NONLINEAR INELASTIC RESPONSE OF SLENDER REINFORCED CONCRETE BRIDGE COLUMNS By Ata Babazadeh-Naseri A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Civil Engineering—Doctor of Philosophy 2017 ABSTRACT NONLINEAR INELASTIC RESPONSE OF SLENDER REINFORCED CONCRETE BRIDGE COLUMNS By Ata Babazadeh-Naseri The earthquake-resistant design of bridges requires inelastic deformations in columns to dissipate seismic energy. Reinforced concrete (RC) bridge columns are thus designed to exhibit a stable ductile inelastic response when subjected to earthquakes. Previous research on the inelastic response of RC bridge columns have led to current seismic design specifications. Yet, the slenderness effects on the inelastic response of RC bridge columns have not been adequately addressed. In this research, the second-order effects of slenderness, namely, P-Δ and P-δ moments on the inelastic response of slender RC bridge columns are experimentally evaluated. Four largescale RC columns with aspect ratios (shear span length to section width) of 10 and 12 were tested under axial and lateral loads. The destabilizing effect of P-Δ and the contribution of P-δ to the extent of the critical plastic region along columns height were experimentally evaluated. It was found that the test columns exhibited stable cyclic response beyond the conventional stability limit indices, despite the destabilizing effects of P-Δ moments. Experimental results also showed that the second-order P-δ moments lead to a significant increase in the length of the plastic region (Lpr). Experimental Lpr values were compared against North American, European, and New Zealand design guidelines. It was found that all design codes considered in this study significantly underestimated Lpr, except for the seismic design provisions provided by Caltrans. Previous expressions for the length of the plastic region were reexamined in light of the new experimental data and strong disagreement between experimental and predicted Lpr was observed. A study on the capability of continuum-based finite element (FE) models for predicting seismic damage in reinforced concrete (RC) bridge columns is presented. Experimental data from 4 largescale columns, under quasi-static reversed cyclic loading, were used to verify the FE results at two levels of global and local structural responses. Statistical measures for goodness-of-fit were utilized to quantitatively evaluate the accuracy of the FE models in addition to the conventional method of visual comparison of overlaid plots. Seismic damage states were predicted from the simulation results and compared against experimental observations from tests. 3D FE simulations were proved to be capable of simulating seismic damage in slender RC columns. Analytical expressions and closed-form solutions to the effects of slenderness and second-order moments on the inelastic response of RC columns are presented. A mathematical expression for the length of the plastic region (Lpr) in slender RC columns, which is significantly affected by nonlinear second-order moments, is presented. Nonlinear beam-column theory in conjunction with a bilinear inelastic moment-curvature response was utilized to determine the magnitude by which the inelastic response of RC columns is affected by their slenderness. Also, a slenderness limit for RC bridge columns is defined in terms of design variables, beyond which the second-order effects on the inelastic response of RC bridge columns are significant and can no longer be ignored. A parametric study was conducted on all the possible design configurations that are permitted by current seismic design guidelines for bridge columns to evaluate the sensitivity of RC bridge columns to second-order effects. Recommendations are made to update the current design guidelines for consideration of the second-order effects. Furthermore, simple design formulas are proposed to predict the maximum P-δ moment and its impact on the length of the plastic region. To my parents, Gholamreza and Shiva, and my brother, Rasa, for their endless and unconditional love, support, and encouragement iv ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Prof. Rigoberto Burgueño, for his tremendous help, support, and encouragement over the course of my studies at Michigan State University (MSU). I am very grateful of him for providing me with the chance to conduct research at the highest levels of technical sophistication. I learnt a lot from his wise advices and priceless feedbacks. In addition to providing guidance on the technical aspects of my research, Prof. Burgueño was always available to discuss life issues. His understanding and considerations of my personal situations along with his discipline provided me with a joyful and productive journey at MSU. Also, I would like to thank the members of the doctoral committee, Profs. Parviz Soroushian, Alejandro Diaz, and Mahmood Haq for their time and valuable feedback. In addition, I would like to thank Dr. Nizar Lajnef who previously served on my doctoral committee. I am very grateful to Dr. Pedro F. Silva and his colleagues at the George Washington University for their help and support during this collaborative research project. I would like to express my gratitude to everyone who helped me with the immense task of the experimental program. Special thanks to CIL manager, Mr. Siavosh Ravanbakhsh, for his superb help. I would also like to thank my friends and former colleagues, Mansour Alturki, Lauren K. Fedak, Nan Hu, David Stringer, Yi Sun, and Caroline Williams who helped me throughout the construction of the test units. v TABLE OF CONTENTS LIST OF TABLES ....................................................................................................................... xiii LIST OF FIGURES ...................................................................................................................... xv KEY TO ABBREVIATIONS .................................................................................................... xxiv CHAPTER 1 ................................................................................................................................... 