Constitutive relationship of plain concrete under uniaxial load

CONSTITUTIVE RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD LIU DING NATIONAL UNIVERSITY OF SINGAPORE 2005 CONSTITUTIVE RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD LIU DING (M.Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 ACKNOWLEDGEMENT The author wishes to express his most sincere appreciation and deep gratitude to his academic supervisors, Assoc. Professor W.A.M. Alwis and Assoc. Professor Quek Ser Tong for their invaluable guidance, patient encouragement and comments throughout the course of this study. In addition, special appreciation to National University of Singapore for the support granted for this period of study. Finally, but never the least, extremely thanks are presented to his family members for their endless love and understanding. TABLE OF CONTENTS ACKNOWLEDGEMENT ………………………………………………………………………. i TABLE OF CONTENT …………………………………………………………………………. ii LIST OF TABLES……………………………………………………………………………… v LIST OF FIGURES…………………………………………………………………………… viii NOMENCLATURE…………………………………………………………………………… . xvii SUMMARY……………………………………………………………………………………… xx CHAPTER INTRODUCTION . 1.1. 1.2. General Literature Review 1.2.1. 1.2.1.1. 1.2.1.2. 1.2.1.3. 1.2.1.4. 1.2.1.5. 1.2.1.6. 1.2.2. Constitutive Modeling on Deformational Behaviour of Concrete Introduction .4 Viscoelastic Models .5 Elastic / Viscoplastic Models Endochronic Models Damage Models Discrete-Element Method (DEM) .10 Empirical Relationships .10 1.3. Aims and Objectives 13 1.3.1. 1.3.2. Scope of Work .13 Thesis Layout 14 CHAPTER LOADING RATE SENSITIVE BEHAVOUR OF CONCRETE AND CREEP MODELS . 20 2.1. 2.2. General 20 Comparison Between Creep and Dynamic Response of Concrete . 20 2.2.1. 2.2.2. 2.2.2.1. 2.2.2.2. Phenomenological Similarities 20 Similarities in Mechanism .22 The Microcracking Theory 22 Similarities .23 Table of Content iii 2.3. 2.4. 2.5. 2.6. 2.7. Creep Models . 24 B3 Model . 30 CEB 1990 Model . 33 The Sakata Model 34 Elastic Deformation . 35 2.7.1. 2.7.2. Initial Modulus 35 Determination of Asymptotic Modulus .37 CHAPTER APPLICATION OF CREEP MODELS FOR RAPID AXIAL COMPRESSION 42 3.1. 3.2. General 42 Creep Strain Due To Varying Stress History 43 3.2.1. 3.2.2. 3.2.3. 3.2.4. Effective Modulus Method (EM Method) .43 Method of Superposition .44 Rate of Creep Method (RC Method) .45 Rate of Creep Compliance Method (For the B3 Model Only) 47 3.3. Rapid Axial Compression Modelling 49 3.3.1. 3.3.2. 3.3.3. Computational Model 49 Time-Marching Calculation Algorithm .49 Convergence of the Simulation Results .53 CHAPTER SIMULATION AND DISCUSSION . 59 4.1. 4.2. General 59 Variation of Secant Modulus . 59 4.2.1. 4.2.2. 4.2.3. 4.2.4. 4.2.5. 4.2.6. General 59 Stress-Time Paths 60 Characteristic Variation of Secant Modulus 61 Effects of Strains .61 Effect of Stress Path 62 Discussion .62 4.3. Rapid Axial Compression Experiments 63 4.3.1. 4.3.1.1. 4.3.1.2. Experimental Data .63 Deformational Characteristics .64 Strength Characteristics .64 4.4. Nonlinearity Of Stress-Strain Curves 65 4.4.1. 4.4.2. 4.4.3. Nonlinearity of Concrete Creep .65 Nonlinear Amplification of Creep Strain 66 Material Constatant λ .68 4.5. Simulation Results . 70 4.5.1. 4.5.2. 4.5.3. Performance of Creep Models .71 Performance of Strategies for Creep Calculation 72 Performance of λ Expression .73 Table of Content iv 4.5.4. CHAPTER Comparison with CEB Emprical Formula .73 APPLICATION OF CREEP MODELS FOR CYCLIC LOADING . 99 5.1. 5.2. General 99 Simulation On Cyclic Loading Test 99 5.2.1. 5.2.2. 