 Average Stress Rate and Flexural Strength of Unconfined Beams 142 Table 6.5 Calculated n-value for Both Unconfined and Confined Beams 145 Table 6.6 Fracture Energy (30mm) of Unconfined and Confined Beams 151 x i i i Subjected to Static and Impact Loading Table 6.7 Impact/Static and Confined/Unconfined Fracture Energy Ratio 151 Table 7.1 Peak Loads of Plain and F R C Plates under Impact Loading 180 Table 7.2 Measured Tup Loads and Inertial Loads of Unconfined Plates 182 Table 7.3 Measured and inertial loads of plain and F R C plates with Uniaxial Confinement (Tested at 250mm Hammer Drop Height) 185 Table 7.4 Measured and inertial loads of plain and F R C plates with Biaxial 186 Confinement (Tested at 250mm Hammer Drop Height) Table 7.5 Measured and Inertial Loads of Plain and F R C Plates with 189 Uniaxial Confinement (Tested at 500mm Hammer Drop height) Table 7.6 Measured and Inertial Loads of Plain and F R C Plates with 190 Biaxial Confinement (Tested at 500mm Hammer Drop Height) Table 7.7 Fracture Energy of Unconfined Specimens under Static and 192 Impact Loading Table 8.1 Physical Properties of Each Specimen Type 202 Table 8.2 Average Strength and Fracture Energy 204 Table 8.3 Average Loading Rate for Each Type of Specimen 208 Table 8.5 The Impact/Static Rate Sensitivity of Unconfined Specimens 209 Table 9.1 Peak Load of 0.625MPa Confined Concrete Prisms with Steel 218 and Rubber Platens Table 9.2 Peak Load of 2.5 M P a Confined Concrete Beams with Steel and 226 Rubber Platens Table 9.3 Peak Load of 2.5 M P a Confined Concrete Plates with Steel and 228 Rubber Platens Table 9.4 Peak Loads of 2.5 M P a Confined Plates Tested with 100mm and 234 150mm Dia . Tup xiv L I S T O F F I G U R E S Fig. 2.1 Pneumatic Hydraulic Testing Device 5 Fig. 2.2 Schematic Illustration of the Hopkinson Pressure Bar (Compression) 6 Fig. 2.3 Mode 1 Opening Mode 7 Fig. 2.4 Mechanism of Microcracks under Impact Loading 9 Fig.2.5 Relationship between Strength and Stress rate based on Macro Level- 21 L E F M Model Fig. 2.6 Geometry of Cracked Concrete 22 Fig.2.7 Relationship between Strength and Stress rate based on Micro Level- 23 L E F M Model Fig. 2.8 Linked Elements Model for Concrete 24 Fig 2.9 Zielinski 's Fracture Surface Model 25 Fig. 2.10 A n early version of an instrumented confinement apparatus developed 32 at the University of British Columbia, Mindess and Rieder Fig. 3.1 Schematic V i e w of 578 kg Impact Machine 35 Fig. 3.2 Instrumented Impact Testing Machine with 578 kg Hammer 36 Fig. 3.3 Circular Tup Load Cel l (not to scale) 38 Fig. 3.4 Tup Load Ce l l Calibration Curve 3 8 Fig. 3.5 Piezoelectric Accelerometer (mm.) 39 Fig. 3.6 Controller unit for A R O M A T Laser sensor 40 Fig. 3.7 Sensor Unit for A R O M A T Laser Sensor 41 Fig. 3.8 A R O M A T Laser Sensor Calibration Curve 41 Fig . 3.9 Schematic V i e w of A n Instrumented Stresses Confinement Apparatus 42 Fig. 3.10 A n Instrumented Confinement Apparatus 43 Fig. 3.11 Load Cells Calibration Curve 44 Fig. 3.12 Experimental Setup for the Impact Tests of Unconfined Prisms 48 Fig. 3.13 Experimental Setup for Impact Tests of Unconfined Beams and Plates 49 Fig. 3.14 Experimental Setup for the Impact Test of Confined Prisms 52 Fig. 3.15 Top V i e w of the Test Setup for Confined Prism Tests 52 Fig . 3.16 Experimental Setup for the Impact Test of Confined Beams and Plates 57 X V Fig. 3.17 Top V i e w of the Test Setup of Confined Beam Test 57 Fig. 3.18 Top V i e w of the Test Setup of Confined Plate Test 58 Fig. 4.1 Set of Unfiltered Data for an Impact Test 61 Fig. 4.2 Non-event Impact Data 62 Fig. 4.3 Portion of the Data during the Impact Event 62 Fig. 4.4 Processed Data-after Three Cycles of Moving Average Analysis 63 Fig. 4.5 Plot between the Existing Electronically Filtered Data and the Processed 63 Data Fig. 4.6 Displacements as Measured by the Accelerometer and the Laser Sensor 66 Fig. 4.7 Load-Deflection Curves Determined Using the Accelerometer 66 Fig. 4.8 The Generalized Inertial Load and Assumed Acceleration Distribution 71 for an Unconfined Beam Fig. 4.9 The Generalized Inertial Load for a Confined Beam 71 Fig. 4.10 The Generalized Inertial Load and Assumed Acceleration Distribution 74 for Unconfined Plates Fig. 4.11 Failure Pattern and the Generalized Inertial Load for Biaxially 77 Confined Plate Fig. 4.12 Failure Pattern and the Generalized Inertial Load for Uniaxially 78 Confined Plates Fig. 4.13 Relationship of K and crack velocity 80 Fig. 4.14 Plot between log(stress) vs log(stress rate) 81 Fig . 5.1 Failure Patterns of Plain Concrete Prisms Subjected to Static 83 Compressive Loading Fig. 5.2 Failure Patterns of Unconfined F R C Prisms Subjected to Static Loading 83 Fig. 5.3 Failure Patterns of Unconfined Prisms Subjected to Impact Loading 85 from a 250mm Drop Height Fig. 5.4 Failure Patterns of Unconfined Prisms Subjected to Impact Loading 85 from a 500mm Drop Height Fig. 5.5 Failure Pattern of Confined Prisms Subjected to Impact Loading 86 Fig. 5.6 Schematic Sketch Indicate Non-Uniform Stress Distribution 87 Fig. 5.7 Stress-Strain Curves Unconfined Prism under Static Loading 89 xv i Fig. 5.8 Stress-Strain Curves of Plain Concrete Prisms Subjected to Static and 90 Impact Loading Fig. 5.9 Stress-Strain Curves of 0.5%HE F R C Prisms Subjected to Static and 90 Impact Loading Fig. 5.10 Stress-Strain Curves of 1.0%HE F R C Prisms Subjected to Static and 91 Impact Loading Fig. 5.11 Comparison of Stress-Strain Curves of Plain Concrete and the Three 91 Types of F R C under Impact Loading (250mm Drop Height) Fig. 5.12 (a) Effect of Fibre Content on the Static and Impact Post-Peak 93 Response, and (b) Effect of Fibre Type on the Impact Post Fig. 5.13 Effect O f Confinement on Impact Response of Plain Concrete 94 (250 mm) Fig. 5.14 Effect of Confinement on Impact Response of 0 .5%HE F R C Prisms 94 (250 mm) Fig. 5.15 Effect of Confinement on Impact Response of 1% H E F R C Prisms 95 (250 mm) Fig. 5.16 Splitting force between four particles due to compressive force 99 Fig. 5.17 Confined/Unconfined Strength Ratio of (a) Plain and F R C , and (b) 100 Comparison between three types of fibre Fig. 5.18 Splitting and fibre pullout forces between four particles 101 Fig. 5.19 Effect of Confinement on Relative Ultimate Strain (250mm) 103 Fig. 5.20 Effect of Confinement on Relative Ultimate Strain (500mm) 104 Fig. 5.21 Impact/Static Modulus Ratio of Plain and F R C Prisms 105 Fig. 5.22 Effect of Strain Rate on E s (Ult) of Plain Concrete and F R C 106 Fig. 5.23 Confined E s (Ult) under Impact Loading (250mm Drop Height) 107 Fig. 5.24 Effect of Strain Rate on Strength of Plain Concrete and F R C 110 Fig. 5.25 Effect of Fibre Type on Strain Rate Sensitivity 110 Fig. 5.26 Relationship between Stress Rate and Stress of (a) Plain Concrete, (b) 112 0 .5%HE F R C and (c) 1.0% H E F R C Fig. 5.27 Relationship between Stress Rate and Strength of Confined (a) Plain 113 Concrete, (b) 0 .5%HE F R C and (c) 1.0%HE F R C xv i i Fig. 5.28 Effect of Fibres on n-values in Confined Concrete 114 Fig . 5.29 Fracture Energy of Unconfined Plain Concrete and F R C (250mm) 116 Fig. 5.30 Fracture Energy of Unconfined Plain Concrete and F R C (500mm) 116 Fig. 5.31 Effect of Loading Rate on Shape of the Energy vs Time Curves 117 Fig. 5.32 Fracture Energy of Confined Plain Concrete (250mm) 117 Fig. 5.33 Fracture Energy of Confined 0.5% Hooked End F R C (250mm) 118 Fig. 5.34 Fracture Energy of Confined 1.0% Hooked End F R C (250mm) 118 Fig. 5.35 The Stress-Time and Hammer Velocity-Time Response of Unconfined 120 and 0.625-MPa Confined Plain Concrete Prisms Fig. 5.36 Fracture Energy of 0.625 M P a Confined Concrete (250mm) 121 Fig. 5.37 Fracture Energy of Confined Concrete at 500 mm Drop Height 121 Fig. 5.38 Shape of the Energy vs Time for Different Drop Heights 122 Fig. 6.1 Failure Patterns of Unconfined Beams under Static Loading: 125 Fig. 6.2 Failure Patterns of Unconfined Beams Subjected to Impact Loading 125 Fig. 6.3 Failure Patterns of Confined Beams under Impact Loading (150mm) 126 Fig. 6.4 Static Response of Unconfined Plain and F R C Beams 129 Fig. 6.5 Impact Response of Unconfined Plain and F R C Beams (150mm) 129 Fig. 6.6 Effect of Hammer Drop Height on Unconfined Plain Concrete Beam 130 Fig. 6.7 Effect of Hammer Drop Height on Unconfined 0.5%HE F R C Beams 130 Fig. 6.8 Effect of Hammer Drop Height on Unconfined 1.0%HE F R C Beams 131 Fig. 6.9 Response of Confined Plain Concrete Beams under Impact Loading 132 (150mm) Fig . 6.10 Response of Confined 0.5%HE F R C Beams under Impact Loading 132 (150mm) Fig. 6.11 Response of Confined 0.5% F R C Beams under Impact Loading 133 (150mm) Fig . 6.12 Effect of Hammer Drop Height on Response of 2 .5MPa Confined 133 Beams Fig. 6.13 Confining Load-Time Response of Beam under Impact Loading with 139 Different Loading Platen Fig. 6.14 Typical Inertial Load Responses of Unconfined Beams at Different 140 XV111 140 Drop Height Fig. 6.15 Typical Inertial Load Responses of Confined Beams at Different Drop Height Fig. 6.16 Typical Inertial Load Responses of Confined Beams at Different 141 Levels of Confinement Fig. 6.17 Relationship between Stress Rate and Impact/Static Flexural Strength 142 Ratio of Plain Concrete and Hooked End F R C Beams (Log Scale) Fig. 6.18 Relationship between Stress Rate and Impact/Static Flexural Strength 142 Ratio Comparing between Three Types of F R C Beams (Log Scale) Fig. 6.19 Relationship between Confining Stress and Stress Rate of Plain and 144 F R C Beams Fig. 6.20 Relationship between Stress Rate and I/S Stress Ratio of Unconfined 145 and Confined Beams Fig. 6.21 Relationship between Log(Stress) and Log(Stress Rate) of Unconfined 147 (a) Plain Concrete, (b) 0 .5%HE F R C and (c) 1.0%HE F R C Beams Fig. 6.22 Relationship between Log(Stress) and Log(Stress Rate) of Confined 148 (a) Plain Concrete, (b) 0 .5%HE F R C and (c) 1.0%HE F R C Beams Fig. 7.1 Typical Failure Patterns of Plates Tested under Static Loading 153 Fig. 7.2 Typical Failure Patterns of Plates Tested under Impact Loading 154 Fig. 7.3 Flexural Failure Pattern of a) Plain Concrete and b) F R C 155 Fig. 7.4 The Development of Punching Shear Failure with the Increase of 155 Confining Stress Fig. 7.5 Unfractured Specimens: a) with High Fibre Content and b) tested under 156 High Biaxial Confinement Stress Fig. 7.6 Flexural Failure of Specimen Tested under Uniaxial Confinement: 156 a) Plain Concrete and b) 0.5% Flattened End F R C Fig. 7.7 Response of (a) Plain Concrete and (b) F R C Plates under Static Loading 157 Fig. 7.8 Typical Load-Deflection Curves of 0.5% F R C Plates Subjected to 158 Impact Loading (250 mm Drop Height) Fig. 7.9 Typical Load-Deflection Curves of 1.0% F R C Plates Subjected to 159 Impact Loading (250mm Drop Height) xix Fig. 7.10 Typical Load-Deflection Curve of Plain Concrete Plate Subjected to 159 Impact Loading (250mm Drop Height) Fig. 7.11 First Peaks of Plain and 0 .5%FRC Plates Subjected to Impact 161 Loading(Vh=2.21 m/s) Fig. 7.12 First Peaks of Plain and 1.0%FRC Plates Subjected to Impact Loading 161 (V h =2.21 m/s) Fig. 7.13 Effect of Impact Velocity on First Peaks of Plain Concrete and 162 0 .5%FRC Plates Fig. 7.14 Effect of Impact Velocity on First Peaks of Plain Concrete and 162 1.0%FRC Plates Fig. 7.15 Schematic Representation of Multiple Peaks in a F R C Plate under 163 Impact Loading (250 mm Drop Height) Fig. 7.16 Schematic Representation of Multiple Peaks in a F R C Plate under 164 Impact Loading (500 mm Drop Height) Fig. 7.17 Effect of Loading Rate on the Occurrence of Multiple Peaks on the 164 Load-Deflection Curve Fig. 7.18 Relative Peak Heights for the Two Different Drop Height 165 Fig. 7.19 Effect of Fibre Content on Number of Peaks 166 Fig. 7.20 More Rapid Strength Recovery of Fibre Reinforced Concrete with 167 Higher Fibre Volumes Fig. 7.21 Effect of Biaxial Confinement of the Response of Plain Concrete Plate 168 Fig. 7.22 Effect of Biaxial Confinement on the Response of 0 .5%HE F R C Plates 168 Fig. 7.23 Effect of Biaxial Confinement on the Response of 0.5%FE F R C Plates 169 Fig. 7.24 Effect of Biaxial Confinement on the Response of 0.5%CP F R C Plates 170 Fig . 7.25 Effect of Uniaxial Confinement on the Response of Plain Concrete 171 Plates Fig. 7.26 Effect of Uniaxial Confinement on the Response of 0 .5%HE F R C 172 Plates Fig. 7.27 Impact Load-Deflection Curves of Biaxially Confined 1%HE F R C 173 Plates Fig . 7.28 Impact Load-Deflection Curves of 5 M P a Biaxially Confined 1 % F R C 173 X X Plates Fig. 7.29 Impact Load-Deflection Curves of Uniaxially Confined 1% F R C Plates 174 Fig. 7.30 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly 175 Confined Plain Concrete Plates Fig. 7.31 Effect of Hammer Drop Height on the Response of 5MPa Biaxial ly 175 Confined 0.5%HE F R C Plates Fig. 7.32 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly 176 Confined 0.5%FE F R C Plates Fig. 7.33 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly 176 Confined 0.5%CP F R C Plates Fig. 7.34 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly 177 Confined 1%HE F R C Plates Fig. 7 35 Effect of Loading Rate on Peak Loads of Plain Concrete Plates 178 Fig. 7.36 Effect of Loading Rate on Peak Loads of 0 .5%FRC Plates 179 Fig. 7.37 Effect of Loading Rate on Peak Loads of 1.0 % F R C Plates 179 Fig. 7.38 Typical Tup Load and Inertial Load vs Time Curves for Plain Concrete 183 Plates Fig. 7.39 Typical Tup Load and Inertial Load vs Time Curves for F R C Plates 183 Fig. 7.40 Effect of Confinement Stress on Peak Load of Plain and 0 .5%HE F R C 187 Fig. 7.41 Effect of 5 M P a Biaxial Confinement on Peak Load 187 Fig. 7.42 Effect of 2.5 M P a Biaxial and Uniaxial Confinement on the Measured 188 Peak Load Fig. 7.43 Effect of 5 M P a Biaxial and Uniaxial Confinement on the Measured 188 Peak Load Fig. 7.44 Peak Load of 5MPa Biaxial and Uniaxial Confined Plate Subjected to 190 Impact from 500 mm Drop Height Fig. 7.45 Effect of Confinement on Inertial Load 191 Fig. 7.46 Effect of Fibre Content on Energy Lost by the Hammer and Fracture 194 Energy of Unconfined Plates Fig. 7.47 Effect of Fibre Type on Energy Lost by the Hammer and Fracture Energy of Unconfined Plates 195 Fig. 7.48 Load/Deflection vs Time Curves of 0.5% and 1 % Hooked End F R C 196 Plates Fig . 7.49 Relationship between Load Response and Energy Absorption 197 Fig. 7.50 Effect of Confinement on Energy Lost by Hammer and Fracture 198 Energy of Plain Concrete Plates Fig. 7.51 Effect of Confinement on Energy Lost and Fracture Energy of 0.5% 198 F R C Plates Fig. 7.52 Effect of Confinement on Energy Lost and Fracture Energy of 1%FRC 199 Plates Fig. 7.53 Energy Lost and Fracture Energy of 2.5 M P a Biaxially Confined F R C 199 Plates Fig. 7.54 Energy Absorption: Comparison Between Biaxial and Uniaxial 200 Confinement Fig. 8.1 Relationship Between Stress Rate and I/S Strength Ratio of Unconfined 209 Concrete Fig. 8.2 Comparing Load-Deflection Response of Plain Concrete under 211 Compression, I D and 2D Bending Fig. 8.3 Comparing Load-Deflection Response of 0 .5%HE F R C under 212 Compression, I D and 2D Bending Fig. 9.1 Distribution of Stress and Strain at the Contact Surfaces 214 Fig. 9.2 Stress-Strain Relationship of Rubber Platen 215 Fig. 9.3 Test Setup for a Confined Prism with Rubber Platens (Top View) 216 Fig. 9.4 Tensile Failure of Confined Plain Concrete with (a) Steel and (b) Rubber 216 Platen, and (c) Diagonal Shear Failure of Confined 1%HE F R C with Rubber Platen Fig. 9.5 Typical Load-Deflection Response of 0.625 M P a Confined Prisms 217 with Steel and Rubber Platen Fig. 9.6 Effect of Loading Platens on the State of Stress at the Contact Surfaces 219 Fig. 9.7 Load-Time Response of Confined Plain Concrete with Steel and Rubber 220 Platen Fig. 9.8 Test Setup for Confined Beam with Rubber End Platens 221 XXII Fig. 9.9 Schematic Illustrate the Formation of Shear Crack in Steel Platen 222 Confined Beam Compared with Actual Failure Fig. 9.10 Schematic Illustrate the Formation of Flexural Crack and Compressive 223 Crushing in Rubber Platen Confined Beam Compared with Actual Failure Fig. 9.11 Acceleration Distribution of Steel Platens Confined 0.5%HE F R C 224 Beam Fig. 9.12 Acceleration Distribution of Rubber Platens Confined 0 .5%HE F R C 224 Beam Fig. 9.13 Typical Load-Deflection Response of 2.5 M P a Confined Beam with 225 Steel and Rubber Platen Fig. 9.14 Effect of Platen Type on Load-Time Response of Confined Beams 226 Fig. 9.15 Test Setup for Confined Plates with Rubber Platens (Top View) 227 Fig. 9.16 Failure Pattern of 2.5 M P a Confined Plain Concrete Plate with 227 (a) Steel Platen and (b) Rubber Platen, and 0.5%HE F R C Plate with (c) Steel Platen and (d) Rubber Platen Fig. 9.17 Typical Load Responses of Confined Plates with Steel and Rubber 228 Platen Fig. 9.18 Typical Response of Confining Force with Time of Confined (a) 230 Prisms, (b) Beams, and (c) Plate Tested with Steel and Rubber Platens Fig. 9.19 Failure Patterns of Confined Plain Concrete Plates with 150mm 231 Diameter Tup Fig. 9.20 Failure Patterns of Confined 0.5%HE F R C Plates with 150mm 231 Diameter Tup Fig. 9.21 Mid-Plate Acceleration Response of Concrete Plate Tested with 233 100 and 150 mm-Diameter Tup Fig. 9.22 Schematic Illustration of the Mid-Plate Acceleration Distribution of 233 Plate Tested with 100 and 150mm Diameter Tup Fig. 9.23 Typical Load-Deflection Response of Confined Concrete Plates Tested 235 with 100 and 150mm Tups Fig. 9.24 Typical Confinement Force Response of Plain Concrete Plates Tested 235 x x i i i with 100mm and 150mm Tups Fig. 10.1 A Damaged Body and R V E , after Lemaitre (4) 238 Fig. 10.2 Variation of E with Damage of Plain Concrete under Static Compression 240 Fig. 10.3 Measurement of E for (a) Ductile Material and (b) Brittle Material 241 Fig . 10.4 Actual and Predicted Damage of (a) Plain Concrete, (b) 0 .5%HE F R C , 244 and (c) 1.0%HE F R C Prisms under Static Compressive Loading Fig. 10.5 Actual and Predicted Response of (a) Plain Concrete, (b) 0 .5%HE F R C 244 and (c) 1.0%HE F R C Prisms under Static Compressive Loading Fig. 10.6 Response of Plain Concrete-Comparison of Proposed Model and 246 Loland's Model Fig. 10.7 Relative Stress Comparison between the Two Models 246 Fig. 10.8 Change of Strain Rate with Time for Plain Concrete under Impact 248 Compressive Loading Fig. 10.9 Schematic Illustration of the Initial and the Ultimate Strain Rate, 248 and the Strain Rate at Any Time t Fig. 10.10 Actual and Predicted Response of Plain Concrete under Impact 250 Compressive Loading at 250mm Drop Height Fig. 10.11 Actual and Predicted Response of Plain Concrete under Impact 251 Compressive Loading at 500mm Drop Height Fig. 10.12 Predicted Impact Strength of Unconfined Plain Concrete 252 Fig . 11.1 Relationship Between Strain and Time of Unconfined and Confined 255 Concrete Subjected to Impact Compressive Loading (250mm) Fig. 11.2 Relationship Between Relative Strain Rates and Confining Stress 255 Fig . 11.3 Actual vs Predicted Response of 0.625 M P a Confined Concrete 256 Fig. 11.4 Actual vs Predicted Response of 1.25 M P a Confined Concrete 256 XXIV A C K N O W L E D G E M E N T S First o f all , I would like to thank the Royal Thai Government for financially supporting my family and me during the study period. Even during the difficult time of Asian economic crisis in 1997-2000, the support was never short. Recognition is also due to the Natural Sciences and Engineering Research Council of Canada ( N S E R C ) for their contribution to the research project. I wish to deeply thank my thesis advisor, Dr. Sidney Mindess, for his trust, excellent supervision, kind instruction, unconditional and continuous support, and freedom that I am given and benefit from. His advice is always clear and direct to the point not only in the issue of academic but also in the way of life and patience. His trust, support, and freedom that are given to me to explore the world of material testing with no limitation are really pleasant and very enjoyable. Working with Dr. Mindess makes me feel not enough. It teaches me that every tiny piece of work is important and valuable in itself. I would like to extend sincere gratitude to my co-thesis advisor, Dr. Nemkumar Banthia. Without his expertise on impact testing and valuable comments on the analytical part, this thesis would not be as complete as it is. Dr. Banthia is always easy to approach and supportive; he is both an excellent advisor and a good friend to me at the same time. I would like to thank Dr. Frank Lam from the Department of Wood Science, for serving in my thesis committee and for his several valuable comments. Thank is also due to the opportunity given to me to experiment the impact behaviour of Parallel Strand Lumber (PSL) in 1999. Thank is also due to Dr. Perry Adebar for serving both in my comprehensive and thesis committees, and for his several key pieces of advise. For their outstanding works, acknowledgement is owed to the technical staff at the Civil Engineering Workshop, especially Max Nazar for his excellent final design and construction of the confinement apparatus, Herald Schrempp, Douglas Smith, John Wong, and Douglas Hudniuk. Also appreciated are the friendship and support of people in the material group who are like a second family to me, particularly Cheng Yan, Andrew Boyd, Vivek Bindiganavile and Nanda Natarajan. Finally, I also would like to thank Canada, the lovely country where I come to pursue my highest educational degree and where my dear daughter, Charissa, was born. To my wife and my daughter: 1 CHAPTER 1 INTRODUCTION AND SCOPE OF STUDY The measurement of the mechanical properties of concrete depends upon several factors: the test setup (i.e., machine stiffness, type of loading platens, strain measurement method, etc.), the rate o f loading, the specimen geometry, the direction o f loading, the moisture content, the curing technique, and so on. Each of these contributes differently to the measured properties of concrete. In the case of loading rate, there are two extremes: slow (i.e., static or quasi-static) and rapid (i.e., impact). A t low rates of loading, the apparent strength of concrete decreases; at high rates, it increases. In this study, the effect of a particular type of high rate loading (impact) was investigated. The tests were carried out using a drop weight testing method, during which the loading event lasted about 1-3 milliseconds. The study of concrete under impact loading has been carried out extensively for several decades. A t the University of British Columbia, impact loading was first carried out by Mindess and his PhD students: Banthia (1) and Yan (2). They successfully separated the inertial load from the measured load, by determining the acceleration distribution along the beam length. In these studies, many aspects of the behavior of concrete under impact loading were studied, including compression, tension, and flexure (except for shear behavior). It is well known that concrete is more rate sensitive when tested under flexure (beams) than under compression (cylinders or prisms). However, the published test results are largely incomparable, due to differences in test setup, the method of testing, the impact energy, the applied strain rate, and so on. In the present study, to eliminate these differences, a complete series of tests on uniaxial compression, ID-bending and 2D-bending were conducted using the same configuration (drop weight machine), with similar drop height, hammer weight and impact energy, leading to more comparable results. A s a result, material properties such as compressive strength and elastic modulus under both static and impact loading determined from a uniaxial compression test could be used to develop an analytical model to predict the flexural response of concrete. 2 The second objective was to investigate the impact behavior of concrete under multiaxial loading. In most standard tests, the properties of concrete specimens are assessed by subjecting the concrete to a simple loading condition, such as four-point bending. However, in practice, concrete may be subjected to much more complicated multiaxial loading. While the behavior of confined concrete under static loading is quite well established, very few studies of impact loading under confined stress have been done. Thus, in this study, the impact behavior of plain concrete and F R C under confinement was examined with a focus on the failure pattern, response of the material, mechanical properties, and energy absorption capacity. In general, there are two possible configurations in a confined test: 1) passive confinement (e.g., by spiral steel), and 2) active confinement (e.g., by external pressure from fluid or from a machine). Under passive confinement, the lateral confining stress changes with the axial load. On the other hand, in an active (or true) confinement test using external pressure, the lateral stress is applied first and kept constant while the axial load is applied. In this study, an active confinement test was selected. The confinement tests were carried out on the three types of loading: compression, 1 - D flexure, and 2-D flexure. In addition, the effect of boundary restraint on the confinement test was also studied. A s with others tests, the multiaxial test can easily be affected by the test setup. Parameters such as boundary restraint and friction are two of the most pervasive influences. In the present study, the effect of boundary restraint was determined. Two types of loading platens were used: hard (steel) and soft (rubber). The hard platens provide concrete a more uniform strain but a less uniform stress. On the other hand, the soft platens provide a more uniform stress but a less uniform strain. Also , the soft platens, which are more deformable, allow more end rotation to occur. In the analytical part of this study, a scalar damage mechanics ( S D M ) model was used to predict both the static and impact behaviour of concrete, under uniaxial compression and tension. The theory of damage mechanics is based mainly on the state of degradation (damage) of the material, without considering the microstructure of the material (3). It is believed that most materials exhibit similar behaviors under stress, including: elastic behavior, yielding, plastic strain, damage by monotonic or fatigue 3 loading, and crack growth under static and dynamic loading. This may mean that the common mesoscopic properties for most materials can be explained using the same theory (4). This then permits the possibility of studying material behavior at the meso-scale level using the mechanics of continuous media without considering in detail the complexity of the microstructure. Under static compressive loading, the damage was defined in terms of the degradation of the elastic modulus (as compared to initial elastic modulus) with increasing load. In the case of impact, in order to relate the strain rate variable to the damage, the concept of time dependent damage was introduced; i.e., the damage was defined by the change of strain rate with increasing load. In addition, as noticed by Banthia et al (5), the high rate sensitivity of the flexural behavior of concrete when tested by beam specimens can, in some cases, provide unrealistic results in tests with high impact energy. They used plate specimens instead of beams to study the flexural behavior of shotcrete. They found that the rate sensitivity of the concrete was reduced considerably when using plate specimens. Therefore, one of the main objectives here was to continue investigating the behaviour of plate specimens in flexural impact with more focus on cast plain and fibre reinforced concretes (FRC) . Comparison of these different tests has led to a tentative conclusion that the beam specimen is less suitable for flexural impact tests than are plate specimens. 4 C H A P T E R 2 L I T E R A T U R E S R E V I E W 2.1 Impact Testing Methods A number of different methods have been used to measure the impact properties of concrete. The simplest one is the drop weight impact machine (6,7). A mass is raised to a predetermined height and then dropped on to the target. The number of drops required to cause a prescribed level of damage is counted and then used as a measure of the amount of energy absorbed by the specimen to reach this level of damage. There are several limitations for this kind of impact machine. For instance, the energy measurement is only an estimate; no information on load-time history is obtained, and values such as deflection or acceleration are also not obtained. A n instrumented drop-weight impact machine (8), initially developed for metals, provides much more information that the simple drop-weight machine described above. It can be used to obtain reliable time histories of parameters such as load, deflection and acceleration. Different levels of applied energy can be obtained by increasing either the drop height or the mass. The machine constructed at University of British Columbia (1 and used in this study is capable of dropping a 578 kg mass from heights of up to 2.5m (Chapter 3). Another type of instrumented drop weight machine is the pendulum machine, which is a modified version of the Charpy impact testing device. The pendulum machine is also capable of measuring impact load, support reactions and inertial loads. For instance, a machine constructed at the University of Stellenbosch (7) is capable of dropping a mass of 650 to 1450kg from 2.7m. A 500kg machine built in Japan (9), used for testing concrete slabs, is able to produce impact loads of up to 350.1 k N . A n outdoor impact testing pendulum built at Pennsylvania State University (10), is believed to be the largest one of this type, capable of dropping a maximum weight of 4536kg from a 15m height. Even though the instrumented impact machine is able to provide quite a lot of information, it also has some limitations. For instance, the amount of "noise" due to the 5 specimen and striking tup vibrations can be quite substantial, and it may require a noise removal or filtering technique. However, the use of a filter (either software or hardware), i f done improperly, can lead to an under- or overfiltered result. Also , the instrumented impact machine is an energy-controlled test, not a stress rate-controlled test, and therefore the stress rate is an unknown factor prior to the test. It requires a trial and error process (on drop height, hammer weight and specimen geometry) to obtain a desired stress rate. Another type of impact testing machine is the pneumatic hydraulic testing machine (11,12) primarily developed for testing composite materials. This machine is a strain rate-controlled machine (Fig. 2.1) and consists of a unique open/closed loop testing system. The closed-loop mode incorporates a feedback system with a function generator for monitoring load and displacement. The open loop system used fast acting valves with various orifice sizes, which when coupled with adjustable piston strokes insures a controlled displacement rate. Even though the pneumatic type of machine can help in solving the problem of uncontrollable strain rate (as in the drop weight machine), there are still some limitations. For instance, it cannot test large specimens, and the range of strain rates is limited to the medium range (1 to 50 s"1). The small specimen size may make it unsuitable for testing concrete. Orifice Diaphragm—— | Movable Piston Valve High Pressure Gas Line Valve 2 Test Specimen Strain Gauges —| Displacement Transducer Elastic Stress Bar Fig. 2.1 Pneumatic Hydraulic Testing Device (7) 6 One of the most widely used tests for evaluating high strain rates invented by Hopkinson (13), is the Split Hopkinson Pressure Bar, or S H P B (Fig. 2.2). The machine is able to obtain information on duration of event, maximum stress associated with the impact event, strain rate sensitivity, damage propagation, and failure mechanisms. It consists of three major parts: the striking tup, the incident bar and the transmitter bar. During the test process, the specimen is placed between the incident bar and transmitter bar. The striking bar is accelerated by means of the force supplied by a spring mechanism or a gas gun launch system. Upon impact, the compression pulse from the striking bar travels along the incident bar to the specimen. When the pulse reaches the specimen, a part of the incident pulse is absorbed by the specimen, a part is transmitted to the transmitter bar and a part is reflected back to the incident bar; all of the pulses are recorded and used for analysis. Though the S H P B pressure bar is able to test materials at high strain rates and provides a large amount of information, it also has some limitations. For example, the specimen length and diameter must be kept as small as possible to ensure a uniform stress distribution. The effect of friction between the end surface of the specimen and the incident bar is quite high. The S H P B Bar can be used to test materials under various load types, such as compression, tension, bending, and shear (12). Counter for Velocity Measure Striker Bar-Incident Pressure Bar-77 Trigger—/ Temperature Control Unit Furnace Specimen Strain gauge Integrator Amplifier Storage Scope .Transmitter Pressure Bar. Strain gauge I Amplifier 1 -Dashpot Fig . 2.2 Schematic Illustration of the Hopkinson Pressure Bar (Compression) (12) 7 2.2 Concrete under H i g h Strain Rate 2.2.1 Exper imental evidence (a) Mechanica l properties and Fracture parameters Concrete is a strain rate sensitive material; its properties depend not only on the test setup but also on the rate of loading. While a great deal of research has been carried out to try to characterize this behaviour, it is still far from being completely understood. Early evidence on mode I (opening mode) fracture (Fig. 2.3) was obtained by Birkimer and Lindeman (14), they found an increase in both strength and fracture strain with increasing rate of loading. The increase in fracture strain was partly attributed to the increasing microcracking that occurred at the higher rates o f loading. Reinhardt (15) reported about a 20% increase in uniaxial tensile strength when the displacement rate increased from 85xl0" 4 to 1000 mm/sec. Further study by Reinhardt (16) on the modulus of rupture of beams under three-point bending also indicated that the modulus of rupture increased with increasing strain rate. It should be noted, however, that the inertial load (see section 2.2.1 (b)) must be taken into account in interpreting these types of test data. The earliest available data on compressive impact of concrete go back to 1917; Abrams (20) found that the compressive strength of concrete under impact loading was higher than the static strength. Watstein (21) later conducted compression tests with strain x Fig. 2.3 Mode 1 Opening Mode 8 rates varying from 10" to 10 sec" , and found a substantial increase in dynamic compressive strength over static strength. The failure strain and the secant modulus also increased with the increasing strain rate. He found no relationship between concrete strength and the ratio between dynamic and static strength. Green (22), however, reported opposite findings to those of Watstein. Green indicated that the dynamic/static strength ratio increased with increasing compressive strength. The texture of the aggregate also affected the dynamic/static strength ratio, as the angular aggregates showed higher strengths than rounded aggregates. In the case of tensile and flexural impact strengths, Kormeling (23) had conducted direct tensile tests on concrete at different rates of loading and by adopting a stochastic approach, he obtained the relationship between strength and stress rate as follows: Tensile strength cp7 f . -^0.043 (7 V ^Stat J (2.1) Compressive strength ^ + 10 ln(f) s< 0.191 p = 1.3 + 0.131n(ff) s> 0.191 where cp = increasing factor, & = stress rate, and e = strain rate Reinhardt (24) found an increase of tensile strength from about 4 M P a to 5 M P a with an increase of displacement rate from 85*10"6 mm/s to 1000 mm/s. Hawkins et al. also found that the tensile strength obtained from bending tests increased up to 88% when the strain rate increased to 0.2 /s. Gopalaratnam and Shah (17) found that, in flexure tests, the peak bending load increased from about 2.47 k N to 4.27 k N as the strain rate increased from 1.0* 10"6 to 0.30 Is. A plot of strain rate vs. tensile strength shows a gradually increase of strength proportional to the increasing of strain rate from quasi-static loading up to about 10'Vs. Beyond this point, a steep increase of strength is obtained. It should be noted that the effect of strain rate on strength of concrete is most pronounced in tension, followed by flexure and compression, respectively. Zielinski (25) proposed an empirical relationship between tensile strength and stress rate based on his test results using the S H P B : In ^ - = ,4+ 5 I n — (2.2) fo °~o where / and fo = impact and static strength (3.05 N / m m ), respectively 9 & and cr0 = stress rate and the static stress rate (10" N / m m /ms) A and B = constant equal to 0.082 and 0.042, respectively In his study, the fracture of aggregate particles was found more commonly in impact tests than in static tests. His explanation was that under impact loading, a large number of microcracks were forced to extend (propagate) simultaneously due to the rapid increase in stress of the whole volume. With the extremely short time to fracture, the stress relief and redistribution (that occurred under static loading) could not occur, and so the cracks were essentially forced to propagate through the aggregate particles. Since the aggregates were denser and tougher, an increase in strength and fracture energy was observed. In terms of the ultimate strain, there seems to be no definite agreement on whether the ultimate strain increases or decrease with increasing rate of loading: values have been reported in the range of a decrease of 30% to an increase of 40%. On one side, it is believed that the reduction of creep strain under high rates of loading is responsible for the decrease in ultimate strain (26,27,28). On the other side, the increase in ultimate strain is believed to be caused by the simultaneous occurrence of multiple cracks in the test specimens (Fig. 2.4) (21,29,30). The specimens tested under high rates of loading were fractured into more pieces than those tested under static loading (25). Static Impact Existing microcracks Unloading Growing cracks Crack begin to initiate Fracture occur Fig . 