1 1. INTRODUCTION .................................................................................................................. 1 1.1. General ............................................................................................................................. 1 1.2. Knowledge Gap and Research Motivation....................................................................... 2 1.2.1. Scarcity of Experimental Evidence for Inelastic Response of Slender RC Columns 2 1.2.2. Prevalence of Slender RC Bridge Columns in Structures ........................................ 3 1.2.2.1. Slender RC Bridge Columns as Vital Elements in Modern Infrastructure........ 3 1.2.2.2. Increased Slenderness due to Soil-Structure Interaction ................................... 5 1.2.2.3. Slender RC Columns due to Use of High-Strength Concrete ......................... 12 1.3. Slenderness Effects on Inelastic Response of RC Bridge Columns............................... 13 1.3.1. Amplified Second-Order Effects ............................................................................ 13 1.3.1.1. Destabilizing Effect of Second-order P-Δ Moments ....................................... 14 1.3.1.2. Nonlinear Second-order Moment Profiles due to P-δ Effect........................... 15 1.3.2. Damage States for Performance-Based Seismic Design ........................................ 17 1.3.3. Lengths of the Plastic Region (Lpr) and Plastic Hinge (Lp) .................................... 18 1.4. Research Significance and Contributions ...................................................................... 21 1.4.1. Significance............................................................................................................. 21 1.4.2. Main Contributions ................................................................................................. 22 1.4.2.1. Expanded Structural Performance Database of RC Bridge Columns ............. 22 1.4.2.2. Evaluated Slenderness Effects on Inelastic Response of RC Columns ........... 22 1.4.2.3. Finite Element Models to Capture Seismic Damage in RC Columns ............. 23 1.4.2.4. Derived Analytical Expressions for Second-order Effects .............................. 23 1.4.2.5. Evaluated Sensitivity of Second-order Effects to Design Parameters ............. 23 1.4.2.6. Proposed Guidelines for Seismic Design of Slender RC Columns ................. 24 1.5. Research Hypothesis ...................................................................................................... 24 1.6. Research Methods and Dissertation Outline .................................................................. 24 1.6.1. Experimental Studies .............................................................................................. 25 1.6.2. Numerical Studies ................................................................................................... 26 1.6.3. Analytical Studies ................................................................................................... 27 1.6.4. Parametric Studies .................................................................................................. 28 CHAPTER 2 ................................................................................................................................. 30 2. LITERATURE REVIEW ..................................................................................................... 30 2.1. Previous Experimental Studies on the Inelastic Response of RC Columns ................... 30 2.1.1. Tests on Slender Columns ...................................................................................... 30 2.1.2. Tests on Shear-Dominated Columns ...................................................................... 31 2.1.3. Tests on High Axial Load Columns........................................................................ 31 2.2. Previous Experimental Studies on Column Slenderness Effects ................................... 32 vi 2.3. Previous Numerical Studies on Predicting Seismic Damage in RC Columns ............... 33 2.3.1. 1D and 2D Finite Element Simulations .................................................................. 33 2.3.2. 3D Continuum-Based Finite Element Simulations ................................................. 34 2.4. Previous Analytical Studies on Second-order Effects.................................................... 35 2.4.1. Large-Deflection Solutions to Elastic Columns ..................................................... 35 2.4.2. Nonlinear Beam-Column Theory ........................................................................... 36 2.5. Previous Models for Lpr and Lp ...................................................................................... 36 2.5.1. Current Seismic Design Specifications for Lpr ....................................................... 37 2.5.2. Previous Lpr and Lp Models .................................................................................... 38 2.6. Slenderness Effects in Previous Lpr and Lp Models ....................................................... 39 CHAPTER 3 ................................................................................................................................. 41 3. EXPERIMENTAL STUDY.................................................................................................. 41 3.1. General ........................................................................................................................... 41 3.2. Test Units ....................................................................................................................... 42 3.2.1. Geometry................................................................................................................. 42 3.2.2. Test Parameters ....................................................................................................... 