5.2.3. Variation of Stress and Strain 99 Tentative Analysis .100 Simulations On Experiments .100 5.3. Discussions 101 5.3.1. 5.3.2. Creep Model Performances .101 Creep Calculation Strategies Performances 103 CHAPTER CONCLUSION AND RECOMMENDATION . 113 6.1. 6.2. 6.3. General 113 Conclusions . 113 Recommendations . 114 REFERENCES . 116 PUBLICATION . 130 APPENDIX 131 LIST OF TABLES Table 3.1 Convergence of Simulation Results – B3 Model, Rate of Creep 55 Compliance Method Table 3.2 Convergence of Simulation Results – CEB Model, RC Method 55 Table 3.3 Convergence of Simulation Results –Sakata Model, RC Method 55 Table 4.1 Labels of Loading Cases For Secant Modulus Study 75 Table 4.2 Figures Numbers for Different Loading Paths For Secant Modulus Study 75 Table 4.3 Mix Details and Basic Information for Bischoff & Perry Test 75 Table 4.4 Test Results for Bischoff & Perry Test 76 Table 4.5 Mix Details and Basic Information for Ahmad Test 77 Table 4.6 Test Results for Ahmad Test 77 Table 4.7 Summary of The Experiments Selected For Simulation 78 Table 4.8 Specimen Parameters 79 Table 4.9 Best Fit Material Parameter λ 80 Table 4.10 Coefficients for Expression of λ 81 Table 4.11 Results of Simulation – Logarithmic λ Expression, B3 Model, Rate of 82 Creep Compliance Method Table 4.12 Results of Simulation – Logarithmic λ Expression, B3 Model, Rate of 83 Creep Method Table 4.13 Results of Simulation – Logarithmic λ Expression, B3 Model, 84 Superposition Method Table 4.14 CEB Empirical Formula Results and Creep Model Simulation (B3Model, Rate of Creep Compliance Method) 85 List of Table vi Table 4.15 CEB Empirical Formula Results and Creep Model Simulation (CEB Model, Rate of Creep Method) 86 Table 4.16 CEB Empirical Formula Results and Creep Model Simulation (Sakata Model, Superposition Rate of Creep Method) 87 Table 5.1 Summary of the Repeated Loading Tests 170 Tables listed in Appendix App-1 Results of Simulation – Logarithmic λ Expression, CEB Model, Rate of 133 Creep Method App-2 Results of Simulation – Logarithmic λ Expression, CEB Model, 134 Superposition Method App-3 Results of Simulation – Logarithmic λ Expression, Sakata Model, Rate of 135 Creep Method App-4 Results of Simulation – Logarithmic λ Expression, Sakata Model, 136 Superposition Method App-5 Results of Simulation – Linear λ Expression, B3 Model, Rate of Creep 137 Compliance Method App-6 Results of Simulation – Linear λ Expression, B3 Model, Rate of Creep 138 Method App-7 Results of Simulation – Linear λ Expression, B3 Model, Superposition 139 Method App-8 Results of Simulation – Linear λ Expression, CEB Model, Rate of Creep 140 Method App-9 Results of Simulation – Linear λ Expression, CEB Model, Superposition Method 141 List of Table App-10 vii Results of Simulation – Linear λ Expression, Sakata Model, Rate of Creep 142 Method App-11 Results of Simulation – Linear λ Expression, Sakata Model, Superposition 143 Method App-12 Results of Simulation – λ =5, B3 Model, Rate of Creep Compliance 144 Method App-13 Results of Simulation – λ =5, B3 Model, Rate of Creep Method 145 App-14 Results of Simulation – λ =5, B3 Model, Superposition Method 146 App-15 Results of Simulation – λ =7, CEB Model, Rate of Creep Method 147 App-16 Results of Simulation – λ =7, CEB Model, Superposition Method 148 App-17 Results of Simulation – λ =6, Sakata Model, Rate of Creep Method 149 App-18 Results of Simulation – λ =6, Sakata Model, Superposition Method 150 LIST OF FIGURES Figure 1.1 Magnitude of Stain Rates with Respect to Different Loading Cases 15 Figure 1.2 Typical Stress-Strain Relationship of Concrete under Different Loading 