2.4 Mechanism of Microcracks under Impact Loading 10 In flexural, the evidences show that an increase of the fracture strain (or deflection) was usually found with an increase of loading rate. Banthia et al. (1) shown an increasing in the midpoint fracture strain of concrete beams under impact from 2.7 to 3.5xl0" 4 as the hammer dropped height (strain rate) increased from 0.15 to 0.5m. This increase was more pronounced when fibers were incorporated in the concrete. Results from Zielinski (31) indicated higher ultimate strains in mortar, micro concrete and concrete under impact loading than under static loading. The ultimate strain o f concrete is higher than that of mortar due to the presence of aggregate particles. He also observed that multiple fractures of concrete beam under impact loading. Shah (32) also found increasing mid point deflection from tests of unnotched plain concrete beams (from 45,720 u..mm to 76,200 u..mm) as the strain rate increased from quasi-static to 0.3 /s. This increase is believed to come from the simultaneous occurrence of multiple cracks (Fig. 2.4). (b) Inertial effect on impact testing The inertial load is defined as that portion of the applied (measured) impact load used to accelerate the specimen from rest. In quasi-brittle material like concrete, this load cannot be directly measured separately; it is a part of the total measured load, and is one of the major sources of errors in impact testing i f not identified. The inertial load was first considered in 1962 by Cotterel (33), in tests on metallic specimens. In 1979, Hibbert (34) reported a 10 to 15 fold strength increase due to rate effects. However, as indicated by Suaris and Shah (35), Hibbert failed to taken into account the inertial effect. In Hibbert's tests, the specimens failed within the first oscillation in which the inertial load was a maximum, and contributed to the measured tup load to provide an unrealistic apparent increase in strength. The first attempts to separate the inertial load from the measured load were made by Saxton et al.. (36) and Server et al.. (37), in which they stated that a reliable measure of the load should be made only after three oscillations of inertial load. However, for a brittle material such as concrete, the impact event is much shorter than for metallic materials and the failure usually occurs within the first oscillation. Thus, it is impossible to isolate the inertial load visually from the measured load. For concrete and 11 other brittle materials, the analysis of the inertial load must be approached differently from metallic materials. Gopalaratnam et al.. (38) suggested that the inertial load could be obtained by subtracting the support reaction load from the measured tup load. Sauris and Shah (35) used a rubber pad between the striking tup and the beam to minimize the inertial load. However, the rubber pad significantly reduces the rate of loading and also tends to absorb large amounts of energy, which must be accounted for when calculating the energy absorption. In 1987, Banthia (1) mounted three accelerometers along the beam length. B y extrapolating between the measured accelerations along the beam, the acceleration distribution could be determined, and thus also the inertial load. The distributed inertial load was then generalized to a point inertial load acting at the centre of beam, using the principle of virtual work. The "true" bending load was obtained by subtracting from the measured tup load the generalized inertial load. This concept has been widely accepted, and is the one used in this study. He found that the inertial load could be as much as 60% to 70% of the measured load. (c) Fracture process zone, fracture toughness and fracture energy A t quasi-static or low strain rates, the occurrence of a large fracture process zone (due to subcritical crack growth) at the crack tip is usually found. However, in the case of dynamic loading, it is found that the fracture process zone decreases with an increasing rate of loading (35,39). Shah (32) measured the decrease of pre-peak crack growth (process zone) of concrete and mortar with increasing of strain rate; his results are given in the Table 2.1. The fracture toughness (the critical stress intensity factor (Kic)) was found to increase with increasing rate of loading. Evans (40) proposed, a relationship between critical stress intensity factor (fracture toughness) and loading rate as shown below, log(Klc) = —^—log(x) + B (2.3) (l + n) where n = subcritical crack growth parameter, x = loading rate and B = constant 12 Mindess et al. (41) tested concrete beams using an instrumented drop weight machine and found an increase of fracture toughness from less than 1 MPa(m)" (static, Kic) up to around 8 MPa(m)" l / 2 with increasing hammer dropped height from 0.15 to 0.5m (strain rate). The calculation of dynamic fracture toughness was based on the relationship given by Broek (42) without any correction for pre-peak crack growth; the value of the peak bending load was modified to account for the effect of inertial loading. They also concluded that high strength concretes are more brittle than normal strength concretes at high rate of loading, due to the decrease of K ID and fracture energy with increasing strength. Table 2.1 Reduction of Pre-peak Crack Growth with the Increasing of Strain Rate. Strain Rate /s Crack Extension at Peak Load (in) Mortar Concrete 0.2x10"4 0.075 0.12 0.1 0.03 0.05 0.2 - 0.028 0.4 0.01 0.016 The neglect of a pre-peak crack growth correction for fracture toughness calculations seems to be appropriate for concrete under high rates of loading. A s indicated by Shah, the values o f the uncorrected K | and modified K i (to account for pre-peak crack growth) were not significantly different under high rates of loading (compared to static loading). A s discussed earlier, the increase of both ultimate strain and strength indicates that materials can sustain higher energy before fracture. The increase in energy absorption capacity is due primarily to the strength increase but also, in part, to the increase of deformation. The latter may, or may not, be significant depending on the extent o f the increase in the ultimate strain. The mechanism of fracture energy under dynamic loading can be explained by Zielinski 's model (31) (see section 2.2.2e). Results predicted by his model for both 13 mortar and concrete agreed very well with the experiment results and show the increasing strength as the strain rate increases. Mindess et al. (41) also found an increase of fracture energy with increasing rate of loading. About 90% of the fracture energy is absorbed in beyond the post-peak region of the curve. The difference between fracture energies in the dynamic and static cases is believed to come at least in part from the testing machine. If it does not have enough rigidity, this results in an underestimate of the fracture energy in the static case. The dynamic fracture energy of normal strength concrete is higher than that of high strength concrete due to the less brittle nature of normal concrete. Oh et al. (43) concluded that fracture energy depended not only on the strain rate but also on the.specimen geometry. Based on their experimental results on various beam sizes, they found the relationships between the dynamic (Gjd) and the static (G/0) fracture energy as follows: - ^ = 3.2 + 0.3941og(£) For large size beam Gf<> —— = 1.9 + 0.1571og(i-) For medium size beam (2.4) G f » —^- = 1.85 + 0.1621og(e) For small size beam (d) Dynamic crack propagation and velocity In the experimental work by Mindess et al. (44), the propagation of cracks under impact loading was captured using a high speed camera at about 10,000 frames per second; the crack velocity was determined by measuring the crack propagation distance between the successive frame. They found that the crack velocity increased with increasing strain rate and decreased dramatically when fibers were incorporated in to the concrete (115 m/s for hardened cement paste and 74 m/s for fiber reinforced concrete). Yan et al. (45) found crack velocities between 132 and 250 m/s. Ross et al. (46) found that crack velocity increased linearly with increasing strain rate on log scale; they also measured crack velocities of 100 m/s at strain rates greater than 1 Is. Takeda et al. (30) 14 has reported crack velocities as high as 1000 m/s under extremely high rates of explosive loading. Kobayashi et al. (47) using two different finite element models to predict the fracture time and crack velocity of plain concrete subjected to three point bending under impact loading. The first model was constructed without considering the fracture process zone, while in the second model, the fracture process zone was included. The crack-closing stress was modeled as being trilinear, with the lines intersecting at three critical crack widths. In first segment, the crack-closing stress is equal to the tensile strength of concrete. The crack closing stress decreases in the second segment, followed by a long trailing wake of moderate crack closing stress. His results from the finite element predictions were in good agreement with the experimental results when the fracture process zone was included in the model. 2.3 Steel F ib re Reinforced Concrete The use of fibres to reinforced brittle materials goes back at least to Egyptian times. The fibres used then were natural fibres, such as horsehair, straw, etc. In the early 1900s, the first commercialized asbestos fibres were introduced. Now, there are numerous fibre types available for commercial use, the basic types being steel, glass, synthetic materials (polypropylene, carbon, nylon, etc.) and some natural fibres. In this study, since the focus is on steel fibres, the literature review also is mostly confined to steel fibres. The earliest tests on steel fibres date back to the 1910s, when wire and metal clips were used to improve the properties of concrete. Serious research on steel fibres began in the 1960's (48,49), and since then substantial amounts of research, development, application and commercialization have occurred (33-38, 50-52). Currently, steel fibre reinforced concrete (SFRC) is used mainly in road construction, slabs on grade, refractory materials, shoterete, and some pre-cast structural components. In general, steel fibres are very high in both strength and modulus of elasticity. The bond strength is dependent upon the surface characteristics (shape, roughness) of fibres, and upon the aspect ratio (ratio of length to diameter of the fibre). The bond can be 15 enhanced by increasing the mechanical anchorage or surface roughness of fibres, or by increase the aspect ratio. The minimum tensile yield strength of steel fibre is 345 M P a ( A S T M A820). Typically, the fibres volume fraction in S F R C is in the range of 0.5% to 1.5%. The use of steel fibres may reduce the slump by about 25 to 100 mm depending on the type, volume fraction, and shape of the fibre (53,54). Therefore, some adjustments to the fresh mix (in term of aggregate size and gradation, water content, or the use of admixture) are required in order to obtain adequate workability with minimal segregation and bleeding, and to provide a uniform distribution of fibres. 2.3.1 Properties of hardened SFRC under static loading It is well known that using moderate volumes of fibres in concrete is not expected to increase the strength of the concrete matrix, but instead should inhibit crack growth and widening. Since the fibres only become active beyond the point of matrix cracking, the first crack strength is always similar for given strength of concrete. For instance, the compressive strength is found to change very little with fibre additions, the increases ranging from 0 to 15% for volume fractions up to about 1.5% (55,56). For tensile strength, the increases are more significant with percentage increases of up to about 30 to 40% (57). A t low volume fractions (<1%), fibres have no real effect on the shear strength of concrete (58). However, at higher volume fraction (>1%), the shear strength of S F R C has been found to increase by up to 30% (59) depending on the test technique, specimen geometry and the alignment of fibres at the fracture surface. For instance, fibre pullout and bridging effects can increase the frictional shear strength, and the close fibre spacing can lead to a more uniformly distributed cracking with reduced crack widths. Even though there is potential for using steel fibres as a replacement for some shear reinforcement in reinforced concrete beam (vertical stirrups) as proposed by several researchers (60-63), it would required more extensive studies to come up with the right solution. 16 In terms of flexural strength, the volume fraction and fibre geometry play an important role. A t low volume fractions (<1%) steel fibres increase only a small fraction of the flexural strength (first crack strength); the real benefit of using fibre is the increase in energy absorption capacity or toughness (area under the load-deflection curve). With higher volume fractions (> 1%), the increase in flexural strength of S F R C may become substantial. Because o f the increased tensile strength o f F R C , fibres permit the tension face of the beam to undergo further pseudo-plastic deformation while the compression face is still in the elastic range. A t 2.0% volume fraction, the flexural strength o f S F R C tested under three-point bending was found to increase by about 50 to 70% compared to plain concrete (64). However, the use o f steel fibres at such high volume fraction is generally unrealistic because of the economical issue. In all cases, the increases in flexural strengths are higher than the increases in either compressive or tensile strength. A t the practical volume fraction used in S F R C (<1%), the increase in compressive, tensile, or flexural strength is small because the matrix cracks essentially at the same stress and strain as in plain concrete. The real advantage of adding fibres is that, after matrix cracking, fibres bridge these cracks and restrain them. In order to further deflect the beam, additional forces and energies are required to pull out or fracture the fibres. This process, apart from preserving the integrity of concrete, improves the load-carrying capacity beyond cracking. This improvement creates a long post-peak descending portion in the load-deflection curve. The area under the curve is usually referred as the energy absorption capacity or toughness (not to be confused with the fracture toughness, Kic) . In the process of fibre pullout, the fibre geometry, the matrix strength and the fibre volume fraction are the key factors in improving the toughness (65,66). Fibres with a deformed shape exhibit significant improvements in toughness compared to straight fibres. Amongst the deformed fibres, the end-deformed fibres are generally more effective than fully deformed fibres (67). The increase in toughness is also proportional to the aspect ratio (length/diameter) (68). Balaguru (69) showed that the response of steel fibre reinforced concrete beams was essentially linear up to 90% of the first crack load. After the first crack, a drop in peak load occurred, followed by an increase in post-peak load (load recovery). The drop in peak load and the load recovery portion depended mainly on the fibre content and 17 geometry. Wi th higher fibre contents, the drop in peak load decreased and the increase in post-peak load became more significant, as did the increase in toughness. However, at very high volume fractions, quite apart from economic consideration, the mixing became very difficult. The properties of fibre are often characterized by the single fibre pullout (bond-slip) test. For a straight fibre, the load transfer is entirely through bond. However, in the case of a deformed fibre, due to its irregular shape, anchorage bond must be added to the mechanism of load transfer. Both fibre fracture and matrix splitting are commonly found in the bond-slip test of a deformed fibre. Banthia et al. (70) found that the peak bond-slip load was affected significantly by the shape and the inclination angle o f the fibre, and matrix strength. With increasing angle of inclination with respect to the load direction, the peak bond-slip load decreased. Premature failure such as matrix splitting or fibre fracture became more common with increasing matrix strength. 2.3.2 Properties of hardened S F R C under impact loading The addition of steel fibres to concrete leads to substantial increase in impact strength. However, the increase is governed mainly the fibre geometry and volume fraction, the matrix strength, and the test method. There is considerable evidence that fibres can greatly enhance the impact properties of concrete. A s long ago as 1977, Bhargava and Rehnstrom (71) tested plain, polymer concrete and polypropylene concrete under high rate of compressive loading, and showed that the strengths of both polymer cement concretes and polypropylene F R C were higher than those of plain concrete by 30% and 15%, respectively. Ramakrishnan et al. (72) performed comparative tests on end deformed and straight fibres and found higher impact strengths with the deformed F R C than with either the straight F R C or plain concrete. Using the repeated drop weight test on F R C , Jamrozy and Swamy (73) found that F R C required an increase number of blows to reach the predetermined level of damage. The increase depended on the volume fraction and type of fibres. However, there was an optimum volume fraction that gave the highest efficiency. The repeated drop weight test 18 was also used by Knab and Clifton (74), who defined the damage by measuring the crater depth at the point of impact. Similar to Jamrozy and Swamy (73), the use of fibres was found to increase the number of blows to "failure". The number of blows to failure of 0.3% hooked end fibre reinforced shotcrete was over 1000 blows, compared to 100 blows for plain shotcrete (75). However, in shotcrete, since most of the fibres are aligned two-dimensionally (lengthwise), this provided better impact resistance. Briggs' (76) test results on F R C using Izod or Charpy machines showed a linear relationship between the impact fracture energy and fibre volume fraction. The increase in fracture energy o f F R C was found to depend strongly on the fibre volume fraction, not the fibre orientation. However, Brown (77) and Harris (78), using the same technique, indicated no change in the toughness of F R C at the initiation of the crack as compared to plain concrete. Majumdar (79) speculated that the volume fraction of the F R C used in their studies was relatively small and could not be used to represent the behaviour of high volume fraction F R C . A n extensive study by Gopalaratnam (17) on plain mortar and notched S F R (1.5%Vf) mortar beams using the instrumented Charpy test showed an increase in peak load, linearity (in the load-deflection curve) and deflection at the peak load with increasing rate of loading. At a strain rate of 0.3 sec"1, the peak load of SFR mortar was about 80%o higher than that of plain mortar. The ratio of dynamic to static modulus of rupture was also found to increase from 1.4 to 2.0 for plain mortar and 1.8-2.6 for S F R mortar. Banthia et al. (80) performed tensile tests on three types of S F R C using a pendulum type impact testing machine. In all cases, increases in tensile strength and fracture energy of both plain concrete and S F R C under impact loading were observed. For F R C , the increase was fibre geometry and matrix strength dependent. The impact/static strength ratio was found to increase with increasing matrix strength. Using a vertical drop weight test, Naaman and Gopalaratnam (81) tried to relate the strain rate sensitivity of S F R C to the aspect ratio and volume fraction by testing S F R C with three volume fractions, three different aspect ratios and two matrix strengths. They found that the strain rate sensitivity increased with increasing aspect ratio and, for a 19 given aspect ratio, the strain rate sensitivity increased with increasing volume fraction. The increase was attributed to the strain rate sensitivity of the bond between matrix and fibre. Banthia et al. (82,83) reported the peak bending load of 1.5%Vf-SFRC was about 40% higher than that of plain concrete under impact loading. Similar increases were also observed in high strength S F R C . The fracture energy of S F R C was also higher than that of plain concrete by a factor of about 2.5 to 3.5, depending on matrix strength and the type and volume of fibres. However, the increase in flexural strength and fracture energy of S F R C under impact loading was relatively smaller than the increase under static loading, perhaps because of the fracture of fibres under a high rate of loading. It was also suggested that, apart from fibre fracture, under impact loading a number of fibres might be pulled out simultaneously. Therefore, a smaller number of fibres was left to bridge across the cracks. According to Mindess et al. (41), similar to plain concrete, the fracture toughness (KID) (calculated without considering crack growth prior to the crack) and fracture energy of 0.5% polypropylene F R C also increased with increasing hammer drop height. However, they indicated that the dramatic increase in fracture energy was due to the underestimated of static fracture energy. 2.4 Analytical Models for Concrete under High Rates of Loading 2.4.1 Continuum damage mechanic (CDM) model The C D M model was first developped by Loland (84) and Mazars (85). The concept of C D M makes use of the idea that every material is imperfect, containing defects (initial damage) such as flaws, pores, or pre-existing cracks. Under an applied load, the damage begins to accumulate due to the propagation and coalescence of these defects, causing degradation of the specimen. When the damage reaches a critical value, failure occurs. The C D M stress-strain relationship is given: a = E(\ - a>)s (2.5) 20 where E is the tangent modulus of elasticity of the undamaged material and co is the accumulated damage variable and can be expressed as a linear function of strain: co = As where A is a constant (2.6) The combination of equations 2.5 and 2.6 yields: ar = E(l- Ae)e (2.7) According to Equation 2.7, strain softening is clearly indicated as a part o f in stress-strain relationship. In the case of dynamic loading, the damage equation can be express as a function of strain rate as follows: kco + cb-As = 0 (2.8) The first term represents the inertial resistance for microcrack growth; at low strain rates, this term becomes zero. Combining equation 2.5 and 2.8, yields the relationship between stress and strain: af cc -Jl (2.9) The prediction of the response of concrete using this model is appropriate for the elastic region up to the peak load but beyond the peak, the predicted values underestimate the response. 2.4.2 Linear fracture mechanics model (a) Macro level (scale of meter) In this model, concrete is treated as a large, homogeneous and isotropic structure containing a single crack subjected to impact loading (86). The use of a single parameter, K (from linear fracture mechanics) is applicable. The crack starts to propagate when K reaches a critical value, Kc- However, for the case in which the crack propagates at a constant velocity, the value o f K decreases gradually. When the crack velocity reaches its critical value, K decreases to zero. The reduction of dynamic stress intensity factor with the increasing crack velocity can be explained by the view that, at low crack velocity, the strain store in the body (in front of and behind the crack tip) can still be transferred to the crack tip faster than the crack grows so that a high localized stress can still occur. But at higher crack velocity, the 21 crack face may not move fast enough to provide the strains at the crack tip necessary for a high stress intensity factor, which result in less stress localization at the crack tip as well as a reduced stress intensity factor. If the crack velocity reaches it critical value (Rayleigh wave speed) the stress intensity factor drops to zero. Failure occurs when K reaches its critical value, K o A s a result of the lower stress intensity factor at the same displacement compared to the static case, this means that the load capacity increases under dynamic loading. Kipp et al. (87) derived the stress intensity factor for a penny-shaped crack in a material under a constant strain rate or stress rate. 3V7Z" 3/2 (2.10) where a = geometry coefficient (1.12 for the penny-shaped crack) cs = shear wave velocity t = duration of loading. Substituting Kic for K i , the fracture stress becomes 9nEK IC 16a c \ 1/3 1/3 (2.11) The cube root indicates that there is a linear relationship (when plotted in the log scale) between the strength and the stress rate with a slope of 1/3. log f_ fo 1 / 3 log 10 Fig.2.5 Relationship between Strength and Stress rate based on Macro L e v e l - L E F M Model-Kipp et al. (87) 22 (b) M i c r o level (scale of cm) A t this level, concrete is schematized as a material containing penny-shaped cracks of a single size and uniformly distributed (88) (Fig. 2.6). Each flaw has the diameter of 2a with a symmetrically distributed distance of 2b between each flaw over the entire cross section. Under an applied load, a crack begins to coalesce from these flaws. B y assuming a certain value of stress (rjc) as the critical stress at which unstable crack propagation starts, the critical flaw size (ac) can be estimated: T2 (2.12) Fig. 2.6 Geometry of Cracked Concrete (88) Once the critical flaw size is known, the kinetic energy during crack propagation from ai to a2 can be determined by subtracting from the total energy (Gi) the fracture toughness (Gic), which is then related to the tensile strength. The results from this model show a bilinear relationship between strength and strain rate. There is gradual increase in strength up to a stress rate = 10 1 1 N / m 2 , with a much more rapid increase at higher stress rates (Fig. 2.7). 23 4.5 f/fo 0 Fig.2.7 Relationship between Strength and Stress rate based on Micro L e v e l - L E F M Model 2.4.3 Thermodynamic model This approach can be considered a stochastic approach or a thermally activated flaw growth model. The main idea is that concrete is assumed to consist of a large number of atoms. These atoms are assumed to be in the state of continuous motion, attracting and repelling each other all the time. When external energy is applied to system either by external loading or heating, the atoms gain a higher energy. Once the energy exceeds the bond strength, they begin to break away. With a continuous supply of energy to the system, more bonds are broken than re-established. Mihashi et al. (89) applied this approach with fracture mechanics to predict the effect of loading rate on concrete strength. They stated that the fracture of concrete could occur by a series of failures in any preexisting cracks or flaws at any time, once the failure criterion was satisfied. A concrete specimen with defects distributed uniformly over the body was modeled as a group of concrete elements linked together like a chain in series, with each element containing one crack (Fig. 2.8). 24 Fig. 2.8 Linked Elements Model for Concrete Based on this model, a simple relationship between the rate of crack initiation (r) and stress is: kT r = — exp h 1 kT (qcr) ,hkT (2.13) where k = Boltzmann constant, h = Planck constant, T = absolute temperature, U 0 = activated energy, q = local stress intensity factor and nb = material constant. 2.4.4 Zie l inski ' s model (25, 31) In this model, concrete is considered as a composite material consisting of spherical aggregates dispersed and embedded in the cement matrix (Fig.2.9). A t the meso level, the fracture surface is assumed to consist of three fracture surface phases i.e., matrix, interface and aggregate particle. The occurrence of fracture either at the interface or in the aggregate particle is controlled by two parameters: the angle of the xy plane of crack approaching the surface of the aggregate and the rate of loading. It is believed that fracture wi l l occur at the aggregate particle when the angle between the xy plane of the crack and the normal to the aggregate surface is smaller than <j)c. The value of <j)c is determined by a statistical approach, and is equal to 8° for static loading and 45° for impact loading. 25 Fracture interfacial bond zones Fracture matrix Fracture aggregate particles F ig 2.9 Zielinski 's Fracture Surface Model The fracture energy determined by this model is assumed to be the summation of the products of specific surface energy and fracture surface area of each phase: where U c = fracture energy associated with a single crack; ac = multiple crack coefficient; Am , A*a, A*b = area of fracture of matrix, aggregate and interface, interface, respectively. In normal strength concrete, the specific surface energy of the aggregate is one order of magnitude higher than that of the matrix, and the matrix surface energy is twice as higher as that of the interface (Table 2.2). If the fracture area of the aggregate phase increases, the fracture energy wi l l increase dramatically due to very high specific surface energy of aggregate. A t high rates of loading, fracture through aggregate particles is more pronounced than through the bond phases, and therefore the fracture energy under dynamic loading is much higher than under static loading. Zielinski also closely related the energy absorbed during the fracture process as: U = acUc = 2ac (Amym + Aju + A'hyb) (2.14) respectively; and ym,ya,yb = specific surface energies of matrix, aggregate and (2.15) 26 where = shape factor of the stress-strain relationship; co = coefficient taking into account the descending branch of stress-strain curve; V = volume of stressed material; se = elastic strain. Using the empirical relationship (Eq. 2.15) obtained from test results, the effect of stress rate can be related to fracture energy as: / = / ( * ) = ( 2 1 6 ) /„ f(&0) ac(&0)Uc(&0Mf) Table 2.2 Fracture Toughness, Elastic Modulus and Surface Energy of Concrete, Steel and Aggregates Material Fracture Elastic Surface Toughness (MPaJm) Modulus (GPa) Energy (Jm 2 ) Concrete r 1.0-1.5 30 5 Cement Paste r 0.5 . 20 15-40 Steel r 20-100 200 25-25000 Granites 4 ' ' 0 0.38-2.87 70-75 -Basalts 4" 0 1.7-2.5 35-85 -Limestones 4 ' ' 0 0.66-3.05 29-56 -Sandstones4"'0 0.28-5.29 18-19 -2.5 Concrete under Multiaxial Stress 2.5.1 Testing techniques In general, there are two basic types of confined tests: 1) passive confinement (using spiral steel), and 2) active confinement (using external pressure from a fluid or a r Lawn, B., "Fracture of Brittle Solids 2"d Edition," Cambridge University Press, 1993 T Whittaker, B.N., Singh, R.N., and Sun, G., "Fracture Mechanics: Principles, Design and Applications," Elsevier, 1992 * Goodman, R.E., "Introduction to Rock Mechanics 2 n d Edition," John Wiley & Sons, 1989 27 machine). With passive confinement, the lateral confining stress changes with the axial load. The ratio of confining stress to uniaxial compressive stress using spiral steel is usually less than 0.15. On the other hand, in a triaxial test using external pressure, the lateral stress is applied first and kept constant while the axial load is applied. The external pressure provides the test with a more active and constant confining stress. Even though Richart et al. (90,91) indicated that both types o f test have similar effects on the behaviour of concrete, Bazant and Tsubakl (92) later concluded that specimens tested with external pressure appeared to be stiffer and to suffer less damage, because the early application of pressure tended to close the cracks and to prevent the growth of microcracks. However, there was agreement that the behavior of concrete under both types of confinement was similar i f the ratio of confining stress to uniaxial compressive stress was less than 0.15. 2.5.2 Static loading In general, the behavior of concrete under multiaxial stress is characterized by its increase in strength, deformation and toughness with increasing confinement stress. Back in the 19 t h century the influence of combined stresses on the strength of structural materials was first considered by Grant (93). He proposed a cube test for measuring the quality of concrete. This was followed by a report by Foppl (94) at about the same time on the behavior of cement-based materials under multiaxial compression, which he studied by loading cylinders surrounded by thick steel jackets. Interestingly, at that early stage, he also recognized the problem of the frictional restraint at the boundary between the testing machine and the specimen. However, not until 1905 did Considere (93) propose an equation for the relationship between concrete strength and lateral confining stress, based on his experimental results on mortar cylinders. Axial stress at failure = K/ +K2(Confining pressure) (2.16) where K i and K.2 are empirical constants In the late 1920s, Richart et al. (90,91) published their classic studies on the behavior of concrete loading axially in compression through rigid platens, with lateral confining pressures applied both hydraulically and with spiral reinforcing steel. They 28 found that the axial compressive strength of confined concrete was considerably higher than that of unconfined concrete. Their well-known equation for the ultimate strength envelope, which has been used and revised by later investigators is: fcc=f:+Aala, (2.17) where fcc is the axial compressive strength of confined concrete, fc is the axial compressive strength of unconfined concrete, A is a constant, and <jkll is the lateral confining stress. According to Richart et al., the constant A is equal to 4.1 (for concrete with / c ' o f 5 to 25 MPa) . Recently, in 1998, Ansari et al. (95) proposed a value of A equal to 3 for concrete with fc of 100 M P a . The study of concrete under multiaxial loading decreased during World War II and then decreased further even after the war was over. One of the reasons for this was the emphasis on simplifying the number of mechanical properties of concrete required for structural design proposes. The philosophy of those times was that, for design proposes, the mechanical properties of concrete should be expressed under relatively simple loading conditions: either uniaxial compression, or flexure. Results from such tests have, since then, been used as design parameters in most structural codes and are, for ordinary structures, quite adequate. However, we know that the state of stress in concrete is usually not as simple as it is assumed, and thus the results obtained under simple loading states may not accurately reflect the true state of the concrete under load in an actual structure. Therefore, using only such data as a source of design information may be limiting. A s concrete structures have become more complex and sophisticated, the need for better information on concrete behaviour under complex states of stress has increased. B y the late 1950s, the multiaxial test was reactivated, and has been used since then by research workers in many countries. One of the problems regularly found in the multiaxial test is the effect of boundary restraint. Richart et al. (90) found that the strength of confined mortar and concrete specimens was greater than that of unconfined specimens by a factor o f 1.3. However, in their study the effect of boundary restraint was not considered. Not until the late 1930s were the effects of boundary restraint on multiaxial tests examined (96-99). 29 Chinn and Zimmerman (96) reported on increase in strength of confined concrete by a factor of 1.25. In their study ^  the effect of axial restraint was estimated roughly and taken into account. Kupfer et al. (97) attempted to reduce the friction induced by end boundary restraints by using steel brush platens; they found an increase in strength of a factor of 1.16 to 1.27. M i l l s and Zimmerman (98) studied the effects of boundary restraint by using different kinds of platen materials such as teflon, polyester, teflon-grease-teflon, and polyester-grease-polyester. They found a decrease in strength under uniaxial, biaxial and triaxial loading as compared to specimens tested using steel platens. A n extensive study by Newman (99) on the effects of boundary conditions indicated an increase in strength of 1.10 to 1.17 compared to unconfined concrete, depending on the boundary condition. He also proposed a non-linear relationship, which provided a better fit than the Richart et al. (90) relationship. f'cc=\f'c+AcTlal\B (2.18) where B is a constant depending on concrete mix characteristics. Several investigators have attempted to relate the failure of both unconfined and confined concrete to volumetric strain (Imran et al. (100), van Mier (101), Smith (102), and Pantazopoulou et al. (103)). They believed that the damage in concrete, whether confined or unconfined, was related to microcracking and manifested by volumetric expansion. For unconfined specimens, the volumetric strain usually decreased slightly in the pre-peak region, before the volumetric expansion occurred that led to subsequent strength decrease and failure. However, for confined concrete, Imran et al. (100) found that the volumetric strain depended on the degree of confinement. Wi th increasing confinement the decrease in the volumetric strain at the beginning of loading was greater before expansion occurred; this provided the confined concrete with a greater strength due to the delay in the occurrence of the expansion strain. They also concluded that, at lower levels of confinement, the restraint provided by the confining system delayed the softening and degradation of the matrix leading to a less brittle failure than in the unconfined case. A t high levels of confinement (40%fc') almost no strength degradation was observed beyond the peak load, the response being pseudo-ductile (resembling 30 plastic flow). In this case, no visible macrocracks could be seen and the failure was believed to be caused by the collapse and compaction of the internal pore structure. Experiments conducted by Ansari and L i (95) on the behaviour of confined high strength concrete under compression also led to similar conclusions: confinement increased both strength and strain at the peak. However, they concluded that the effect of confinement was more pronounced in normal strength concrete than in higher strength concrete. The effect of pore water pressure on confined specimens was studied by both Imran (100) and Akroyd (104). Imran (100) indicated that the pore water pressure had a negative effect on the mechanical behavior of confined concrete, as shown by the decrease in strength of the saturated specimen. The pore pressure in the saturated concrete reduced the confining effectiveness of the applied lateral stress; the weakening effect was more significant for concretes with higher w/c ratio. On the other hand, Akroyd (104) reported that the effect of pore pressure was totally negligible at low levels of confining pressure. Very little research on the behavior of F R C under multiaxial stresses has been carried out (105-107). A recent study by Chern (107) on the behavior of steel fibre reinforced concrete (SFRC) concluded that the mechanical behaviour of S F R C is affected by the confining stress in a similar fashion to that of plain concrete. Even at low confining pressure, S F R C show more ductility than plain concrete, 2.5.3 Impact loading While the understanding of the behavior of concrete under static multiaxial loading is now quite well established, very little research has been carried out under confined impact loading. Most of the available experimental evidences is from partially pre-confined (prestressed) concrete rather than from true confinement. The research on the impact strength of prestressed concrete has been carried out mostly in military applications, and in the nuclear industry. On the military side, an impact event often consists of a high velocity impact of a relatively small impactor compared to the target. Barr et al. (108) showed that the failure 31 patterns of concrete tested under high velocity impact were quite localized, because the target had no time to respond. However, there were some signs of bending and shear failure involved. For very high velocity impact (about lOOOm/s), test results obtained by Wason et al. (109) revealed that the failure of concrete under hypervelocity impact is highly localized. In the case of c iv i l engineering structures, impact is most commonly of a low velocity, with a relatively large impactor. Impact tests on prestressed slabs with a lOm/s impact velocity by Perry et al. (110) showed that the failure of prestressed slabs changed from a flexural mode under static loading to a mixed mode (flexure and shear) under impact loading. However, the mode of failure was strongly influenced by the impact velocity and geometry of the test specimen. Scabbing failure was also found by Fuji and Miyamoto (111); the scabbing could be reduced significantly by using fibres. Studies o f multiaxially loaded concrete under impact loading at the University of British Columbia started in the 1990's, with tests of prestressed concrete railroad ties by Wang (112). Later, the first "true" confinement tests were carried out by Mindess and Rieder (113,114) in order to study the properties of laterally confined concrete under impact loading. Their version of a confinement apparatus (Fig. 2.10) consisted of a steel frame with four instrumented load cells located in both the x and y axis directions. The confinement force was applied manually by the jacks on both axes. However, due to insufficient stiffness of the frame, the side plates tended to deform outwards during the impact event, causing some loss of confining stress. For the present study, a new confinement apparatus was constructed in an attempt to eliminate the loss of confinement stress during the impact event (details are given in Chapter 3). 32 Steel frame Tfc l Steel rod with threat dia 38 mm Top View -Steel handle (for applying stress) Specimen Tup load cell Steel " plates 4 ) Steel rod with threat dia 38 mm \_Support load cell 402 Strain gauges Sec t ion 1 — 1 Fig. 2. 10 A n early version of an instrumented confinement apparatus developed at the University of British Columbia, Mindess and Rieder (113,114) 33 CHAPTER 3 EXPERIMENTAL PROCEDURES 3.1 Introduction Tests on plain and fibre reinforced concrete were conducted on three different shapes of specimens. For uniaxial compression tests, concrete prisms were used; for 1-D bending tests, concrete beams were used; and for 2-D bending tests, concrete plates were used. Each type of specimen was subjected to both static and impact loading (with and without confining stress). A l l tests were carried out in the Structures Laboratory, Department of C i v i l Engineering, University of British Columbia. For all o f the impact tests, a 578 kg drop weight impact machine was used. For the confined tests, the confinement apparatus was attached to the base of the impact machine. B y introducing a confining stress in the impact tests, a significant change of both the failure mechanism and the crack pattern occurred, leading to some modifications in the analysis of the impact data (Chapter 4). In addition, the effects of the testing devices themselves were studied, by changing or modifying part of the impact machine and the confinement apparatus; details are described below in the chapters related to each particular study. In this chapter, the specimens, testing machine, confinement apparatus and associated devices are described. 3.2 Specimen Preparation The specimens were cast using the following materials: Cement: C S A Type 10 Normal Portland Cement ( A S T M Type I) Fine aggregates: Clean river sand with a fineness modulus of about 2.7 Coarse aggregates: Gravel with a 10 mm maximum size Steel fibers: Three different types of steel fibers were used (Table 3.1) at two different volume fractions, 0.5% and 1.0% 34 Table 3.1 Geometry of Fibres Type Shape Length (mm) Section Dia. (mm) Hooked End ^ s- 30 • d = 0.50 Flattened End • < 30 • d = 0.75 Crimped 35 0 .8x3 .0 The mix proportions shown in Table 3.2 were used, providing an average compressive strength of 44.5 M P a for the plain concrete and 45.1 M P a for the fibre reinforced concrete. Table 3.2 M i x Proportions, by weight Cement Water Fine Agg. Coarse Agg . 1 0.5 2 2.5 The dimension of the prisms, beams and plates were 100x100x175mm, 100x100x350mm and 75x400x400mm, respectively. Prior to the mix, the water content in sand and aggregates was determined to adjust the amount of mixing water. The concrete was mixed using a pan type mixer, placed in oiled P V C forms in a single layer, then roughly compacted with a shovel, and finally vibrated on a vibrating table before being covered with polyethylene sheets. P V C forms were used to ensure smooth side surfaces to help reduce localized stresses at the contact surfaces in the confined tests. After 24 hours, the specimens were demoulded and transferred to storage in a water tank for 30 days. Average compressive strength of concrete randomly taken from most batches is given Table A l (Appendix A ) . 