42 3.2.2.1. Aspect Ratio .................................................................................................... 42 3.2.2.2. Longitudinal Reinforcement Ratio .................................................................. 42 3.2.2.3. Axial Load Ratio ............................................................................................. 43 3.2.3. Naming Convention ................................................................................................ 43 3.2.4. Design Specifications.............................................................................................. 44 3.2.4.1. Longitudinal Reinforcement ............................................................................ 44 3.2.4.2. Transverse Reinforcement ............................................................................... 44 3.2.5. Components and Reinforcement Layouts of Test Units ......................................... 45 3.2.5.1. Column Element .............................................................................................. 45 3.2.5.2. Footing ............................................................................................................. 46 3.2.5.3. Loading Block ................................................................................................. 47 3.3. Test Setup ....................................................................................................................... 48 3.3.1. Steel Loading Frame ............................................................................................... 49 3.3.2. Axial Loading Setup ............................................................................................... 50 3.3.3. Lateral Load Application ........................................................................................ 51 3.3.4. Resultant Applied and Reaction Forces .................................................................. 52 3.4. Instrumentation............................................................................................................... 53 3.4.1. Displacement and Rotation Measurements ............................................................. 55 3.4.1.1. Lateral Displacement ....................................................................................... 55 3.4.1.2. Top Rotation .................................................................................................... 55 3.4.2. Deformation and Strain Measurements .................................................................. 56 3.4.2.1. Flexural Curvatures ......................................................................................... 56 3.4.2.2. Shear Strains .................................................................................................... 57 3.4.2.3. Reinforcement Strains ..................................................................................... 59 3.4.3. Force Measurements ............................................................................................... 59 3.5. Material Properties ......................................................................................................... 59 3.5.1. Compressive Strength of Concrete ......................................................................... 59 3.5.2. Split Tensile Strength of Concrete .......................................................................... 60 3.5.3. Uniaxial Stress-Strain Response of Reinforcement Steel ....................................... 61 vii 3.5.3.1. Longitudinal Reinforcement No. 8 Bars.......................................................... 61 3.5.3.2. Longitudinal Reinforcement No. 6 Bars.......................................................... 62 3.5.3.3. Transverse Reinforcement No. 4 Spirals ......................................................... 63 3.6. Experiment Design ......................................................................................................... 64 3.6.1. Preliminary Analyses .............................................................................................. 64 3.6.1.1. Section Moment-Curvature Analysis .............................................................. 64 3.6.1.2. Yield and Ultimate Points................................................................................ 66 3.6.1.3. Anticipated Force-Displacement Response ..................................................... 68 3.6.2. Loading Protocol ..................................................................................................... 69 3.6.2.1. Reversed Cyclic Loading Protocol .................................................................. 69 3.6.2.2. Pseudo-Dynamic Loading Protocol ................................................................. 72 3.7. Test Observations and Damage States ........................................................................... 75 3.7.1. Tensile Cracking ..................................................................................................... 75 3.7.2. First Yielding .......................................................................................................... 76 3.7.3. Initiation of Spalling ............................................................................................... 77 3.7.4. Significant Spalling ................................................................................................. 77 3.8. Test Results .................................................................................................................... 78 3.8.1. Direct Measurement Results ................................................................................... 79 3.8.1.1. Top Rotation Response .................................................................................... 79 3.8.1.2. Accidental Moment at Column Top ................................................................ 80 3.8.1.3. Axial Force Response ...................................................................................... 