15 Rates Figure 1.3 Basic Mechanical Constitutive Model Units 16 Figure 1.4 Maxwell Model 16 Figure 1.5 Deformation Response of Maxwell Model 17 Figure 1.6 Kelvin Model 17 Figure 1.7 Deformation Response of Kelvin Model 18 Figure 1.8 Burgers Model 18 Figure 1.9 Deformation Response of Burgers Model 19 Figure 1.10 Typical Stress-Strain Relation For Concrete 19 Figure 2.1 Strain Development of Concrete under Creep Phenomena 39 Figure 2.2 Components of Concrete Deformation at Creep 39 Figure 2.3 Components of Concrete Deformation Under Rapid Load 40 Figure 2.4 General form of Strain-time Curve for Concrete Subjected to Creep 40 Figure 2.5 Initial Modulus of Creep Models 41 Figure 2.6 Flow Chart For Determination of Asymptotic Elastic Modulus 41 Figure 3.1 Stress History Effect On Effective Modulus Method 56 Figure 3.2 Scheme for Superposition Method 57 Figure 3.3 Displacement of Elements 58 . Appendix . 13.000 12.000 Values Computed From Tests 11.000 Logarithmic Regression Curve 10.000 9.000 8.000 7.000 6.000 y = 2.2912Ln(x) - 11.661 R2 = 0.002 5.000 4.000 3.000 2.000 1.000 0.000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 Density (kg/m3) Figure App-B13 Relation Between λ and Density for Sakata Model (Logarithmic) 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 6.000 5.000 4.000 3.000 y = -0.0858x + 8.651 R2 = 0.5665 2.000 1.000 0.000 10 20 30 40 50 60 Compressive Strength f c (MPa) Figure App-B14 Relation Between λ and Compressive Strength for B3 Model (Linear) - 182 - 70 80 . Appendix . 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 6.000 5.000 4.000 3.000 y = -0.0054x + 7.3553 R2 = 0.4309 2.000 1.000 0.000 200 300 400 500 600 700 800 900 1000 1100 Cement Content c (kg/m ) Figure App-B15 Relation Between λ and Cement Content for B3 Model (Linear) 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 6.000 5.000 4.000 3.000 y = -0.0089x + 6.8576 R2 = 0.1219 2.000 1.000 0.000 100 150 200 250 300 350 400 Water Content w (kg/m ) Figure App-B16 Relation Between λ and Water Content for B3 Model (Linear) - 183 - 450 . Appendix . 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 y = -0.489Ln(x) + 8.3029 R2 = 0.0046 6.000 5.000 4.000 3.000 2.000 1.000 0.000 300 500 700 900 1100 1300 1500 Sand Content s (kg/m ) Figure App-B17 Relation Between λ and Sand Content for B3 Model (Linear) 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 y = -0.0025x + 10.879 R2 = 0.0189 6.000 5.000 c 4.000 3.000 2.000 1.000 0.000 2050 2100 2150 2200 2250 2300 2350 2400 2450 Density (kg/m ) Figure App-B18 Relation Between λ and Density for B3 Model (Linear) - 184 - 2500 2550 . Appendix . 14.000 13.000 Values Computed from Tests 12.000 Lineaer Regression Curve 11.000 10.000 9.000 8.000 7.000 6.000 5.000 4.000 y = -0.1264x + 12.264 R2 = 0.5547 3.000 2.000 1.000 0.000 10 20 30 40 50 60 70 80 Compressive Strength f c (MPa) Figure App-B19 Relation Between λ and Compressive Strength for CEB Model (Linear) 10.000 9.000 Values Computed from Test 8.000 Lineaer Regression Curve 7.000 6.000 5.000 4.000 y = -0.0072x + 9.9472 R2 = 0.402 3.000 2.000 1.000 0.000 200 300 400 500 600 700 800 Cement Content c (kg/m ) Figure App-B20 Relation Between λ and Cement Content for CEB Model (Linear) - 185 - 900 1000 . Appendix . 10.000 9.000 Values Computed from Tests 8.000 Lineaer Regression Curve 7.000 6.000 5.000 y = -0.0092x + 8.6614 R2 = 0.066 4.000 3.000 2.000 1.000 0.000 100 150 200 250 300 350 400 450 Water Content w (kg/m ) Figure App-B21 Relation Between λ and Water Content for CEB Model (Linear) 14.000 13.000 Values Computed from Tests 12.000 Lineaer Regression Curve 11.000 10.000 9.000 8.000 y = -0.001x + 7.5912 R2 = 0.007 7.000 6.000 5.000 4.000 3.000 2.000 1.000 0.000 300 500 700 900 1100 1300 Sand Content s (kg/m ) Figure App-B22 Relation Between λ and Sand Content for CEB Model (Linear) - 186 - 1500 . Appendix . 