3.3 Testing Equipments and Apparatus 3.3.1 The 578 kg impact machine A n instrumented, drop-weight impact apparatus designed and constructed in the Department of C i v i l Engineering, University of British Columbia, and having the capacity of dropping a 578 kg mass from heights of up to 2500 mm on to the target 35 specimen, was used to carry out the impact tests. The velocity o f the falling mass was obtained by: vA=V2(0.91g)/j (3.1) where v n = the velocity of the falling hammer g = the gravitational acceleration h = the drop height From the work of Banthia (1), a correction factor of 0.91 was applied to g to account for the frictional effects between the guide columns and the falling hammer. The impact machine is shown schematically in Fig. 3.1; Fig. 3.2 is a photograph of the machine. Hammer Test Specimen-Hoist 1 TON Hoist Chain -Guide Rails -Machine Columns Air Brake Unit -Tup Load Cell -Support Machine Base Fig . 3.1 Schematic V i e w of 578 kg Impact Machine Fig. 3.2 Instrumented Impact Testing Machine with 578 kg Hammer 37 3.3.2 Tup load cell A circular striking tup (diameter 100 mm) with four electric resistance strain gauges mounted within it was rigidly connected to the 578 kg drop hammer. The shape and dimensions of the striking tup are shown in Fig. 3.3. The tup, made of heat-treated high carbon steel, consists of two parts: 1) an outer (or contact) part and 2) an inner part. The outer part makes full contact with the surface of the specimen during an impact test and then transfers the load to the inner part. There is no instrumentation on the outer part. The inner part, however, is instrumented with four strain gauges mounted on its outer surface. The cross-sectional area at the center of the inner part was reduced by making it hollow; this amplified the signals from the reduced cross-sectional area. A t the "unload" condition, the circuit is balanced and the output is zero. During an impact event, the inner part deforms; the strain gauge circuit becomes unbalanced and emits an output voltage. Briefly, the strain gauges* measured both axial and Poisson deformation. The axial component of the gauge was responsible for the deformation in the vertical direction, and the Poisson component was responsible for the transverse deformations due to the Poisson effect. It also compensated for any unbalanced force due to an uneven contact surface between the tup and the specimen. The general specifications of the strain gauge are given in Appendix B I . The tup was calibrated statically using a universal testing machine so that the proportionality constants for the load cell could be obtained. A calibration factor of 1 m V = 49.233 k N was obtained; the calibration curve is shown in Fig. 3.4. * JP Technologies, Inc. USA, Model # PA06-200TR-350 38 B INNER OUTER 100 A J B; -Strain gauge Impact Surface Side View P - Strain gauge Contact Area Inner & Outer Section A - A Inner Part Outer Part Section B-B 1600 1400 1200 1000 ta 800 « o —I 600 400 200 0 Fig. 3.3 Circular Tup Load Cel l (not to scale) F(kN) = 49.233 * U(mV) E x . : ± 1 0 V D . C . Offset: -0.1 mV Max. capacity: 1000 kN (220.000 lb) +• 10 15 Output U (mV) 20 25 Fig. 3.4 Tup Load Cel l Calibration Curve 39 3.3.3 Accelerometer The accelerometer (Fig. 3.5) used in this study was a Piezoelectric ICP accelerometer (the dynamic performance specifications of the accelerometer are given in Appendix B2). During the test, the accelerometer was attached to the specimen using a threaded plastic base glued to the specimen surface, and was connected to the signal conditioner by a coaxial cable. The base was 22 mm in diameter and 10 mm in depth. Before fixing a base on to the desired location, the surface of the specimen was smoothed using sand paper to prevent an uneven surface. The base was fixed to the surface of the specimen using an epoxy adhesive. Connected to Coaxial Cable Plastic Base 13 .Connected to Plastic Base Fig. 3.5 Piezoelectric Accelerometer (mm) * PCB Piezotronics, Inc. USA, model # 350A14 40 3.3.4 A R O M A T laser sensor A n analog laser sensor* with measurable range of 30 to 50 mm was used in conjunction with the accelerometer for measuring the deformation of the prisms in the uniaxial compression test. It consisted of two major parts: controller section (Fig. 3.6) and sensor section (Fig. 3.7) (the standard specifications are given in Appendix B3). Timing indicator LED_ Operation indicator L E D -Luminious volume excess indicator LED Luminious volume shortage indicator L E D -Sensitivity switch Laser emission indicator L E D -o o o o o o o L J OPERATION //^ -l-^ VJ MQ LASER ANALOG SENSOR LASER ON -110-120V AC j-IOOV AC -100V AC —i =1 LZ. REMOTE INTERLOCK o o o o o o Level monitor meter Criterion position adjuster -Criterion position output switch -Key switch Fig. 3.6 Controller unit for A R O M A T Laser sensor The detection principle employed with this equipment is the optical triple beam distance measurement system. A s shown in Fig. 3.7, the laser beam emitted from the light-emitting element passes through the projector lens and is projected on to an object. A t this time, a part of the diffuse reflected light passes through the receiver lenses to make a light spot on the position sensitive device. The position of the light spot varies according to the detection distance. If this variation is detected, the distance to the object can be determined. * AROMAT, Matsushita Electric Work Ltd., model MQ-LA-C4L-AC120V-S14 class LA 40 41 Within the measurable range, the voltage output is proportional to the distance from the sensor as shown by the calibration curve in Fig. 3.8. Fig. 3.8 A R O M A T Laser Sensor Calibration Curve 42 3.3.5 Instrumented confinement apparatus A n instrumented confinement apparatus (Figs. 3.9 and 3.10) was designed and constructed in the Department of C i v i l Engineering, University of British Columbia, using a total of 4000 kg of steel. The apparatus consists of two-50 ton hydraulic jacks and two load cells located opposite to each other on both the x and y-axes. Each hydraulic jack could exert a maximum pressure of 70 M P a . Two steel load cells with strain gauges mounted on them were placed opposite to the hydraulic jacks on each axis. Four solid steel blocks aligned in the x- and y-axis directions, rigidly connected to the base plate to prevent both vertical and horizontal movements, were used to hold both the hydraulic jacks and the load cells. Between the specimen and the hydraulic jacks or load cells, steel plates of dimensions 400*75*50 mm (width*depth*thickness) were placed in order to distribute the load uniformly to the specimen. 921 Hydraulic Jack Steel Plate TBI Co ^ Spepimerii. jx< 2> 730 < ° •>> • B-Detail 1-1 1250 Load Cell Detai l 1-1 Locking Nut Strain Gauges -Section A - A Fig . 3.9 Schematic V i e w of A n Instrumented Stresses Confinement Apparatus 43 Each load cell, made from high carbon steel, had 4 strain gauges mounted around its outer surface. The output signal was amplified using the same principle as with the tup load cell (deformation of the reduced section). The load cell was calibrated statically using a universal testing machine so that the proportionality constants for the load cell could be obtained. The calibration factors of 1 m V = 67.8 k N and 67.2 k N for load cell No . 1 and No . 2, respectively, were obtained. The calibration curves are shown in Fig. 3.11 (a-b). (b) Side View Fig . 3.10 The Instrumented Confinement Apparatus 44 Fig. 3.11 Load Cells Calibration Curve 45 3.3.6 Data acquisit ion system The data acquisition system consisted of a 16-channel analog to digital (A/D) plug-in interface (ISC-16 card), data acquisition software and a signal conditioner. The A / D board, which was commercially produced*, is capable of digitizing 16 analog data channels with a maximum sampling rate of up to 1 M H z i f only one channel is activated. The sampling rate decreased with the number of active channels by 1/N M H z , where N is the number of active channels. For example, i f four channels were used, the sampling rate was then reduced to V* M H z or 4 Lisec per data point per channel. There were several trigger mode options for the A / D board. In this study, slope and single sweep were used. The voltage output from channel 1 was set at a certain level as a triggering point of the system. The different triggering voltage levels were selected depending on the kind of test (beam, cube or plate) and the strength and stiffness of the material being tested. The setting level also depended on the electrical noise, which changed every day. Obviously, zero or a very small voltage could not be selected because the system would be trigged prior to the test due to the electrical noise. The data points were acquired after the triggering. However, the pre-triggering data could be recorded as well by setting up a delay. The signal conditioner acted as a noise filter as well as a data amplifier. The maximum capacity of the signal conditioner to amplify the signal was lOOOx the original output. In this study, two amplification factors were adopted: 500x for prism and plate tests and lOOOx for beam tests. The data conditioner also acted as a noise filter. However, it is believed that the output signal can easily be distorted by the use of filter. Therefore both types of data (filtered and unfiltered) were recorded; the unfiltered results were then put through a noise removing process (using the filtering technique "Moving Average", details in Chapter 4). The filtered data (raw data from filtered channel) were then compared with the processed results to come up with the best solution that, later, was adopted and used for the entire set of impact data (details in Chapter 4). * RC Electronics, C A , USA Designed and built at the University of British Columbia 46 3.4 Testing Program Three test series were carried out in this study: unconfined static tests, impact tests without confinement, and impact tests with confinement. 3.4.1 Static tests Static tests were carried out on a 1784 k N universal testing machine*. A cylindrical loading head, 250 mm in diameter, was used for the prism compression tests. Two L V D T s (Linear Variable Displacement Transducers) located on opposite sides of the specimen were used to measure the displacement. For the beams, the specimen was tested in 3-point loading with a clear span of 300 mm. For the plate tests, the 400 x 400 x 75 mm plates were simply supported on all four edges with a clear span of 300mm in both directions. Load was applied at the center and an L V D T was placed on the underside at the center to measure the displacement. The data were collected by a PC-based data acquisition system. The static tests were conducted only in the unconfined condition. The testing program is summarized in Table 3.3. 3.4.2 Impact testing 3.4.2.1 Unconfined tests a) Prism tests The specimen was placed vertically on a 100*100 mm rigid steel base located at the center of the impact machine as shown in Fig 3.12. The hammer was dropped from 250 and 500 mm heights to provide two different striking velocities of 2.21 and 3.13 m/s, respectively, and impact energies of 1417 and 2835 J , respectively. The P C B accelerometer and the Aromat Laser Analog Sensor ( A L A S ) described above were used to measure the specimen deformation. The P C B accelerometer was mounted on the * Baldwin Model GBN, manufactured by Statec System Inc., USA 47 hammer, while the A L A S was mounted on the machine with the laser beam pointing vertically at an extension part of the hammer (Fig. 3.1). The differences in test results from the two different instruments were also studied. The testing program is summarized Table 3.4. Table 3.3 Static Testing Program (For prisms, beams and plates) Designation Description Fiber Type V f No . of (%) Specimens P L N S T Plain Concrete - 3 05HEST Fiber Reinforced Plate Hooked End 0.5 3 05FEST Fiber Reinforced Plate Flattened End 0.5 3 05CPST Fiber Reinforced Plate Crimped 0.5 3 1HEST Fiber Reinforced Plate Hooked End 1.0 3 Table 3.4 Impact Testing Program of the Unconfined Prisms Designation Description Steel Fibers Drop Number of Type Content (%) Height (mm) Specimen CPL250 Plain Concrete - - 250 3 CPL500 Plain Concrete - - 500 3 C05HE250 F R C Hooked End 0.50% 250 3 C05HE500 F R C Hooked End 0.50% 500 3 C05CP250 F R C Crimped 0.50% 250 3 C05CP500 F R C Crimped 0.50% 500 3 C05FE250 F R C Flattened End 0.50% 250 3 C05FE500 F R C Flattened End 0.50% 500 3 C1HE250 F R C Hooked End 1.00% 250 3 C1HE500 F R C Hooked End 1.00% 500 3 48 ACCELEROMETER 1 y Accelerometer Signal HAMMER TUP M A C H I N E BASE Fig. 3.12 Experimental Setup for the Impact Tests of Unconfined Prisms b) Beam tests The specimens were placed on a 300 mm support anvil, which provided simple support at both ends. Since concrete is more stress rate sensitive in flexural than in compression, to obtain approximately the same stress rate as that of the prism (for later use in the analytical model), the beam was tested from smaller drop heights, 150 and 300 mm, providing two different striking velocities (1.71 and 2.42 m/s, respectively), and impact energies (850J and 1701, respectively). A n accelerometer was mounted on a plastic base glued to the underside of the specimen at the center to measure the acceleration of the specimen under the impact event. The testing program and experimental setup are given in Table 3.5 and F ig 3.13, respectively. c) Plate tests Specimens were placed on a 400 mm x 400 mm support anvil, which provided simple support at the four edges with a clear span of 300 mm. The hammer, with a 49 cylindricaliy shaped tup load cell, was dropped from two different heights: 250 and 500 mm, to provide different striking velocities (2.21 and 3.13 m/s, respectively) and impact energies of 1417 and 2835 J, respectively. Load was recorded by a PC-based high-speed data acquisition system. The testing program and setup are given in Table 3.6 and Fig. 3.13, respectively. Table 3.5 Impact Testing Program for the Unconfined Beams Designation Description Steel Fibers Drop Number of Type Content (%) Height (mm) Specimen BPL150 Plain Concrete - - 150 3 BPL300 Plain Concrete - - 300 3 B05HE150 F R C Hooked End 0.50% 150 3 B05HE300 F R C Hooked End 0.50% 300 3 B05CP150 F R C Crimped 0.50% 150 3 B05FE150 F R C Flattened End 0.50% 150 3 B1HE150 F R C Hooked End 1.00% 150 3 B1HE300 F R C Hooked End 1.00% 300 3 100/75 / (Beam/Plate) HAMMER TUP v/M\ 350/400 (Beam/Plate) SPECIMEN BEAM/PLATE, _25 Tup Signal ACCELEROMETER MACHINE BASE 350 Accelerometer Signal Data Acquisition System Personal -»-| Computer and Controls Fig. 3.13 Experimental Setup for Impact Tests of Unconfined Beams and Plates 50 Table 3.6 Impact Testing Program of Unconfined Plate Designation Description Steel Fibers Drop Height (mm) Number of Specimen Type Content (%) PPL250 Plain Concrete - - 250 3 PPL500 Plain Concrete - - 500 3 P05HE25 F R C Hooked End 0.50% 250 3 P05HE500 F R C Hooked End 0.50% 500 3 P05CP250 F R C Crimped 0.50% 250 3 P05CP500 F R C Crimped 0.50% 500 3 P05FE250 F R C Flattened End 0.50% 250 3 P05FE500 F R C Flattened End 0.50% 500 3 P1HE250 F R C Hooked End 1.00% 250 3 P1HE500 F R C Hooked End 1.00% 500 3 P1CP250 F R C Crimped 1.00% 250 3 P1CP500 F R C Crimped 1.00% 500 3 P1FE250 F R C Flattened End 1.00% 250 3 P1FE500 F R C Flattened End 1.00% 500 3 3.4.2.2 Confined tests a) P r i s m tests Concrete prisms with and without fibre were tested under biaxial confining stress conditions. The confining stresses were varied from 0 to 1.25 M P a . A similar setup to that of the unconfined tests was used. The concrete prism was place in the centre of the impact machine, and then the confining pressure was applied. The hammer, with a 150mm diameter tup, was dropped from two different drop heights: 250mm and 500mm. Two types of instrument were used to measure the deformation: an accelerometer and a laser sensor. The testing program and setup are given in Table 3.7 and Figs. 3.14 and 3.15, respectively. Table 3.7 Impact Testing Program for Confined Prisms Designation Description Fiber Type V f Drop Confinement Number of (%) Height (mm) Stresses Specimen CPL25B125S Plain - 250 1.25 M P a 3 CPL25B625S Plain - 250 0.625 M P a 3 CPL25B0S Plain - 250 O M P a 3 CPL50B625S Plain - 500 0.625 M P a 3 CPL50B0S Plain - 500 O M P a 3 C05H25B125S FRC Hooked End 0.5 250 1.25 M P a 3 C05H25B625S FRC Hooked End 0.5 250 0.625 M P a 3 C05H25B0S FRC Hooked End 0.5 250 O M P a 3 C05H50B625S FRC Hooked End 0.5 500 0.625 M P a 3 C05H50B0S FRC Hooked End 0.5 500 O M P a 3 C05F25B625S FRC Flattened End 0.5 250 0.625 M P a 3 C05F25B0S FRC Flattened End 0.5 250 O M P a 3 C05F50B625S FRC Flattened End 0.5 500 0.625 M P a 3 C05F50B0S FRC Flattened End 0.5 500 O M P a 3 C05C25B625S FRC Crimped 0.5 250 0.625 M P a 3 C05C25B0S FRC Crimped 0.5 250 O M P a 3 C05C50B625S FRC Crimped 0.5 500 0.625 M P a 3 C05C50B0S FRC Crimped 0.5 500 O M P a 3 C1H25B625S FRC Hooked End 1 250 0.625 M P a 3 C05H25B0S FRC Hooked End 1 250 O M P a 3 C1H50B625S FRC Hooked End 1 500 0.625 M P a 3 C05H50BOS FRC Hooked End 1 500 O M P a 3 52 A C C E L E R O M E T E R HYDRAULIC JACK 1 Accelerometer Signal H A M M E R TUP 100 Z 1 U tu OH Tup Signal Signal from Confinement Load Cell Data Acquisition System MACHINE BASE A R O M A T - L A S E R SENSOR Personal Computer and Controls Output to Disk Fig . 3.14 Experimental Setup for the Impact Test of Confined Prisms Fig. 3.15 Top View of the Test Setup for Confined Prism Tests 53 b) Beam tests Both plain and F R C beams were tested under uniaxial confinement; the confining stresses were varied from 0 to 5 M P a . The same setup as for the impact testing without confinement was used (span length, loading head, and hammer drop height) except for the confining stress, which was applied by an instrumented confinement apparatus (Figs.3.16 and 3.17). In addition, the effects of frictional restraint at the contact surfaces, and the influence of the diameter of the tup load cell were also studied. For the frictional restraint effect, two types of materials were used: rubber and polypropylene sheets; results were then compared with those of the plain steel plate. For the effects of tup diameter, the tup size was changed from 100mm to 150mm. Details are given in the chapters related to each. The testing program is summarized in Table 3.8. c) Plate tests Two series of plate tests were carried out: impact testing under both biaxial confining stresses and uniaxial confining stresses. In both studies, stresses were varied from 0 to 5 M P a , providing both symmetrical and asymmetrical confining stress conditions. Again, the hammer was dropped from heights of 250 and 500 mm. Similar to the beam tests, the effects of frictional restraint at the contact surface and of the diameter of the tup load cell were studied. Details are given in the chapters related to each particular study. The testing program and experimental setup are shown Tables 3.9 and 3.10 and Figs. 3.16 to 3.18, respectively. 54 Table 3.8 Impact Testing Program for Confined Beams Designation Description Fiber Type V f Drop Confining Number of (%) Height (mm) Stress Specimen BPL15B5S Plain 150 5 M P a 3 BPL15B2.5S Plain - 150 2.5 M P a 3 BPL15B1.25S Plain - 150 1.25 M P a 3 BPL15B0S Plain - 150 O M P a 3 BPL30B5S Plain - 300 5 M P a 3 BPL30B2.5S Plain - 300 2.5 M P a 3 B05H15B5S FRC Hooked End 0.5 150 5 M P a 3 B05H15B2.5S FRC Hooked End 0.5 150 2.5 M P a 3 B05H15B125S FRC Hooked End 0.5 150 1.25 M P a 3 B05H15B0S FRC Hooked End 0.5 150 O M P a 3 B05H30B5S FRC Hooked End 0.5 300 5 M P a 3 B05H30B2.5S FRC Hooked End 0.5 300 2.5 M P a 3 B05F15B5S FRC Flattened End 0.5 150 5 M P a 3 B05F15B2.5S FRC Flattened End 0.5 150 2.5 M P a 3 B05C15B5S FRC Crimped 0.5 150 5 M P a 3 B05C15B2.5S FRC Crimped 0.5 150 2.5 M P a 3 B1H15B5S FRC Hooked End 1 150 5 M P a 3 B05H15B2.5S FRC Hooked End 1 150 2.5 M P a 3 B1H30B5S FRC Hooked End 1 300 5 M P a 3 B05H30B2.5S FRC Hooked End 1 300 2.5 M P a 3 55 Table 3.9 Impact Testing Program of Biaxially-Confined Plates Designation Description Fiber Type V f Drop Confining Number of (%) Height (mm) Stress Specimen PPL25B5S Plain 250 5 M P a 3 PPL25B2.5S Plain - 250 2.5 M P a 3 PPL25B125S Plain - 250 1.25 M P a 3 PPL25B0S Plain - 250 O M P a 3 P05H25B5S FRC Hooked End 0.5 250 5 M P a 3 P05H25B2.5S FRC Hooked End 0.5 250 2.5 M P a 3 P05H25B125S FRC Hooked End 0.5 250 1.25 M P a 3 P05H25B0S FRC Hooked End 0.5 250 O M P a 3 P05F25B5S FRC Flattened End 0.5 250 5 M P a 3 P05F25B2.5S FRC Flattened End 0.5 250 2.5 M P a 3 P05C25B5S FRC Crimped 0.5 250 5 M P a 3 P05C25B2.5S FRC Crimped 0.5 250 2.5 M P a 3 P1H25B5S FRC Hooked End 1 250 5 M P a 3 P1H25B2.5S FRC Hooked End 1 250 2.5 M P a 3 P1F25B5S FRC Flattened End 1 250 5 M P a 3 P1C25B5S FRC Crimped 1 250 5 M P a 3 PPL50B5S Plain - 500 5 M P a 3 P05H50B5S FRC Hooked End 0.5 500 5 M P a 3 P05F50B5S FRC Flattened End 0.5 500 5 M P a 3 P05C50B5S FRC Crimped 0.5 500 5 M P a 3 P1H50B5S FRC Hooked End 1 500 5 M P a 3 Table 3.10 Impact Testing Prog] ;ram for Uniaxially Confined Plates 56 Categories Description Fiber Type V f Drop Confining Number of (%) Height (mm) Stresses Specimen P1F50B5S FRC Flattened End 1 500 5 M P a 3 P1C50B5S FRC Crimped 1 500 5 M P a 3 PPL25U5S Plain - 250 5 M P a 3 PPL25U2.5S Plain - 250 2.5 M P a 3 P05H25U5S FRC Hooked End 0.5 250 5 M P a 3 P05H25U2.5S FRC Hooked End 0.5 250 2.5 M P a 3 P05F25U5S FRC Flattened End 0.5 250 5 M P a 3 P05F25U2.5S FRC Flattened End 0.5 250 2.5 M P a 3 P05C25U5S FRC Crimped 0.5 250 5 M P a 3 P05C25U2.5S FRC Crimped 0.5 250 2.5 M P a 3 P1H25U5S FRC Hooked End 1 250 5 M P a 3 P1H25U2.5S FRC Hooked End 1 250 2.5 M P a 3 P1F25U5S FRC Flattened End 1 250 5 M P a 3 P1C25U5S FRC Crimped 1 250 5 M P a 3 PPL50U5S Plain - 500 5 M P a 3 P05H50U5S FRC Hooked End 0.5 500 5 M P a 3 P05F50U5S FRC Flattened End 0.5 500 5 M P a 3 P05C50U5S FRC Crimped 0.5 500 5 M P a 3 P1H50U5S FRC Hooked End 1 500 5 M P a 3 P1F50U5S FRC Flattened End 1 500 5 M P a 3 P1C50U5S FRC Crimped 1 500 5 M P a 3 57 Personal Computer and Controls Output to Disk Fig. 3.16 Experimental Setup for the Impact Test of Confined Beams and Plates Fig. 3.17 Top V i e w of the Test Setup of the Confined Beam Test Fig. 3.18 Top V i e w of the Test Setup of the Confined Plate Test 59 C H A P T E R 4 D A T A A N A L Y S I S 4.1 Introduction This chapter describes the filtering technique adopted for the data analysis o f the impact tests. Generally, several different output signals were obtained, depending on the particular test setup. For the unconfined tests, there were three sources of signals: 1) Tup load cell, 2) Accelerometer and 3) Aromat laser sensor. For the confined tests, two additional sources of signals were the two confinement load cells, in the X and Y-axis directions. The "true" signals, without any amplification, ranged from 0 to 6.00 m V for the tup load cell, 0 to 0.50 m V for the confinement load cell and -0.50 to 0.50 m V for the accelerometer. The duration of the impact event was generally 30 ms or less, which produced about 4000 data points for each test at the sampling rate of 8 us. Since the precise effects of the electronic filtering system built into the data conditioner board were unknown, both the unfiltered and electronically filtered data were recorded. The unfiltered data were analyzed to remove the "noise" as described below, and were then compared with the electronically filtered data. In addition to the filtering technique, this chapter also describes the determination of the velocities and displacements from the accelerometer and Aromat laser sensor data, the determination of the inertial forces corresponding to the different types of failure patterns; and the calculation of the potential energies, kinetic energies, fracture energies and energies lost by the falling hammer at any given time. 4.2 Data Fi l te r ing The noise that occurred in the impact data is mostly electrical noise and could be divided into two main parts: 1) noise prior to the impact event, or "non-event" noise and 2) noise during the impact event. The non-event noise is essentially an "electrical" noise and could easily be detected prior to the beginning of the impact event; it ranged from about -0.05 to 0.05 volts (the full range of output was about 2 volts at 500 m V 60 amplification) or less. The average non-event noise could be subtracted from the data collected for the entire impact event as an initial adjustment. Once the non-event noise was removed, then a smoothing process, using the technique described below, was undertaken. 4.2.1 M o v i n g average analysis The data filtering technique employed in this study was a moving average analysis. The main idea behind this analysis is that for a set of data that shows high fluctuations, but in practice is bounded (i.e., has lower and upper limits), one can smooth out the fluctuations in the data set. The fluctuation within the data set is usually measured by the standard deviation (s): The magnitude of s clearly depends not only on x , but also on n; the larger n is, the smaller s becomes. In other words, given a noisy but bounded measurement data set, the larger the number of data points recorded, the better estimate one can get of its true value. Suppose that at any instant k, the average of the previous n samples of data, x, is 1 k given by: x k = - £ x , (4.3) n i=k-n+\ Also , at the previous instant of time, k-1, the average of the previous n samples is: (4.1) where x is the mean of the n measurements: (4.2) x k-\ 1 k-\ (4.4) Then, k xk-\ ~ n 1 k k-] | •k-n+\ i-k-n J f t (4.5) Rearranging 4.5 gives: 61 **=**-!+-[** -**-,.] ( 4- 6) B y repeating the same sequence throughout the time interval of interest, one can smooth out the fluctuations in the data and show the trends more clearly. 4.2.2 Example : Data filtering analysis Step 1: Given a set of unfiltered data (Fig. 4.1), first determine the "non-event" noise by looking in detail at the portion of the curve prior to the onset of impact loading (Fig. 4.2). Then, determine the average value of this particular non-event noise and use it as an initial value to be subtracted from the entire impact event. In this example, the initial value is 0.0202. Step 2 Consider the data during the impact event (Fig. 4.3). Applying the moving average analysis up to the 3 r d cycle (as described in the previous section) to these data, the results shown in Fig. 4.4 indicate a smoother curve after the third cycle. Step 3 Correct the entire data set by the initial value from step 1. Results are then plotted against the existing electronically filtered data in Fig. 4.5 1.6 -r Time (uSec) Fig. 4.1 Typical Set of Unfiltered Data for an Impact Test 62 Time (uSec) Fig. 4.2 Non-event Impact Data 0.25 -0.05 Time (uSec) Fig. 4.3 Portion of the Data during the Impact Event 63 0.25 i _ 0 _ 0 5 J Time (uSec) Fig. 4.4 Processed Data after Three Cycles of Moving Average Analysis 1.60 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Time (uSec) Fig . 4.5 Comparison of the Electronically Filtered Data and the Processed Data 64 It can be seen that the data recorded through the electronic filtering system were very similar to the processed data; however, the processed data seem to lie slightly below the filtered data. In practice, either of these two curves may be used to represent the "true" signal without significantly affecting the final results. It must be noted that data filtering is sensitive to the method employed to filter the data and must be carried out with caution. Careless use of electrical "noise removers" or mathematically noise remover techniques can lead to a loss of important information. One way to check the filtering technique is by using data obtained from other devices, which can then be compared with the test data. For example, in the case of the beam test, in order to check either the unfiltered or the filtered data from a tup load, one might try to obtain information such as support reactions from the support anvil, and then check whether the reaction force is in agreement with the tup force. 4.3 Accelerat ion, Veloci ty, and Deflection For the cube tests, due to the nature of the confinement, the entire outer surfaces of the specimen were in contact with steel plates, and so direct measurements of the deformation of the specimen with any kind of attached strain gauges were not possible. Thus, two different indirect techniques were used (accelerometer and Aromat laser sensor) to measure acceleration and displacement, respectively. The accelerometer was placed on the impact hammer, while the Aromat laser sensor was placed on the impact machine base, about 500 mm away from the impact location for safety, with the laser aimed at the extension part of hammer. Since the accelerometer was attached to the hammer instead o f the specimen, the recorded output was, in fact, the acceleration o f the hammer. However, since there was no other means of direct measurement, it was assumed that the hammer displacement was essentially equal to the specimen deformation. Then, the velocity and the displacement could be calculated using the equations below. From the initial potential energy, the hammer velocity, uh (0), at beginning of the impact event is uh(0) = j2g*h (4.7) 65 where g = corrected gravitational acceleration (0.91 g) h = hammer drop height A t any time "t" after impact, the velocity iih (t) is equal to i uh(t) = uh(0)+ \uh(t)dt (4.8) 0 where uh (t) = recorded acceleration Then, the displacement of the hammer, uh (t), at any time t becomes uh(t)=\ uh(0) + \uh(t)dt .dt (4.9) For the Aromat laser sensor, the output was a displacement measurement and did not require any conversion. When the two displacements were compared (Figs. 4.6 and 4.7), it was found that the displacements obtained from the Aromat laser sensor were higher than those obtained from the accelerometer, due to the cantilever deflection effect of the hammer extension arm during the impact event. Therefore, the displacements obtained by the Aromat laser sensor were considered to be an overestimate; they were instead, used as an upper bound for checking those obtained by the accelerometer. For the beams and plates, the accelerometer was placed on the underside of the specimen at the centre to measure the mid-point acceleration. Somewhat different from the cube test, the velocities and displacements at the center of the beams and plates could be calculated using Eqs. 4.10-4.12. u0(t) = AgU(t) (4.10) i u0(t)=ju0(t)dt . (4.11) o t u{t) = ju0(t)dt (4.12) 0 where uQ (t) - acceleration at time t (m/s ) g = gravitational acceleration (9.81 m/s 2) U(t) - output from the accelerometer (Volts) * The factor 0.91 (1) accounts for the friction in the hammer guides; it was obtained by direct measurement. uo (0 = velocity at time t (m/s) uQ (t) = displacement at time t (m) A = amplification factor given by manufacturer = 1000 c £ o ra a. •Laser Sensor Accelerometer 1000 2000 3000 Time (jisec) 4000 5000 6000 Fig . 4.6 Displacements as Measured by the Accelerometer and the Laser Sensor 70.00 60.00 50.00 S. 40.00 % 30.00 20.00 10.00 Results from Accelerometer •Results from Laser Sensor 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Strain Fig . 4.7 Stress-Strain Curves Determined Using The Accelerometer and The Laser Sensor 67 4.4 Energies 4.4.1 Energy lost by hammer In all cases (cubes, beams and plates), the energy lost by the hammer could be calculated as follows. According to Newton's law, the impulse is equal to the change of momentum. \P(t)dt = mhuh(0)-mhuh(t) (4.13) where mh = hammer mass and P(t) = measured load from the tup load cell Solving for iih (t) uh(t) = iih(V>)-\P(t)dt 0 The kinetic energy lost at time t is AE(t) = ^mh[uh(0)-uh(t)] Substituting 4.7 and 4.14 into 4.15 yields (4.14) (4.15) A£(0 = ^ m , 2gh- Jlgh \P,(t).dt 4.16 According the Equation 4.16, i f the load-time curve is known, we can find the energy lost by the hammer at any given time t. 4.4.2 Fracture Energy or energy absorbed by the specimen The energy absorbed by the specimen at any particular displacement, Eab, is equal to: 4.17 Eab(t)= \Ph(t)du0 where PB(f) = corrected load (See next section 4.5) and uo = mid-plate displacement 68 4.5 True L o a d (1) Under impact loading, the hammer, with a particular momentum, strikes the specimen (beam or plate) which is at rest on its supports. This causes the unsupported portion of the specimen to gain momentum suddenly and thus to undergo acceleration in the same direction as the falling tup. This gives rise to an inertial load (D'Alambert force), acting in the opposite direction to the impact load. In order to obtain the true bending load, this inertial load must be subtracted from the measured tup load. There is a difference in the nature of the two loads; the measured tup load is a point load, while the inertial load is a body force distributed throughout the body of the specimen. Therefore, in order for this inertial load to be subtracted from the measured load, it must be replaced by an equivalent or generalized inertial load acting at the striking location. The data from the accelerometer placed at the center of the beam or the plate were used to calculate the inertial load. However, since the exact distribution of the acceleration along the specimen was not known, an assumption regarding the acceleration distribution was required in order to simplify the calculation. Once the inertial load is calculated, the true bending load, PD(f), can be determined using Eq. 4.18. Pb{t) = P,{t)-P,(t) (4-18) In this study, the acceleration distribution was assumed to depend upon the failure patterns observed. Several different types of failure patterns occurred in these tests, depending on the type of specimen, the test setup, and the confining pressures. Therefore, the inertial load was divided into two main categories based on the type of specimen, and several sub-categories based on the test setup. 69 4.5.1 Inert ial loads 4.5.1.1 Beams a) Unconfined beams Based on acceleration distribution data obtained using the same impact machine as in this study, Banthia (1) concluded that at any instant of time, the acceleration distribution along the length of either plain concrete or F R C beams without conventional reinforcement could be approximated as linear. Therefore, the acceleration distributions of both the plain concrete and F R C beams in the present study were also assumed to be linear, and symmetric about the mid point (Fig. 4.8). Using the recorded mid-span acceleration, the acceleration at any point from x = 0 to x = 1/2 could then be obtained: W ( x , 0 = ^ ^ x (4.19) u(xj) = ^ ± x (4.20) du(x,t)=^^-x (4.21) where u(x, t) = deflection at any point, x, on the beam at a given time t UQ (t) - mid-span deflection of the beam specimen at any time t ii(x, t) = acceleration at any point, x, at any time t UQ (t) = measured mid-span acceleration of the beam specimen at time t Su(x, t) = virtual deflection at any point x at time t SUQ (t) - mid-span virtual deflection of the beam specimen at time t The given by inertial load, dl, of the beam segment dx, with an acceleration u\x,t), is dl(x,t) = pAu\x,t)dx (4.22) 70 where p - density of concrete A = cross-sectional area of the beam Using the principle of virtual work, the virtual work done by the distributed inertial load acting over the distributed virtual displacement should be equal to the virtual work done by the generalized inertial load Pj(t) acting over the virtual displacement at the center: Pi(t)SuQ =2 ^pAu(x,t)Su(x,t)dx (4.23) where / = clear span length If, = overhanging length of the beam Substituting eq. 4.20 and 4.21 into 4.23, and solving for Pt(t), yields: Pi(t) = pAuh0(t) 3 3/ 2 (4.24) b) Confined beams For confined beams, the failure mode changed from flexure to shear with a crack inclination of approximately 40° to 45°. The inertial load was then the product of the mass of the broken piece and the centre-point acceleration, in which the mass of the broken piece was estimated from the measured angles of failure on both sides and the acceleration was assumed to be the same over the entire broken piece (Fig. 4.9): Pi(t) = mX (4.25) where mt = mass of the broken out piece of concrete UQ = acceleration at the centre of the plate Pt(t) Accelerometer Assumed Linear Distribution Recorded Acceleration 4.8 The Generalized Inertial Load and Assumed Acceleration Distribution for Unconfined Beam Confining stress T Confining stress P,(t) i Confining stress .Accelerometer P,(t) Confining stress Mb Recorded Acceleration Pi(t) Fig. 4.9 The Generalized Inertial Load for a Confined Beam 72 4.5.1.2 Plates a) Unconfined plates The unconfined plates failed primarily in the flexural mode. The acceleration and displacement distribution along the diagonal from the centre of the specimen to the corners was also assumed to be linear along the diagonal line and symmetric about mid-plate (Fig. 4.10). If the plate was divided into four segments, for the segment-1, the acceleration and displacement distributions were given as follow {x + y) where u(x,y,t) = u0° 1 L, ii(x,y,t) = u£ Su(x,y,t) = Su (x + y) I (x + y) (4.26) (4.27) (4.28) u(x, y, t) = deflection at any coordinate, x-y, on the plate at a given time t UQ (t) = mid-plate deflection at any time t u\x, y,t) = acceleration at any coordinate, x-y, at any time t UQ (t) - measured mid-plate acceleration at time t Su(x, y, t) = virtual deflection at any coordinate x-y at time t 5UQ (t) = mid-plate virtual deflection at time t Ld = half-diagonal length of the plate If the segment of plate, dx*dy*h, (length*width*thickness) undergoes a central acceleration il(t), the inertial force acting on it is given as dl(x, y, t) = phu\x, y, i)dx.dy (4.29) Using the principle of virtual work, the virtual work done by the distributed inertial load acting over the distributed virtual displacement should be equal to the virtual 73 work done by the generalized inertial load P,(t) acting over the virtual displacement at the center (Eq. 4.30). For the whole plate (four segments), the Pj(t) is given as £„(/-,/-r) / » ( 0 . < 5 w 0 = 4 j " \phu\t)du(t)dxdy (4.30) 0 0 Substituting eqs. 4.27 and 4.28 into eq. 4.30 and solving for Pj(t), yields: J > ( 0 = ^ 4 (4.31) where p = density of the concrete h = thickness of the plate Ld = half-diagonal length of the plate Thus, once the acceleration at the center of the plate is known, the generalized inertial force can be calculated. b) Biax ia l ly confined plates When the plate is biaxially confined, it fails in a punching shear mode, where pieces of the specimen separate completely. Assuming that the whole broken out piece of concrete has the same acceleration (Fig. 4.11), the inertial force can be given as: W) = mX (4-32) where mi, = mass of the broken out piece of concrete UQ = acceleration at the centre of the plate 74 x/y 4(0 2Ld —Assumed Linear -Recorded Acceleration Fig. 4.10 The Generalized Inertial Load and Assumed Acceleration Distribution for Unconfined Plates c) Uniaxially confined plates When a specimen is under uniaxial confinement, it fails in a flexural mode along the edge of the support anvil. The generalized inertial force is then calculated as for a simply supported beam under three-point bending with the width equal to 300 mm (clear span length). Due to the stiffening effect that limits the rotation at the edges of specimen, the overhang portion can be ignored. 75 The displacement and acceleration distribution are assumed to be linear along the x axis (Fig. 4.12). Using the recorded mid-span acceleration, the acceleration at any point from x = 0 to x = 1/2 could then be obtained u(XJ) = ^iS!lx (4.33) u(x,t)= ° x (4.34) Su(x,t) = ^ j ^ - x (4.35) where u(y, t) = deflection along the fracture plane (y-axis) of the plate at time t u\y, t) = acceleration at any point, x, at time t Su(y, t) = virtual deflection at any point x at time t / = clear span length (300 mm) Again, by applying the principle of virtual work, the work done by the distributed inertial load over the distributed displacement must be equal to the work done by the generalized inertial load over the virtual displacement (Eq. 4.23) in P,(t)-Su0 =2 ^pAu\x,t)5u(x,t)dx (4.36) 0 Substituting eqs. 4.34 and 4.35 into eq.4.36, and solving for Pj(t), yields P,(t) = Apu'(t)t (4.37) where A = cross-sectional area / = clear span length p = density of concrete d) Unfractured Plates When the specimen does not fail, or shows no signs of significant failure, the response of the specimen remains in the elastic region. The inertial force is then assumed to have a sinusoidal shape (115), and is given as: 76 w(x ,^ ,0 = < ( / ) s i n — s i n - y ^ - (4.38) u(x,yj) = u^(t)sm—sm~ (4.39) 5u{x, y, t) = 5u% (t) sin ~ sin ^  (4.40) Substituting eqs. 4.39 and 4.40 into eq. 4.30, yields Pi(t) = phupa(t)1-^ (4.41) where / = clear span length of the concrete plate h = thickness of the plate 77 Fig. 4.11 Failure Pattern and The Generalized Inertial Load for Biaxial ly Confined Plate 78 x Assumed Linear Recorded Acceleration Fig. 4.12 Failure Pattern and The Generalized Inertial Load for Uniaxially Confined Plates 79 4.6 Stress (or Strain) Rate and Its Relationship with Strength (n) The stress and strain rates are calculated by assuming a linear relationship from zero load to peak load as follow: Stress rate (<r) = — (4.42) Ku Strain rate (e) = — (4.43) Kit where o~c = ultimate stress (MPa) sc = ultimate strain tull = time to reach the ultimate stress (seconds) The value of "n" (1,116,117), which may be used to relate strength and stress rate, is derived based on a linear elastic fracture mechanics approach. According to the Griffith theory (118), at the crack or flaw edge of the stressed material, the stress is considerably greater than the far-field stress due to the stress concentration. Even when the applied stress is smaller than the strength o f the material, the stress at the crack tip may reach a critical value, at which failure eventually occurs. The stress at the crack tip is defined by a parameter called the stress intensity factor (Ki), which is a function of applied stress, and crack geometry (Fig. 4.13). For a brittle material, K i exhibits a critical value, K ic . K, = Yo-fa (4.44) where Y = specimen geometry constant a = crack length a = applied stress In accordance with the concept of subcritical crack growth, for a stressed material containing a crack, the rate (or velocity) of subcritical crack growth is associated with K i (Fig. 4.13): 80 where logVj V = AK", V = crack velocity K i = stress intensity factor A , n = constants (4.45) a n K = Yo-4a V = da/dt = AK; a CJ logK. Fig. 4.13 Relationship of K and crack velocity A t a low rate of loading, the subcritical crack may propagate slowly until K reaches its critical value, at which failure occurs. However, at higher rates of loading, there is not sufficient time for subcritical crack growth to occur. Therefore, a higher stress is needed to make K i equal to K ic . According to Eq . 