82 3.8.2. Lateral Force-Displacement Response ................................................................... 83 3.8.2.1. Cyclic Response .............................................................................................. 83 3.8.2.2. P-Δ Moments ................................................................................................... 85 3.8.3. Response Profiles .................................................................................................... 85 3.8.3.1. Lateral Displacement Profile ........................................................................... 85 3.8.3.2. Deformation Profile ......................................................................................... 87 3.8.3.3. P-δ Moment Profile ......................................................................................... 88 3.8.3.4. Curvature Profile ............................................................................................. 89 3.8.3.5. Total Nonlinear Moment Profile ..................................................................... 92 3.8.3.6. Base Moment-Curvature (M-ϕ) Response....................................................... 94 3.8.4. Shear Deformations ................................................................................................ 96 3.9. Analysis of Test Results ................................................................................................. 99 3.9.1. Effect of P-Δ Moments ........................................................................................... 99 3.9.2. Decomposition of Moment Profile ....................................................................... 101 3.9.3. Plastic Region Length (Lpr)................................................................................... 104 3.9.3.1. Methods of Evaluating Lpr ............................................................................. 107 3.9.3.2. Components of Lpr ......................................................................................... 109 3.9.3.3. Effect of Tension Shift on Lpr ........................................................................ 111 3.9.3.4. Effect of P-δ Moment on Lpr ......................................................................... 112 3.9.3.5. Effects of Pseudo-Dynamic Loading on Lpr .................................................. 115 3.9.4. Plastic Hinge Length (Lp) ..................................................................................... 118 CHAPTER 4 ............................................................................................................................... 122 4. NUMERICAL STUDY ...................................................................................................... 122 4.1. General ......................................................................................................................... 122 viii 4.2. Methodology and Background ..................................................................................... 122 4.2.1. Intermediate Damage Limit States........................................................................ 122 4.2.2. Methods for Numerical Studies ............................................................................ 123 4.3. Experimental Database ................................................................................................. 124 4.3.1. Geometry............................................................................................................... 124 4.3.2. Material Properties ................................................................................................ 125 4.3.2.1. Concrete ......................................................................................................... 125 4.3.2.2. Reinforcement Steel ...................................................................................... 127 4.3.3. Instrumentation ..................................................................................................... 128 4.3.4. Test Unit Components .......................................................................................... 128 4.4. Finite Element Model Description ............................................................................... 129 4.4.1. Parts....................................................................................................................... 129 4.4.1.1. Concrete Parts ................................................................................................ 129 4.4.1.2. Steel Reinforcement Parts ............................................................................. 129 4.4.2. Mesh ...................................................................................................................... 131 4.4.3. Boundary Conditions ............................................................................................ 131 4.4.4. Material Models .................................................................................................... 133 4.4.4.1. Elastic Properties ........................................................................................... 133 4.4.4.2. Concrete Plasticity Properties ........................................................................ 133 4.4.4.3. Concrete Damage Properties ......................................................................... 134 4.4.4.4. Steel Plasticity Properties .............................................................................. 136 4.4.5. Solution Algorithms .............................................................................................. 137 4.4.6. Identification of Optimum Loading Rate .............................................................. 137 4.5. Model Evaluation and Validation ................................................................................ 139 4.5.1. Evaluation Levels.................................................................................................. 