14.000 13.000 Values Computed from Tests 12.000 Lineaer Regression Curve 11.000 10.000 9.000 8.000 y = -0.0029x + 13.328 R2 = 0.0128 7.000 6.000 5.000 4.000 3.000 2.000 1.000 0.000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 Density (kg/m3) Figure App-B23 Relation Between λ and Density for CEB Model (Linear) 13.000 12.000 Values Computed From Tests 11.000 Lineaer Regression Curve 10.000 9.000 8.000 7.000 6.000 5.000 4.000 3.000 y = -0.0893x + 9.8885 R2 = 0.2792 2.000 1.000 0.000 10 20 30 40 50 60 70 Compressive Strength f c (MPa) Figure App-B24 Relation Between λ and Compressive Strength for Sakata Model (Linear) - 187 - 80 . Appendix . 13.000 12.000 Values Computed from Tests 11.000 Lineaer Regression Curve 10.000 9.000 8.000 7.000 6.000 5.000 4.000 y = -0.009x + 9.8966 R2 = 0.5284 3.000 2.000 1.000 0.000 200 300 400 500 600 700 800 900 1000 Cement Content c (kg/m ) Figure App-B25 Relation Between λ and Cement Content for Sakata Model (Linear) 13.000 12.000 Values Computed from Tests 11.000 Lineaer Regression Curve 10.000 9.000 8.000 7.000 6.000 5.000 4.000 3.000 y = -0.0151x + 9.2436 R = 0.2145 2.000 1.000 0.000 100 150 200 250 300 350 Water Content w (kg/m ) Figure App-B26 Relation Between λ and Water Content for Sakata Model (Linear) - 188 - 400 450 . Appendix . 13.000 12.000 Values Computed From Tests 11.000 Lineaer Regression Curve 10.000 9.000 8.000 7.000 6.000 5.000 4.000 3.000 y = -0.0052x + 10.763 R2 = 0.2574 2.000 1.000 0.000 300 500 700 900 1100 1300 1500 Sand Content s (kg/m ) Figure App-B27 Relation Between λ and Sand Content for Sakata Model (Linear) 13.000 12.000 Values Computed From Tests 11.000 Lineaer Regression Curve 10.000 9.000 8.000 7.000 6.000 y = 0.0011x + 3.5333 R2 = 0.0025 5.000 4.000 3.000 2.000 1.000 0.000 2050 2100 2150 2200 2250 2300 2350 2400 2450 Density (kg/m3) Figure App-B28 Relation Between λ and Density for Sakata Model (Linear) - 189 - 2500 2550 . Appendix . 75 70 Low Loading Rate Computed 65 High Loading Rate 60 Lower Loading Rate 55 Experiment High Loading Rate Stress (MPa) 50 45 40 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C1 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of Creep Method, Test 75 70 Low Loading Rate Computed 65 High Loading Rate 60 Lower Loading Rate 55 Experiment High Loading Rate Stress (MPa) 50 45 40 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C2 Results of Simulation - Logarithmic λ Expression, Sakata Model, Superposition Method, Test - 190 - 0.0035 . Appendix . 50 45 40 Stress (MPa) 35 30 25 Low Loading Rate Mid Loading Rate 20 Computed High Loading Rate 15 Lower Loading Rate 10 Mid Loading Rate Experiment High Loading Rate 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C3 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of Creep Compliance Method, Test 60 Low Loading Rate 55 Mid Loading Rate 50 Computed High Loading Rate 45 Lower Loading Rate Stress (MPa) 40 Mid Loading Rate 35 Experiment High Loading Rate 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C4 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of Creep Method, Test - 191 - 0.0035 . Appendix . 