4.44, K i is proportional to V . Substituting Eq. 4.44 into Eq. 4.45, yields — = AY"a"an/2 dt The stress rate is, Substituting Eq . 4.47 into Eq. 4.46 da , da a - — => = dt a (4.46) (4.47) A V " annda = ^—anda a (4.48) 81 Integrating Eq . 4.48 from the initial condition (i) to the final condition (f) yields, (n-2) ( " - 2 ) ' a, 2 -af 2 AY" d{n +1) (4.49) ( 2 a = and B = [YCT) 2(n + l)K (2-n) ( » - 2)^r 2 a5."+l)=5a(CT/("-2)-£75.B-2)) , Eq . 4.49 is rewritten as (4.50) log(af) = [\l{n + Yy}\ogB(J + [\/(n + l)]log(ar 2 -<?n/2) (4-51) Rearranging Eq.4.51 logcr , = log & + C n + \ where C = constant (4.52) Thus a plot of log 07 vs log 6 (Fig. 4.14) would yield a straight line with a slope of l/(n+l). The observed results (details in chapter 5, 6 and 8) indicate that the value of n decreases with increasing stress rate under test conditions used here. t/2 <L> 00 o log(stress rate) Fig. 4.14 Plot between log(stress) vs log(stress rate) 82 C H A P T E R 5 R E S U L T S A N D DISCUSSION-PRISMS 5.1 Introduction The experimental results described in this chapter deal with the behavior of both unconfined and confined plain and F R C rectangular prisms under static and impact loading. The effects of fibre type and content, type and degree of confinement, and hammer drop height are also described. For the unconfined tests, the static responses of both plain concrete and F R C were quite similar, except that the plain concrete failed in a more catastrophic manner than the F R C . A "shear-cone" failure pattern (Fig. 5.1) was generally found under both static and impact loading, though a splitting mode of failure (section 5.2) was found for some F R C specimens. The ultimate strength ( / c . ) , strain at ultimate stress (svH) and the elastic modulus (E) were found to be proportional to the rate of loading. For the confined tests, only impact loading was carried out. In compression, the shear cone failure pattern was replaced by a splitting failure pattern as the magnitude of the confinement increased. Both strength (/J) and ultimate strain (suh) were found to increase with increasing confinement. However, the elastic modulus (E) of confined specimens was essentially unchanged, or, even a little bit lower than that of unconfined prisms. In addition, the relationship between stress and stress rate, and " n " (see Chapter 4 for the derivation of n) were also determined. It was found that, with confinement, the materials became more rate sensitive as indicated by the decrease in the value of n. 5.2 Failure Mode 5.2.1 Unconfined prisms a) Static loading The typical failures modes of unconfined prisms under static loading are shown in Figs. 5.1 and 5.2. Since the tests were carried out using steel loading platens with a 83 relatively high frictional restraint at the contact surface between the loading platens and the specimen, the plain concrete specimens clearly showed an "hour glass (101) or "shear-cone (119)" failure mode (Fig. 5.1). For F R C prisms, either tensile splitting, shear, or combined shear-tensile splitting failure modes were generally found (Fig. 5.2). Fig. 5.1 Failure Patterns of Plain Concrete Prisms under Static Compressive Loading (a) (b) (c) Fig. 5.2 Failure Patterns of Unconfined F R C Prisms Subjected to Static Loading: (a) Splitting (0.5%HE) (b) Shear (1.0%HE) (c) Combined shear-splitting (0.5%CP) For plain concrete subjected to uniaxial compression in a machine in which there is significant and friction on the specimens due to the stiff steel platens on the top and bottom, a triaxial state of stress occurs near the contact zones which is in turn due to the differences in Poisson's ratio between the steel (v = 0.33) and the concrete ( v = 0.20). This end restraint "strengthens" the specimen ends, leading to failure initiating near the mid-height of the specimen. This results in the typical shear-cone type of failure, as was found in this study. Two main diagonal cracks initiate, intersecting at the mid-height of the specimen. Fewer (but longer) cracks are found in plain concrete than in F R C prisms. 84 The failure is catastrophic after the applied load reaches its peak value for the hydraulic machine used in these tests. In F R C prisms, more and finer cracks are observed than in plain concrete prisms. Due to the effects of fibre bridging, most of the fractured pieces were held together by the fibres. Hence, the failure pattern could not always clearly be classified into a single type. Thus, a mixed mode type failure, between shear and splitting, was often observed to be the failure mode o f F R C . b) Impact loading The failure patterns of plain and F R C prisms subjected to impact loading from a 250 mm drop height are shown in Fig. 5.3. In plain concrete, the failure was catastrophic and only the bottom portion of the prism remained on the base after impact; clearly, it was a shear cone type of failure (Fig. 5.3a). A n increased loading rate did not alter the failure pattern of plain concrete, though, of course, failure then occurred more rapidly. In F R C prisms, a mixed mode failure pattern occurred: a combination of shear and tensile splitting (Fig. 5.3b). The ability of the fibres to bridge across the cracks held the broken pieces together and prevented violent failure. However, this was also dependent on the type and content of the fibres. In contrast to the 0.5%HE and 1%HE prisms, considerable volumes of the 0.5%FE and 0.5%CP F R C prisms spalled off the outer surface (Fig. 5.3c). The efficiency of hooked end fibres in preventing damage was higher than that o f the other two fibres. Also , as the fibre content increased, the damage became less severe. Wi th a higher rate of loading (500 mm drop height), the failure mode was catastrophic even for the F R C prisms. Larger volumes of the specimens were broken out compared to those tested at the lower drop height, and only the bottom portion of the specimens remained intact. The failure of both the plain and 0.5%HE F R C prisms was of the shear cone type (Fig. 5.4a). When the fibre content increased to 1%, the specimens were able to sustain loads at larger deformations, and diagonal shear and splitting-shear (with severe damage) failures were observed (Fig. 5.4b and c). (a) (b) (c) (d) Fig. 5.3 Failure Patterns of Unconfined Prisms Subjected to Impact Loading from a 250mm Drop Height: (a) Shear failure (plain), (b) M i x e d mode failure (0 .%HE F R C ) , and (c)-(d) Spalling type failure (0.5% C P and F E F R C ) "s. (a) Shear cone failure (plain and 0.5%HE F R C ) (b) Diagonal shear failure (1%HE F R C ) (c) Mixed mode failure (1%HE F R C ) F ig 5.4 Failure Patterns of Unconfined Prisms Subject to Impact Loading from a 500 mm Drop Height 86 5.2.2 Confined prisms The general failure pattern for both plain and F R C confined prisms subjected to impact loading was of a columnar or splitting mode, combined with some spalling (Fig. 5.5). The state of triaxial stress, which was limited to the area near the contact zone in the unconfined tests, now extended throughout the entire length of the specimen due to the applied confining stress, preventing the occurrence of shear failure, and forcing the cracks to propagate vertically from the top to the bottom of the prism (Fig. 5.5a and b). The number of these tensile splitting cracks decreased with increasing confinement and fibre content. It was possible to increase the confinement stress to such a level that the tensile splitting cracks would not occur unless a substantially larger amount of impact energy was applied. The same phenomenon also applied to F R C at higher fibre contents. As the fibre content increased, the formation of tensile splitting cracks was inhibited by the effect of the bond between the fibres and the matrix. A t increased rates of loading (i.e., increased hammer drop heights), the failure pattern remained that of tensile splitting. A larger amount of spalling at the top and bottom of the specimens, and a larger number of cracks also developed (Fig. 5.5c). The occurrence of spalling at the top and bottom of the specimens was the result of some non-uniformity in the distribution of confinement stress which could not be avoided due to the test setup. This resulted in somewhat less confinement near the specimen ends than at the middle portion of the specimen (Fig. 5.6). (a) (b) (c) Fig . 5.5 Failure Pattern of Confined Prisms Subjected to Impact Loading: (A) Splitting Failure (Plain), (B) Splitting Failure (0.5%HE F R C ) , and (C) Combined Splitting-Spalling Failure (0.5%CP F R C ) 87 Concrete prism Hydraulic jack Prior to applying stress Fig. 5.6 Schematic Sketch Indicate Non-Uniform Stress Distribution 5.3 Stress-Strain (a - s) Response 5.3.1 Unconfined prisms (a) Static loading The typical stress-strain responses of plain and F R C prisms subjected to short-term static loading are shown in Fig. 5.7. In plain concrete, it was found that the linear portion of the cr - e curve extended up to about 40-60% of peak load. Although a small amount of creep or other non-linearity might occur in this linear portion, the deformation was essentially recoverable. A t about 80-95% of the peak load, internal disruption began to occur, and small cracks might appear on the outer surface. These small cracks 8 8 continued to propagate and interconnect until the specimen was completely fractured into several separate pieces. The formation and coalescence of the small cracks has long been recognized as the prime cause of fracture and failure of concrete, and of the marked non-linearity of the stress-strain curve (120). The deformation associated with cracks is completely irrecoverable; and may in some cases be considered as a quasi-plastic deformation. The load (or stress) at which the cracking became severe enough to cause a distinct non-linearity in the a — e curve is often referred to as the point of "discontinuity" (121). A t the peak load, the failure of concrete occurred in catastrophic manner, due to the large amount of energy that was released from the testing machine; thus, the full descending branch of the a - s curve could not be obtained. In F R C prisms, the pre-peak response was essentially the same as for plain concrete. However, because of the ability of the fibres to bridge across the matrix cracks, only a limited amount of strain energy was released at the ultimate load; the failure was much less catastrophic than that of the plain concrete, since the pieces of concrete were held together by fibres even at very large deformations. The descending branch, or post-peak response of the stress-strain curve, could thus be obtained. A s expected, the F R C prisms were considerably "tougher" than the plain concrete prisms. However, the post-peak response depended on the type (or geometry) and content of fibres. A t 0.5% fibre volume fraction, the mildly deformed H E fibres exhibited a tougher post-peak response than the other two types of fibres, F E and C P (Fig. 5.7). Toughness also increased when the fibre content was increased to 1.0%. The aspect ratio (length/diameter) and number of fibres also played a role in this increase (122,123). A t the same volume fraction, fibres with a higher aspect ratio resulted in a larger number of fibres in the mix and thus, a greater probability that a fibre was available to bridge over any particular crack. 89 50 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig . 5.7 Stress-Strain Curves Unconfined Prism under Static Loading (b) Impact loading The unconfined concrete subjected to impact loading exhibited a quite different behavior from that subjected to static loading. The material behaved in a more brittle manner, and increases in strength, toughness, and modulus of elasticity were found as the rate of loading increased. Typical curves for plain, 0.5% H E , 1%HE and the comparison between the three types of fibre are shown in Figs. 5.8 to 5.11. For the specimens tested under impact loading, the linear portion of the a - s curve extended to higher stress values than for specimens tested under static loading. This linear portion was also found to increase with increased rates of loading, as seen in the specimen tested at the highest rate of loading (Figs. 5.8 to 5.10, 500 mm drop height). Theoretically, the small cracks that cause the non-linearity in the static case are forced to propagate much more quickly under impact loading. Thus, the cracks tend to propagate through rather than around aggregate particles, leading to an increase in strength and toughness, and a decrease in the non-linear portion of the a — e curve. 90 80 70 — Static loading — Impact loading (250mm) — Impact loading (500mm) 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.8 Stress-Strain Curves of Plain Concrete Prisms Subjected to Static and Impact Loading 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.9 Stress-Strain Curves of 0.5%HE F R C Prisms Subjected to Static and Impact Loading 91 90 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.10 Stress-Strain Curves of 1.0%HE F R C Prisms Subjected to Static and Impact Loading 70 - l : , : 0.005 0.010 0.015 0.020 Strain Fig. 5.11 Comparison of Stress-Strain Curves of Plain Concrete and the Three Types of F R C under Impact Loading (250mm Drop Height) 92 In addition, it was found that the post-peak response of both plain and F R C prisms decreased more rapidly with the increased rate of loading. To show this, the relative stress (<J/O~C ) and the relative strain (e/eull ) o f the post-peak portions o f the curves are plotted in Figs. 5.12a and b (due to the inability to capture the post-peak response of plain concrete under static loading, only results for the F R C specimens are plotted). In F R C , this can be explained as follows: under impact loading, instead of the individual fibre pulling out slowly, it is probable that larger numbers of fibres were either pulled out or fractured at the same time. Since most of the fibre-matrix bonding is consumed at the same time, a rapid decrease of post peak response results. This effect also depends on the fibre type and content, since different types of fibre have different bond rate sensitivities (that is different sensitivities to strain rate effects during pullout tests). A t a fibre content of 1.0%, the post-peak response was better than at a content of 0.5% (Fig. 5.12a). However, there was no significant difference among the three fibre types (Fig. 5.12b). 5.3.2 Confined prisms The confinement stresses used in this study were 0, 0.625, and 1.25 M P a . The typical responses of the plain and F R C prisms are given in Figs 5.13 to 5.15. The response of the confined concrete was quite similar to that of the unconfined concrete; the pre-peak response consisted of both linear and non-linear portions, followed by a post-peak response. The confined specimens behaved in a more ductile manner as the degree of confinement increased, as indicated by the increase in peak stress, strain (at peak), and toughness. A t 0 M P a confinement, the increase in peak stress was small; as the degree of confinement increased, a more significant increase in peak stress was observed. There was no direct relationship between the confinement stress and the elastic modulus. The responses were quite similar for the plain and F R C concretes, except that the toughness of the plain concrete was found to be less than that of the F R C . 1-00 2.00 3.00 4.00 5.00 6.00 Relative Strain (Strain/Ult.Strain) (a) 1.00 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Relative Strain (Strain/Ult.Strain) (b) Fig. 5.12 (a) Effect of Fibre Content on the Static and Impact Post-Peak Response (b) Effect of Fibre Type on the Impact Post-Peak Response 94 90 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.13 Effect O f Confinement on Impact Response of Plain Concrete (250 mm) 90 i 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.14 Effect of Confinement on Impact Response of 0 .5%HE F R C Prisms (250 mm) 95 90 0.005 0.010 0.015 0.020 0.025 0.030 Strain Fig. 5.15 Effect of Confinement on Impact Response of 1% H E F R C Prisms (250 mm) Note: Failure did not occur at the confinement level of 1.25 M P a 5.4 Ma te r i a l Properties 5.4.1 Compressive strength (f'c) (a) Unconfined prisms The static compressive strengths of the unconfined prisms were in the range o f 40 to 50 M P a , depending on the type of specimen, as shown in Table 5.1. Even though increased strength is not the main purpose of adding fibres to concrete, a small increase of compressive strength with fibre additions was noted. 96 Table 5.1 Average static compressive strength of plain and F R C prisms Concrete Type Average strength C V (MPa) (%) Plain Concrete 40.1 3.6 0.5%HE 44.4 6.6 0.5% CP 41.1 4.7 0.5% F E 42.4 12.9 1.0% H E 51.5 13.3 A s previously mentioned, many microcracks develop when the specimen is loaded up to about 60% of the peak load, and at 85-90%) of the peak load, significant crack propagation begins. In plain concrete, the cracks propagate with hardly any restraint. However, in F R C , many of these cracks are intercepted by fibres, which offer some resistance to further crack growth. A s a result, somewhat higher compressive strengths were obtained. In low (0.5%>) fibre content F R C , when the matrix was ruptured, the concrete lost most of its ability to carry load. The fibre-matrix interaction helped the composite to carry at least some load in the post-peak region. However, with only a small volume of fibres, the bonding was not sufficient to pick up a significant amount of load before final fracture occurred. When the fibre content was increased to 1%, the bonding was sufficient for the fibres to contribute considerable load-carrying capacity even after matrix cracking had occurred. . Under high rates of loading, the compressive strength increased by 1.4 to 2.0 times the static strength (Table 5.2). The plain concrete showed higher rate sensitivity than the F R C . A s the impact energy increased from 1417J to 2835J, the impact/static strength ratio increased from was 1.6 to 2.0. In F R C , the rate sensitivity decreased at the higher fibre contents; for the H E F R C , when the fibre content was increased from 0.5%> to 1.0%, the impact/static strength ratio decreased from 1.5 to 1.4 for a 250 mm drop height, and from 1.8 to 1.7 for a 500mm drop height. 97 It must be noted that the data presented here cannot really be compared to other researchers data due to differences in testing machines, specimen geometries, data acquisition systems, and so on. Table 5.2 Impact strength of plain and F R C prisms in comparison with static strength Average Strength Concrete type Static Impact Ratio Impact Ratio (MPa 250 mm (MPa) I250/ST 500 mm (MPa) I500/St Plain concrete 40.1 65.8 1.6 79.0 2.0 0 .5%HE 44.4 66.3 1.5 80.9 1.8 0.5%CP 41.1 65.2 1.6 77.9 1.9 0.5%FE 42.4 63.6 1.5 81.3 1.9 1.0%HE 51.5 70.5 1.4 85.8 1.7 (b) Confined prisms The compressive strength was found to increase with increasing confinement, as shown in Tables 5.3 and 5.4. For passive confinement (OMPa) and an impact energy of 1417 J, the increase in strength was only about 0%-5%; for the plain concrete, there was no sign of strength increase. At higher levels of confinement (0.625 and 1.25 MPa) , the apparent compressive strength level increased by l%-26% and 23%-40%, respectively, of the unconfined strength. When the impact energy was increased to 2835 J, the confined/unconfined (C/U) strength ratio did not change much: l % - 9 % and 15%-24% for 0 M P a and 0.625 M P a confinement, respectively. The increase in apparent strength of confined specimens is due in part to the friction between the fractured surfaces (124). A s well , concrete also contain preexisting cracks and pores. Under stress, these cracks begin to propagate, coalesce, and form several large cracks. At load levels above about 40% of the peak, these cracks become wider, the material is partially fractured and the surface of the material begins to spall. 98 However, with the confining pressure holding the cracks together, friction between the fractured surfaces continues to support an axial load. Table 5.3 Strength of confined plain concrete and F R C under impact loading Concrete Type 250 mm drop height 500 mm drop height Unconf. Confinement stress (MPa) Unconf. Confinement stress (MPa) 0 0.625 1.25 0 0.625 Plain 65.8 62.7 66.5 80.9 81.0 86.0 90.8 0.5% HE 66.3 66.9 81.4 82.6 80.9 83.8 99.1 0.5% CP 65.2 66.1 74.8 * 78.0 81.1 89.4 0.5% FE 63.6 66.6 79.9 * 81.3 81.9 100.5 1.0% HE 70.5 72.4 86.5 * 85.8 88.0 105.0 *No test has been carried out Table 5.4 Confined/Unconfined strength ratios of plain concrete and F R C under impact Drop height Concrete Type 250 mm 500mm 0-C/U 0.625-C/U 1.25-C/U 0-C/U 0.625-C/U Plain 0.95 1.01 1.23 1.06 1.12 0.5% H E 1.01 1.23 1.25 1.04 1.22 0.5% C P 1.01 1.15 * 1.04 1.15 0.5% F E 1.05 1.26 * 1.01 1.24 1.0% H E 1.03 1.23 * 1.03 1.22 Note: 0-C/U= Ratio between OMPa Confined and Unconfined strength 0.625-C/U= Ratio between 0.625MPa Confined and Unconfined strength 1.25-C/U= Ratio between 1.25MPa Confined and Unconfined strength * N o test was carried out Another explanation comes from observation of the failure modes of the confined specimens. A s shown in the previous section, the typical failure mode of a confined 99 concrete prism was of a splitting or columnar type, with cracks running parallel to the direction of applied load. This suggests that the tensile strain (due to Poisson's ratio) was the principal strain that caused failure. The reason for this tensile splitting failure was, in the confined test, the state of triaxial stress (which occurred only near the contact surfaces in the unconfined test) was applied to the entire specimen by the external force; the easiest way for these confined specimens to fail was by tensile failure (which is the weakest mechanical property of concrete). However, in order to create sufficient transverse strain to cause a splitting failure, sufficient compression stress was required both to do that and also to overcome the confining stress. To make this explanation clearer, let examine the four-particle model (102) in Fig. 5.16. The model in Fig.5.16 represents four aggregate particles in a concrete composite, interacting with each other under the applied load P i . Ideally, when the uniaxial compressive force P i is applied, the splitting tensile force F t starts to develop and becomes the major driving force that leads the concrete to fracture. However, in the case of a biaxial test, when the confinement forces (P2) are applied laterally, they act opposite to the splitting force F ' t caused by the vertical force P i . A s a result, the local tensile splitting force is reduced, which means that more forces are required to fail the concrete. Uniaxial Compression Biaxial Compression Fig. 5.16 Splitting force between four particles due to compressive force (van Mier (1)) P i P i P i P i The difference in fibre contents and types also provided different degrees of strength enhancement (Fig. 5.17 a and b); 1%HE F R C exhibited a higher C / U strength 100 ratio (Fig. 5.17a) than both plain and 0.5% H E at any confinement level. A t 1.25 M P a , the 1.0%HE F R C became so strong that the failure eventually did not occur. Comparing the three types of fibre, F E and C P fibres both exhibited higher C / U ratios than the H E fibre (Fig. 5.17b). The F E fibres seemed to be the most effective. Again, in the case of F R C , the 4-particle model can be used to explain the strength increase as well . Assume that there are fibres located horizontally between the particles on the left and right hand sides (Fig. 5.18). The fibres act as a passive confinement force (F'f), which become active only when there is force acting on them. Similar to plain concrete, when the compression force Pi is applied, the tensile splitting force (F \ ) begins to develop and creates a crack (or propagates a pre-existing crack). However, in the case of F R C , fibres w i l l try to prevent the crack from forming (or propagating) and so does the confinement force P2. Thus, in order to cause failure, a higher compressive force Pi is required to pull out (or fracture) the fibres and overcome the confinement force. o "5 1-40 0.80 Unconfined I 1.80 s: g> 1.60 OMPa-C/U 0.625MPa-C/U (a) 1.25MPa-C/U c o o 0.5%HE •0.5%CP 0.5%FE 0.80 Unconfined OMPa-C/U 0.625MPa-C/U 1.25MPa-C/U (b) Fig. 5.17 Confined/Unconfined Strength Ratio of (a) Plain and F R C , and (b) Comparison between three types of fibre 101 • P i Fig . 5.18 Splitting and fibre pullout forces between four particles 5.4.2 Ultimate strain (sull) (a) Unconfined prisms The term "ultimate strain" is used here to represent the strain corresponding to the peak stress (strength). Concrete under static compressive loading normally exhibits ultimate strains of about 0.002-0.004. In this study, the average ultimate static strain was found in the range of 0.0026 to 0.0036. A s the rate of loading increased, the ultimate strain also increased, though not very much (Table 5.5). Table 5.5 Ultimate strains of plain concrete and F R C at different rates of loading Average Ultimate strain Concrete type Static CV (%) Impact 250 CV (%) Impact 500 CV (%) Plain 0.0035 13.4 0.0027 . 35.0 0.0037 22.9 0.5%HE 0.0036 13.2 0.0037 21.1 0.0042 6.6 0.5%CP 0.0028 9.8 0.0038 14.5 0.0045 12.5 0.5%FE 0.0035 8.9 0.0037 17.4 0.0044 18.8 1.0%HE 0.0040 10.4 0.0044 18.0 0.0047 14.6 102 Clearly, there is an increase trend in ultimate strain with increasing impact energy (or loading rate). However, the increase did not appear to be directly proportional to the impact energy; different systems showed different magnitudes of increase. The increase in ultimate strain is due to the occurrence of multiple cracking (125). Under static loading, cracks develop in a highly localized manner, initiating at areas in which the stress intensity is very high. These cracks interconnect as the load increases, leading to the fracture of the specimen. The failure phenomenon is very different under high rates of loading. Because the impact event occurs over a very short time period, the entire specimen is subjected to similar high stresses instantaneously; this results in the formation of a large number of cracks throughout the specimen, leading to larger strains at failure. (b) Confined prisms The ultimate strains of the confined prisms subjected to impact loading were found to increase with the degree of confinement roughly by about 1.0 to 2.6 times its unconfined strain, as shown in Table 5.6. However, the relationship between ultimate strain and confinement degree was highly dependent on the type of fibre, fibre content and impact energy. Because the results showed a high degree of scatter, there is no clear indication as to what type of fibre, or what fibre content would lead to a particular ultimate strain at a given confinement and impact energy. In general, it was found that confined plain concrete exhibited the highest ultimate strain, up to 2.6 times its unconfined strain, for both drop heights (Figs. 5.19 and 5.20). A t the same degree of confinement, the ultimate strain increased with the rate of loading for both plain concrete and F R C . 103 Table 5.6 Ultimate strains of confined plain and F R C under impact loading Concrete Type H a m m e r Drop height A v g . Unconf. Strain A v g . Confined strain O M P a 0 .625MPa 1 .25MPa Plain 250 0.00212 0.00407 0.00519 0.00543 Plain 500 0.00239 0.00505 0.00601 * 0.5%HE 250 0.00375 0.00429 0.00593 0.00686 0.5%HE 500 0.00383 0.00549 0.00631 * 0.5%CP 250 0.00442 0.00496 0.00665 * 0.5%CP 500 0.00451 0.00605 0.00616 * 0.5%FE 250 0.00443 0.00460 0.00539 * 0.5%FE 500 0.00437 0.00542 0.00684 * 1%HE 250 0.00413 0.00535 0.00648 0.00702 1%HE 500 0.00398 0.00729 0.00703 * * N o test has been carried out 1.00 Unconfined OMPa 0.625MPa 1.25MPa Plain250 •0.5%HE250 • 0.5%CP250 - * -0.5%FE250 --•© 1%HE250 Fig . 5.19 Effect of Confinement on Relative Ultimate Strain (Impact from 250mm) 104 2.80 r c 2.60 n E 2.20 •o 2.00 c c 1.80 o o I 1.60 •a § 1.40 c o ° 1.20 L O O tr^' Unconfined OMPa 0.625MPa —©— Plain500 -A-0.5%HE500 X 0.5%CP500 " Q •0.5%FE500 - o- 1%HE500 Fig. 5.20 Effect of Confinement on Relative Ultimate Strain (Impact from 500mm) 5.4.3 Elastic modulus (a) Unconfined prisms In this study, the values of secant elastic modulus (E s ) was determined up to 40% of ultimate load-E s(0.4Ult). The calculated values of both E s static and impact are given in Table 5.7. The static values of E s(0.4Ult) of both plain concrete and F R C were slightly different and ranged from about 31-40 GPa, respectively. However, both were sensitive to the loading rate, as the rate of loading increased, they increased by a factor of about 3 The relationships with strain rate are as shown in Fig. 5.22. Even though the results show a high degree of scatter, as previously stated, some indication of strain rate sensitivity for E s (0.4Ult) was observed. Plain concrete appear to be more strain rate sensitive than F R C as indicated by the highest impact/static (I/S) ratio of E s(0.4Ult).The (Fig. 5.21). sensitivity decreased with increasing fibre content for both values of Es, as shown in 5.24. Table 5.7 E s o f plain concrete and F R C subjected to static and impact loading Concrete Type Es(0.4Ult) (GPa) Static Impact250 Impact500 Plain 30.8 108.3 122.9 0 .5%HE 39.0 116.0 121.6 0.5%CP 40.7 82.4 85.4 0.5%FE 35.4 59.3 85.8 1.0%HE 33.8 67.5 68.1 • Es(0.4 Ult) Im250/St • Es(0.4 Ult) Im500/ST Fig . 5.21 Impact/Static Elastic Modulus Ratios of Plain and F R C Prisms 106 The higher rate sensitive of plain concrete (in terms of E s ) is probably due to the increase in strength and in brittleness under high strain rates. Under impact loading, increases in both strength and corresponding strain were usually found. In the case of F R C , which is less rate sensitive than plain concrete, a smaller E s was clearly observed. It is suspected that the decrease in E s of F R C with increasing fibre content is due to the differences in compaction since F R C becomes more difficult to compact with increasing fibre content. Plain concrete 0.5%Hooked End 1%Hooked End & » A * • / A li m 0.00001 0.0001 0.001 0.01 0.1 Strain rate (1/Sec) 7.00 6.00 5.00 4.00 3.00 2.00 1.00 10 ro a. 3 T3 O LU O in o ro Q. E Fig . 5.22 Effect of Strain Rate on E s(0.4Ult) of Plain Concrete and F R C (b) Confined prisms For the confined prisms, E s(0.4Ult) was found to range widely, from 50-120 GPa, and decreased with increasing confinement. As seen in Fig . 5.23, both the lower bound and the upper bound were somewhat decreased at different degrees of confinement. The effect of confinement on E s observed in this study is opposite to what was found by Vi l e (126). He found that E s increased with confinement, and suggested that it might perhaps come from the increase of Poisson's ratio with increasing confinement. His experimental evidence indicated a slight increase of Poisson's ratio under static biaxial stress. If such a change is applied to Hooke's law, then a larger E s may be observed. However, in the present, a small decrease of E s with confinement was found in general, perhaps due to the increase of corresponding strain with increasing confinement. The difference may be differences in the applied rate of loading, since V i l e ' s tests were 107 carried out under static loading, and the results could be different under impact. Since, under a high rate of loading, an increase in ultimate strain was generally found, as E is inversely proportional to the strain, the increase in strain would decrease E s . The differences with Vi le ' s results might also due to the different test set-up. In the present impact study, the measurement of deflection was based solely on the acceleration data obtained from the accelerometer. In the regular beam and plate tests, the accelerometer was placed on the specimen to measure its acceleration and deflection, and results obtained from such tests were quite reliable. However, for the confined prism tests, since it is impossible to mount an accelerometer directly on the specimen, the accelerometer was mounted on the hammer, with the assumption that the movement of the hammer was the same as the deformation of the specimen, one could then derive the specimen deformation from the hammer acceleration. A s discussed earlier in Chapter 4, however, the displacement obtained using this method might be larger than the true displacement. With such a large apparent deformation, a decrease in E would result. 140.0 120.0 Upper bound Lower bound 20.0 Unconfined OMPa 0.625MPa 1.25MPa Plain250 0.5HE250 0.5CP250 — 0.5FE250 1HE250 Fig. 5.23 Confined E s(0.40Ult) under Impact Loading (250mm Drop Height) 108 5.4.4 Relationship between stress (or strain) rate and strength B y assuming a linear relationship between stress or strain and time up to the peak load, the stress and strain rates can be calculated as follows: Stress rate (cr) = (5.6) Ku Strain rate (s) =— (5.7) Ku where ac = ultimate stress (MPa) ec = ultimate strain tuh = time to reach the ultimate stress (second) (a) Unconfined prisms The relationship between the strain rate and strength is shown in Figs. 5.24 and 5.25, by graphs of impact/static strength ratio and log(strain rate). The rate sensitivities of plain concrete and F R C were found to be quite similar to each other. A n empirical relationship can be given as: g c ( f o P ) = e * W ( 5 . 8 ) where ac (imp) = impact strength (MPa) <j'c(st) = static strength (MPa) a Jimp) . , , . — = impact/static strength ratio ac(st) e = strain rate (1/sec) k = material constant (0.17-0.26) The material constant, k, was found to depend on the type of material. The suggested k-values obtained from this study are: k = 0.26 for plain concrete, k = 0.23-0.26 for 0 .5%FRC depending on the type of fibre, and k = 0.17 for 1.0%FRC. Plain 109 concrete seemed to be somewhat more sensitive to strain rate than F R C . However, since k is limited to a very narrow range, the average value at 0.20 could be used adequately to represent both plain concrete and F R C . The impact/static strength ratio was found to lie between 1.75 and 2.10 for strain rates of 1.0-5.0 sec"1. Neither plain concrete nor F R C exhibited large differences in strain rate sensitivities. The three types of fibres exhibited quite similar strain rate sensitivities, though the crimped and hooked end fibres seemed to be slightly more rate sensitive than flattened end fibre (Fig. 5.25). Value of "n" (1,127), which can be used to relate strength and stress rate, is given as follows (See Chapter 4 for the derivation of n): logcr,. = C + — ^ - l o g c r (5.9) n +1 where af= failure stress (MPa) a = stress rate (MPa/sec) C = constant In order to determine n, the relationships between log(stress) and log(stress rate) were plotted (Fig. 5.26). Due the lack of data in the intermediate range between static and impact loading, the plots were done using impact data only. The straight line at the high stress rate range in each curve was used to determine the slope and hence "n". The n-values found in this study ranged between 4.61 and 4.79, and both plain concrete and F R C exhibited essentially the same n-values. Theoretically, the n-value indicates the rate sensitivity of material; as the slope of the log( cr )-log( & ) plot increases, the n-value decreases, and the lower the n-value, the higher the rate sensitivity. According to Banthia (1), n-values as low as 1.25 have been obtained. However, his work was carried out on beam specimens and it is generally found that bending tests are more rate sensitive than compression tests. Thus, higher values of n were found in the present work. 110 0.0 0 0 0 1 0.001 0.01 0.1 S t r a i n r a t e (1 IS e c ) Fig . 5.24 Effect of Strain Rate on Strength of Plain Concrete and F R C 2.5 0 0 . 5 % H o o k e d E n d • 0 . 5 % C r i m p e d • 0 . 5 % F la tte n e d E n d 1.5 0 o> u 3 1.0 0 E 0.5 0 0.0 0 0 0 1 0.0 0 0 1 0 .001 0.01 0.1 S t r a i n r a t e (1 /S e c ) 1 0 Fig . 5.25 Effect of Fibre Type on Strain Rate Sensitivity (b) Confined prisms The relationship between stress rate and strength of confined concrete is shown in Fig. 5.27. Confined concrete is more stress rate sensitive than unconfined concrete, as may be seen by the steeper slopes and the lower n-values of the curves, as given in Table 5.8. I l l Table 5.8 Slopes and n-values of unconfined and confined concrete prisms Concrete Type Unconfined Confined at OMPa Confined at 0.625MPa Slope n-value Slope n-value Slope n-value Plain 0.173 4.79 0.704 0.42 0.898 0.11 0.5%HE FRC 0.175 4.70 0.541 0.85 0.743 0.35 1%HE FRC 0.178 4.61 0.355 1.82 0.714 0.40 A s previously stated, the smaller the value of n, the higher the rate sensitivity. It was here that the n-values dropped significantly with increasing confinement stress, especially for plain concrete. This suggests that confined concrete is more rate sensitive than unconfined concrete. The n-value decreased from 4.78 to 0.42 and then to 0.11 when the plain concrete was confined at 0 M P a and 0.625 M P a , respectively. Unlike the unconfined tests, fibres had a substantial effect on the n-values in the confined tests. It was found that the rate of decrease in n-values for F R C was slightly slower than in plain concrete and decreased with increasing fibre content (Fig 5.28). This indicates that confined F R C is less rate sensitive than confined plain concrete. In addition, it was also observed that the n-value of F R C at each confined state of stress was higher than that of plain concrete. The higher rate sensitivity of confined concrete as represented by the smaller values of n, can be explained by examining the failure patterns. A s described earlier, unconfined concrete usually fails in shear, while confined concrete usually fails in splitting tension, with cracks running parallel to the direction of loading. The value of n, which represents the rate sensitivity of the concrete, also represents the crack velocity. Since the cracks in unconfined specimens are primarily shear cracks, while the cracks in confined specimens are predominantly tensile cracks, it may be possible that the shear cracks travel at a slower velocity than the tensile cracks (see Chapter 6 for shear and tensile crack velocities). In addition, the length of the crack path is also a key factor. In the unconfined specimens, the cracks actually travel in the diagonal direction, which thus involves a longer crack path than for the vertical cracks in the confined specimens. With 112 a slower speed and longer path, a much higher n in the unconfined concrete can be expected. 0.0 1.0 2.0 3.0 4.0 log(stress rate) (a) 5.0 1.0 2.0 3.0 4.0 log(stress rate) (b) 5.0 r 2.10 2.00 1.90 O) c 1.80 <D 42-1.70 o 1.60 1.50 6.0 6.0 r 2.10 2.00 si 1.90 c 1.80 V 2. 1.70 o 1.60 1.50 1.0 2.0 3.0 4.0 5.0 6.0 log(stress rate) (c) Fig. 5.26 Relationship between Stress Rate and Stress of (a) Plain Concrete, (b) 0 .5%HE F R C and (c) 1.0% H E F R C 113 -Unconfined Confined at OMPa -Confined at 0.625MPa 1.0 2.0 3.0 4.0 Log (stress rate) (a) 5.0 •Unconfined •Confined at OMPa -Confined at 0.625MPa 1.0 2.0 3.0 4.0 Log(stress rate) (b) 5.0 1.0 5.0 2.10 2.00 1.90 1.80 c 1.70 o 1.60 1.50 6.0 2.10 2.00 1.90 1.80 S 1.70 1.60 1.50 ro o 6.0 Unconfined Confined at OMPa Confined at 0.625MPa /, y jit 2.10 2.00 1.90 1.80 1.70 1.60 1.50 ro c OJ ro o 6.0 2.0 3.0 4.0 log(stress rate) (c) Fig. 5.27 Relationship between Stress Rate and Strength of Confined (a) Plain Concrete, (b) 0.5%HE F R C and (c) 1.0%HE F R C 114 5.00 Unconfined OMPa Confined 0.625MPa Confined 1%FRC 0.5%FRC Plain concrete Fig . 5.28 Effect of Fibres on n-values in Confined Concrete For F R C , the larger n as compared to plain concrete is the direct result of the fibres themselves. The effect of fibres in tying the cracks together leads to lower crack velocities; thus a higher n is observed. With increased fibre content, the value of n increases. 5.4.5 Fracture energy a) Unconfined prisms The fracture energy (calculated out to complete failure) of unconfined prisms tested at a 250mm drop height was approximately 336J for plain concrete, 480-600J for 0 .5%FRC and up to 900J for 1.0% H E F R C (Fig. 5.29). When the drop height increased to 500mm, the fracture energy increased to about 1000J for plain concrete, 1400J for 0 .5%FRC and 2250J for 1.0%FRC (Fig. 5.30). The fracture energies in both cases were quite low compared to the applied impact energy (1417J for the 250mm drop height and 115 2835J for the 500mm drop height). Significant portions of the applied energy appeared to have been dissipated in various parts of the system (machine, sound, heat, vibration); only a portion of the applied energy was absorbed by the specimens. In plain concrete, the fracture energy was about 1/5 to 1/3 o f the impact energy. Wi th fibres, the fracture energy was increased to about 1/3 to 2/3 o f the impact energy. The fracture energy for 1.0%FRC was about 1.7 times that of 0 .5%FRC at the 250mm drop height. Comparing the three types of fibres, the hooked end and flattened end F R C could absorb more energy than the crimped F R C . In general, the energy absorbed by the specimen increased with time until it reached the point of specimen rupture, at which time the energy become constant (no more energy being absorbed, i.e., fracture energy). It was found that the rate o f increase of the absorbed energy was similar in most types o f concrete for the same impact energy. But the starting times (when the curves began to rise) were somewhat different depending on impact energy (Fig. 5.31). A t the an applied impact energy of 1417J (250mm drop height), the curves began to rise at about 1,000 usee after the impact, whereas at an impact energy of 2835J (500mm drop height), the curves began to rise at around 300 to 800 usee after impact. This is probably because the higher impact energy (i.e., the higher rate of loading) caused the specimen to start picking up load earlier than at the lower rate o f loading. b) Confined prisms For the confined prisms, the fracture energy was found to increase with increasing confinement stress. The results from the 250mm drop height are shown in Figs. 5.32-5.34. The fracture energy of the confined concrete was about 1/2 to 2/3 of the applied impact energy at OMPa confinement and up to about 4/5 at 1.25MPa confinement. For confined concrete, the loss of energy to the system decreased with increasing confinement; most of the applied impact energy was absorbed by the specimen. 