139 4.5.1.1. Global Responses .......................................................................................... 140 4.5.1.2. Local Responses ............................................................................................ 145 4.5.1.3. Selected Responses for Comprehensive Evaluations .................................... 149 4.5.2. Evaluation Methods .............................................................................................. 151 4.5.2.1. Visual Qualitative Evaluation ........................................................................ 151 4.5.2.2. Quantitative Evaluation ................................................................................. 155 4.5.3. Results of Evaluation and Validation ................................................................... 162 4.5.3.1. Global Level .................................................................................................. 162 4.5.3.2. Local Level .................................................................................................... 163 4.6. Determining Intermediate Damage Limit States .......................................................... 163 4.6.1. First Yield of Longitudinal Reinforcement Bars .................................................. 163 4.6.2. Tracking the Spalling of Cover Concrete ............................................................. 165 4.6.1. Onset of Spalling of the Cover Concrete .............................................................. 167 4.6.2. Significant Spalling of the Cover Concrete .......................................................... 169 4.6.3. Engineering Design Limits States......................................................................... 171 4.6.3.1. Displacement Ductility .................................................................................. 171 4.6.3.2. Drift Ratio ...................................................................................................... 172 CHAPTER 5 ............................................................................................................................... 174 5. ANALYTICAL STUDY .................................................................................................... 174 5.1. General ......................................................................................................................... 174 ix 5.2. Linear Elastic Solution to Lpr ....................................................................................... 175 5.3. Nonlinear Elastic Solution to Moment Profile ............................................................. 175 5.3.1. Coordinate Transformation ................................................................................... 175 5.3.2. Member Deformation Profile (δ) .......................................................................... 177 5.3.3. Bending Moment Profile (Mt)............................................................................... 179 5.4. Nonlinear Elastic Solution to Displacement Profile .................................................... 180 5.4.1. Cross-Section Moment-Curvature Response ........................................................ 180 5.4.1.1. M-ϕ as a Curvilinear Response...................................................................... 181 5.4.1.2. M-ϕ as an Idealized Bilinear Response ......................................................... 181 5.4.2. Solution to Curvature Profiles (ϕ)......................................................................... 182 5.4.3. Solution to Rotation Profile (θ) ............................................................................. 183 5.4.4. Solution to Lateral Displacements (Δ) .................................................................. 183 5.4.5. Solution to Length of the Plastic Region (Lpr) ...................................................... 184 5.5. Verification of the Nonlinear Elastic Solutions ........................................................... 185 5.5.1. Experimental Database for Evaluating the ‘Elastic Solution’ .............................. 185 5.5.2. Solution Parameters for Test Columns ................................................................. 186 5.5.2.1. Yield Moment (My) and Yield Curvature (ϕy) .............................................. 187 5.5.2.2. Ultimate Moment (Mu) and Ultimate Curvature (ϕu) .................................... 188 5.5.2.3. Flexural Stiffness of Cracked RC Section (EIel)............................................ 189 5.5.3. Experimental Data vs. Closed-Form Solution ...................................................... 191 5.5.3.1. Bending Moment Profiles .............................................................................. 191 5.5.3.2. Curvature Profiles .......................................................................................... 192 5.5.3.3. Displacement Profiles .................................................................................... 193 5.5.4. Quantitative Evaluation of the Proposed Solution ................................................ 194 5.6. Application of the Nonlinear Elastic Solution ............................................................. 195 5.6.1. Maximum P-δ Moment ......................................................................................... 196 5.6.2. Nonlinear Solution to the Length of the Plastic Region (Lpr, NL) .......................... 200 5.6.3. Slenderness Parameter .......................................................................................... 205 5.7. Nonlinear Inelastic Solution to Moment Profiles ......................................................... 207 5.7.1. Methods and Assumptions for Deriving the Nonlinear Inelastic Solution ........... 207 5.7.1.1. Nonlinear Beam-Column Theory .................................................................. 208 5.7.1.2. Column Partition............................................................................................ 208 5.7.1.3. Coordinate Transformation............................................................................ 