55 Low Loading Rate 50 Mid Loading Rate Computed 45 High Loading Rate Stress (MPa) 40 Lower Loading Rate 35 Mid Loading Rate 30 High Loading Rate Experiment 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C5 Results of Simulation - Logarithmic λ Expression, B3 Model, Superposition Method, Test 55 Low Loading Rate 50 Mid Loading Rate Computed 45 High Loading Rate Stress (MPa) 40 Lower Loading Rate 35 Mid Loading Rate 30 High Loading Rate Experiment 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C6 Results of Simulation - Logarithmic λ Expression, CEB Model, Rate of Creep Method, Test - 192 - 0.0035 . Appendix . 55 Low Loading Rate 50 Mid Loading Rate Computed 45 High Loading Rate Stress (MPa) 40 Lower Loading Rate 35 Mid Loading Rate 30 High Loading Rate Experiment 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C7 Results of Simulation - Logarithmic λ Expression, CEB Model, Superposition Method, Test 80 75 70 Low Loading Rate 65 Mid Loading Rate 60 Computed High Loading Rate Stress (MPa) 55 Lower Loading Rate 50 45 Mid Loading Rate 40 High Loading Rate Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C8 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of Creep Method, Test - 193 - 0.0035 . Appendix . 80 75 70 Low Loading Rate 65 Mid Loading Rate 60 Computed High Loading Rate Stress (MPa) 55 Lower Loading Rate 50 45 Mid Loading Rate 40 High Loading Rate Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C9 Results of Simulation - Logarithmic λ Expression, Sakata Model, Superposition Method, Test 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C10 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of Creep Compliance Method, Test 17 - 194 - 0.0035 . Appendix . 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C11 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of Creep Method, Test 17 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C12 Results of Simulation - Logarithmic λ Expression, B3 Model, Superposition Method, Test 17 - 195 - 0.0035 . Appendix . 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain Figure App-C13 Results of Simulation - Logarithmic λ Expression, CEB Model, Rate of Creep Method, Test 17 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 Strain Figure App-C14 Results of Simulation - Logarithmic λ Expression, CEB Model, Superposition Method, Test 17 - 196 - 0.003 . Appendix . 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Strain Figure App-C15 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of Creep Method, Test 17 60 55 Low Loading Rate 50 High Loading Rate 45 Lower Loading Rate 40 High Loading Rate Computed Stress (MPa) Experiment 35 30 25 20 15 10 0 0.0005 0.001 0.0015 0.002 0.0025 Strain Figure App-C16 Results of Simulation - Logarithmic λ Expression, Sakata Model, Superposition Method, Test 17 - 197 - 0.003 [...]... Mix B under Loading Case LM-A 155 App-A10 Graph for Mix B under Loading Case LM-B 155 App-A11 Graph for Mix C under Loading Case L0-A 156 App-A12 Graph for Mix C under Loading Case L0-B 156 List of Figures xii App-A13 Graph for Mix C under Loading Case L1-A 157 App-A14 Graph for Mix C under Loading Case L1-B 157 App-A15 Graph for Mix C under Loading Case S0-A 158 App-A16 Graph for Mix C under Loading... App-A1 Graph for Mix B under Loading Case S0-A 151 App-A2 Graph for Mix B under Loading Case S0-B 151 App-A3 Graph for Mix B under Loading Case S1-A 152 App-A4 Graph for Mix B under Loading Case S1-B 152 App-A5 Graph for Mix B under Loading Case IS0-A 153 App-A6 Graph for Mix B under Loading Case IS0-B 153 App-A7 Graph for Mix B under Loading Case IS1-A 154 App-A8 Graph for Mix B under Loading Case IS1-B... Graph for Mix C under Loading Case S1-A 159 App-A18 Graph for Mix C under Loading Case S1-B 159 App-A19 Graph for