116 E z >> S» c LU 900 800 700 600 500 400 300 200 100 Plain concrete@250mm 0.5%CP@250mm —0.5%FE@250mm 0.5%HE@250mm 1%HE@250mm 2000 8000 4000 6000 Time (Lisec) Fig . 5.29 Fracture Energy of Unconfined Plain Concrete and F R C (250mm) 10000 E z >. D) i_ O) c 111 — Plain concrete@500mm — 0.5%HE@500mm — 1%HE@500mm 2000 8000 4000 6000 Time (u.sec) Fig. 5.30 Fracture Energy of Unconfined Plain Concrete and F R C (500mm) 10000 2,000 1,800 1,600 1,400 1,200 1,000 800 600 400 200 — Plain concrete@250mm — 0.5%HE@250mm Plain concrete@500mm 0.5%HE@500mm 2000 8000 4000 6000 Time (usee) Fig . 5.31 Effect of Loading Rate on Shape of the Energy vs Time Curves 10000 1,400 1,200 1,000 800 600 400 200 — Unconfined Confined-OMPa — Confined-0.625 MPa —Confined-1.25 MPa / \ — — i -I , . 2000 8000 4000 6000 Time (usee) Fig. 5.32 Fracture Energy of Confined Plain Concrete (250mm) 10000 2000 4000 6000 T ime (Lisec) 8000 10000 Fig . 5.33 Fracture Energy of Confined 0.5% Hooked End F R C (250mm) 1,400 1,200 1,000 — Unconfined Confined-OMPa — Confined-0.625MPa 2000 8000 10000 4000 6000 T ime (|_isec) Fig. 5.34 Fracture Energy of Confined 1.0% Hooked End F R C (250mm) 119 To explain the increase in fracture energy of confined concrete, we must look at two things: first, the ability of concrete increases with the increasing confinement and second, the energy availability for the specimen to absorb. Obviously, the increase in fracture energy is due to the highly effectiveness of the confining stress that able the specimen to withhold together and increase the load-carrying capacity at the large deflection (see Section 5.4.1b). Another speculation is the energy availability. The energy supplied by the hammer can be determined directly from the energy lost by the hammer. The more energy lost by the hammer, the more energy that becomes available to the system. In the confined tests, the energy lost by the hammer is larger than that in the unconfined tests, as shown by looking at the hammer's velocity after impact. The energy lost by the hammer (Eq. 4.14 Chapter 4) is given as AE(t) = ±mh[uh(P)-u„(t)] (5.10) According to Eq. 5.10, for any given drop height and hammer weight, the initial hammer velocity at time t = 0 is the same. The factor which determines the energy lost is the hammer velocity at the end of impact event. If the final velocity is large, the energy lost by the hammer is low. On the other hand, i f the final velocity is small, the energy lost is high. In general, the hammer velocity at time t = 0 (221 cm/sec) was the same for both the unconfined and confined tests. However, as the impact event proceeded, the hammer velocity decreased due to the momentum transfer to the specimen. B y the end of the impact event (specimen fractured), the hammer velocity reached a constant value (Fig. 5.35). However, the final hammer velocities for the unconfined and confined specimens were not the same, as seen in Fig. 5.35. The average hammer velocity at the end of the impact event of unconfined specimen (180cm/sec) was still higher than for the confined specimen (150cm/sec), which means that the hammer in the unconfined test lost a smaller amount of energy to the system. According to these results, the energy lost by the hammer for the confined specimen could be as high as about 1.7 times that for the unconfined specimen. 120 The effects of fibre type and content (Fig. 5.36), were not as significant as in the case of unconfined concrete, About a 20-30% increase in fracture energy was found in 0.5% F R C . When the fibre content increased to 1%, only about a further 10% increase in the fracture energy was observed. There were small difference in the fracture energy between the three types of fibres, but these were not very significant. The reason behind the small increase in the fracture of the F R C (as compared to plain concrete) is perhaps due to the performance of the confinement apparatus that is so dominant (due to its rigidity) that it is able to absorb a large portion of energy, and increase the energy absorption capacity of the plain concrete to be almost the same as that of F R C . A s the hammer drop height increased to 500mm, the fracture energy increased up to about 2/3 to 4/5 of the applied impact energy (2835J) depending on the type of concrete (Fig. 5.36). Similar effects as found in the unconfined tests was also observed, in term of the change in shape of the energy vs time curves (Fig. 5.38). Fig. 5.35 The stress-time and hammer velocity-time response of unconfined and 0.625-M P a confined plain concrete prisms 121 1,400 I 0 2000 4000 6000 8000 10000 Time (uSec) Fig. 5.36 Fracture Energy of 0.625 M P a Confined Concrete (250mm) i 1 1 1 Plain 0.5%HE 0.5%FE 0.5%CP 1.0%HE ill OMPa •0.625MPa Fig . 5.37 Fracture Energy of Confined Concrete at 500 mm Drop Height 122 2,800 2,400 2,000 | 1,600 > g 1,200 LU 800 400 Confined Plain@500-0MPa — Confined Plain@500-0.625MPa — Confined Plain@250-0MPa — Confined Plain@250-0.625MPa / / /-'; • ^^^^^^ ' " 1 1 — -2000 4000 6000 Time (uSec) 8000 10000 Fig. 5.38 Shape of The Energy vs Time for Different Drop Heights (Loading Rates) 123 C H A P T E R 6 R E S U L T S A N D D I S C U S S I O N - B E A M TESTS 6.1 Introduction In this chapter, the results of unconfined and confined beam tests are discussed. It was found that confinement played a significant role in the behaviour and properties of both plain and F R C beams. In general, the failure pattern was found to change gradually with increasing confinement, from a flexural to a mixed mode (flexure and shear) and, finally, to a shear mode. The rate of change in failure pattern also depended on the material type; plain concrete still maintained a mixed mode of failure at confinements up to 2.5 M P a . Confined concrete was stronger and tougher than unconfined concrete, as indicated by the increase in both peak load and fracture energy. However in terms of strength, confined concrete beams were found to be less stress rate sensitive than unconfined concrete beams, as indicated by the higher n-value (Chapter 4). This was due to the change in the failure mode from flexure to shear. Since the shear crack velocity was slower than the flexural crack velocity (Section 6.5c), a higher value of n was found. 6.2 Failure Mode 6.2.1 Unconfined beams Unconfined beams under both static (Fig.6.1) and impact loading (Fig.6.2) failed primarily in the flexural mode. In plain concrete, a single crack initiated from the bottom surface at the centre of the beam (where the tensile stress was a maximum) and then propagated straight to the top to cause failure. There was no sign of crack branching or crack arrest (Fig. 6.1a). For F R C beams, cracks also initiated from the bottom, but not always at the centre of the beam. In F R C , even though the tensile strength of the concrete was somewhat enhanced by the fibres, there were still some relatively weak points in parts of the specimens, due to non-uniform fibre distribution. Thus, the cracks did not 124 necessarily started at the point of maximum tensile stress, but at the weakest point; instead of running straight up to the top, they followed the weakest path (Fig. 6.1b). Therefore, some signs of crack deviation, branching (Fig. 6.1c) and crack arrest were observed. Under impact loading, the failure patterns of the plain concrete beams were essentially the same as for those tested under static loading. For F R C beams, crack deviation and branching were still found (Fig 6.2c); however, these decreased with increasing rate of loading (Fig. 6.2d- 0.5%HE F R C , 300mm drop height). Under still higher rates of loading, the crack had no time to select the weakest path, and instead propagated by the shortest path, resulting in a straighter crack path. 6.2.2 Confined beams The failure modes of both plain and F R C beams changed gradually from flexural to shear with increasing confinement. A t very low (passive) confinement, a mixed mode of failure was observed (Figs. 6.3a, d and g). Two types of cracks were found: (1) a straight crack at the centre due to bending, and (2) diagonal cracks on both sides of the beam due to shear. A s the level of confinement increased, the flexural cracks gradually diminished and at the highest confinement stress (5 MPa) , the flexural cracks were completely eliminated (Figs. 6.3c, f and i). The level of confining stress at which the failure mode changed from mixed-mode (flexural-shear) to a shear mode was not the same for each type of concrete. In particular, at 2.5 M P a , plain concrete still exhibited mixed-mode failure (Fig. 6.3b), while F R C beams showed no sign of flexural cracks (Fig. 6.3e and h). This may be because the fibres themselves acted as internal passive confinement within the specimen and, helped to prevent a crack from initiating. The change of failure mode from flexure to shear was, in part, due to the fixed-end effect from the confining machine. A s the confinement increased, the end fixity increased and began to limit the rotation of both ends of the beam under bending. At a small confinement, the fixed-end effect was small enough that the bending of the beam (even though limited by the confining stress) was still sufficient to cause a combined 125 flexure-shear failure. However, at a high confinement, the bending of the beam was essentially eliminated and the failure became shear dominated. Fig . 6.1 Failure Patterns of Unconfined Beams under Static Loading: (a) Straight Crack in Plain Concrete, (b) Crack Deviation in 0.5%CP F R C and (c) Crack Branching in 0.5%FE F R C (c) (d) Fig. 6.2 Failure Patterns of Unconfined Beams Subjected to Impact Loading: Straight Crack in Plain Concrete Beam Subjected to Impact from (a) 150mm and (b) 300mm, (c) Highly Deviated Crack in F R C Beam (150mm), and (d) Less Deviated Crack in F R C Beam (300mm) 126 (n) 2.5 M P a (o) 5.0 M P a Fig . 6.3 Failure Patterns' of Confined Beams under Impact Loading from 150 mm Drop Height (a)-(c): Plain Concrete, (d)-(f) 0 .5%HE F R C , and (g)-(i) 1.0HE F R C ; from 300 mm Drop Height: fl)-(k) Plain Concrete, (l)-(m) 0 .5%HE F R C , and (n)-(o) 1.0%HE F R C 1 It must be noted here also that the cracks patterns may depend on the exact geometry of the test setup (i.e., tup geometry and span length). The stress flow or spread between contact points could be one of the factors affecting crack alignment. 127 6.3 Load-Deflection Response 6.3.1 Unconfined beams Under static loading, the responses of unconfined beams were as shown in Fig . 6.4. The pre-peak responses of both plain and F R C beams were much the same: the load increased linearly with deflection up to about 60%-80% of the peak before any significant non-linearity occurred. The non-linearity is largely controlled by toughening mechanisms in the fracture process zone, as observed by Mindess and Diamond (128) using a scanning electron microscope. Examples of the toughening mechanisms in the fracture process zone of plain concrete (118) included aggregates bridging, friction between the crack faces, microcracking, and so on. In the case o f F R C , the toughening mechanism would also include fibres bridging across the crack. In addition, the bend-over point (BOP) was also considered in the response of F R C . A t the onset of loading, the cracks were initiated randomly in the matrix. A t the B O P , the cracks became localized, the matrix failed, and the fibres began to take over. A t lower fibre contents, the B O P was the same as the peak load, as seen in this case for all 0 .5%FRC beams. However, at the higher fibre content (1.0%), the B O P and the peak load were not necessarily the same, as seen for the 1.0% hooked end F R C beam. It was the post-peak response that really differentiated the plain concrete from the F R C . For plain concrete, once the strain energy was high enough to cause the crack to self-propagate, fracture occurred almost instantaneously once the peak load was reached due to the tremendous amount of energy being released. For F R C , the fibre bridging effect helped to control the rate of energy release. Thus, F R C maintained its ability to carry load after the peak. However, this also depended on the type and content of fibres; hooked end fibres performed better than crimped or flattened end fibres at 0.5%Vf. The responses of both plain and hooked end F R C beams subjected to impact loading are illustrated in Fig. 6.5. Plain concrete still exhibited no post-peak response, though increases in strength, and deflection at peak were generally found. For the F R C beams, the effects of the fibres were more pronounced under impact, as even the low fibre content F R C beams exhibited much higher peak loads than did the plain concrete 128 beams. Apparently, under impact loading, the point at which matrix cracking and fibre pullout occurred were essentially the same. Thus, the matrix strength and the fibre pull-out force were combined and resulted in higher F R C strengths under impact loading compared to plain concrete. The bend-over point (BOP) in F R C beams subjected to impact loading was generally found to coincide with the point of peak load. A s previously stated, for a beam tested under impact loading, most fibres were pulled out almost simultaneously at the peak. There were not enough fibres left to permit a strength recovery or a second peak that was higher than the first peak. (However, as w i l l be seen in detail in Chapter 7 this was different for the plate specimens. A t the peak, there were several macro and micro cracks in the F R C plates. Since fibre bridging and stretching only start at cracking, i f there are enough crack surfaces (as in the plate specimen), the effect of fibre bridging could become large and could drive the plate response to have a second or even a third peak higher than the first peak). The post-peak response for beams did include the occurrence o f some small peaks (multiple-peak) due to the effect of fibre bridging and stretching. However, the multiple-peak characteristic found in the beam tests was not as significant as that found in plate tests (Chapter 7). The effects of loading rate (drop height) on both plain and F R C beams are given in Figs. 6.6 to 6.8. When the hammer drop height increased from 150 to 300mm, the plain concrete clearly exhibited a more brittle behavior, as indicated by the increase in the linear portion of the load-deflection response. Also , a slight increase in stiffness (indicated by the steeper slope of the pre-peak response) was found in both plain concrete and F R C . There was, as well , an increase in strength, deflection at peak load and toughness as the drop height increased. 129 Fig. 6.4 Static Response of Unconfined Plain and F R C Beams 60.00 50.00 40.00 •a 30.00 n o 20.00 10.00 0.50 1.00 1.50 2.00 Deflection (mm) 2.50 3.00 Fig. 6.5 Impact Response of Unconfined Plain and F R C Beams (150mm) 130 50 40 ^ 30 T3 ro O 20 10 0.20 — Static Impact at 150mm — Impact at 300mm 2T 0.40 0.60 Deflection (mm) 0.80 1.00 Fig. 6.6 Effect of Hammer Drop Height on Unconfined Plain Concrete Beam Fig. 6.7 Effect of Hammer Drop Height on Unconfined 0 .5%HE F R C Beams 131 80.00 0.50 1.00 1.50 2.00 2.50 3.00 Deflection (mm) Fig. 6.8 Effect of Hammer Drop Height on Unconfined 1.0%HE F R C Beams 6.3.2 Confined beams The responses of confined beams are given in Figs. 6.9 to 6.12. The responses of both plain and F R C beams were quite similar: an increase in strength and toughness with increasing confinement was found in general for both plain and F R C beams (Figs. 6.9 and 6.10). The B O P and the peak load were also found to occur at about the same point similar to those of the unconfined beams. A significant post-peak response of the confined plain concrete occurred because the confinement stresses prevented catastrophic disintegration of the concrete. However, the performance of plain concrete was quite different from that of F R C . A l l F R C beams (0.5% fibre content) performed better than plain concrete at 2 .5MPa confinement and 150 mm hammer drop height (Fig. 6.11). Hooked end F R C exhibited a slightly better post-peak response than the other two FRCs . When the hammer drop height was increased from 150mm to 300mm, the responses of both the plain and the F R C beams did not change much. Even though the impact energy was doubled, there was less effect on the 132 confined beam than on the unconfined beams; slightly increased peak load and toughness were found. 160 T ! T : 1 1.00 2.00 3.00 4.00 5.00 Displacement (mm) Fig. 6.9 Response of Confined Plain Concrete Beams under Impact Loading (150mm) 200 2.00 4.00 6.00 8.00 10.00 Deflection (mm) Fig. 6.10 Response of Confined 0.5%HE F R C Beams under Impact Loading (150mm) 133 180 1.00 2.00 3.00 4.00 5.00 Deflection (mm) Fig . 6.11 Response of Confined 0.5% F R C Beams under Impact Loading (150mm) 200 1.00 2.00 3.00 4.00 5.00 Displacement (mm) Fig. 6.12 Effect of Hammer Drop Height on Response of 2 .5MPa Confined Beams 134 6.4 True Peak L o a d The "true" peak load is the load actually acting on beam; it can be obtained by subtracting from the measured tup load (PT) the inertial load (1). Details of how to determine the true load (P t r) and the inertial load (Pi) were described previously in Chapter 4. 6.4.1 Unconfined beams The peak loads and the inertial loads of the unconfined beams subjected to both static and impact loading, and the impact strength to static strength ratios are given in Table 6.1. The static peak loads typically ranged from 13 to 25 k N , depending on the fibre type and content; the peak load increased somewhat with increasing fibre content. Normally, for plain concrete and 0.5%Vf F R C beams, the peak coincided largely with matrix failure; the peak loads of these concretes were quite similar at around 13-16 k N . Comparing the three types of fibres at 0.5%Vf, hooked end F R C exhibited a higher increase in peak load than the other two fibres, though this was very small (2 kN). In the case of high fibre content (1.0%Vf) specimens, the peak load occurred either at or beyond the B O P , wherever the effect of fibre bridging became a maximum, which depended on the type of fibre. In this study, for the 1.0% hooked fibre, the B O P was at about 20 k N , while the peak load occurred at 25 k N , beyond the B O P (Fig. 6.4 and Table 6.1). This showed that fibre bridging could be effective enough to lead to recovery of all o f the strength that was lost after the matrix first cracked. 135 Table 6.1 Peak Load of Unconfined Beams Subjected to Static and Impact Loading Designation Concrete Drop height True peak Inertial load Im/St No. Type (mm) load (kN) (kN) %Peak Ratio Specimen Static Tests BPL Plain - 14.28 - - - 3 B05HE 0.5%HE FRC - 16.42 - - - 3 B05CP 0.5%CP FRC - 13.14 - - - 3 B05FE 0.5%FE FRC - 14.36 - - - 3 B1HE 1.0%HEFRC - 25.16 - - - 3 Impact Tests BPL150 Plain 150 30.02 6.24 21% 2.10 3 BPL300 Plain 300 45.31 12.82 28% 3.17 • 3 B05HE150 0.5%HE FRC 150 37.08 5.13 14% 2.26 3 B05HE300 0.5%HE FRC 300 45.35 8.04 18% 2.76 3 B05CP150 0.5%CP FRC 150 31.26 7.23 23% 2.38 3 B05FE150 0.5%FE FRC 150 32.75 5.21 16% 2.28 3 B1HE150 1.0%HE FRC 150 49.81 4.82 10% 1.98 3 B1HE300 1.0%HEFRC 300 68.37 10.13 15% 2.72 3 Under impact loading, the peak loads ranged from about 30 to 70 k N ; however, the increase depend upon hammer drop height, and fibre content and type. The increase in peak load at higher rates of loading was more pronounced in plain concrete than in F R C . If rate sensitivity is defined as the ratio between impact and static strengths, then the rate sensitivity of plain concrete was in the range of 2.1 to 3.2, while for the F R C , the rate sensitivity was in the range of 2.0 to 2.7 regardless of fibre type and content. Even though the impact strength of the F R C was somewhat higher than that of the plain concrete, the rate sensitivity of F R C was often lower than that of plain concrete. However, the differences were not really significant. It is believed that the higher static strength of the F R C led to the lower rate sensitivity (as defined by the impact/static strength ratio). 136 The increase in peak load can be explained in different ways. It is well known that the interfacial zone between the matrix and the aggregate is the weak link in ordinary concrete under static loading. However, it appears that under impact loading, the interfacial bond strength ceases to be the weak link (Mindess and Rieder (129)). The crack propagates through both the aggregate and the matrix, following a less tortuous path. Since aggregates tend to be both stronger and tougher than the matrix (130), a higher peak load is required for crack propagation. In the case of F R C , the fibre-matrix bond plays an important role in the effectiveness of fibre bridging across the cracks. According to Bindiganaville and Banthia (131), the bond strength between a single steel fibre and the matrix increased by a factor of two when tested at a crack opening displacement rate (COD) of 2000mm/s, compared to that pulled out at slower rate ( C O D of 0.3mm/s). In addition, the total pullout energy was also found to increase by about 60%, while the slip at peak pullout load decreased by about 38%>. The smaller slip means that under high rates of loading, the steel fibres can easily reach their maximum load-carrying capacity at a very small crack opening. This is advantageous when the matrix cracks begin to accumulate and become localized (mostly at the peak). A t higher loading rates, the fibre-matrix bond strength reaches its maximum at almost the same time as the matrix cracks. This adds to the matrix strength itself and leads to higher peak loads. 6.4.2 Confined beams The peak and inertial loads of the confined beams are given in Table 6.2 and the comparisons between confined and unconfined beams are given in Table 6.3. Since the confined beams failed mostly by shear, the peak load measured here was essentially the peak shear load. Thus, the peak load on the beams increased dramatically with increasing confining stress. The increase ranged from 2.3 to 5.3 times that of the unconfined beams (Table 6.3). In all cases, it was found that plain concrete beams exhibited a higher sensitivity to confining stress than F R C beams did, as seen by their higher confined/unconfined (C/U) load ratios (3.4 to 5.3). 137 6.5 Inertial Load In the previous section, the inertial loads of the unconfined and confined beams were given in Tables 6.1 and 6.2. The typical beam responses with respect to time are shown in Figs. 6.14 to 6.16. Typically, for the unconfined beams, the inertial load was very sensitive to the rate of loading. The plain concrete beams seemed to develop a higher inertial load than did the F R C beams. The inertial loads of the unconfined beams was in the range of 5 to 13 k N ; this represent about 10% to 28% of the true peak load. In the confinement test, the confining stress and the end boundary condition played a very important role in the strength increase. To fracture the specimen, sufficient force both to fracture the concrete and to overcome the pre-existing confining stress is required. However, the force required also depends on the end effects. A s wi l l be described in Chapter 9, on changing contacting face of the loading platen from steel to rubber, the rate of strength increase due to the confining stress changes significantly. With a steel platen, the friction at the specimen ends produces a stiffening effect that does not allow rotation to occur and minimizes the bending. Therefore the beam is prevented from failing in a flexural mode, and instead fails by shear, which requires a higher load. With a rubber platen, however, the lower stiffness o f rubber allows more rotation at the ends, and hence the beam is able to bend more under load. This may suggest that the failure should be in flexure. However, the experimental results indicate that the confining stress never decreases to zero (Fig. 6.13), which means that even though the beam is cracked in tension at the bottom, it is still able to carry load through its remaining uncracked section. This resulted in failure by compressive crushing at the top surface and a higher peak load than in the unconfined beams (see also Figs. 9.7b-d, Chapter 9). 138 OH o o IT; CS CU s q i n c i i n t N t N O c i i \© « n — o >n o i T f oo vo c i CN r-^  o t N (N ^ p i \ei r i v~i so oo Os T f >n T f t N C l 0 0 s o t N CN O — rN tN T f —; —; c s t N >n T f tN T f t N tN tN o PH c« s "5 ( S c o U PH >5 CN cs I*- T J - r-~ oo i n t N T f T f T f o i n so oo t N OS CN oo Tf' t N Os i n OO C l s o i n o © t N Os O- JS O O M E 0) u s o U <n T f tN —; i ' i ©' <n Os t N CN t N o q T f c i r - ; C ) s o r ~ o s © ' c i r - ~ c i c i T f ° i n • * <n tN od i r i o s T f oo o s o o o o o o o o l o o m o i n m i n o PH OH U CJ O U O O pi pi psj ^ oi ttc IX, HH ft, ft, ix, i a a & s a a N ? X ? X ? X ? N? s ? i n i n i f i it~i o O O O O O _ J 'cfl o in OH CQ o in in o CQ o o m o in o i n o m o i n W i n o CQ o i n CQ CQ •a S3 o cs OH -a u e a o o c T3 C w T3 <D e ie c o U S3 « cS P< so es H ^ ci ^ ' <t n l O C T s m a N m r ^ T f T f O T t r f O N f N i n r ^ t T s n"> c i T f c i i n Tf' c i t N t N T f CN so JS OX) a a u Q tu s-u e o U T f C l T f o CN o o o o o o o o m o i n o m i n > n o .s . g a a & e a a cS I I s » S ° S » s O - o — — 0 S C S 0 X O X Q X ~ \ C ^ D n i o i n i n i n o o o o o o ^ X e c *s cS X U II OH o x •a a u T 3 <D O O I I w TT cs o 60 a 'S e (D X CS a. o 139 - — S t e e l plate [ A Rubber plate 0 5000 10000 15000 20000 25000 Time (mSec) Fig . 6.13 Confining Load-Time Response of Beam under Impact Loading with Different Loading Platen For the confined beams, there was no clear relationship between the inertial load and the confining stress. The inertial loads of the confined beams were almost negligible, about 1 % to 10% of the measured load, which was far less than those of the unconfined beams (Table 6.1). This is probably because the load carrying capacity o f the confined beams was significantly enhanced by the confinement stress, and due to the end restraint the movement (or acceleration) was also controlled by the confinement. A s a result, much lower inertial loads (as a percentage of the tup load) were observed. A s the hammer drop height increased from 150mm to 300 mm, the hammer velocity at impact was increased as well . Therefore for the same impact duration, beams tested from a 300 mm drop height were accelerated to a higher velocity than the ones tested from a 150 mm drop height. With higher hammer drop heights, higher accelerations and inertial loads were found. 140 16.00 14.00 12.00 10.00 z v 8.00 rs O _l 6.00 4.00 2.00 Plain 150mm DH Plain 300mm DH 0.5%HE 150mm DH 0.5%HE 300mm DH r s V 500 1000 1500 Time (usee) 2000 2500 Fig. 6.14 Typical Inertial Load Responses of Unconfined Beams at Different Drop Heights 6.00 5.00 4.00 z T3 3.00 ro o 2.00 1.00 500 1000 1500 Time (usee) 2000 2500 Fig. 6.15 Typical Inertial Load Responses of Confined Beams at Different Drop Heights 141 3.00 Plain-OMPa 2.50 Plain-5MPa 0.5%HE-0MPa 0.5%HE-5MPa 0 500 1000 1500 2000 2500 Time (usee) Fig . 6.16 Typical Inertial Load Responses of Confined Beams at Different Levels of Confinement 6.6 Relationship between Stress and Stress Rate 6.6.1 Unconfined beams In this section, the relationship between the flexural stress (modulus of rupture), and the stress rate is discussed. For the unconfined beams, the stress rates achieved in this study were in the range of 0.05 to 35,000 MPa/sec, while the flexural strength ranged from about 6 to 27 M P a (Table 6.4). The strengths of the plain concrete beams under impact were less than those of the F R C beams. However, there was also a higher stress rate for the plain concrete beams than for the F R C beams, due to the much shorter impact event for plain concrete. The relationships between stress rate and impact/static (I/S) strength ratio are shown in Figs. 6.17 and 6.18. Table 6.4 Average Stress Rate and Flexural Strength of Unconfined Beams Concrete type Load ing type Stress rate (MPa/sec) F lexura l strength ( M P a ) P l a in Static 0.049 6.43 Impact 34,867 16.49 0 . 5 % H E Static 0.048 7.39 Impact 21,660 18.97 0 . 5 % C P Static 0.054 5.91 Impact 18,617 14.44 0 . 5 % F E Static 0.052 6.46 Impact 17,052 14.81 1 . 0 % H E Static 0.048 11.32 Impact 21,548 27.16 Plain concrete beam •0.5%HE FRC beam 1.0% HE FRC beam 10 100 Stress rate 4.00 o 2 3.00 g g M 2.00 o 8 co 1.00 ^ ro a. E 1,000 10,000 100,000 Fig. 6.17 Relationship between Stress Rate and Impact/Static Flexural Strength Ratio of Plain Concrete and Hooked End F R C Beams (Log Scale) 0.5%CP FRC beam • — 0.5%HE FRC beam r — 0.5%FEFRC beam I 1 1 1 . 1 1 r -r 3.50 o 3.00 s rat 2.50 w o 2.00 </> o 1.50 ro 1.00 ct/Sl 0.50 ro D. E 10 100 Stress rate 1,000 10,000 100,000 Fig. 6.18 Relationship between Stress Rate and Impact/Static Flexural Strength Ratio for Three Types of F R C Beams (Log Scale) 143 The relationship between strength and stress rate can be expressed using the same empirical equation described in Chapter 5 (Eq. 5.3), but with the k value changed to 2.56xl0" 5 for plain concrete, and 4.00-4.85xl0" 5 for F R C . Even though the plain concrete underwent a higher stress rate than the F R C , it appears that the plain concrete beams were less sensitive to the stress rate than the F R C beams. A t the same stress rate, F R C beams exhibited slightly higher flexural strengths than plain concrete beams. 6.6.2 Confined beams For the confined beams, since the failure was primarily in the shear mode, the shear stress was used for calculating the stress, given as: r = (6.1) 2{bd) where r = shear stress P == peak bending load (N) b and d = width and depth of beam (mm) 6 = angle of the shear failure plane (use 45° in all cases) It was found that the shear stresses and shear stress rates of the confined beams were much smaller than the flexural stresses and flexural stress rates of the unconfined beams. The relationship between the shear stress rate and the level of confinement showed clearly that the stress rate increased with increasing confinement, due to the increase of strength and stiffness with confinement (Fig. 6.19). The average stress rates obtained in the confined tests ranged from 800 to 3,500 MPa/sec, which were much less than those obtained in the unconfined tests. Thus, the shear stress rate of the concrete was much less than the flexural stress rate for the same applied load and duration. From the plot in Fig . 6.19, it appears that the stress rate of the F R C beams approached the limiting value at around 2,100 MPa/sec when the confinement reached 5 M P a (as seen by the flatter curve at high confinement). However, this did not seem to happen with the plain 144 concrete beams because the trend was still one of increase even at the highest confinement of 5MPa . The relationship between stress rate and impact/static stress ratio of confined beams is given in Fig . 6.20. The shear strength of plain concrete under static loading was assumed at 0.2 •y[f^; for F R C , as stated by A C I Committee 544, the shear strength of 1% volume fraction F R C about 30% higher than that of plain concrete. In this case, shear strengths of 15% and 30% higher than that of plain concrete were assumed, conservatively, for 0.5%>FRC and 1.0%>FRC, respectively. This could also be expressed empirically using the same relationship as given for the unconfined beams. However, the value of k must, again, be changed. For plain concrete, k = 6.67 x 10"4 could be used, and for F R C , a value of k in the range of 5.78 to 6.62 x 10' 4 could be used. Unlike the unconfined tests, confined plain concrete was found to be more rate sensitive than confined F R C . 2,500 £ 1,500 S w 1,000 o d> ro 2,000 a 0.5%HE 500 *>-1.0%HE 0 1 2 3 4 5 Confinement (MPa) Fig . 6.19 Relationship between Confining Stress and Stress Rate of Plain and F R C Beams 145 Confined Plain concrete beam Unconfined Plain concrete beam Confined 1.0%HE FRC beam Unconfined 1.0% HE FRC beam •*/ • / • * • / " i 1 —r 1 '— i ~ — , -. 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 10 100 Stress rate 1,000 10,000 100,000 Fig. 6.20 Relationship between Stress Rate and I/S Stress Ratio of Unconfined and Confined Beams .2 "•a CO w (0 (A CD i _ +-» (/) o •*> u CO a. E 6.6.3 n-value The value of "n" may be used to describe the relationship between the stress and stress rate (see Chapter 5 for a detailed derivation o f n). In order to obtain n, the relationships between log(stress) and log(stress rate) of both unconfined and confined beams were plotted as shown in Figs. 6.21a-c and 6.22a-c, respectively, and " n " was calculated from the slope of each curve as given in Table 6.5. Table 6.5 Calculated n-values for Both Unconfined and Confined Beams Concrete type Slope n Unconfined Confined Unconfined Conf ined P la in 0.9508 0.5593 0.052 0.79 0.5%HE 0.9404 0.5203 0.063 0.92 1.0%HE 0.9142 0.4351 0.094 1.30 The value of n is related to the velocity of the crack; the lower the value of n, the higher the crack velocity. In all cases, it was found that the n-values increased with the 146 fibre volume, which means that fibres, because of their crack bridging, also slow down the velocity of crack propagation. For the unconfined beams, the n-value ranged from 0.05 to 0.09 whereas for the confined beams, the n-values were much larger, and ranged from 0.80 to 1.30. However, it must be noted that the cracks in the unconfined beams were flexural cracks, while the cracks for the confined beams were essentially shear cracks. The shear wave velocity (132) is given as: (6.2) where G = the shear modulus = E/(l + v) p = the specific density For v = 0.25, G is equal to 0.4E and c s becomes ^JOAEjp . If the flexural wave velocity (cf) is assumed equal to the longitudinal wave velocity, ^E/p , this yields a ratio between shear and flexural crack velocities of: = & 0.632 (6.3) cf 1 c, =0.632c y. According to Eq . 6.3, the shear crack velocity was about 60% of the flexural crack velocity. With the much slower crack velocity in the confined beams, much larger n-values were observed. 147 R2 = 0.3571 Slope = 0.9404" 3.0 3.5 4.0 Log (stress rate) (b) 4.5 R2 = 0.9474 Slope^0T9142 3.0 3.5 4.5 1.70 1.50 1.30 1.10 0.90 5.0 1.70 1.50 1.30 1.10 0.90 0.70 0.50 ro c 3 X 0.70 ™ 0.50 O D) C 0> k. m re 3 X O o 5.0 4.0 Log(stress rate) (c) Fig . 6.21 Relationship between Log(Stress) and Log(Stress Rate) of Unconfined (a) Plain Concrete Beam, (b) 0.5%HE F R C Beam and (c) 1.0%HE F R C Beam 148 2.0 2.5 3.0 Log (stress rate) (b) 3.5 ro 0.40 2 0.20 g> 4.0 Fig . 6.22 Relationship between Log(Stress) and Log(Stress Rate) of Confined (a) Plain Concrete, (b) 0 .5%HE F R C and (c) 1.0%HE F R C Beams 149 6.7 Fracture Energy The fracture energy may be defined as the area under the load-deflection curve calculated up the end of loading event (load equal to zero-Chapter 4). However, beams usually failed at very small deflections (less than 5 to 10mm); therefore a centre point deflection of 3 mm was selected as a reference point for calculating fracture energy. The fracture energy term (used interchangeably with "energy absorption") in this chapter is actually the energy absorption up to a centre point deflection of 3 mm. The energy absorbed up to 3 mm-centre point deflection is given in Tables 6.6 and 6.7. Since the failure of the plain concrete (with 300 clear span length) occurred at a very small deflection (much less than 3 mm), the energy absorption of plain concrete was actually calculated up to the end of the loading event. Under static loading, the fracture energy increased dramatically with the inclusion of fibres; the post-peak response contributed most to this increase. Without fibres, plain concrete specimens were not able to carry load beyond a deflection of 0.5 mm. With fibres, due to fibre bridging across the cracks, F R C was able to sustain loads at the much larger deflections and hence absorbed much more energy. A t 0.5% Vf, hooked end F R C was able to absorb more energy than either the crimped or flattened end F R C (Table 6.6). When the fibre content increased to 1.0%, the ability to absorb energy increased by a further 50% (see 0.5%>HE and 1.0%HE-Table 6.6). Under impact loading, for the unconfined specimens, the fracture energy of the plain concrete beams was still low compared to the F R C beams. The fracture energies of the F R C specimens were about 17 to 40 times higher than those of the plain concrete specimens, depending on the fibre type and volume fraction (Table 6.6). The effect of fibre bridging across the cracks still played a significant role in this increase. However, when comparing static and impact behaviour, the increase in fracture energy of F R C was not as high as for plain concrete (Table 6.7). A s seen by the ratio between static and impact fracture energy (I/S ratio), the I/S ratio of plain concrete was quite high, about 13 to 16, while for F R C , the I/S ratio was only about 1 to 3. This is perhaps because most fibres were pulled out simultaneously during the impact event (resulting in higher strength at the peak). However, this also reduced the efficiency of the fibres in bridging 150 over the cracks in the post-peak regime (because fewer fibres remained to participate in bridging at the larger deflections), and hence only a small increase in fracture energy resulted. This was not the case for high volume fraction F R C (i.e., 1.0%) where most of fibres were still intact and effective (even though many fibres had pulled out at the peak load). A t 0.5%, all three fibres performed similarly under impact (Table 6.6), though the crimped fibres were slightly worse than the other two fibres. For the confined beams, the ability of both F R C and plain concrete to absorb energy increased dramatically with confinement (Table 6.7), especially for the plain concrete beams, where the increase in energy absorption ability as indicated by the Confined/Unconfined (C/U) fracture energy ratio was around 21 to 32 (Table 6.7). For F R C beams, the C / U ratio was in the range of 2.5 to 6.9. The increase in fracture energy of confined concrete was the direct result of the increase in strength and toughness with the confining stress. For F R C , the effect of the confining stress also depended on the fibre type and content. Hooked fibres exhibited the highest increase among the three fibres ( C / U ratio ranged from 5.4 to 6.9). In addition, it was observed that the efficiency of the confining stress dropped with increasing hammer drop height, since the material exhibited a more brittle behavior at higher rates of loading. 151 00 g '-3 cd O +-> O Cd T3 Cl cd cd » O -4—• T3 <D -*-» O <u -C? 00 cd <D PQ CD CI U H cl o U T3 CD d UH cl o o Cl o ? a co DO > H CD a W CD >-< -4—» o cd i -IX, vo J D cd H T 3 a « a o u a Si a * e H i n O N i n co co co N O co CN T f O ON CN CN CN Tt"' CN CN O N i n oo N O co - H CN C O m CN —^ O N C O r n —< C O C O C O C O o r o co r>- r-~ -—< r - H K O N 0 0 co o r ^ i n - H CN CN CN CN r~- oo N O T l -- H CN t--CN - H CN r-I O i n co r - H CN CN N O O N * i n od oo O N C O m N O CN T f ON O C O C O O O i n oo V O N O oo CN oo i n r -O C O CN W CH y W ,S B U h B 3 £ $ £ ^ 1 i n i n i / i 9 M 6 s = PH in W ITi © © ~ © r - ; c > at) .5 T 3 crj _o <4H o a CO T 3 <D CS O 3 U3 cd g 00 s »H CD a w o cd i H IX <U Cl UH1 Cl o o Cl -a Cl « Cl o U cd -t-» C/3 O cd r--IH E = © E a © © ro s-T3 GO * H O * H « S I U O N CN CN O N S ^ O co CN CN CO N O N O N O CO O O ^ © CN O N O CN N O i n co i n CN 0 0 i n CN CN CN m r2 < N o co ° ^ co O N m r--< N ^ ^ ^ W PH .s a u w vo s» s« vo M 0 S . 0 S 0 S 0 S i/-) i r , >r< © © © © 152 C H A P T E R 7 R E S U L T S A N D D I S C U S S I O N S - P L A T E S 7.1 Introduction The experimental results described in this chapter deal with the impact tests of plain and fibre reinforced concrete plates, with and without confinement. The unconfined F R C plates exhibited different fracture characteristics from those of plain concrete plates. These differences are discussed in terms of the peak loads, the load-deflection curves, and the fracture energies. Steel fibres improved the fracture energy of the plates, and also led to the occurrence of multiple peaks in the load vs. deflection curves. For the confined plates, it was found that confinement had a major effect on the failure patterns as well as on the mechanical properties of both plain and fibre reinforced concretes. Wi th different types and degrees of confinement, the failure changed from a flexural mode to a punching shear mode or a combined flexural-shear mode. The apparent strength of the concretes increased by up to two or three times and the inertial load decreased to a small enough fraction of the measured impact load that it could be ignored in the analysis. The ability of the F R C to absorb energy also increased with increasing confinement. 