209 5.7.2. Derivation of the Nonlinear Inelastic Solution for Moment Profiles ................... 210 5.7.2.1. Equilibrium of the Elastic and Inelastic Segments ........................................ 210 5.7.2.2. Section Moment-Curvature (M-ϕ) Response ................................................ 211 5.7.2.3. Elastic Segment ............................................................................................. 211 5.7.2.4. Inelastic Segment ........................................................................................... 212 5.7.2.5. Transition Point ............................................................................................. 213 5.8. Verification of the Nonlinear Inelastic Solution .......................................................... 216 5.8.1. Experimental Database ......................................................................................... 216 5.8.2. Moment-Curvature Response ............................................................................... 216 5.8.3. Flexural Stiffness (EI) Profile ............................................................................... 218 5.8.4. Moment, Curvature, and Displacement Profiles ................................................... 221 5.8.5. Force-Displacement Response .............................................................................. 224 5.9. Comparing Different Solutions (Linear, Nonlinear Elastic, and Nonlinear Inelastic) . 225 x 5.9.1. 5.9.2. 5.9.3. 5.9.4. Second-Order P-δ Moment Profile ....................................................................... 225 Maximum P-δ Moment ......................................................................................... 228 Length of the Plastic Region (Lpr) ........................................................................ 229 Effect of P-δ on Lpr ............................................................................................... 230 CHAPTER 6 ............................................................................................................................... 234 6. PARAMETRIC STUDY .................................................................................................... 234 6.1. General ......................................................................................................................... 234 6.2. Parametric Study Based on the Nonlinear Elastic Solution ......................................... 235 6.2.1. Key Variables of the Nonlinear Elastic Solution .................................................. 235 6.2.1.1. Slope Sign of Post-yield M-ϕ Branch (EIin) .................................................. 235 6.2.1.2. Moment Overstrength Ratio (Mu / My).......................................................... 236 6.2.1.3. Elastic Slenderness Parameter (κel) ............................................................... 236 6.2.2. Methods................................................................................................................. 237 6.2.2.1. Structural Parameters ..................................................................................... 237 6.2.2.2. Parametric Study Method .............................................................................. 238 6.2.2.3. Section Analyses............................................................................................ 238 6.2.2.4. Regression Models ........................................................................................ 239 6.2.2.5. Factor of Safety ............................................................................................. 239 6.2.3. Parametric Study Results for Nonlinear Elastic Solution ..................................... 240 6.2.3.1. Sign of Post-yield Flexural Stiffness (EIin) ................................................... 240 6.2.3.2. Moment Overstrength Ratio (Mu / My).......................................................... 244 6.2.3.3. Elastic Slenderness Parameter (κel) ............................................................... 249 6.2.3.4. Effect of Second-order P-δ Moments on Lpr ................................................. 253 6.2.4. Regression Models and Design Formulas............................................................. 255 6.3. Parametric Study Based on the Nonlinear Inelastic Solution ...................................... 256 6.3.1. Methods................................................................................................................. 257 6.3.1.1. Study Parameters ........................................................................................... 257 6.3.1.2. Domain for Parameters κel, κin, and Mu/My ................................................... 258 6.3.2. Parametric Study Results for the Nonlinear Inelastic Solution ............................ 259 6.3.2.1. Stability of the Inelastic Solution .................................................................. 259 CHAPTER 7 ............................................................................................................................... 263 7. SEISMIC DESIGN IMPLICATIONS ................................................................................ 263 7.1. General ......................................................................................................................... 263 7.2. Seismic Design Implications of Experimental Studies ................................................ 264 7.2.1. Effect of Increased Strength on P-Δ Moments ..................................................... 264 7.2.2. Stability of Cyclic Response ................................................................................. 266 7.3. Seismic Design Implications of Analytical Studies ..................................................... 267 7.3.1. Comparison of Experimental Lpr with Previous Research .................................... 267 7.3.1.1. Comparison with Current Design Guidelines ................................................ 