Mix C under Loading Case IS0-A 160 App-A20 Graph for Mix C under Loading Case IS0-B 160 App-A21 Graph for Mix C under Loading Case IS1-A 161 App-A22 Graph for Mix C under Loading Case IS1-B 161 App-A23 Graph for Mix C under Loading Case LM-A 162 App-A24 Graph for Mix C under Loading Case LM-B... Graph for Mix II under Loading Case L0-A 163 App-A26 Graph for Mix II under Loading Case L0-B 163 App-A27 Graph for Mix II under Loading Case L1-A 164 App-A28 Graph for Mix II under Loading Case L1-B 164 App-A29 Graph for Mix II under Loading Case S0-A 165 App-A30 Graph for Mix II under Loading Case S0-B 165 App-A31 Graph for Mix B under Loading Case S1-A 166 App-A32 Graph for Mix II under Loading Case... II under Loading Case IS0-A 167 App-A34 Graph for Mix II under Loading Case IS0-B 167 App-A35 Graph for Mix II under Loading Case IS1-A 168 App-A36 Graph for Mix II under Loading Case IS1-B 168 App-A37 Graph for Mix II under Loading Case LM-A 169 List of Figures xiii App-A38 Graph for Mix II under Loading Case LM-B 169 App-A39 Graph for Mix IV under Loading Case L0-A 170 App-A40 Graph for Mix IV under. .. for Mix IV under Loading Case L0-B 170 App-A41 Graph for Mix IV under Loading Case L1-A 171 App-A42 Graph for Mix IV under Loading Case L1-B 171 App-A43 Graph for Mix IV under Loading Case S0-A 172 App-A44 Graph for Mix IV under Loading Case S0-B 172 App-A45 Graph for Mix IV under Loading Case S1-A 173 App-A46 Graph for Mix IV under Loading Case S1-B 173 App-A47 Graph for Mix IV under Loading Case IS0-A... prediction of long-term deformation Potential exploitation of similarities between the dynamic and creep behaviour of concrete is the focus of the present work An attempt is made here to extend the application of creep theories to modelling of concrete behaviour under rapid uniaxial loading Three creep models, namely CEB model, B3 model and Sakata Model, were chosen for simulating the behaviour of concrete under. .. response of concrete under uniaxial rapid loading more conveniently was investigated in the present study A - 13 - Chapter 1 Introduction systematic way to apply creep theories in modelling concrete behaviour under uniaxial rapid loading was developed Three creep models, namely CEB model, B3 model and Sakata Model, were chosen for simulating the behaviour of concrete under various rates of rapid loading...List of Figures ix Figure 3.4 Discretized Representation of the Specimen 58 Figure 3.5 Flow Chart of Simulation 59 Figure 4.1 Loading Paths of Analysis for Secant Modulus 88 Figure 4.2 Graph for Mix B under Loading Case L0-A 89 Figure 4.3 Graph for Mix B under Loading Case L0-B 89 Figure 4.4 Graph for Mix B under Loading Case L1-A 90 Figure 4.5 Graph for Mix B under Loading Case L1-B 90... interest of this study is to investigate the potential use of the established knowledge on the long terms behaviour of concrete for predicting behaviour of concrete over very short time periods under dynamic conditions The loading rate sensitive behaviour of concrete and its constituents has been under investigation for a long time Initially, most researchers studying high strain-rate effects of concrete . CONSTITUTIVE RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD LIU DING NATIONAL UNIVERSITY OF SINGAPORE 2005 CONSTITUTIVE. RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD LIU DING (M.Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF. LIST OF FIGURES Figure 1.1 Magnitude of Stain Rates with Respect to Different Loading Cases 15 Figure 1.2 Typical Stress-Strain Relationship of Concrete under Different Loading Rates