7.2 Fa i lure M o d e 7.2.1 Unconfined plates The plain concrete plates under both static loading (Fig. 7.1a) and impact loading (Fig. 7.2a) failed primarily in a brittle mode. The specimens fractured completely into 4 or 5 pieces. For the fibre reinforced concrete plates, failure was mostly in a more ductile manner under both static (Figs. 7.1b-7.1d) and impact loading (Figs. 7.2b and 7.2c). The F R C specimens fractured into more pieces then the plain concrete, and multiple macrocracking and microcracking occurred. After failure, the pieces of concrete were 153 still held together by the fibres near the top surface. However, some specimen failed in a mixed mode between ductile and brittle failure, depending on fibre type, volume fraction and loading rate. In mixed mode fracture, the concrete fractured into 4 or 5 pieces, but some pieces were totally separated from the others while some were still held together. There were some signs of microcracking. For example, 0.5% crimped F R C failed in the mixed mode pattern (Fig. 7.2e and 7.2f). A t higher rates of loading, localized failure patterns could be observed (Fig. 7.2d). The localized failure was circular, in the shape of the tup load cell. 7.2.2 Confined plates Several different types of failure patterns were observed, depending on the test arrangement. a) Flexural Failure (Fig. 7.3). This type of failure occurred most commonly in the specimens tested without confinement. The specimen was free to bend under the applied load without any restraint at the ends. Several cracks resulted, running randomly from one edge to the other edge. (a) (b) Fig . 7.1. Typical Failure Patterns of Plates under Static Loading: (a) Plain Concrete, (b) 0.5% H E F R C , (c) 0.5% F E F R C , and (d) 0.5% C P F R C 154 Fig. 7.2. Typical Failure Patterns of Plates under Impact Loading: (a) Plain Concrete, (b) 0.5 % H E F R C , (c) 1.0% H E F R C , (d) Localized Failure of 0.5% F E F R C , (e) and (f) Mixed Mode Failure of 0.5% C P F R C b) Punching Shear Failure (Fig. 7.4). With biaxial confinement, the failure mode changed gradually from flexural to punching shear. A t the lowest level of confinement, or passive confinement (0 MPa) , the failure was a combination of punching shear and flexure (a punching shear crack formed along the supports, while several flexural cracks ran from one edge to the other). A s the confinement stress increased, the flexural cracks began to disappear and the failure mode was dominated by a punching shear crack. There was one case in which "failure" did not occur; no large cracks developed, but several hairline cracks on the bottom surface, or a very small permanent deflection, were noted (Fig. 7.4d bottom row and Fig. 7.5). This mode only occurred in F R C that contained a high fibre content (i.e., 1% Vf) , or F R C specimen tested under high confinement (5 M P a Biaxial) . 155 c) Flexural, or 3-point Bending Mode (Fig 7.6). For uniaxial confinement, the failure was in a flexural mode, as in a beam subjected to three-point bending. A t both sides of the confined plane, cracks occurred along the edges of the supports; a main crack causing failure formed at the middle of plate parallel to the unconfined plane. (a) (b) (C) (d) (a) (b) (c) (d) Fig. 7.4 The Development of Punching Shear Failure with the Increase of Confining Stress: a) 0 M P a , b) 1.25 M P a , c) 2.5 M P a , and d) 5 M P a Top row: plain concrete; Bottom row: F R C (0.5% hooked end fibres) 156 (a) (b) Fig. 7.5 Unfractured Specimens: a) With High Fibre Content and b) Tested Under High Biaxial Confinement Stress (Fine cracks have been highlighted by black dots) II ' : A ''"hi ' ? ' f i k / j m * ; I />' i m ^ S j H H B n M H B B j I ^v (a) (b) Fig . 7.6 Flexural Failure of Specimen Tested under Uniaxial Confinement: a) Plain Concrete and b) 0.5% Flattened End F R C 7.3 Load-Deflection Response 7.3.1 Unconfined plates (a) Load-deflection curves Typical load-deflection curves of plain and F R C plates subjected to static loading are shown in Fig. 7.7. It may be seen that steel fibres increased not only the load carrying capacity but also the toughness of the concrete plates. For plain concrete, the first cracks occurred at about 0.02 mm central deflection and led immediately to the failure o f the specimens. For the F R C , first cracks did not lead to instantaneous failure, which only occurred at a central deflection of about 1-3 mm. 157 The three types of fibres were very much the same in terms of load carrying capacity. However, in term of toughness, the hooked end fibres seemed to perform better than either the flattened or the crimped fibres. Typically, the static load response o f F R C plates was not characterized by the occurrence of multiple peaks. The load vs deflection response was quite linear up to about 80% of the peak, followed by strain hardening and finally failure. There was some evidence of double peaks, as seen in the 0.5% hooked end F R C plate (Fig. 7.7b), where a significant drop of load occurred before the peak. However, because of the fibre bridging effect, the load started to increase again. 40.00 30.00 •a 20.00 re o 10.00 100.00 80.00 60.00 40.00 20.00 Static Test — Plain Concrete 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Deflection (mm) (a) Static Test •0.5% Hooked End Fiber 0.5% Flattened End Fiber -0.5% Crimped Fiber 0.00 2.00 4.00 6.00 Deflection (mm) 8.00 10.00 (b) Fig . 7.7 Response of (a) Plain Concrete and (b) F R C Plates under Static Loading 158 For impact loading at 250 mm drop height (2.21 m/s), the typical response of the steel F R C plates (Fig. 7.8 and 7.9) was characterized by the appearance o f multiple peaks in the load vs deflection curves; these were more numerous and pronounced than those observed for plain concrete (Fig. 7.10). The first peak was caused by matrix cracking, while the subsequent peaks were caused by fibres bridging across the cracks. These multiple peaks w i l l be discussed in depth below. 250 i 200 j - 1 100 50 0 4 Impact Test from 250mm \ 0.5% Hooked End Fibre 0.5% Flattened End Fibre 0.5% Crimped Fiber \ X \ 1 1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Mid-Plate Deflection (mm) Fig. 7.8 Typical Load-Deflection Curves of 0.5% F R C Plates Subjected to Impact Loading (250 mm Drop Height) 159 250 200 Impact Test from 250mm — 1% Crimped Fiber — 1% Hooked End Fiber — 1% Flattened End 2 3 Mid-Point Deflection (mm) Fig . 7.9 Typical Load-Deflection Curves of 1.0% F R C Plates Subjected to Impact Loading (250mm Drop Height) 200 160 120 "D ro O 80 40 Impact Testing at 250mm 2 3 Mid-Plate Deflection (mm) Fig . 7.10 Typical Load-Deflection Curve of Plain Concrete Plate Subjected to Impact Loading (250mm Drop Height) 160 (b) Mul t ip l e peaks in F R C plates under impact loading A s stated above, the first peak in the load-deflection curve was the result of concrete matrix cracking. Concrete plates of the same compressive strength, whether plain concrete or F R C , subjected to the same rate of loading, exhibited similar loads at the first peak. Figs. 7.11 and 7.12 show the first peaks for plain concrete, 0.5% and 1.0% F R C plates subjected to impact loading of 2.21 m/s (250 mm drop height). Except for the crimped fibre, the other concretes exhibited similar first peaks which ranged between 110-120 k N . These first peaks in the F R C plates, theoretically, should be sensitive to the rate of loading, similar to those of the plain concrete plates, because they represent the matrix strength. The results are shown in Figs. 7.13 and 7.14, in which the first peaks o f the 0.5%) and 1.0% F R C plates subjected an impact velocity of 2.21 m/s (250mm drop) are compared to those obtained from an impact velocity of 3.13 m/s (500mm drop). The observed first peaks of the F R C plates were as sensitive to the rate of loading as were those of the plain concrete. A t 0.5% volume fraction (Vf), flattened end fibres were the most sensitive, while at 1.0% volume fraction, the hooked end fibre was the most sensitive. After the first peak, when the concrete matrix had cracked, the fibres were mobilized (stretching and anchorage effects) and strength began to increase again. A s this process continued, and with the overlap of successive cycles of fibre behavior (further cracking, fibre bridging, fibre stretching, then strength recovery), the subsequent peaks occurred. The occurrence of multiple peaks in the load-deflection curves is shown schematically in Fig. 7.15, where the first peak is due to matrix cracking, while the subsequent peaks result from the effects of fibres bridging and stretching across the cracks. The occurrence of the fibre bridging-stretching phenomenon was random, depending on the number of fibres intercepted by each crack branch, and the rate of loading. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Deflection (mm) Fig. 7.11 First Peaks of Plain and 0.5%FRC Plates Subjected to Impact Loading (V h=2.21 m/s) 180 160 140 120 z 100 •D CO O 80 _J 60 40 20 I Impact Testing at 250 mm r vi i i i : Plain Concrete - - 1% Crimped Fiber 1% Hooked End Fiber 1% Flattened End Fiber / /« « TV 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Deflection (mm) Fig . 7.12 First Peaks of Plain and 1.0%FRC Plates Subjected to Impact Loading (V h=2.21 m/s) 162 300 250 200 Plain Concrete Vh=2.21m/s 0.5% Hooked End Fibre Vh=2.21m/s 0.5% Flattened End Fibre Vh=2.21 m/s Plain concrete Vh=3.13m/s - - 0.5% Hooked End Vh=3.13 m/s 0.5% Flattened End Vh=3.13 m/s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Mid-Plate Displacement (mm) Fig . 7.13 Effect of Impact Velocity on First Peaks of Plain Concrete and 0 .5%FRC Plates 300 250 200 z ^ 150 A O - J 100 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Mid-Plate Displacement (mm) Fig. 7.14 Effect of Impact Velocity on First Peaks of Plain Concrete and 1.0%FRC Plates 163 (c) Effect of loading rate (hammer drop height) The effect of rate of loading on the load-deflection curves is shown in Figs. 7.16 and 7.17. Wi th an increased rate of loading, fewer peaks (two peaks) occur, the first peak due to concrete fracture and the second peak due to the fibre bridging effect. A t the higher rate of loading, the fracture is more sudden, and the fibre bridging-stretching mechanisms in the different crack branches occur at almost the same time. The relationship between the rate of loading and the maximum value of each subsequent peak is shown in Fig. 7.18. The relative peak load (p r) is plotted against the mid-point deflection; the relative peak load is given as the ratio of subsequent peaks (p n) to the first peak (pi); where n = 2, 3.. . n (7.1) "4< "A Concrete matrix fracture Beginning of the 1" cycle of fiber anchoring and stretching. The 2 cycle of fiber anchoring and stretching. The 3 cycle of fiber anchoring and stretching. 1 2 3 4 5 Mid-plate Deflection (mm) Fig . 7.15 Schematic Representation of Multiple Peaks in a F R C Plate under Impact Loading (250 mm Drop Height) 164 o 0 1 2 3 4 5 Mid-plate Displacement (mm) Fig. 7.16 Schematic Representation of Multiple Peaks in a F R C Plate under Impact Loading (500 mm Drop Height) 300 250 200 150 100 50 0 — 250 mm Drop Height — 500 mm Drop Height 1 JJ V*ri,/ « V —2-. ; 1 2 3 4 5 Deflection (mm) Fig. 7.17 Effect of Loading Rate on the Occurrence of Multiple Peaks on the Load-Deflection Curve 165 1.20 Xi 1 0 0 fc •3 0.80 -* Q) a. 0.60 s -2 a: 0.40 0.20 f sf Pea/c • 2 / 7 d Pea* • 3rd Peak 2nd Peak m 3rd Peak 500 mm Drop Height 250 mm Drop Height 2 4 6 8 Mid-plate Deflection (mm) 10 Fig. 7.18 Relative Peak Heights for the Two Different Drop Height A s shown in Fig . 7.18, the relative peak loads decreased more rapidly at higher rates of loading, perhaps as a result of the rate sensitivity of concrete. A s discussed earlier, the first peak was the result of matrix cracking alone and since concrete is a rate sensitive material, as the rate of loading increased, much higher first peaks could occur. However, the subsequent peaks (which were dominated by the mechanism of fibre bridging, anchoring and stretching) seemed to be less affected by the rate of loading. It may also be that the bond between matrix and fibres is less sensitive to the rate of loading than is the matrix strength. (d) Effect of fibre content The number of peaks, the strength recovery (the lower bound), and the decreased magnitude of the subsequent load peaks were affected by parameters such as the hammer drop height (the loading rate), and the fibre content and geometry. For example, as the fibre content increased, a larger number of distinct peaks could be observed (Fig. 7.19) 166 The higher fibre content also enhanced the strength recovery process; after the load dropped, a more rapid strength recovery was usually observed in the concretes with higher fibre contents (Fig. 7.20). This might result from the reduction in fibre spacing with increasing fibre content, as more fibres participated in bridging and anchoring the cracks. The development of multiple cracks and microcracks is related to these multiple peaks. A n increase in the number of microcracks and macrocracks means that a larger number of fibres would be intercepted by the crack. 200 160 _ 120 z -a ro o 80 40 0.00 Impact Testing at 250mm — 0.5% Flattened End Fibre — 1% Flattened End Fibre l/ll i i l l i l l / \ / I i 1 1.00 2.00 3.00 Deflection (mm) 4.00 5.00 Fig . 7.19 Effect of Fibre Content on Number of Peaks 167 180 160 140 120 z 100 -o TO o 80 _l60 40 20 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Deflection (mm) Fig. 7.20 More Rapid Strength Recovery of Fibre Reinforced Concrete with Higher Fibre Volumes 7.3.2 Confined plates The response of confined plain concrete and F R C plates to impact loading depended on several factors: confinement type, confinement stress, fibre type and content, and rate of loading. Generally, it was found that strength and toughness increased with increasing confinement, but the peak deflection and the inertial load decreased. A s the fibre content increased, the material could carry increased loads, and in some cases, did not fail at all under a particular impact loading. Al so , confined plates were found to be less sensitive to the rate of loading than unconfined plates. A detailed discussion of each of these effects follows. 168 a) Effect of confinement Biaxial confinement Typical load-deflection curves of biaxially confined plain concrete, 0.5% hooked end, 0.5%) flattened end and 0.5% crimped F R C plates subjected to impact loading from a 250mm hammer drop height are shown in Figs. 7.21 to 7.24, respectively. In all cases, it was found that the response of the material changed with the degree of confinement. In plain concrete, the load-deflection response did not change much with increasing confinement, except for an increase in peak load and toughness (Fig.7.21). For F R C plates, the response of material with and without confinement was quite different (Figs.7.22-7.24). Apart from the increases in peak load, toughness and peak deflection, the multiple peak characteristics found in unconfined studies were clearly eliminated. Impact Testing from 250mm Mid-plate deflection (mm) Fig. 7.21 Effect of Biaxial Confinement of the Response of Plain Concrete Plate 169 350 300 250 200 to 3 150 100 50 Impact Testing from 250mm 4 6 Mid-plate deflection (mm) 8 10 Fig . 7.22 Effect of Biaxial Confinement on the Response of 0 .5%HE F R C Plates 350 CO O 300 250 200 150 100 50 Impact Testing from 250mm —2.5 MPa — Unconfined n KJ 4 6 Mid-plate deflection (mm) 10 Fig . 7.23 Effect of Biaxial Confinement on the Response of 0 .5%FE F R C Plates 170 350 4 6 Mid-plate deflection (mm) 10 Fig . 7.24 Effect of Biaxial Confinement on the Response of 0.5%CP F R C Plates The difference between the response of the biaxially confined plate (characterized by a single peak in the load vs deflection curve) and that of the unconfined plate (characterized by multiple peaks) might be the result of the single macro-shear crack which formed around the edges of the confined specimen which caused failure to occur. In the unconfined plates, several macro-cracks formed from one edge to the other, and it is these macro-cracks, which ended up causing the "multiple peaks" response. For the F R C tested under high confinement (5 MPa) , except for the crimped fibre, the response of the material was found to be elastic-plastic (Fig. 7.23 and 7.24). The load increased up to the peak, and then, without fracturing the specimen, fell back to zero with only a small amount of damage (or permanent deformation). Un iax i a l confinement For the specimens with uniaxial confinement, the responses of the plain concrete and the 0.5% hooked end F R C plates subjected to impact loading from a 250mm hammer 171 drop-height are shown in Figs. 7.25 and 7.26. Both the uniaxially confined plain and F R C plates had much the same response to impact loads as the biaxially confined plates. The increase in peak load was modest. In one cases (i.e., 2.5 M P a uniaxially confined plain concrete plate), a reduction in peak load was observed. The "single peak" response of the uniaxially confined plates was the direct result of the formation of only a single crack at the middle of the plate, perpendicular to the confined plane. However, several macro-cracks also developed in the diagonal direction, which resulted from the non-uniformly distributed confinement load. These macro-bending cracks were probably the cause of the numbers of minor peaks along the load-deflection curve (mostly found in F R C plates, Fig. 7.26). 200 i Mid-plate deflection (mm) Fig . 7.25 Effect of Uniaxial Confinement on the Response of Plain Concrete Plates b) Effect of fibre type and content Most biaxially confined plates were found to be substantially stronger than unconfined plates. A s stated earlier, the strength increased as the degree of confinement increased. In some cases, the confinement was high enough to prevent failure from 172 occurring. However, the latter phenomenon only occurred in high V f F R C plates; thus, it appears to be related not only to the confinement but also to the fibre content. — 2.5 MPa — 5 MPa — Unconfined 4 6 Mid-plate deflection (mm) 1 0 Fig. 7.26 Effect of Uniaxial Confinement on the Response of 0 . 5 % H E F R C Plates In order to confirm this, tests on higher fibre content F R C plates were conducted, with the results shown in Figs. 7.27 and 7.28. The 1% hooked end F R C plates were tested at confinement pressures of 0, 2.5 and 5 M P a , and the 1% flattened end and crimped F R C plates were tested at a confinement of 5 M P a . At 0 M P a , the specimen failed completely in shear. However, at 2.5 M P a , the failure modes were different for the 0.5% hooked end and 1%> hooked end plates. In Fig. 7.28, the responses of the various types of F R C plates are plotted against each other. With all three types of fibres, at 5 M P a , failure did not occur at all . The responses of the specimens were elastic-plastic, with a small amount of residual damage at the end of impact event. For the uniaxially confined F R C plates (7.29), the confining stress did not prevent failure from occurring in most types of specimens. Only with the 1%> hooked end F R C did failure not occur. With the other types of fibres, the specimens completely failed under the applied impact load. 173 400 350 300 250 z ^ 200 ro O _l 150 100 50 0 Impact Testing from 250mm — 5 MPa — 2.5 MPa — 0 MPa 1 II • / '/ 1 i • 4 6 Mid-plate deflection (mm) 8 10 Fig. 7.27 Impact Load-Deflection Curves of Biaxially Confined 1%HE F R C Plates 400 350 300 250 z v 200 ro o _ i 150 100 50 0 Impact Testing from 250mm — 1%CP@250mm-5MPa — 1%FE@250mm-5MPa — 1%HE@250mm-5MPa -i i i 4 6 Mid-plate deflection (mm) 10 Fig. 7.28 Impact Load-Deflection Curves of 5 M P a Biaxial ly Confined 1%FRC Plates 174 240 Mid-plate deflection (mm) Fig . 7.29 Impact Load-Deflection Curves of Uniaxially Confined 1% F R C Plates c) Effect of loading rate (hammer drop height) The effects of hammer drop height on the responses of 5 M P a biaxially confined plain, 0.5% H E , 0.5% F E , 0.5% C P and 1% H E F R C plates are given in Figs. 7.30 to 7.34, respectively. In general, an increased rate of loading decreased the toughness and the peak deflection of both biaxially confined plain and F R C plates, though a slight increase in the peak load was found. The increase in peak load of confined plates with increasing hammer drop height was not as great as for the unconfined plates; this w i l l be discussed in greater detail in the next section. Plain concrete plates fractured at both 250mm and 500mm drop height. For the F R C plates, some (0.5%HE, 0.5%FE, and 1.0%HE) did not fracture at the 250mm drop height, and thus their responses were elastic-plastic. However, all were fractured at both the intermediate (317.5 and 375 mm) and 500mm drop height (500mm). It must be noted here that the results described in this section are for the 5 M P a biaxially confined specimens; the conclusions might be different for different values of confining stress. 175 0 1 2 3 4 5 6 7 8 Deflection (mm) Fig . 7.30 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly Confined Plain Concrete Plates 500 400 — 300 ro o 200 100 0 1 2 3 4 5 Deflection (mm) Fig . 7.31 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly Confined 0.5%HE F R C Plates 176 400 2 3 4 Deflection (mm) 6 Fig. 7.32 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly Confined 0.5%FE F R C Plates 400 5 MPa Biaxial Confinement 0 1 2 3 4 5 6 Deflection (mm) Fig. 7.33 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly Confined 0.5%CP F R C Plates 177 400 0 1 2 3 4 5 6 Deflection (mm) Fig . 7.34 Effect of Hammer Drop Height on the Response of 5 M P a Biaxial ly Confined 1%HE F R C Plates 7.4 Peak Bending L o a d and Inertial L o a d 7.4.1 Unconfined plates a) Peak bending load Plain concrete is highly strain rate sensitive. In the present study, the failure stress of plain concrete under impact loading from a 500 mm drop height was about 1.5 times that from a 250 mm drop height (Table 7.1, Fig. 7.35). The increase in the strength of plain concrete under high rates of loading is, in part, due to the fact that cracks tend to propagate through the aggregate particles (rather than around them and through the interfacial zone) when the rate of loading is increased. Since aggregates are denser, tougher and stronger than ordinary cement paste, this leads to an increase in measured strength. In addition, Reinhardt (133) has shown that under 178 dynamic loading, the stress intensity at the crack tip is lower than that under static loading. Thus, in order to cause the material to fracture, a higher stress is required. In the case of the F R C plates, the strength also increased with an increased rate of loading (Table 7.1, F ig 7.36 and 7.37). This is a direct result of the bonding mechanisms of the deformed fibres. Under an applied load (either static or impact), in order to fracture the specimen, the load must be sufficient not only to fracture the matrix, but also to pull the fibres out of the matrix (or to fracture the fibres). Since the concrete matrix and the bond between matrix and fibre (131,134,135) are both rate sensitive, the fibre reinforced concrete is also rate sensitive. However, the rate sensitivity of the fibre reinforced concrete was found to be highly dependent on the fibre type and content. The impact strength of fibre reinforced concrete using a 500 mm drop height ranged from about 1.0 to 2.8 times as high as that using a 250 mm drop height. The geometry of the fibres also has an effect on the plate bending strength. Wi th a properly deformed shape, the effects of anchorage are more pronounced. Banthia and Trottier (136) found higher bond strengths in single fibre pullout tests for crimped fibres than for hooked end fibres. However, in the present tests, the increase in bending strength was more pronounced for mildly deformed (hooked end and flattened end) fibres than for fully deformed fibres (crimped). This is because the lower aspect ratio (1/d) and lower number of fibres per unit volume made the crimped fibres less effective at equivalent fibre volumes. 250 — Static loading — Impact Loading (2.21m/s) Impact loading (3.13 m/s) 0.00 0.20 0.40 0.60 0.80 1.00 Deflection (mm) Fig.7.3 5 Effect of Loading Rate on Peak Loads of Plain Concrete Plates 300 250 200 150 100 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Mid-Plate Displacement (mm) Fig.7.36 Effect of Loading Rate on Peak Loads of 0 .5%FRC Plates 300 250 200 150 100 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Mid-Plate Displacement (mm) 0.35 — 1% Hooked End FRC-250mm — 1% Hooked End FRC-500mm 1% Flattened End FRC-250mm — 1% Flattened End FRC-500mm if \ \ \ \ V I 1 i i — 0.40 Fig.7.37 Effect of Loading Rate on Peak Loads of 1.0 % F R C Plates 180 Table 7.1 True Peak Loads of Plain and F R C Plates under Impact Loading Concrete type Description j Fibre V f True Peak No. of Type (%) Load (kN) Specimen Static loading PPLNST Plain Concrete 34 3 P05HEST Fibre Reinforced Plate Hooked End 0.5 81 3 P05FEST Fibre Reinforced Plate Flattened End 0.5 89 3 P05CPST Fibre Reinforced Plate Crimped 0.5 69 3 Impact at 250mm hammer drop height PPL250 Plain Concrete 137 3 P05HE250 Fibre Reinforced Plate Hooked End 0.5 160 3 P05FE250 Fibre Reinforced Plate Flattened End 0.5 161 3 P05CP250 Fibre Reinforced Plate Crimped 0.5 112 3 P1HE250 Fibre Reinforced Plate Hooked End 1 177 P1FE250 Fibre Reinforced Plate Flattened End 1 166 3 P1CP250 Fibre Reinforced Plate Crimped 1 106 3 Impact at 500mm hammer drop height PPL500 Plain Concrete 197 3 P05HE500 Fibre Reinforced Plate Hooked End 0.5 187 j P05FE500 Fibre Reinforced Plate Flattened End 0.5 255 3 P1HE500 Fibre Reinforced Plate Hooked End 1 263 3 P1FE500 Fibre Reinforced Plate Flattened End 1 212 3 181 b) Inertial loads The values of the generalized inertial loads for both plain and F R C plates are given in Table 7.2, and typical inertial load-time curves are shown in Figs. 7.38 and 7.39. It may be seen that the inertial loads of unconfined plates (both plain concrete and F R C ) , ranged between 10% and 20% of the measured tup load. In general, the inertial load increased with increasing rate of loading. For example: in plain concrete, the inertial load increased by 42% with an increase of hammer drop height from 250mm to 500mm. The effect of rate of loading was more pronounced in plain concrete than in F R C . In F R C , when the hammer drop height increased from 250mm to 500mm, the inertial load increased by as little as 7%, and up to a maximum of 33%, depending on fibre type and content. Most F R C plates exhibited slightly smaller inertial loads than plain concrete plates (approximately 9%-14%> of the measured tup load). A s the fibre content increased, the inertial load decreased (comparing by percentage of measured peak load). This is perhaps due to less movement of the F R C specimen under impact loading. A s we know, the inertial load is proportional to the acceleration; i f the specimen is permitted to move freely under impact loading, a higher acceleration is found. On the other hand, i f the specimen is partially (or fully) restrained, a smaller acceleration is expected. In F R C , fibres themselves act like internal passive confinement, which partially restrains the specimen from moving freely, hence, smaller accelerations and inertial loads occur. 182 Table 7.2 Measured Tup Loads and Inertial Loads of Unconfined Plates Specimen Description Fiber Type Vf Drop Measured Inertial Load Type (%) Height (mm) load (kN) (kN) % of Peak Static loading PPLST Plain - 34 P05HEST F R C Hooked End 0.5 81 P05FEST F R C Flattened End 0.5 89 P05CPST F R C Crimped 0.5 69 Impact at 250 mm tiammer drop height PPL250 Plain - 250 157 24 15% P05HE250 F R C Hooked End 0.5 250 160 22 14% P05FE250 F R C Flattened End 0.5 250 168 24 14% P05CP250 F R C Crimped 0.5 250 116 22 19% P1HE250 F R C Hooked End 1 250 177 23 13% P1FE250 F R C Flattened End 1 250 180 25 14% P1CP250 F R C Crimped 1 250 120 18 15% Impact at 500 mm liammer drop height PPL500 Plain - 500 221 34 16% P05HE500 F R C Hooked End 0.5 500 212 26 12% P05FE500 F R C Flattened End 0.5 500 280 32 11% P1HE500 F R C Hooked End 1 500 289 25 9% P1FE500 F R C Flattened End 1 500 240 30 12% Note: Three specimens o each type were tested 183 1000 2000 3000 time (mSec) 4000 5000 Fig. 7.38 Typical Tup Load and Inertial Load vs Time Curves for Plain Concrete Plates 200 2000 4000 6000 time (mSec) 8000 10000 Fig. 7.39 Typical Tup Load and Inertial Load vs Time Curves for F R C Plates (HE) 184 7.4.2 Confined plates a) Measured peak load Uniaxial and biaxial confinement affected the measured peak loads of the plates to different degrees (Tables 7.3 and 7.4). The effect of biaxial confinement was more pronounced than that of uniaxial confinement. A s shown in Figure 7.40, for plain and 0 .5%HE F R C plates subjected to 250 mm hammer drop height, the increase of confining stress from 0 to 5 M P a increased the peak loads from about 20% to 300% compared to those of unconfined specimens. However, this increase was also dependent on the type of material. Regardless of fibre type, all of the 0.5% F R C plates with 5 M P a biaxial confinement exhibited similar increases in peak load, though the flattened end fibres seemed to be slightly better than the other two (Fig. 7.41). However, at high fibre volume fractions (1%), the "peak load" decreased for most types of fibres (Fig. 7.41) because the 1% F R C plates with 5 M P a confinement did not fail, and therefore the fracture strength of material was not reached. For the uniaxially confined specimens subjected to the same rate of loading (250mm drop height), the increase in peak load was not as great as for biaxial confinement (Table 7.3, Figs. 7.42 and 7.43). The increase in peak load was about 10%-20%) at 2.5 M P a confinement and 15%-50% at 5 M P a confinement, compared to the unconfined tests. This is because of the Poisson's ratio effect in the plates under uniaxial confinement. When uniaxial confining stress was applied, a tensile force was induced in the direction perpendicular to the applied load (on the unconfined side) due to the effect of Poisson's ratio. With this pre-existing tensile force on the unconfined side, a smaller load was required to fracture the plates. However, these loads were still higher than those required to fracture the unconfined specimens. This is because the partially fractured portions of plate are held intact by the confined pressure and the friction between the fractured surfaces is still high and continues to support load. 185 Table 7.3 Measured and inertial loads of plain and F R C plates with Uniaxial Confinement (Tested at 250mm hammer drop height) Specimen Type Description v f (%) Confining Stresses (MPa) Measured Peak load (kN) Inertial load (kN) % peak PL25U5 Plain Concrete 5 M P a 170 39 23% PL25U2.5 Plain Concrete - 2.5 M P a 126 35 28% 05HE25U5 Hooked End F R C Plate 0.5 5 M P a 237 64 27% 05HE25U2.5 Hooked End F R C Plate 0.5 2.5 M P a 181 31 17% 05FE25U5 Flattened End F R C Plate 0.5 5 M P a 185 24 13% 05FE25U2.5 Flattened End F R C Plate 0.5 2.5 M P a 178 28 16% 05CP25U5 Crimped F R C Plate 0.5 5 M P a 179 23 13% 05CP25U2.5 Crimped F R C Plate 0.5 2.5 M P a 138 32 23% 1HE25U5 Hooked End F R C Plate 1 5 M P a 219 33 15% 1HE25U2.5 Hooked End F R C Plate 1 2.5 M P a 211 34 16% 1FE25U5 Flattened End F R C Plate 1 5 M P a 195 27 14% 1CP25U5 Crimped F R C Plate 1 5 M P a 137 18 13% Note: Three specimens were tested for each test 186 Table 7.4 Measured and inertial loads of plain and F R C plates with Biaxial Confinement (Tested at 250mm hammer drop height) Specimen Type Description v f (%) Confining Stresses (MPa) Measured Peak Load (kN) Inertial load (kN) % Peak PL25B5 Plain Concrete 5 M P a 255 20 8% PL25B2.5 Plain Concrete - 2.5 M P a 240 22 9% PL25B1.25 Plain Concrete - 1.25 M P a 186 21 11% PL25B0 Plain Concrete - 0.00 M P a 136 9 * 7% 05HE25B5 Hooked End F R C Plate 0.5 5 M P a 290 21 7% 05HE25B2.5 Hooked End F R C Plate 0.5 2.5 M P a 280 25 9% 05HE25B1.25 Hooked End F R C Plate 0.5 1.25 M P a 245 21 9% 05HE25B0 Hooked End F R C Plate 0.5 0.00 M P a 197 8* 4% 05FE25B5 Flattened End F R C Plate 0.5 5 M P a 321 21 6% 05FE25B2.5 Flattened End F R C Plate 0.5 2.5 M P a 268 22 8% 05CP25B5 Crimped F R C Plate 0.5 5 M P a 315 26 8% 05CP25B2.5 Crimped F R C Plate 0.5 2.5 M P a 291 30 10% 1HE25B5 Hooked End F R C Plate 1 5 M P a 319 18 6% 1HE25B2.5 Hooked End F R C Plate 1 2.5 M P a 271 20 7% 1FE25B5 Flattened End F R C Plate 1 5 M P a 301 19 6% 1CP25B5 Crimped F R C Plate 1 5 M P a 243 31 13% Note: Three specimens were tested for each test * Inertial load was calculated based on flexural type of failure Eq . 4.31 187 0.5% HE • without confinement DO H1.25 B2.5 1 5 Fig . 7.40 Effect of Confinement Stress on Peak Load of Plain and 0.5% H E F R C 400 -, 350 Plain 0.5%HE 0.5%FE 0.5%CP 1%HE 1%FE 1%CP • Without Confinement •With 5 MPa Biaxial Confinement Fig. 7.41 Effect of 5MPa Biaxial Confinement on Peak Load 188 300 • Without Confinement 2.5 MPa Biaxial 2.5 M P a Uniaxial Fig . 7.42 Effect o f 2.5 M P a Biaxial and Uniaxial Confinement on the Measured Peak Load 350 Plain 0.5%HE 0.5%FE 0.5%CP 1%HE 1%FE 1%CP • Without Confinement I With 5 Mpa Biaxial iWith 5 Mpa Uniaxial Fig . 7.43 Effect of 5 M P a Biaxial and Uniaxial Confinement on the Measured Peak Load 189 In order to study the effect of hammer drop height on the measured peak loads of uniaxially and biaxially confined plates, an additional series of tests was conducted at 500 mm drop height. The results are given in Table 7.5, Table 7.6 and Fig . 7.44. It was found that the biaxially confined specimens tested at the 500 mm drop height generally exhibited higher peak loads than those tested at a 250 mm drop height. On the other hand, for the uniaxially confined specimens, the peak load decreased to values even lower than those of the unconfined specimens. This is, again, due to the Poisson's ratio effect in the uniaxially confined specimens. Table 7.5 Measured and Inertial Loads of Plain and F R C Plates with Uniaxial Confinement (Tested at 500mm Hammer Drop Height) Specimen Description Fibre V r Confining Peak Load Inertial load Type Type (%) Stresses (kN) (kN) % of Peak PL50U5S Plain 5 M P a 150 48 32% P05HE50U5S F R C H E 0.5 5 M P a 151 39 26% P05FE50U5S F R C F E 0.5 5 M P a 178 38 21% P05CP50U5S F R C CP 0.5 5 M P a 213 30 14% P1HE50U5S F R C H E 1 5 M P a 247 36 15% P1FE50U5S F R C F E 1 5 M P a 166 39 24% P1CP50U5S F R C C P 1 5 M P a 206 59 29% 190 Table 7.6 Measured and Inertial Loads of Plain and F R C Plates with Biaxial Confinement (Tested at 500mm Hammer Drop Height) Specimen Descript ion F ibe r v f Conf in ing Peak L o a d Inert ial load Type Type (%) Stresses (kN) (kN) % o f Peak P L 5 0 B 5 S Plain _ 5 M P a 280 25 9% P 0 5 H E 5 0 B 5 S F R C H E 0.5 5 M P a 369 19 5% P05FE50B5S F R C F E 0.5 5 M P a 373 28 7% P05CP50B5S F R C CP 0.5 5 M P a 345 26 8% P 1 H E 5 0 B 5 S F R C H E 1 5 M P a 366 30 8% P 1 F E 5 0 B 5 S F R C F E 1 5 M P a 379 27 7% P1CP50B5S F R C C P 1 5 M P a 270 22 8% 400 Plain 0.5%HE 0.5%FE 1%HE 1%FE • Without Confinement •With 5 Mpa Biaxial BWith 5 Mpa Uniaxial Fig. 7.44 Peak Load of 5MPa Biaxial and Uniaxial Confined Plate Subjected to Impact from 500 M m Drop Height 191 b) Inertial load For the inertial loads (see also Tables 7.3 to 7.6), which are used to correct the "measured" loads, the unconfined specimens exhibited somewhat higher inertial forces as a percentage of the peak load (about 9-20% of the measured peak load). However, when confinement stresses were applied, the inertial forces decreased. Except for the case o f uniaxial confinement (which the inertial force was in the similar range o f those unconfined plates at about 13-28% of the measured peak load), the inertial load of biaxial confined specimens decreased to only 6-13%> of the measured peak load. Assuming that confinement prevents the specimen from moving freely under impact loading, the increase in confinement stress from 0-5 M P a led to a significant drop in the inertial force, from 16%> to 8% for plain concrete and 14% to 7% for 0.5% Hooked end F R C plates (Fig. 7.45). In addition, it was also observed that the inertial force not only decreased with increasing confinement but also decreased with increasing fibre content. In both unconfined and confined tests, the inertial loads of F R C with l % V f were less than those of F R C with 0.5%Vf by 1-15%) This is perhaps due to the fibres acting as passive confinement inside the specimen. 18% 5 MPa 2.5 MPa 1.25 MPa 0.00 MPa Unconfined Note * Inertial load was calculated based on flexural failure mode Fig. 7.45 Effect of Confinement on Inertial Load 192 7.5 Energy Lost by the Hammer (Impact Loading) and Fracture Energy 7.5.1 Unconfined plates a) Static loading For the plate specimens, the failure usually took place at a mid-plate of deflection about 10-15 mm (which is about three times larger than for beams); thus, for a comparison purposes, a reference value of 10mm was selected for calculating fracture energy. Again, the words "fracture energy" and "energy absorption" wi l l be used interchangeably during the discussion in this section. The 10 mm-fracture energy was found to increase significantly when steel fibres were incorporated into the concrete (Table 7.7). The two primary reasons for this were: 1) the effect o f fibre bridging and anchoring; and 2) the number of fibres dispersed in the matrix. The effect of fibre bridging was to reduce the stress intensity at the crack tip; thus more energy was required to open the cracks. A s the number o f fibres increased, the fibre spacing decreased, and thus more fibres were intercepted by the crack, and again more energy was required to open the cracks. Table 7.7 1 Omm-Fracture energy of Unconfined Specimens under Static and Impact Loading Categories Description Fiber Type Fracture Energy (N-m) (10mm) Static Impact (250) Impact (500) PPLST Plain concrete - 0.35 12 13 P05HEST 0.5%FRC Hooked End 334 541 512 P05FEST 0.5%FRC Flattened End 258 395 -P05CPST 0.5%FRC Crimped 171 378 -193 A t very small deflections (i.e., 0.5mm), the energy absorptions of the F R C plates under static loading were not much different for the different fibre types. The real difference occurred in the post-peak fracture energy at large deflections (i.e., 10mm). Hooked end and flattened end fibres (334 N - m and 258 N - m , respectively) gave higher post-peak fracture energies than did crimped fibres (171 N-m), due to their higher aspect ratio and larger number of fibres contained in the concrete. b) Impact loading The fracture energies of unconfined plates calculated out to a 10 mm deflection are given in Table 7.7. The fracture energy was higher under impact loading than under static loading. The increase in fracture energy was due to the rate sensitive behaviour of concrete, in which the peak load increased with the rate of loading. The effect of fibre type on impact fracture energy was the same as in static fracture energy. Hooked end fibres were the most effective primarily due to their lower aspect ratio. For the impact testing, besides the fracture energy, the energy lost by the hammer also needed to be considered. During the impact event, different amounts of energy were dissipated from the hammer (energy lost) to the system (specimen, machine, sound, temperature etc), depending on several factors. The amount of energy dissipated through the specimens and the amount of energy absorbed by the specimens depended on specimen type. F R C plates permitted more energy to dissipate, and also absorbed more energy than plain concrete plates. The plots of fracture energy and energy lost by the hammer for unconfined plates with respect to time are shown in Figs 7.46-7.47. For plain concrete (Fig. 7.46), the energy lost by the hammer was very small as was the fracture energy (about 190 J and 100 J for energy lost and fracture energy, respectively). For the 0 .5%V f F R C (Fig. 7.22), the energy lost by the hammer was higher (about 800J) and the fracture energy also increased up to about 550J. Although the gap between energy lost by the hammer and fracture energy decreased, it was still wide, which means that a considerable amount of energy was dissipated during the impact event. B y increasing the fibre content to 1%, the energy absorption capacity of the 194 concrete was enhanced to 790J, and the gap between energy lost and energy absorption was reduced. The increase in the fracture energy of F R C derived from the effects of fibre bridging across the cracks. Plain concrete is a brittle material, and so very small deflections and cracks can lead to instantaneous failure of the specimen. F R C plates, on the other hand, because of the effects of fibres stretching and bridging across the cracks, can sustain higher deflections prior to the failure. Since the fracture energy is a function of both load and deflection, i f the peak loads are similar for most types of concrete, then the increase in deflection at failure is the principle factor that increases the fracture energy. In F R C with higher fibre contents, as the number of fibres increases, the reduction in fibre spacing helps to promote the fibre bridging effect, which in turn leads to the occurrence of multiple peaks in the load-deflection response, a longer post-peak regime, and an increase in fracture energy. Clearly, since the duration of the impact event for plain concrete is much shorter than that for F R C , a smaller energy loss was observed for the plain concrete. 1000 2000 3000 4000 5000 6000 7000 Time (uSec) Fracture Energy-Plain Energy Lost-Plain Fracture Energy-0.5%HE Energy Lost-0.5%HE Fracture Energy-1%HE Energy Lost-1%HE 8000 Fig. 7.46 Effect of Fibre Content on Energy Lost by the Hammer and Fracture Energy of Unconfined Plates 195 1000 2000 3000 4000 5000 Time (uSec) 6000 7000 8000 — Fracture Energy-0.5%CP — Energy Lost-0.5%CP Fracture Energy-0.5%HE — Energy Lost-0.5%HE Fracture Energy-0.5%FE Energy Lost-0.5%FE Fig . 