268 7.3.1.2. Comparison with Current Expressions .......................................................... 273 7.3.2. Proposed Design Expression for Lpr ..................................................................... 274 7.3.2.1. Lpr due to Linear Moment Gradient (Lpr,L) .................................................... 274 7.3.2.2. Lpr due to Nonlinear Moment Gradient (Lpr,NL)............................................. 276 7.3.2.3. Verification of the Proposed Formula ........................................................... 277 xi 7.4. Seismic Design Implications of Parametric Studies .................................................... 282 CHAPTER 8 ............................................................................................................................... 285 8. SUMMARY AND CONCLUSIONS ................................................................................. 285 8.1. Summary ...................................................................................................................... 285 8.2. Conclusions .................................................................................................................. 286 8.2.1. Experimental Studies ............................................................................................ 286 8.2.2. Numerical Studies ................................................................................................. 288 8.2.3. Analytical Studies ................................................................................................. 288 8.2.4. Parametric Studies ................................................................................................ 289 8.2.5. Seismic Design Implications................................................................................. 290 8.3. Suggestions for Future Research .................................................................................. 291 8.3.1. Inelastic Response of Slender Concrete Columns Built with New Materials ...... 292 8.3.2. Seismic Response of Slender RC columns with Sub-sufficient Reinforcement... 292 8.3.3. Finite Element Models for Damage Analysis ....................................................... 293 8.3.4. Effects of Slenderness on Dynamic Response of RC Columns ............................ 293 8.3.5. Analytical Solution to the Plastic Hinge Length (Lp) ........................................... 294 8.3.6. Probabilistic FE Analysis of RC Structures .......................................................... 294 REFERENCES ........................................................................................................................... 295 xii LIST OF TABLES Table 3-1: Geometry and reinforcement details for test units ...................................................... 47 Table 3-2: Concrete Compressive Strengths................................................................................. 60 Table 3-3: Reinforcing steel properties......................................................................................... 62 Table 3-4: Key parameters of moment-curvature response for the test columns ......................... 68 Table 3-5: Target load or displacement levels for lateral loading using reversed cyclic protocol 73 Table 3-6: Extent of tensile cracking along the height of test columns ........................................ 75 Table 3-7: Onset of initial spalling of test columns ...................................................................... 77 Table 3-8: Onset of significant spalling and height of spalled region at the end of tests ............. 78 Table 3-9: Maximum possible error associated with the non-zero moment at top of the test column M121505 ....................................................................................................................................... 81 Table 3-10: Equivalent plastic hinge length (Lp,MG) for the test columns .................................. 120 Table 4-1: Geometry, reinforcement details and material properties for test units .................... 126 Table 4-2: Accuracy of FE models in predicting the lateral force-displacement response ........ 162 Table 4-3: Accuracy of FE models in predicting the flexural curvature profiles ....................... 163 Table 4-4: Displacement ductility at intermediate damage limit states ...................................... 164 Table 4-5: Drift ratio at intermediate damage limit states .......................................................... 164 Table 5-1: Properties of RC test columns for evaluation ............................................................ 186 Table 5-2: Analysis parameters for the test columns .................................................................. 189 Table 5-3: Key parameters of the proposed solution determined for the test columns .............. 219 Table 6-1: Dimensionless parameters for the parametric study.................................................. 238 Table 6-2: Range of parameters investigated in the parametric study based on the nonlinear inelastic solution ......................................................................................................................... 257 Table 7-1: Comparison between experimental Lpr and design guidelines with variable Lpr ...... 269 xiii Table 7-2: Comparison of experimental Lpr against design guidelines with constant Lpr .......... 272 Table 7-3: Comparison of experimental Lpr against current Lpr and Lp models ......................... 274 Table 7-4: Summary of the error in predicting Lpr in test columns ............................................ 278 xiv LIST OF FIGURES Figure 1-1: Existing experimental data on seismic performance of RC columns .......................... 3 Figure 1-2: Slender RC bridge columns of Judge Harry Pregerson interchange in Los Angeles, CA ......................................................................................................................................................... 