7.47 Effect of Fibre Type on Energy Lost by the Hammer and Fracture Energy of Unconfined Plates A s stated previously, the smaller energy loss and energy absorption of plain concrete is a direct result of the shorter impact event and smaller deflection at failure than for F R C plates. For the F R C plates, comparing the 0.5% V f and 1.0% V f hooked end F R C (Fig. 7.48), the impulse (which is given by the area under the load-time curve) of both F R C plates was very much the same. Since the energy lost is proportional to the impulse, similar energy loses in both 0 .5%V f and 1.0%V f F R C ' s were found (750J and 850J, respectively). However, in terms of fracture energy or energy absorption, the two F R C ' s behaved differently. A s shown in Fig. 7.48, the peak loads were much the same, but the deflection at failure of the 1.0% F R C plate was about twice as high as that of the 0.5% F R C plate. This means that increasing fibre content enhances the ability of F R C to carry load at large deflections; hence, the larger energy absorption or fracture energy of higher fibre content F R C . 196 200 160 2 120 CO O 80 40 2,000 4,000 6,000 8,000 10,000 12,000 Time (uSec) Load-0.5%HE FRC Deflection-0.5%HE FRC •Load-1.0%HE FRC — Deflection-1.0%HE FRC E 3 O o w-<D T5 CD m a. ~o S Fig. 7.48 Load/Deflection vs Time Curves of 0.5% and 1% Hooked End F R C Plates 7.5.2 Confined plates For the confined tests (Figs. 7.49 to 7.52), the energy lost by the hammer as it passed through the specimen increased to well above 1000 J. The confinement apparatus itself played an important role in this. The confined specimens permitted more energy to dissipate through them, and were also able to absorb more energy than the unconfined specimens. Except for the specimens for which the impact energy used in this study was insufficient to cause rupture, the response of material was elastic-inelastic; the load increased up to a peak value then returned back to zero with some inelastic strain. According to Gere and Timoshenko (137), two types of energy must be considered when the response of the material is elastic-inelastic: elastic energy and inelastic energy (Fig. 7.49). For the unfractured specimen, a portion of the elastic energy was recovered and returned back to system; the measured energy was actually an inelastic energy, and not a fracture energy (Fig. 7.50 and 7.51); hence, lower energy absorption was observed. 197 For the confined plain concrete and F R C with 0.5% Vf, the gap between energy absorption and energy lost by the hammer was still relatively large. However, the gap narrowed as the confinement increased (Figs. 7.51 and 7.52) except in the case of high confinement (Fig. 7.51) and high fibre content (Fig. 7.53) where failure did not occur. Comparing the three types of fibres, the performance of each fibre type also depended on the degree of confinement. For example: at 2.5 M P a biaxial confinement, hooked end and flattened end fibres performed better than crimped fibres (Fig. 7.53). Comparing biaxial and uniaxial confinement, both at 2.5 M P a (Fig. 7.54), the energy absorption of biaxially confined specimens (both plain concrete and 0 .5%HE F R C ) was higher than that of uniaxially confined specimens (42% and 78%) for plain concrete and 0 .5%HE F R C , respectively). Comparing plain concrete and F R C , uniaxial confinement increased the energy absorption capacity by very little (580J and 608J for plain concrete and F R C , respectively) while with biaxial confinement, the energy absorption increased significantly (800J and 1100J for plain concrete and F R C , respectively). O - J Inelastic energy -Elastic energy bo u C W r -Elastic energy -Inelastic energy Deflection Time Unfractured Specimen eS O —J -Inelastic energy 6/) c —Inelastic energy Deflection Time Fractured Specimen Fig . 7.49 Relationship between Load Response and Energy Absorption, after Gere and Timoshenko (6) 198 1,600 1,400 1,200 1,000 3 >. E> 800 0) c LLI 600 400 200 2,000 4,000 6,000 8,000 10,000 12,000 14,000 Time (uSec) Fracture Energy at 5 MPa Fracture Energy at 2.5 MPa - Fracture Energy at 0 MPa Energy lost by the hammer at 5 MPa Energy lost by the hammer at 2.5 MPa -Energy lost by the hammer at 0 MPa Fig. 7.50 Effect of Confinement on Energy Lost by Hammer and Fracture Energy of Plain Concrete Plates 1,600 1,400 1,200 — 1,000 3 §i 800 v c 600 400 200 LU 5,000 Elastic Energy at 5 MPa Fracture Energy at 2.5 MPa -Fracture Energy at 0 MPa 10,000 Time (uSec) 15,000 20,000 Energy lost by the hammer at 5 MPa Energy lost by the hammer at 2.5 MPa ^ — E n e r g y lost by the hammer at 0 MPa Fig. 7.51 Effect of Confinement on Energy Lost and Fracture Energy of 0 .5%FRC Plates 199 1600 1400 1200 1000 >< ro i_ 800 <u c LU 600 400 200 * No failure occurred for both types of confinement; elastic-plastic response 2,000 4,000 6,000 8,000 10,000 12,000 14,000 Time (uSec) Elastic Energy at 5 MPa - Elastic Energy at 2.5 MPa Energy lost by the hammer at 5 MPa 'Energy lost by the hammer at 2.5 MPa Fig. 7.52 Effect of Confinement on Energy Lost and Fracture Energy of 1%FRC Plates 2,000 4,000 6,000 8,000 10,000 12,000 14,000 Time (uSec) •Fracture energy of 0.5%HE - Fracture energy of 0.5%CP Fracture energy of 0.5% FE ^ — E n e r g y lost by the hammer at 2.5 MPa ^ — E n e r g y lost by the hammer at 2.5 MPa — Energy lost by the hammer at 2.5 MPa Fig. 7.53 Energy Lost and Fracture Energy of 2.5 M P a Biaxial ly Confined F R C Plates 200 1,400 1,200 2,000 4,000 6,000 8,000 10,000 Time (uSec) Plain-Biaxial Plain-Uniaxial 0.5%HE-Biaxial 0.5%HE-Uniaxial Fig. 7.54 Energy Absorption: Comparison Between Biaxial and Uniaxial Confinement 201 CHAPTER 8 COMPARISON BETWEEN UNCONFINED PRISMS, BEAMS AND PLATES 8.1 Introduction There has been a growing interest in the use of plate specimens rather than beam specimens for impact (and also static) testing of F R C and fibre reinforced shotcrete. For instance, Banthia et al (5) concluded that using plate specimen for impact testing has several advantages over beam tests: plates are able to absorb much more energy than beams, the formation of multiple random cracks in plate allows fibres to more fully exhibit their ability to bridge across the cracks, and the sensitivity (ratio of impact to static strength) of plates is less than that of beams, providing a more logical choice for characterizing the toughness of F R C . In this chapter, the correlations, the similarities and the differences between prism, beam and plate specimens are discussed. The main purpose for using different types of specimens was to study the behaviour of concrete under different kinds of loading. The rectangular prisms were used to study uniaxial compressive behavior, the beams were used to study 1-dimensional flexure ( IDF) , and the plates were used to study 2-dimensional flexure (2DF). Comparisons are made in terms of the magnitude of the measured strengths and fracture energies, the loading rate, the relationship between loading rate and peak load, and the rate sensitivity. In addition, the advantages and disadvantages o f each specimen type are discussed. It must be noted that the discussions in this chapter are limited only to the unconfined tests, in which the failure was quite simple, and characteristic of each type of test. In the confined tests, the failure patterns were more complex (as they were dependent on the confinement), which made it hard to define the exact type of failure: quite often a mixed-mode of failure between shear and bending was found. 202 8.2 Physical Properties Before addressing any of the results, a comparison of the physical properties of each type of specimen is given in Table 8.1. Table 8.1 Physical Properties of Each Specimen Type Specimen Type Dimensions (mm) Weight (kg)* Ratio by Weight Width Length Depth Vol. (m3) Plain 0.5%FRG 1%FRC Prism 100 100 175 0.0018 4.06 4.13 4.20 1.0 Beam 100 350 100 0.0035 8.12 8.26 8.39 2.0 Plate 400 400 75 0.012 27.84 28.31 28.78 6.9 Note: *the unit weight of p ain concrete= 2320 cg/m 3 and steel fibre = 7800 kg/m j The beam and plate specimens were about 2 and 7 times as large and as heavy as the prism specimens (Table 8.1). The beam dimensions of 100*100*350mm were the standard size specified in A S T M C192-90a: Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory. The prism specimens were prepared by cutting the beam specimens in half. The dimensions of the plate specimens were limited to about 400*400 (width x length) to fit within the existing impact machine. 8.3 Strength and Fracture Energy The average strengths and fracture energies obtained from the tests are given in Table 8.2. It was found, as expected, that the uniaxial compressive strength was higher than the flexural strength. The ratios of the I D flexural strength (1DFS) and the 2D flexural strength (2DFS)*(138), to the uniaxial compressive strength were about 0.16-0.22 ' Stress in a plate is given by: <J • 3W (l + v ) l n — + /? where W is the total applied load, t is the thickness of the plate, V is the Poisson' ratio, b is the length of plate, /? is the width/length ratio, and r 0 is the radius of contact for a concentrated load on a very small area. (Roark, R.J., and Young, W.R.C., Formulas for Stress and Strain 5 t h edition, McGraw-Hill, 1975) 203 and 0.19-0.48, respectively. This is due to the nature of concrete itself; concrete is much stronger in compression than in tension, and the flexural failure of a concrete beam is due to a crack which actually initiates at the tension face. Comparing 1DFS and 2DFS, the 1DFS was smaller than the 2DFS under both loading conditions (static and impact). In the case of static loading, the ratio of the 2DFS to the 1DFS (2DFS/1DFS ratio) also increased with fibre content. A s for plain concrete, the 2DFS/1DFS ratios were about 1.2 and for F R C , the 2DFS/1DFS ratios were about 2.5 to 3.1 depending on the fibre type. The higher 2DFS/1DFS ratio of the F R C than of the plain concrete was due to the better improvement in flexural strength under 2D flexural loading. However, under impact loading, the 2DFS of plain concrete was improved significantly to about the same level as F R C , as indicated by the increasing 2DFS/1DFS ratio for plain concrete up to about 2.3 to 2.5, which was close to that o f F R C (2DFS/1DFS ratio = 1.8 to 2.5). 204 S .O x O s P V ? S ? \ 0 o — 1 * - 1 CN T f CN )\ ^? ^  \ ° \ ° \ ° \ ° \= ? £ ° ° X ^ ^ o o i o T f T f T f ' c K o d s ? v P s P N° XP S O X O , O ^ ^ C N O O O s c r i T f C N O s MD OS T f r f 00 O l O i o ' ^ P ^ P ' n O f»' ^ CN T-C 00 ^ * , ^ ; 2 " ' - , t - ^ O s o s i / ' - > <o i o CN m so p--oo m co oo ~ ,~ © <=! o o <=? <=> £ £ £ £ so r> S g 3 °« £ ^ £ £ ?i S — i - H ^ N C N O O S O s O s O O O i O n ^ ^ o r ^ T f o o o s N (N f S | ' f s i 1 H . ( S | r H IH m so .Os 0 0 > n i o r n O s 0 0 r - - O T f i o i o » n m u - ) i o r - ~ o o r ^ t o i o i o c N m c N s o o o o o o o o o V) s o CN s o T f r - 1 c O T f r O T f t N r ' S T f v o T f r n O so O T f T f - I fS] r H ( S | T f 00 CN O (N m ccj X CD Q CN I Q (N c CD U i T3 o in 00 N PH W W .s a u | a JS ^ ^ ^ 0- 1/5 l /s 0 p c n os CN so i n oo l^o oo so oo so so oo IO OO i n 00 00 00 IT, m oo (N co 00 CN C O T f 00 —I C N C/3 CD 5b CD a CD *—» o a a CD CCJ % X <D Q I Q CD CD _> e o o E .2 - H S S ft w S w .2 .5 M W U to B K « S s= sf s= s= xc ,: « - oN oN oN eN Js >s • • • • © © © © ^ ^ CD c3 o o 205 In terms of fracture energy, the beam specimens ( IDF) were able to absorb much less energy (about 0.5% to 10% of the impact energy) than prisms and plates (which could absorb energies of about 0.8% to 49%> and 23% to 86% of the impact energy, respectively). Under both bending conditions ( IDF and 2DF), the plain concrete exhibited the lowest percentage o f energy absorption (0.4%-0.8%), due the low tensile strength and brittleness of plain concrete. With fibres, the percentage of energy absorption increased significantly from 18% to 49% for plate specimens, though for the beam specimens, the percentage increase remained below 10% for all types of F R C . The increase in fracture energy was due to the fact that fibres somewhat increased the tensile strength of the concrete and increased its ability to sustain load after the peak. Under uniaxial compression, the ratio of the static fracture energy to the impact fracture energy (250 mm drop height) was quiet small for both plain concrete and F R C (1.0-1.7). This was because plain concrete itself is very strong under compression, and fibres do not significantly increase the compressive strength. The small magnitude of the flexural strength and absorbed fracture energy for the beam specimens (ID-bending) make the beam test less attractive for high-energy impacts tests, since most o f the energy was lost to the entire system and much less energy was absorbed by the specimen itself. 8.4 Stress Rate ( & ) The average stress rates are given together with the times to peak stress and the calculated stress rates in Table 8.3. B y assuming a linear response up to the peak stress, the stress rate was calculated by: cr = cr/Tp (8.1) where a is the peak stress (MPa) T p is the time to peak load (sec). Regardless of the type of concrete, the average stress rates obtained from the plate tests (47,000 to 230,000 MPa/sec) were the highest among the three tests; the beam tests 206 (17,000 to 37,000 MPa/sec) were about the same order of magnitude as those obtained from the prism tests (22,000 to 65,000 MPa/sec). The drop hammer impact test is a velocity controlled, not a stress rate controlled test; the stress rate is not directly proportional to the drop height. The stress rate obtained from a particular test is the combination of several factors including the hammer drop height, the mass of hammer, and the properties of the tested specimen itself. If the hammer drop height and the mass are fixed, then the stress rate is solely dependent on the specimen properties (both physical and mechanical). In the present tests, it may be seen that the plate specimens have one key advantage over the beams: their obtained stress rates are higher. 8.5 Relationship between Stress Rate and Strength The relationship between the trued peak load and loading rate for each specimen type is shown in Figs. 8.1a-d. For all specimen types, plain concrete was more rates sensitive than F R C . The beam specimens exhibited higher stress rate sensitivity than did the prisms and plates. For plain concrete, beams were the most rate sensitive followed by plates and prisms, respectively (Fig. 8.1a). For F R C , the beams were still the most rate sensitive specimens. However, the rate sensitivity of F R C plates appeared to decrease significantly, in some cases, the plates became less rate sensitive than the prisms. The high rate sensitivity in beam specimens was due to the relatively high impact strength compared to the static strength, unlike the plates and prisms. This could be explained from the failure pattern of each specimen type. A s discussed in the previous chapters, the failure of beam specimens was normally governed by a single crack while for the plates and prisms, large numbers of cracks were found at failure. First, let us compare beams (ID-bending) and prisms (uniaxial compression). According to Bischoff et al (139), with increasing stress rate in uniaxial compression, there may be fewer microcracks at lower stress levels, but there is also an increase in the amount o f cracking at failure. This reflects the increase in the critical compressive strain when concrete is loaded at high strain rates. The large number of cracks indicates that more energy is also dissipated during loading. Even though an increase in strength under 207 high rates of loading is observed, and a large amount of energy is absorbed under compressive impact loading, a large portion of the energy is also dissipated due to the reduced stress concentrations at the crack tips. Hence, the total energy absorption was not directly related to the peak load, since some of the energy was lost during the impact event. For the beam specimens, even though the absorbed energy was very low since there was only a single crack, the energy lost due to stress relief at the crack tip was lower and almost the entire absorbed energy was consumed in propagating a single crack. In other words, the relative total energy lost due to stress concentration relief at the crack tip was higher in the case of concrete with a large number of cracks (uniaxial compression) than in the case of a smaller number of cracks (bending). A s the result, the increase in impact strength as compared with static strength was less in compression than in bending. The same principle also applies to the plate specimen (2D-bending), where larger numbers of cracks at failure were found, thus leading to lower strain rate sensitivity. 208 u PH H^ C/2 S CJ i n OS 00 m o\ oo co 3^-oo r-- m O ^ o MD CN i n O s CN co M3 MD m i n MD ON" i n MD MD C3 CN MD CN 00 o CO CO CO CO MD Os CN o i n CN i n ON o MD r - 1 ' — 1 o i n o" oo" r>-" r-»" CN T - H ON >n CN C O CN t-^ oo" CN O CN OO Tt-" NO oo MD" CN MD , , MD o MD^  o\ oo" i n CN i n CN CN MD ON OS CN CN VO ^O i n " NO o, o - * H H r--CN 00 CN CN O ON i n m co <n r - < MD C O PH W) S H * H 03 OH E CS P3 «H H . PH * S E <u »-H e CU o SH oo CO MD m ON ^ £ © CO c--MD 00 MD CN CO m CN CN OS CN m CN CN — i o o i n T J -CN" —T r—1 ' CN CN o 00 CN T - H i n r - CN CO CN CN" co" o CO oo oo o OS 00 MD i n M3 CN i n MD CO Tf CN CO CN o o o MD i n CO o CN r--MD O —i CN CN i - H CN o CO oo o i n r - H MD oo CO MD MD OS O 00 CN MD i n MD CO o oo i n 00 oo i n oo - - H C O r ^ oo rt-—c CN H i n oo oo oo i n CO r - H ^ H _ « > r j - H - H , M CN r - , r - i _ fN) CU a I "3 H PH C « OH a a ir, PH u IT) M H X ~ IT) o a O ® H H H H 209 4.00 0.1 10 100 1000 10000 Stress rate (MPa/sec) 100000 1000000 0.5%HE FRC •Prism Beam 0.1 10 100 1000 10000 Stress rate (MPa/sec) 100000 1000000 0.5%CP FRC •Prism Beam Plate 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.1 10 100 1000 10000 100000 1000000 Stress rate (MPa/sec) 0.5%FE FRC •Prism Beam Plate 3.50 3.00 o 2.50 2 2.00 1.50 1.00 0.50 co O ro a. E 0.1 10 100 1000 10000 Stress rate (MPa/sec) 100000 1000000 Fig. 8.1 Relationship Between Stress Rate and I/S Strength Ratio of Unconfined Concrete 210 8.6 Rate Sensitivity To simplify the comparisons, the term "rate sensitivity" used here refers to the impact/static ratios of both the peak loads and the fracture energies (Tables 8.4 and 8.5). Table 8.5 The Impact/Static Rate Sensitivity of Unconfined Specimens Categories I/S ratio (Strength) I/S ratio (Fracture energy) Prism Beam Plate Prism Beam Plate Plain-250 1.29 2.00 4.05 1.89 12.60. 7.28 Plain-500 1.59 3.10 5.79 5.81 16.20 7.91 0.5%HE-250 1.17 2.26 1.98 1.01 1.30 1.62 0.5%HE-500 1.43 2.76 2.31 1.84 2.48 1.53 0.5%CP-250 1.25 2.38 1.61 1.25 1.87 2.21 0.5%FE-250 1.18 2.28 1.81 1.30 1.54 1.53 1.0%HE-250 1.07 1.98 - 0.87 1.69 -1.0%HE-500 1.31 2.72 - 1.59 3.03 -Average 1.29 2.46 2.93 1.95 5.09 3.68 In term of strengths, prism (compression) specimens exhibited less rate sensitivity than either beam or plate specimens, as seen by the smaller average I/S ratio for prisms. A s stated in the previous section, the compressive strength was less rate sensitive than the flexural strength, as was shown in Fig . 8.2. Even though the magnitude of the compressive load was much greater than that of the flexural load, it increased far less under impact loading than the flexural load. We know that at low rates of loading, cracks tend to propagate around the aggregate through the interfacial zone, the traditional "weak link" (140) in concrete. However, at high rates of loading, cracks tend to propagate through, rather than around the aggregates; as a result, higher impact strengths were observed. Comparing beam (IDF) and plate (2DF) tests, their I/S ratios were somewhat similar, with averages of 2.46 and 2.93, respectively. In general, plain concrete was more 211 rates sensitive than F R C , since the effect of the fibres is to decrease the velocity of crack propagation, and to reduce the stress intensity at the crack tip. The impact/static fracture energy (FE-I/S) ratio of prism specimens was, again, found to be much less than those of beam and plate specimens. Beam specimens were the most rate sensitive of the three types of specimens. In general, the FE-I/S ratios of plain concrete for all loading cases were larger than for F R C . The increase in fracture energy of plain concrete was primarily the result of the large increase in peak load, because the post-peak responses of plain concrete under both loading conditions were quite small (Fig. 8.2). For F R C , the FE-I/S ratio was found to be much less than that of plain concrete in all cases. This was because the effects of fibre bridging were reduced under impact loading, since large numbers of fibres were apparently pulled out simultaneously at the peak. A s a result, the post-peak response of F R C under impact loading was similar to, or in one case (Fig. 8.3-0.5%HE F R C prism) worse than that under static loading. Again, the higher FE-I/S ratio of F R C was mainly due to the increase in peak load with loading rate. 1.0 2.0 Deflection (mm) 1.0 2.0 Deflection (mm) 3.0 Plain concrete prism — Plain concrete beam — P l a i n Concrete plate Fig. 8.2 Comparing Load-Deflection Response of Plain Concrete under Compression, I D and 2D Bending 212 0.0 1.0 2.0 3.0 0 1 £ Deflection (mm) Deflection (mm) -0.5% HE FRC prism —0.5% HE FRC beam —0.5% HE FRC plate Fig . 8.3 Comparing Load-Deflection Response of 0 .5%HE F R C under Compression, I D and 2D Bending In summary, then: 1. The absolute magnitudes of the strength obtained from each type of test were quite different. Prism (compression) specimens exhibited higher strengths than beam (ID-flexure) and plate (2D-flexure) specimen. This was due the intrinsic nature o f concrete, which is very strong in compression and weak in tension. 2. The highest loading rate was obtained from the plate specimens, while the rate of loading obtained from the beam specimens was the lowest amongst the three specimens. 3. Comparing the three types of loading, uniaxial compression was less rate sensitive then flexure. The rate sensitivity of flexural specimens was due largely to their large increase in impact strength. The increase in impact strength can be explained by the failure pattern of the flexural specimens, in which there were fewer crack than in the uniaxial compression specimens. 213 CHAPTER 9 PARAMETERS AFFECTING CONFINEMENT TESTS 9.1 Introduction There are several factors which affect multiaxial test results in concrete (e.g., Newman (141), van Mier (101), V i l e (126), Rusch (142), Laddie (143) and Kupfer (144)). Perhaps the most important factor is the particular boundary condition (platen size and stiffness, end friction, etc.). According to van Mier (101), there are two major boundary effects: frictional and rotational restraints, and it is important to understand these two effects on the fracture of a brittle and heterogeneous material such as concrete. Over the years, there have been many attempts to develop methods for reducing the frictional restraint at the contact surfaces. These have included lubricated platens or friction-reducing pads (teflon, rubber, polyethylene sheets, polished metal sheets), or various combinations of lubrication and pads. There is a particular risk in using grease (Newman (145)), due to the optimum thickness of grease that must be chosen for application. It must not be too thick or slip may occur. Friction reducing platens also include the so-called "non-rigid" loading platens, such as brush-bearing platens (146), and the combination of steel rods with an elastomeric plate (147). In this study, the frictional boundary effect was studied using two different loading platens: hard (steel) and soft (rubber). Fig. 9.1 illustrates schematically the differences between hard and soft platens. The hard platens provide concrete with a uniform strain but a non-uniform stress due to the friction at the contact surfaces. The friction is due to the difference in Poisson's ratio between the concrete and the loading platens. On the other hand, with soft platens, the friction decreases significantly; this provides concrete with a uniform stress but a non-uniform strain distribution along the contact area. Indeed, with very soft rubber platens, the outward "f low" of the rubber can induce vertical splitting of a cylinder. Tests were carried out on the three types o f specimens: prisms (uniaxial compression), beams (1-D bending) and plates (2-D bending). The discussion below deals with the failure patterns, the load-deflection responses, and the peak loads. 214 In addition, the effect of the loading head (tup) geometry was studied in 2-D bending. In the confined plate tests, a shear cone type of failure was generally found. However, the failure pattern may be influenced by the size of the tup. In order to study this effect, both 100mm and 150mm diameter tups were used for some of the plate tests. Hard platens (Steel) Soft platens (Rubber) J A t Stress Strain t Fig . 9.1 Distribution of Stress and Strain at the Contact Surfaces (101) 9.2 Effect of L o a d i n g Platen on Confined Impact Test Two types of loading platens were used: steel and rubber. The Young's modulus (E) of steel is approximately 200,000 M P a . For the rubber, the E obtained from tests is about 7.2 M P a (Fig. 9.2). Impact tests were carried on plain concrete, 0 .5%HE and 1.0%HE F R C . The drop heights were: 250mm for prism and plate tests, and 150mm for beam tests. The confining pressures were 0.625 M P a for prism tests, and 2.5 M P a for beam and plate tests. 215 1.00 0.02 0.04 0.06 0.08 0.10 0.12 Strain Fig . 9.2 Stress-Strain Relationship o f Rubber Platen 9.2.1 Prism tests The test setup is shown in Fig . 9.3. A piece of rubber of 15 mm thickness was inserted at each contact surface (except on the top and bottom surface where the impact load was applied). Then, for easy comparison with the specimens tested with steel platens, the same confining pressure of 0.625 M P a was also applied on the four sides. The hammer was dropped from a 250 mm height, providing an impact energy of 1417 J and an impact velocity o f 2.21 m/s. a) Failure pattern The failure patterns of plain concrete for hard and soft platens were similar. The typical mode was a tensile splitting or columnar failure (Fig. 9.4a-b) combined with significant spalling at the top and the bottom. The type o f platen did not alter the failure pattern. However, the specimens tested with the rubber platens seemed to undergo more damage than those tested with the steel platens, as evidenced by the larger crack size. 216 This may be due to the lower stiffness of the rubber, which allowed the specimen to expand more under loading. Also because of its lower elastic modulus, the rubber flowed outward and induced a lateral tensile stress, which was then added to the lateral tensile stress due to the vertical load (Poisson's effect) and caused the splitting tensile failure. For the F R C , the pattern was generally that of a diagonal shear crack, as shown in Fig . 9.4c. Fig . 9.3 Test Setup for a Confined Prism with Rubber Platens (Top View) (a) (b) (c) Fig . 9.4 Tensile Failure of Confined Plain Concrete with (a) Steel and (b) Rubber Platen, and (c) Diagonal Shear Failure of Confined 1% H E F R C with Rubber Platen 217 b) Load-deflection response and peak load The typical load-deflection responses of confined plain concrete and F R C prisms tested using steel and rubber platens are given in Fig. 9.5. Impact Testing from 250 mm 2.00 4.00 6.00 Deformation (mm) 8.00 10.00 Fig . 9.5 Typical Load-Deflection Responses of 0.625 M P a Confined Prisms with Steel and Rubber Platen The effect o f platen type was quite substantial. The response of the confined plain and F R C prisms with rubber platens were different from those confined with steel platens: 1) they had a much lower peak load; 2) they had a larger deformation at peak load; 3) they had a more "flexible" response (lower elastic modulus); and 4) they had a longer post-peak response. The use of rubber platens allowed the specimens to expand laterally under load, while the steel platens did not allow such movement. With rubber, the larger lateral deformation under the same applied load produced concrete with a more flexible response. Thus, the confinement effect of the rubber platens was less effective 218 than that of the steel platens. This led to considerable strength reductions, as shown in Table 9.1. Table 9.1 Peak Load of 0.625MPa Confined Concrete Prisms with Steel and Rubber Platens Concrete Type Peak load (kN) Steel platens Rubber platens % Decrease Plain concrete 530 362 32% 0.5%HE 580 437 25% 1.0%HE 679 519 24% A s stated in Chapter 5, the confined prisms typically failed in a tensile splitting mode, and the failure of the confined prisms appeared to be controlled by their lateral strain. If the lateral strain due to the Poisson effect becomes larger than some critical tensile strain, then failure occurs. In general, the applied confining stress primarily induces a lateral compressive strain in the specimen. However, the state of stress in the contact area is more complex than it appears because of the boundary conditions. With steel platens, a large amount of frictions creates regions of triaxial compressive stress with contracting (negative) strains at the contact surfaces (Fig. 9.6). On the other hand, the outward flow of the rubber platens creates regions of compression-tension-tension stress, which induce expanding (positive) strains at the contact surfaces. These different platens thus have opposite effects under uniaxial compression. The negative strain induced by the steel platens w i l l reduce the lateral strain due to vertical compression load, and make concrete apparently stronger. On the other hand, the positive strain induced by the rubber platens w i l l be added to the lateral strain and make concrete apparently weaker. 219 Deformed shape- Steel platens (hard) ///// w i f 2) / / / Confining pressure -Region of triaxial compression Deformed shape Rubber platens (Soft) Confining pressure A / / /A h 'A K y t k A6 f y\ / \ ^ — R e g i o n of triaxial tension-tension-compression Fig. 9.6 Effect of Loading Platens on the State of Stress at the Contact Surfaces (Top View) The effect of the rubber platens on the lateral expansion of concrete can be illustrated by the delay on the load-time response of the confined concrete as shown in Fig . 9.7. The increase in load for the specimen confined with rubber platens was slower than that in contact with steel platens. Because of the lower elastic modulus of rubber, it took more time for the rubber to deform and stiffen sufficiently to transmit load effectively. 220 Fig. 9.7 Load-Time Response of Confined Plain Concrete with Steel and Rubber Platen 9.2.2 Beam tests The setup for the confined beam test is shown in Fig. 9.8. The beam was placed on supports with a 300mm clear span, and then a piece of 25mm thick rubber was inserted at each end, at the contact area between steel and concrete. For easy comparison with the steel platen beams, a confining pressure of 2.5 M P a and hammer drop height of 150 mm were applied (providing an impact energy of 850 J and a velocity of 1.71 m/s). 221 Top view Side view Fig. 9.8 Test Setup for Confined Beam with Rubber End Platens a) Fai lure patterns The failure patterns o f the concrete beams confined with steel or rubber platens are shown in Fig. 9.9. The effect of platen type on the failure patterns of the beams was more marked than for the prisms. For the confined plain concrete beam tested with steel platens, mixed mode failure occurred (combined shear and flexural failure), as indicated by the diagonal cracks at both ends and the vertical crack in the middle (Fig. 9.9b). It is possible that the use o f steel platens, due to their high friction and stiffness, induce a high degree of partial end fixity at both ends of the beam. This partially fixed effect does two things: it stiffens parts of the beam at both ends and, it permits very little end rotation. A s shown schematically in Fig. 9.9a, the steel platen may be modeled as several rigid short columns distributed over the contact surface. These rigid columns do not allow end rotation and the stiffened (end) portions of the beam cannot move as freely as the centre part, which leads to failure in the shear mode. However, due to the low tensile strength o f plain concrete, a vertical crack is still observed. This vertical crack disappears when the tensile strength of the concrete is enhanced by fibre additions (Fig. 9.9c). 222 Applied load I 1 Support JL Steel (hard) platen Stiffening zone Confining stress (b) (c) Fig. 9.9 (a) Schematic Illustrate the Formation of Shear Crack in Steel Platen Confined Beam Compared with Actual Failure (b) Plain Concrete, and (c) 0 .5%HE F R C With the rubber platens, the failures of both plain concrete and F R C were essentially a combination of flexural and compressive failure, as indicated by the vertical crack at the middle and the crushing of the concrete at the top (Figs. 9.10b and c). Wi th the lower elastic modulus of rubber and lower friction restraint at contact surfaces, the degree o f partial end fixity is smaller, allowing the concrete beams to rotate more at the ends, and to bend more under load, sufficient to cause failure primarily in flexure. With less stiffening and more end rotation, the beam fails more in flexure (Fig. 9.10a). The compressive crushing at the top surface of the concrete is perhaps due to, from the confinement pressure that held the cracked concrete together after the peak load. The 223 compressive stress due to the confinement, combined with that due to the applied bending load, was large enough to lead to localized crushing of the concrete. Applied load A •w-Support Rubber (Soft) platen Confining stress Compressive jnlii | | f l imil | | | i l l l l l l l l l l l l l l | 2 5 >VA 1 •1 / i I l l l l l l l l l l l l l l l l (b) (c) Fig. 9.10 (a) Schematic Illustrate the Formation of Flexural Crack and Compressive Crushing in Rubber Platen Confined Beam Compared with Actual Failure (b) Plain Concrete, and (c) 0 .5%HE F R C b) Accelerat ion distribution The existence of the stiffening zone at both ends of the beam can be verified by the measurement of the acceleration distribution along the beam. B y putting two accelerometers on the beam, at the centre and the quarter point of the clear span, a rough 224 estimate of the acceleration distribution can be determined. For a beam confined with steel platens, due to the stiffening effect, the end portions of the beam cannot move freely; failure is essentially in a shear mode, and the measured accelerations, which are about the same represent the acceleration of the broken piece (Fig. 9.11). According to this, the acceleration (in the direction of loading) can be assumed uniform across the broken piece. 800 700 Accelerometer position Fig. 9.11 Acceleration Distribution of Steel Platen Confined 0 .5%HE F R C Beam For the rubber platens confined beam, due to a decreased amount of stiffening, the ends of the beam were able to rotate more than in the previous case, and the failure was more in a flexural mode. The acceleration (Fig. 9.12) was much higher at the centre than at the quarter point (difference of about 30%). The acceleration distribution is more sinusoidal or linear in shape. 600 r Accelerometers Accelerometer position Fig. 9.12 Acceleration Distribution of Rubber Platens Confined 0 .5%HE F R C Beam 225 c) Load-deflection response and peak load The typical load-deflection responses of beams confined with steel and rubber platens are given in Fig . 9.13. The differences in the load-deflection responses were similar to those of the confined prisms: a decrease in peak load, an increase in deflection at peak, and a more "flexible" response when using rubber platens along with a considerable delay in the load-deflection responses. Using a rubber platen increases the length of time before the beam begins to pick up load, as seen in the tup load vs. time (Fig. 9.14) and confinement force vs. time curves. The rate of loading in the beams tested with rubber platens was lower than those tested with steel platens. The peak loads of confined beams with both steel and rubber platens are given in Table 9.2. The use o f a rubber pad reduces the efficiency of confinement (in terms of strength) by about 10% to 22%, though it provides a more flexible response to the beam. 200 Impact Testing from 150mm — 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 Displacement (mm) Fig. 9.13 Typical Load-Deflection Response of 2.5 M P a Confined Beam with Steel and Rubber Platens 226 120 -o 80 40 Impact Testing from 150mm 0 5000 10000 15000 20000 Time (usee) — Plain-Steel platen —Plain-Rubber platen} Fig. 9.14 Effect of Platen Type on Load-Time Response of Confined Beams Table 9.2 Peak Load o f 2.5 M P a Confined Concrete Beams with Steel and Rubber Platens Concrete type Peak load (kN) Steel platens Rubber platens % Decrease P la in concrete 144 129 10% 0 . 5 % H E 165 135 18% 1 . 0 % H E 185 145 22% 9.2.3 Plate tests The test setup for plates is shown in Fig. 9.15. A piece of rubber with dimensions of 400*75*15 mm was inserted on each contact surface between the specimen and the steel plate. The specimen was placed on the 300*300 mm support, the biaxial confining stress of 2.5 M P a was applied and then the hammer was dropped from a 250 mm height, providing an impact energy of 1417 J and a velocity of 2.21 m/s. 227 a) Fai lure patterns Similar to the beam tests, the use of a soft rubber platen changed the mode of failure of both the confined plain concrete and the F R C from shear (or mixed mode) to flexural failure except that the failure was in two dimensions for the plate specimens. With steel platens, the confinement stress and the high friction created a stiffening frame surround the plate which did not allow plate to move freely or rotate at the edges. The failure was essentially a shear failure (Fig. 9.16a-c). With rubber platens, the degree of stiffening was smaller and the edge fixity was reduced. Thus, the plate was able to bend more during loading. The failure was more in the flexural mode (Figs. 9.16b-d) Fig . 9.15 Test Setup for Confined Plates with Rubber Platens (Top View) (a) (b) (c) (d) Fig. 9.16 Failure Pattern of 2.5 M P a Confined Plain Concrete Plate with (a) Steel Platen and (b) Rubber Platen, and 0.5%HE F R C Plate with (c) Steel Platen and (d) Rubber Platen* * crack pattern was highlighted by black lines 228 b) Load-deflection response and peak load The load-deflection responses of the confined plates tested with different platens are shown in Fig. 9.17. The responses of the rubber platen confined plates were similar to those of the beams: they had lower peak loads and larger peak deflections, and their response was more flexible. The use of soft a platen reduced the confinement effect (in terms of strength) by about 8% to 28% (Table 9.3). 2.00 4.00 6.00 8.00 10.00 12.00 14.00 Deflection (mm) Fig. 9.17 Typical Load-Deflection Responses of Confined Plates with Steel and Rubber Platens Table 9.3 Peak Load of 2.5 M P a Confined Concrete Plates with Steel and Rubber Platens Concrete type Peak load (kN) Steel platens Rubber platens % Decrease P la in concrete 240 220 8% 0 . 5 % H E 280 202 28% 1 . 0 % H E 271 198 27% 229 c) Confining force vs time response The change in confinement force with time was measured during the impact loading. To attempt to maintain the confinement effect in the specimen during the test, the load cells and hydraulic jacks were designed to lock themselves in position both prior to and after the impact event. During the impact, the prisms tried to expand, while the plates and beams tried to bend; these reactions created a change (mostly an increase) in confinement load with time. In all three specimen types, the decrease in confinement and slower load response of concrete confined with rubber platens would be demonstrated by a comparison of the confinement force vs time curves with different platens (Fig. 9.18). From Fig. 9.18, we can see that in all three types of tests, the specimens confined with rubber platens experienced both a delayed and a lower confinement load change during the impact event. The delay in confinement load change was due the low stiffness of rubber, which required a longer time to transmit the applied load. The smaller change of confinement force also reduced the measured strength of the rubber platen confined concrete. 9.3 Effect of the Loading Head (Tup) Geometry on Confined Plates A s stated earlier, the failure mode of the confined plates, which was dominated by a punching shear failure, might have been affected by the loading head (tup) geometry itself. B y using a small diameter tup (compared to the specimen size), it is possible that the impact head might have created a "punching through" type of failure. In order to investigate this possibility, a series of tests was carried out using two different tup diameters: 100mm and 150mm (ratios of 1/4 and 3/8 of the specimen clear span). The hammer was dropped from 250mm. 230 —Steel plate [ — Rubber plate 2000 4000 6000 time (uSec) (a) 8000 10000 160 — Steel plate — Rubber plate 5000 10000 15000 Time (mSec) (b) 20000 25000 300 — Steel Plate Rubber Plate 0 2000 4000 6000 8000 10000 12000 14000 Time (uSec) (c) Fig. 9.18 Typical Response of Confining Force with Time of Confined (a) Prisms, (b) Beams, and (c) Plate Tested with Steel and Rubber Platens 231 9.3.1 Fa i lure patterns The failure pattern of the confined concrete was found to be affected by the diameter of the tup, though this depended in part on the specimen type and confining stress, as shown in Figs. 9.19-20. For brittle plain concrete (Fig. 9.19), the tup diameter did not change the mode of failure much at 0 M P a confinement; the plate still failed by a mixed mode of punching shear and flexure. However, at a confinement stress of 2.5 M P a , the mode of failure began to change to flexural failure. In the case of F R C (Fig. 9.20), a similar change in failure mode was observed as well . (a) (b) Fig. 9.19 Failure Patterns of Confined Plain Concrete Plates with 150mm Diameter Tup: (a) M i x e d Mode Failure at 0 M P a and (b) Flexural Failure at 2.5 M P a (a) (b) Fig. 9.20 Failure Patterns of Confined 0.5%HE F R C Plates with 150mm Diameter Tup: (a) M i x e d Mode Failure at 0 MPa* and (b) M i x e d Mode Failure at 2.5 M P a * crack pattern was highlighted with black lines 232 Thus, the tup diameter played a very important role in the failure pattern of confined concrete; the 150mm tup led to totally opposite effects from those found with the smaller diameter tup. With the 100 mm diameter tup, the failure of the confined concrete was dominated by shear as the degree of confinement increased; starting from mixed mode failure (shear-flexure) at a low level of confinement, the failure pattern changed to punching shear at the highest level of confinement. However, with the 150 mm diameter tup, the failure mode headed in the opposite direction. A s the level of confinement increased, the failure, which started from mixed mode at low confinement levels, remained essentially unchanged (though in some cases it became more "flexural"). These different changes in failure pattern are probably due to the decrease in localized damage with increasing tup diameter. With the 150mm tup, a larger volume of concrete was activated by the applied load. The reduction in localized damage can be shown by examining the acceleration and time responses (Fig. 9.21). When using the smaller diameter tup, the load was concentrated at the center of the plate; a smaller volume of concrete at the contact area was under high stress and high acceleration, and the acceleration distribution was close to that of a point acceleration (or linear); hence, a higher mid-plate acceleration was achieved. With the larger diameter tup, the applied load was more widely distributed, as was the acceleration. A larger volume of concrete was under stress, and a lower mid-plate acceleration was achieved. Both scenarios are illustrated schematically in Fig. 9.22. 233 6000 5000 $ 4000 i c £ 3000 2 5 £ 2000 1000 100 Impact Testing from 250mm 100mm Tup 150mm Tup 200 300 Time (uSec) 400 500 Fig . 9.21 Mid-Plate Acceleration Response of Concrete Plate Tested with 100 and 150 mm-Diameter Tup 100mm-Dia Impactor Target i!l!l!lTl'l;. 150mm-Dia Impactor Target > A -<-r-n Mid-plate acceleration .,11 Mid-plate acceleration Acceleration Distribution Acceleration Distribution Fig . 9.22 Schematic Illustration of the Mid-Plate Acceleration Distribution of Plate Tested with 100 and 150mm Diameter Tup 234 9.3.2 Load-deflection response and peak load The typical load deflection curves of the 2.5MPa confined concrete plates are given in Fig . 9.23. The larger diameter tup increased the confinement effect as seen by the increase in peak load (Table 9.4) and toughness of the confined plates. When a larger volume of concrete was involved, the localized damage decreased, and this also provided the concrete with greater load bearing capacity. The increased confining effect can be seen by the increasing confinement force during the impact event, as shown by the confinement force-time response (Fig. 9.24). The increase in confinement force is perhaps due to the behaviour of the specimen during the impact event. With the smaller diameter tup, the applied load was essentially concentrated at the contact area, and the specimen deflected very little prior to failure (also due to the end restraints that prevent rotation). When a larger diameter was used, the applied load and the damage were more widely distributed, and the specimen deflected more than those tested with the smaller diameter tup. A s the specimen bends more, the reaction at the end restraints become higher; hence, it provides a slightly better confinement effect. Table 9.4 Peak Loads of 2.5 M P a Confined Plates Tested with 100mm and 150mm Dia. Tup Concrete Peak load (kN) type Tup diameter 100mm 150mm Plain 239 319 0.5% HE FRC 279 437 235 500 400 g- 300 ro O 200 100 0.5%HE-100mm Tup Plain concrete-150mm Tup Plain concrete-100mm Tup 5.00 10.00 Deflection (mm) ——r—— 15.00 20.00 Fig. 9.23 Typical Load-Deflection Response of Confined Concrete Plates Tested with 100 and 150mm Tups 350 0 2000 4000 6000 8000 10000 12000 14000 Time (uSec) Fig. 9.24 Typical Confinement Force Response of Plain Concrete Plates Tested with 100mm and 150mm Tups 236 CHAPTER 10 DEVELOPMENT OF PRE-PEAK STRESS-STRAIN RELATIONSHIP USING SCALAR DAMAGE MECHANICS 10.1 Introduction In this chapter, the basic theory of scalar continuum damage mechanics is reviewed, and used to predict the pre-peak behavior of unconfined plain concrete and F R C under both static and impact loading conditions. The modeling begins with the determination of a damage-strain (D- s). relationship for loaded unconfined concrete based on the actual damage. The actual damage for the static case is measured directly from the test results using the procedure described in section 10.1.5, and is expressed in terms of the degradation of the initial E . A comparison between the proposed model and other models is also made. The results indicate that the response obtained from the model proposed here agrees well with the actual response. In addition, an alternative method of measuring damage, based on the absorbed energy, is proposed; these results are then compared with those using the first method. In the case of impact loading, a new set of stress-strain relationships is proposed, again based on scalar damage mechanics. However, in the case of impact loading, the variation in strain rate with time is used as an indicator of the damage. To develop the pre-peak stress-strain relationship fully, the model obtained is then extended to describe the pre-peak tensile behavior of both plain concrete and F R C . However, since direct tension tests were not carried out in this study, the values of tensile strength, strain at peak and elastic modulus are assumed, based on values obtained from the literature. 10.2 Kachanov Concept Q48) Different materials deform differently when loaded, depending on factors such as their atomic structure, composition, rate of loading and temperature. To understand the deformation characteristics of each individual material completely would require an extensive knowledge of its atomic and molecular structure. But, in practice, the general 237 constitutive equations of the continuum model relate materials and.their deformations without considering the atomic structure of the real material. Variables such as stress, strain and elastic modulus are used. Under applied loads, the material structure eventually begins to disintegrate and the load carrying capacity is reduced. The state of deterioration of a material was characterized by Kachanov (148) using a dimensionless, scalar variable denoted as damage (D). Lemaitre (149) has stated that the damage of a material is the progressive physical process by which its breaks. There are three stages in the fracture of a material. First, at the micro-scale, damage occurs due to the accumulation of localized microcracks and the breaking of bonds. Then, the growth and coalescence of microcracks can initiate a single crack at the meso-scale. Finally, at the macro-scale, the growth of that crack is considered. The first two stages may be studied by means of the damage of the continuous medium, whereas the third stage is usually studied using fracture mechanics. A l l materials, regardless of their composition and structure, show elastic behavior, yielding, plastic or pseudo-plastic strain, damage by monotonic loading or fatigue, and crack growth under static or dynamic loading. This suggests that the properties common to all materials can somehow be explained using the same theory. This is the main reason why it is possible to study material behavior at the meso-scale using the mechanics of continuous media, which can explain the behavior of the material without considering in detail the complexity of the microstructure. 10.2.1 Macromechanical or one dimensional damage model (150) Consider a damaged body (an apple) and a Representative Volume Element ( R V E (2)) at a point M oriented in a plane defined by its normal h and its abscissa x along the direction « (F ig . 10.1). If SA represents the cross-sectional area of the R V E and SA(I represent the area of microvoids or microcracks that lie in SA, the value o f damage, D ( M , « ) , is: SA D(M,n) = - ^ (10.1) 238 From the above expression, it is clear that the value of the scalar variable D is bounded by 0 and 1: 0 < D < 1 (10.2) where D = 0 corresponds to the intact or undamaged R V E D = 1 corresponds to the completely fractured R V E In the case o f a simple one-dimensional homogeneous material, eq. 10.1 is simplified to: A P P L E R V E Fig. 10.1 A Damaged Body and R V E , after Lemaitre (149) 10.2.2 Effective stress concept (151) One application of continuum damage mechanics is the effective stress concept. For an R V E with a cross-sectional area of A and loaded by a force F, the uniaxial stress ( c r ) i s : a=^ (10.4) If the cracked area on the representative surface of the R V E is Ad, then the effective cross-sectional area becomes A - A Q and the effective stress (a e f f) is: 239 <y„„ =• "f A-Ad (10.5) Substituting the damage variable (D) from 10.3 into 10.5 yields, ^ r a o r °-'=^E (10-6> 10.2.3 Strain equivalence principle The same concept can also be applied to strain. According to Lemaitre (152), the constitutive strain equation for a damaged material may be derived in the same way as for a virgin material, except that the normal stress is replaced by the effective stress. (10.7) 10.2.4 Relationship between strain and damage Combining Eq . 10.7 with Hooke's law, the relationship between strain (se) and damage can be derived as follow: ee= (10.8) 6 E(l-D) where E is the elastic modulus of the undamaged material. The elastic modulus of the damaged material or the effective elastic modulus (Eeff) is then defined as: Eeff=E(\-D) (10.9) The degradation of E (Fig. 10.2) as expressed by Eeff in Eq. 10.9 is the key ingredient to predicting the behavior or response of concrete under load. 240 10.2.5 Damage measurement Damage measurements can be made in by several ways: direct measurements, variations of the elastic modulus (E), energy based methods, variations o f the microhardness, variations of the density, and so on. If the stress-strain curve is known (from tests), methods such as the variation of E and energy based measurements seem to be most practical. In this study, the variation of the elastic modulus (as in Fig . 10.2) is used to determine the damage variable (D). Energy based measurements are described in Appendix C as an alternative method. The variation of the elastic modulus is an indirect measurement based on the influence of damage on E as expressed by Eq . 10.9. Provided that the E of the undamaged material is known, the damage value can be derived from the measurement of Eeff. 0.40 0.20 0.000 0.000 0.001 0.001 0.001 Strain Fig. 10.2 Variation of E with Damage of Plain Concrete under Static Compression 241 10.3 Predicting the Pre-Peak Response of Concrete under Static Loading Using Scalar Damage Mechanics (SDM) 10.3.1 Pre-peak compressive response a) Proposed model This section describes the use of C D M to predict the response of concrete under static compressive loading. For a brittle material such as concrete, a modification o f equation 10.10 is necessary to make it more suitable. In ductile materials, E e f f is basically the change of E with respect to strain (or damage) after the loading goes beyond the linear stage and enters the plastic region (Fig. 10.3a). However, for plain concrete, the response is different; after reaching peak load, failure occurs abruptly (Fig. 10.3b). Therefore, to use C D M in concrete, the undamaged E is replaced by the initial E (Ej n t), while for the damaged E (Eeff), the secant E is used instead. Eq . 10.10 is then rewritten as: The damage curve obtained from the test results can be assumed to have the sec int (10.11) following form: D = A\n(s)+B (10.12) Stress Stress Strain Strain (a) (b) Fig . 10.3 Measurement of E for (a) Ductile Material and (b) Brittle Material 242 Applying the following conditions: at D = 0, s= £0 and at D = D u i t , £ = ec, yields: 0 = Aln(£0) + B (10.13) Duk=A\n{sc)+B (10.14) From Eq . 10.13, B = -Aln(e0) (10.15) Substituting Eq. 10.15 into Eq. 10.14, and solving for A : A = , P* v (10.16) Substituting A in to Eq. 10.15, one obtains B equal to: B= TD"" Ane0 (10.17) where D is the damage £ 0 is the assumed initial strain (>0) D uit is the damage at peak load £ c is the strain at peak load D u | t is the damage value at the peak and can be obtained directly from the test results. In general, two methods of damage measurement are proposed: 1) Elastic modulus degradation (as stated above) and 2) Energy based method (Appendix A ) . For the elastic modulus degradation, D u i t is equal to l-Ec/Em where E c is the secant modulus at peak load and E j n t is the initial modulus. Substituting into Eqs. 10.16 and 10.17, A and B can be rewritten as: 243 5 = In s. (10.19) To obtain A and B , the secant modulus at peak load (E c ) , the initial modulus (Ejnt), the strength (fc) and the strain at peak (s c) must be known. These values are quite easy to obtain as they are well documented in the literature or can be obtained directly from experiments. After obtaining the corresponding damage (D) for any strain, the stress can then be determined as: The predicted damage and pre-peak responses of plain concrete, 0 .5%HE and 1.0%HE F R C are shown in Figs. 10.4 and 10.5, respectively. The following values were assumed (based on actual experimental results): For plain concrete: E i n t = 35.4 GPa, f'c = 41.6 M P a and = 0.0036 For 0 .5%HE F R C : E i n t = 44.8 GPa, f'c = 41.3 M P a and e' = 0.0025 For 1.0%HE F R C : E i n t = 38.8 GPa, f'c = 43.8 M P a and e' = 0.0039 From Fig . 10.4, it may be seen that the damage of plain concrete can be represented by a bi-linear curve, as it increases linearly with strain from 0.0 up to 0.7 when it begins to slow down and then forms another straight line. For the F R C , the damage vs. strain curve is more like a non-linear curve, as it gradually increases from zero to the peak. The proposed model exhibits better damage prediction in F R C than in plain concrete. Even though the proposed model gave a slightly lower value in the middle of the loading event, the overall prediction was in fair agreement with the actual response of both concretes. cr = EinX(\-D)e (10.20) 244 i i - -r 1 1 0.002 0.004 - 0.002 0.004 - 0.002 0.004 (a) (b) (c) I Actual damage Predicted damage Fig. 10.4 Actual and Predicted Damage of (a) Plain Concrete, (b) 0 .5%HE F R C , and (c) 1.0%HE F R C Prisms under Static Compressive Loading 0.002 0.004 - 0.002 0.004 - 0.002 0.004 (a) (b) (c) Actual response Predicted response Fig . 10.5 Actual and Predicted Response of (a) Plain Concrete, (b) 0 .5%HE F R C , and (c) 1.0%HE F R C Prisms under Static Compressive Loading 245 b) Compar ison wi th other models Two other scalar damage models for predicting the compressive response of plain concrete under static loading have been proposed, by Loland (153) and Mazars (154). Here, Loland's model is compared with the present model. According to Loland, the effective E is given as: Eeff =E(l-D0 -A0s A ) 10.21 a Is where Constant, A = • Strain at peak, sc = f i - A > V 4,(A + 1) AE(\-Dn) Strength, ac = ^ °-±ec A +1 Do = assumed initial damage value From measured values of peak stress (c r c ) and strain (sc), and the initial elastic modulus (Eo), the constants A and Ao can be obtained. Then, the stress can be obtained by substituting Eq . 10.13 into Hooke's law. B y assuming D 0 = 0.001, Loland's model and the proposed model may be compared as shown in Fig. 10.6. It may be seen that both models predicted the behavior of concrete reasonably well . However, Loland's model seems to slightly over-predict the response in the middle o f the loading event by 8% to 10%(Fig. 10.7), while the proposed model slightly under-predict the response. 246 50 7 45 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 Strain Actual response - - Purposed Model - - Model by Loland | Fig. 10.6 Response of Plain Concrete-Comparison of Proposed Model and Loland's Model 1.5 1.4 1.3 0.7 0.6 0.5 -I 1 1 i 1 ' 1 ' 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Strain - Actual * - Proposed Model - - Loland Model Fig. 10.7 Relative Stress Comparison between the Two Models 247 10.4 Predicting the Pre-Peak Response of Concrete under Impact Loading Using Scalar Damage Mechanics (SDM) 10.4.1 Pre-peak impact compressive response a) Scalar damage mechanics: strain rate approach In this section, C D M is used to predict the impact compressive response of concrete. The damage parameters A and B used to predict the static response are inadequate, because they do not include the strain rate parameter that becomes important during the impact event. Therefore, a new set of stress-strain models using C D M was proposed to make use of the variation in strain rate during the impact event. For the drop-weight impact test, the increase in strain up to the peak load was assumed to be linear (constant strain rate) in order to simplify the analysis. However, the actual test results indicated that the strain rate was, in fact, not constant (Fig. 10.8). The strain rate was slower at the beginning of the impact event, and then began to increase with the accumulation of damage (2). A s a result, the damage of the material subjected to time-dependent loading can be expressed as (Fig. 10.9): D = \ - int (10.22) where s-int is the minimum or initial strain rate (sec ) s(t) is the strain rate at any time t (sec ) N is a temperature-dependent constant (assume equal to 1) The stress can then be obtained using Eq . impact elastic modulus (E?nt) a = Efnt(\-D)s 10.20, with E i m replaced by the initial (10.23) 0.0060 0.0050 0.0040 250mm drop height 500mm drop height 2 0.0030 55 0.0020 0.0010 500 1000 1500 2000 time (usee) 2500 3000 Fig. 10.8 Change of Strain Rate with Time for Plain Concrete under Impact Compressive Loading 8 t T Fig . 10.9 Schematic Illustration of the Initial and the Ultimate Strain Rate, and the Strain Rate at A n y Time t 249 b) Proposed model According to Fig. 10.8 and 10.9, the general form of the strain-time relationship can be assumed to be a polynomial function: £ = ATi +BT2 +CT + D 10.24 Applying the following conditions: (1) T = 0-> e{t) = 0, (2) T = tu-> s(t) = e'c, (3) T = 0—>• s(t) = eint (f), and (4) T = f„-» £(f) = eu (r), where t u is the time at the peak load and su (t) is the strain rate at the peak load or time t u, yields: A = KK +£JJU -2e'c ^ eu +4, ~ 2 g a v g g _ 3g c - _ 2 f j n t / u _ 3 £ o v g - f a - 2 i i n t r„2 7/ C = e,, and D = 0 where eavg is the average strain rate over the entire impact event equal to — For a plain concrete prism tested under impact loading from a 250 mm drop height (impact energy 1417 J), the initial impact elastic modulus was 125 GPa, the initial strain rate (sint) was 0.6 sec"1, the ultimate strain rate was 2.61 sec"1, the duration of impact up to the peak load (tu) was 0.00214 sec, and ultimate strain (ec) was 0.00257. The predicted response is shown in Fig. 10.10 below. 250 70 j 0.001 0.002 0.003 0.004 0.005 Strain Fig. 10.10 Actual and Predicted Response of Plain Concrete under Impact Compressive Loading at 250mm Drop Height (Impact Energy of 1417 J) For impact loading from a 500mm drop height (impact energy of 2835J), the initial impact elastic modulus was 150 GPa, the initial strain rate (eM) was 1.53 sec"1, ultimate strain rate (su) was 10.16 sec"1, time to peak (tu) was 0.000824 sec, and ultimate strain (sc) was 0.00344. The predicted response, compared to the actual response, is shown is Fig . 10.11. It was found that, for both drop heights, the predicted responses of plain concrete subjected to impact loading exhibited less of a linear portion and a slightly higher peak load than the actual response (actual strength of 63.5 k N and predicted strength of 65.5 kN). The lack of agreement in the shape of the curve result from the micro-mechanisms of failure that are more complex than can be described by a simple scalar damage mechanics model. 251 80 0.001 0.002 0.003 0.004 0.005 Strain Fig. 10.11 Actual and Predicted Response of Plain Concrete under Impact Compressive Loading at 500mm Drop Height (Impact Energy of 2835 J) 10.4.2 Impact strength prediction When the ultimate strength is the only concern, it can be determined using C D M together with the relationship between impact/static strength derived from the test results (Chapter 5). The impact/static strength ratio of concrete is related to the strain rate by the following expression. g c ( f a P ) = e W ) ( 1 0 . 24 ) ac(st) where <Jc. (imp) = impact strength (MPa) crc (st) = static strength (MPa) <j(imp) . . , . — = impact/static strength ratio ac(st) s = strain rate (1/sec) k = material constant (0.17-0.26) 252 From the test results, the strain rate of plain concrete was in the range of 1.38 to 2.96 sec' 1 for hammer drop heights of 250 and 500mm. The material constant, k, was found to depend on the type of material. The suggested k-values obtained from this study are: k = 0.26 for plain concrete, and k = 0.23-0.26 for 0 .5%FRC, depending on the type of fibre. Replacing stress (a) with strength (crc) in Eq. 10.20, and then substituting it into 10.22, yields the impact strength of the material under high strain rate. <j'c(impact) = ek(i)E.M (1 - Dull )sc (10.25) Predicted strengths obtained from 10.23 is plotted against the actual strengths in Fig. 10.12. It may be seen that the strengths obtained from Eq. 10.23 are quite close to the actual strengths obtained. Q . 5 O c 1.00 2.00 3.00 Strain rate (1/sec) o Actual strength (250mm) • Predicted strength (250mm) A Actual strength (500mm) A Predicted strength (500mm) 4.00 Fig . 10.12 Predicted Impact Strength of Unconfined Plain Concrete 253 CHAPTER 11 PREDICTION OF THE IMPACT RESPONSE OF CONFINED CONCRETE 11.1 Introduction In this chapter, a model to describe the compressive and tensile response of confined concrete under impact loading is constructed. Under confined stress, it was found that the strain rate of concrete, for the same impact energy, increased with the degree of confinement. To include this phenomenon, the relationship between the initial, ultimate, and average strain rates and the confining stress is determined, then applied to the model developed in Chapter 10 to predict the full response. 11.2 Determination of the Stress-Strain Response 11.2.1 Proposed compressive model The experimental results indicate that the strain rate effect for concrete increases with the confinement stress (Chapter 5). This is shown also in Fig. 11.1; the relationship between strain and time. B y using curve fitting, the increase in the relative strain rate with the confinement stress can be expressed by the following empirical formulations (Fig 11.2): Initial strain rate, 4, (conf) (11.1) Average strain rate, 4. (unconf) £mMonf) (11.2) e (unconf) Ultimate strain rate, 4/i (conf) (11.3) sull (unconf) where eiat (unconf) is the initial strain rate of unconfined concrete (sec") s^iconf) is the initial strain rate of confined concrete (sec") 254 £avg (unconf) is the average strain rate of unconfined concrete (sec"1) e (conf) is the average strain rate of confined concrete (sec"1) £uh (unconf) is the ultimate strain rate of unconfined concrete (sec"1) £ull (conf) is the ultimate strain rate of confined concrete (sec"1) <jC(mf is the confining stress (MPa) 11.2.2 Comparison with the actual response According to Eqs. 11.1-11.3, i f the strain rates of unconfined concrete are known, then the strain rates of confined concrete can be calculated. Based on the actual test results, the strain rates of unconfined plain concrete tested under impact loading with an impact energy of 1417J are: Unconfined concrete: eim = 0.63 sec"1, £ a v g = 1.02 sec"1 and eult = 2.61 sec'1 Using Eqs. 12.1-12.3, the strain rates of confined concrete are obtained as follow: For 0.625MPa confinement: sint = 0.90 sec"1, e = 1.76 sec"1 and sull = 4.16 sec"1 For 1.25MPaconfinement: eint= 1.18 sec"1, £avg-2A9 sec"1 and sull= 5.71 sec"1 Using the model developed in Chapter 10, and assuming that the initial dynamic elastic modulus of concrete confined at 0.625 M P a and 1.25 M P a are 67 GPa and 51 GPa, respectively (15% larger than E s(0.4 Ult), see Chapter 5 for values of E s(0.4 Ult)). The predicted responses are then shown in Fig. 11.3 and 11.4. The predicted responses at both levels of confinement are quite consistent with the actual responses. Both have slightly higher predicted peak loads than the actual peak loads. The value of strength is influenced by the initial elastic modulus assumed in the model. In this study, the initial elastic modulus is generally assumed to be about 15% larger than the elastic secant modulus calculated up to 40% of the peak load E s (0.4Ult). The secant modulus was found to decrease slightly with increasing confinement as was shown in Chapter 5. However, the non-linear part is still not predicted very well . 255 0.00800 0.00700 0.00600 0.00500 c 0.00400 'ro (0.00100) Unconfinement 0.625 MPa confinement •1.25MPa confinement 500 1000 1500 2000 Time (uSec) 2500 3000 3500 Fig . 11.1 Relationship Between Strain and Time of Unconfined and Confined Concrete Subjected to Impact Compressive Loading (250mm) 3.00 re 2.50 E 2.00 •a o o c •6 CU c c o O 1.50 1.00 0.50 -Initial strain rate -Average strain rate Ultimate strain rate 0.20 0.40 0.60 0.80 1.00 Confining stress (MPa) 1.20 1.40 Fig . 11.2 Relationship Between Relative Strain Rates and Confining Stress (250mm) 256 ro Q-s in m k_ CO —Actual response •"-Predicted response 0.002 0.004 0.006 0.008 0.010 Strain Fig . 11.3 Actual vs Predicted Response of 0.625 M P a Confined Concrete —Actual response —- Predicted response 0.002 0.004 0.006 0.008 0.010 Strain Fig. 11.4 Actual vs Predicted Response of 1.25 M P a Confined Concrete 257 C H A P T E R 12 C O N C L U S I O N S The purpose of this study was to try to understand the impact behaviour of concrete under multiaxial loading, for which very little research has been carried out up to the present. In general, the unconfined impact behavior of plain and fibre reinforced concretes (FRC) loaded either in compression or flexure are quite similar; increases in peak load, peak deflection, and energy absorption are found, though to different degrees. Comparing the three types of loading, uniaxial compression (prism) is the least rate sensitive, followed by 2D flexure (plate) and I D flexure (beam). In most cases, the rate sensitivity of F R C is less than that of plain concrete. The typical response of plain concrete under impact loading is not changed much from its static response (except for the magnitude); a "single peak" response was found in plain concrete tested under compression, I D and 2D bending. However, in the case of F R C , due to the effect of fibre bridging, the impact response is quite different from its static response, especially in the case of beams and plates. In F R C beams, the bend over point (BOP) (usually separated from the peak load point in the high fibre content F R C tested under static loading) is clearly eliminated under impact loading. For the F R C plates, a multiple peak response is generally found. Under confining stress, the impact behavior of concrete changes quite dramatically. In the case of prisms (compression), the mode of failure changes from a shear or shear cone type to a columnar or splitting type. A n increase in strength and peak strain with increasing confining stress is generally found. On the other hand, the elastic modulus of confined concrete is almost unchanged or slightly smaller. In terms of stress rate sensitivity, the confined concrete is more stress rate sensitive, as indicated by the smaller value of n. It is worth mentioning here that the increase in apparent strength is partly due to the machine phenomena and it is often difficult to separate the machine phenomena from the material phenomena. To be able to do this, a much more sophisticated confinement machine that can interact with the specimen (in order to keep the applied confinement 258 force constant) is required. However, under impact loading, since the loading event is extremely short, it would be very difficult to design a machine with sufficiently fast response time. In the case of beams and plates, the increased confining stress gradually changes the mode o f failure from flexure to shear failure. Confined beams and plates are stronger and tougher. The inertial load, on the other hand, becomes smaller due to the stiffening effect. Opposite to the prisms, confined beams and plates exhibit less stress rate sensitivity (as indicated by the larger value of n) due to the change of failure mode from flexure to shear (shear crack velocity is slower than flexural crack velocity). For F R C , under unconfined conditions, H E and F E fibres seem to perform better than C P fibres. In the case of confined concrete, the performances of the different F R C s cannot be clearly distinguished from each other due to the dominant effect o f the confinement apparatus. The type of loading platen also affects the impact behavior of confined concrete significantly. The smaller elastic modulus of rubber decreases the confinement effect (in terms of strength) by about 20-30%. It has less friction and also permits more end rotation, which leads to a larger deformation (for prisms) or deflection at peak load (for beams and plates). In the case of beams (and plates), the more flexible end rotation allows concrete to fail in tension and compression instead of shear. For the confined plate, the geometry of the tup (i.e., diameter) also plays an important role. With a larger diameter tup, the failure pattern of confined plates is no longer dominated by shear; instead, mixed shear and flexure are generally found. The larger diameter also minimized the localized failure and as a result, a higher peak load but a less mid-plate acceleration were found. Scalar damage mechanics (SDM) is a promising approach to predict the pre-peak response of concrete under both static and impact loading. There are several ways to approach the damage of loaded concrete. Under static loading, when the actual test result is available, the degradation of elastic modulus seems to be the best approach. The strain rate variation is more appropriate in the case of impact loading. For both under static and impact loading, the S D M still works fairly well in predicting the confined compressive response. 259 Where do we go from here? It was not the objective of this study to develop specific empirical formulae that could be used in general. However, it would be useful to be able to come up with some good relationships in the near future. Clearly, there are several factors that affect the confinement test, one of the most important being the boundary effect. Differences in loading patterns could lead to quite different conclusions. If we are going to try to better understand this problem, further tests with other types of loading platens are required to determine their effect. Platens such as polypropylene, Teflon, lubricated pads, or brush type pads are recommended. A n extensive study of confined prisms is also recommended to better understand their behaviour under impact compression for further development of the principle stress-confinement stress envelope and some empirical formulae. For instance, there should be further study under uniaxial confinement (of prisms) with various types of loading platens, as well under a wider range of impact energies. The results obtained from the case of end confined beams were very interesting, in which the typical failure pattern was a mixed mode between shear and flexure. With the increasing degree of confinement, the mixed mode failure was gradually eliminated and dominated by a shear mode of failure. It is well known that Mode III [or shear fracture] of concrete is one of the most difficult topics in the study of concrete fracture. It is also well known that confinement is one of the essential parts in the study of shear in concrete. However, a complete and clear relationship between the degree of confinement and Mode III fracture is not yet known. Evidently, based on the results obtained from this study, i f one were able to supply sufficient confinement to prevent end rotation, shear failure would be achieved. Therefore, it would be very interesting to continue this study to try to understand the Mode III failure of concrete and to determine the relationship between the confinement degree and the changing type of failure [from flexure to shear]. Modification of the confinement machine is also required. Even though the confinement machine is capable of holding the specimen without buckling itself during the impact, its high rigidity, which barely allows the specimen to expand or rotate, creates 260 quite a fluctuation (in all case, increases in confinement were found) in confinement stress. In more sophisticated confinement machines (static), the interaction between the specimen and the loading platen is measured and controlled to ensure a constant stress during the test. However, under impact loading, the loading event is extremely short, and it is impossible at this moment to measure and control the confinement stress. In the future, when the technology is available, a modification might be possible. 261 APPENDIX A GENERAL SPECIFICATIONS Al . Strain gauge for the tup load cell A x i a l Poisson Gage factor at 75°F 2.09 +/- 1% 1.95 +/-1% Transverse sensitivity +0.44% 1.85%) Resistance in Ohms at 75°F 350 +/- 0.15% Fatigue life: 1 M i l l i o n cycles at +/- 1500 Micron Inch/ Inch Temperature range: -325 to 400°F Strain limit: +1-2% A2. The dynamic performance specifications of the accelerometer Voltage sensitivity: 1.0 m V / g [mV/(ms"2)] Measurement Range (for +/- 5V output) 5000 +/-g pk [49,050 ms"2] Frequency Range (+/-ldB) 0.4 to 7500 H z Mounted Resonant Frequency >50 k H z Broadband Resolution (1 H z to 10 kHz) 0.02 g rms [0.20 ms"2] Amplitude linearity +/- 1 %> Transverse Sensitivity < 5%> A2. The standard specification for the AROMAT laser sensor Standard distance 40mm ± 1mm Measurable range 30 to 50mm Light source Laser diode, wage length 750nm Pulse duration 25 fisec (50%) duty ratio) Maximum output 4mW (peak radiant power) Laser protection class Class III b g output: Output voltage ± 5V / ± 10mm Output impedance 50 Q Resolution 2.5um Linearity error ± 30um plus ± 1.0% Response time 25ms (90% response) 263 APPENDIX B COMPRESSIVE STRENGTH In this study, for most batches of concrete, three concrete cylinders were cast and tested approximately at or after 28 days. Results are given in Table B I below. Table B I . Compressive Strength of Concrete Batch No. Date cast Concrete type Specimen Average Strength MPa SD #1 #2 #3 lb MPa lb MPa lb MPa 1 08-Jan-98 0.5%CP 87,500 49.6 92,500 52.4 76,000 43.1 48.4 4.8 2 14-Jan-98 0.5%HE 103,500 58.6 99,000 56.1 100,500 56.9 57.2 1.3 3 16-Jan-98 Plain 89,500 50.7 92,500 52.4 - 51.6 1.2 4 19-Jan-98 0.5%HE 85,000 48.2 83,500 47.3 82,000 46.5 47.3 0.8 5 23-Jan-98 0.5%FE 75,000 42.5 76,500 43.3 72,500 41.1 42.3 1.1 6 28-Feb-98 0.5%FE 76,500 43.3 77,500 43.9 72,000 40.8 42.7 1.7 7 21-Apr-98 1%HE 104,500 59.2 110,500 62.6 94,500 53.5 58.5 4.6 8 22-Apr-98 1%CP 94,500 53.5 75,400 42.7 89,500 50.7 49.0 5.6 9 25-Apr-98 1%FE 90,000 51.0 86,000 48.7 82,500 46.7 48.8 2.1 10 20-May-98 Plain 74,500 42.2 82,500 46.7 77,500 43.9 44.3 2.3 11 09-Jun-98 Plain 85,500 48.4 98,000 55.5 92,000 52.1 52.0 3.5 12 23-Jun-98 0.5%HE 80,000 45.3 75,000 42.5 74,500 42.2 43.3 1.7 13 07-Jul-98 0.5%FE 84,500 47.9 82,500 46.7 95,500 54.1 49.6 4.0 14 16-Jul-98 0.5%CP 91,000 51.6 73,000 41.4 74,000 41.9 45.0 5.7 15 17-Aug-98 1%HE 84,500 47.9 83,000 47.0 95,500 54.1 49.7 3.9 16 18-Aug-98 1%FE 90,500 51.3 100,500 56.9 81,000 45.9 51.4 5.5 17 20-Aug-98 1%CP 91,000 51.6 87,500 49.6 97,500 55.2 52.1 2.9 18 21-Aug-98 0.5%HE 68,500 38.8 71,500 40.5 76,500 43.3 40.9 2.3 19 27-Sep-98 Plain 75,500 42.8 68,500 38.8 71,500 40.5 40.7 2.0 20 02-Oct-98 0.5%HE 65,000 36.8 62,500 35.4 80,000 45.3 39.2 5.4 21 03-Oct-98 0.5%FE 74,500 42.2 71,500 40.5 79,500 45.0 42.6 2.3 22 10-Oct-98 0.5%CP 61,500 34.8 83,500 47.3 75,500 42.8 41.6 6.3 23 17-Oct-98 1%HE 100,000 56.7 96,500 54.7 83,500 47.3 52.9 4.9 24 19-Oct-98 Plain 68,500 38.8 75,000 42.5 92,500 52.4 44.6 7.0 25 24-Oct-98 0.5%HE 91,000 51.6 87,000 49.3 82,000 46.5 49.1 2.6 26 12-Nov-98 Plain 71,000 40.2 74,500 42.2 77,000 43.6 42.0 1.7 27 13-N0V-98 0.5%HE 85,500 48.4 79,000 44.8 85,500 48.4 47.2 2.1 28 17-Nov-98 1%HE 99,000 56.1 92,500 52.4 89,000 50.4 53.0 2.9 29 17-Nov-98 Plain 78,000 44.2 76,500 43.3 83,500 47.3 45.0 2.1 30 18-NOV-98 0.5%HE 72,000 40.8 85,000 48.2 79,000 44.8 44.6 3.7 31 19-Nov-98 1%HE 95,000 53.8 100,500 56.9 88,000 49.9 53.5 3.5 264 Batch No. Date cast Concrete type Specimen Average Strength MPa SD #1 .. #2 #3 lb MPa lb MPa lb MPa 32 20-Nov-98 0.5%CP 91,500 51.8 75,500 42.8 85,500 48.4 47.7 4.6 33 20-Nov-98 Plain 75,000 42.5 81,000 45.9 54,000 30.6 39.7 8.0 34 21-NOV-98 0.5%FE 91,000 51.6 85,000 48.2 72,000 40.8 46.8 5.5 35 21-Nov-98 0.5%HE 72,500 41.1 85,000 48.2 70,000 39.7 43.0 4.6 36 22-Nov-98 0.5%CP 97,000 55.0 96,500 54.7 91,500 51.8 53.8 1.7 37 23-Nov-98 0.5%FE 81,000 45.9 90,500 51.3 87,500 49.6 48.9 2.8 38 28-Nov-98 Plain 80,500 45.6 85,000 48.2 77,500 43.9 45.9 2.1 39 01-Dec-98 Plain 66,000 37.4 71,000 40.2 65,000 36.8 38.2 1.8 40 02-Dec-98 0.5%HE 88,000 49.9 92,000 52.1 81,000 45.9 49.3 3.2 41 03-Dec-98 0.5%FE 70,000 39.7 80,500 45.6 69,000 39.1 41.5 3.6 42 04-Dec-98 0.5%CP 72,500 41.1 85,000 48.2 - 44.6 5.0 43 05-Dec-98 0.5%HE 91,000 51.6 73,000 41.4 82,500 46.7 46.6 5.1 44 07-Dec-98 0.5%HE 87,000 49.3 84,500 47.9 - 48.6 1.0 45 08-Dec-98 0.5%CP 72,500 41.1 70,000 39.7 67,500 38.2 39.7 1.4 46 15-Dec-98 0.5%CP 81,500 46.2 79,000 44.8 87,500 49.6 46.8 2.5 47 19-Dec-98 0.5%FE 100,500 56.9 80,500 45.6 81,500 46.2 49.6 6.4 48 11-Dec-98 0.5%CP 77,000 43.6 95,000 53.8 78,500 44.5 47.3 5.7 49 02-Sep-99 Plain 73,500 41.6 78,500 44.5 69,500 39.4 41.8 2.6 50 03-Sep-99 0.5%HE 71,000 40.2 75,000 42.5 77,500 43.9 42.2 1.9 51 04-Sep-99 0.5%FE 86,500 49.0 79,500 45.0 80,500 45.6 46.6 2.1 52 06-Sep-99 0.5%CP 87,500 49.6 89,000 50.4 81,500 46.2 48.7 2.2 53 07-Sep-99 Plain 75,000 42.5 77,500 43.9 69,000 39.1 41.8 2.5 54 09-Sep-99 0.5%HE 81,500 46.2 69,000 39.1 85,000 48.2 44.5 4.8 55 10-Sep-99 1%HE 90,500 51.3 103,500 58.6 - 55.0 5.2 56 13-Sep-99 0.5%FE 90,000 51.0 84,500 47.9 86,500 49.0 49.3 1.6 57 14-Sep-99 0.5%CP 69,000 39.1 77,000 43.6 80,500 45.6 42.8 3.3 58 15-Sep-99 Plain 71,000 40.2 70,500 39.9 72,500 41.1 40.4 0.6 59 16-Sep-99 0.5%HE 73,500 41.6 72,000 40.8 81,000 45.9 42.8 2.7 60 01-Oct-99 0.5%FE 94,500 53.5 90,000 51.0 - 52.3 1.8 61 02-Oct-99 0.5%CP 61,000 34.6 72,500 41.1 73,500 41.6 39.1 3.9 62 03-Oct-99 1%CP 92,500 52.4 90,500 51.3 95,000 53.8 52.5 1.3 63 04-Oct-99 1%FE 94,000 53.3 92,500 52.4 90,500 51.3 52.3 1.0 64 18-Oct-99 1%HE 98,500 55.8 94,000 53.3 97,000 55.0 54.7 1.3 65 19-Oct-99 Plain 75,000 42.5 71,000 40.2 93,500 53.0 45.2 6.8 66 20-Oct-99 0.5%HE 73,500 41.6 76,500 43.3 72,000 40.8 41.9 1.3 67 21-Oct-99 0.5%FE 75,500 42.8 72,500 41.1 70,500 39.9 41.3 1.4 68 02-NOV-99 0.5%CP 69,000 39.1 75,500 42.8 79,000 44.8 42.2 2.9 69 03-Nov-99 Plain 71,500 40.5 70,500 39.9 73,000 41.4 40.6 0.7 70 04-Nov-99 0.5%HE 76,500 43.3 71,000 40.2 76,000 43.1 42.2 1.7 71 05-Nov-99 0.5%FE 72,000 40.8 68,000 38.5 73,000 41.4 40.2 1.5 72 09-NOV-99 0.5%FE 85,000 48.2 81,500 46.2 76,000 43.1 45.8 2.6 73 IO-Nov-99 Plain 72,500 41.1 73,000 41.4 75,500 42.8 41.7 0.9 74 11-Nov-99 0.5%HE 71,000 40.2 74,000 41.9 76,500 43.3 41.8 1.6 265 Batch No. Date cast Concrete type Specimen Average Strength MPa SD #1 #2 #3 lb MPa lb MPa lb MPa 75 06-Dec-99 Plain 71,000 40.2 76,000 43.1 69,000 39.1 40.8 2.0 76 07-Dec-99 0.5%HE 72,500 41.1 74,500 42.2 76,000 43.1 42.1 1.0 77 08-Dec-99 0.5%FE 74,000 41.9 76,500 43.3 70,000 39.7 41.6 1.9 78 03-Feb-00 0.5%CP 68,500 38.8 73,500 41.6 - 40.2 2.0 79 04-Feb-00 1%HE 91,000 51.6 96,000 54.4 85,000 48.2 51.4 3.1 80 06-Feb-00 1%FE 89,000 50.4 79,500 45.0 91,500 51.8 49.1 3.6 81 08-Feb-00 1%CP 89,500 50.7 82,500 46.7 - 48.7 2.8 82 09-Feb-00 Plain 74,500 42.2 70,500 39.9 - 41.1 1.6 83 10-Feb-00 Plain 71,500 40.5 74,500 42.2 76,500 43.3 42.0 1.4 84 15-Feb-00 0.5%HE 79,000 44.8 71,500 40.5 70,000 39.7 41.6 2.7 85 16-Feb-00 0.5%HE 76,500 43.3 77,000 43.6 70,500 39.9 42.3 2.0 266 APPENDIX C ENERGY BASED DAMAGE MEASUREMENT A n energy-based measurement of damage was proposed by Najar (8) as an alternative way of measuring damage in brittle materials. The damage parameter D is represented by the ratio o f the strain energy deviation (AW£) to the strain energy o f the perfect or undamaged specimen (Wparf ). Consider an elastic body of volume V subjected to tensile strain e. The specific strain energy is given as: W=\cjs.de C l For a material in an undamaged state, the elastic modulus is characterized as an initial elastic modulus (Ej n t); with a linear increase of stress and strain, the specific strain energy of a perfect body is equal to: 1 9 W = — F F C2 r y perf 2 i n t However, for a damaged material, the elastic modulus is characterized by the secant modulus (E s ) , so the specific strain energy of the damaged body is equal to: WD = \os.ds =2~Ese2 C3 The strain energy deviation is defined as the difference between the specific strain energies of perfect and damaged bodies. = Wperf - WD C4 Then, the damage becomes: D = AW£/WperJ C5 The energy based damage measurement is illustrated schematically in F ig . C l . 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