4 Figure 1-3: Map of active faults near judge Harry Pregerson interchange shown by the blue star 5 Figure 1-4: Effect of base flexibility on the effective length of double-curvature bridge columns with fully fixed top.......................................................................................................................... 8 Figure 1-5: Effect of base flexibility on the effective length of double-curvature bridge columns with partially fixed top .................................................................................................................... 9 Figure 1-6: Effect of base flexibility on the effective length of single-curvature bridge columns10 Figure 1-7: Effect of base rotational fixity on the effective aspect ratio of double-curvature columns with: (a) fully fixed top (Rtop = 1), and (b) partially fixed top (Rtop = 0.5) ................................... 11 Figure 1-8: Effect of base rotational fixity on the effective aspect ratio of single-curvature cantilever columns ........................................................................................................................ 11 Figure 1-9: Global lateral displacement (Δ) and member deformation (δ) in deformed configuration of a cantilever column ............................................................................................ 14 Figure 1-10: Effect of P-Δ moment on force-displacement response of typical well-confined ductile RC columns ....................................................................................................................... 15 Figure 1-11: P–δ effects on the spread of plasticity along cantilever columns ............................ 16 Figure 1-12: Main framework for performance-based seismic design (PBSD) of structures ...... 18 Figure 1-13: Effect of P-δ moments on the length of critical plastic region (Lpr) and the momentgradient component of plastic hinge length (Lpm) ........................................................................ 21 Figure 3-1: Test columns’ geometry and reinforcement layout.................................................... 45 Figure 3-2: Footing Design as shown in (a) plan and (b) elevation.............................................. 48 Figure 3-3: Typical test setup and fixtures to apply axial and lateral loads as shown in (a) 3D and (b) side views ................................................................................................................................ 49 Figure 3-4: Test setup for columns (a) M1230 and (b) M1215 as shown at their deformed configuration ................................................................................................................................. 50 xv Figure 3-5: Axial load application setup ....................................................................................... 51 Figure 3-6: Lateral load application setup .................................................................................... 52 Figure 3-7: Schematic diagram of (a) loading and boundary conditions for test columns; (b) free body diagram of the applied and reaction forces on test columns ................................................ 53 Figure 3-8: Instrumentation layout on column M1215 ................................................................. 54 Figure 3-9: Horizontal Displacement Transducers on Reference Frame ...................................... 55 Figure 3-10: Instrumentation for curvature measurement ............................................................ 56 Figure 3-11: Instrumentation setup for measurement of average curvatures on a segment of columns ......................................................................................................................................... 58 Figure 3-12: Shear instrumentation .............................................................................................. 58 Figure 3-13: Compressive response of concrete specimens from M1230 .................................... 60 Figure 3-14: Stress-strain response of no. 8 reinforcing steel ...................................................... 61 Figure 3-15: Schematic stress-strain curve of reinforcing steel with key parameters .................. 62 Figure 3-16: Stress-strain response of no. 6 reinforcing steel ...................................................... 63 Figure 3-17: Stress-strain response of no. 4 transverse steel ........................................................ 64 Figure 3-18: Moment-curvature response of test columns from analysis ..................................... 65 Figure 3-19: Schematic representation of the first, nominal, and ideal points ............................. 66 Figure 3-20: Moment-curvature response for column M1215 with bilinear idealization options 67 Figure 3-21: Expected force-displacement response of test columns obtained from nonlinear structural analysis.......................................................................................................................... 69 Figure 3-22: Reversed cyclic lateral loading pattern for column M1230 ..................................... 71 Figure 3-23: Reversed cyclic lateral loading pattern for column M1215 ..................................... 71 Figure 3-24: Reversed cyclic lateral loading pattern for column M1015C .................................. 72 Figure 3-25: Displacement history of pseudo-dynamic loading protocol for testing M1015D .... 74 xvi Figure 3-26: Pseudo-dynamic lateral loading pattern for column M1015D ................................. 74 Figure 3-27: Tensile cracking pattern of column M1230 at µΔ=3 ................................................ 76 Figure 3-28: Maximum observed extent of spalling region on columns (a) M1230, (b) M1215 and (c) M1015C ................................................................................................................................... 78 Figure 3-29: Rotation measured at the top of the test columns .................................................... 79 Figure 3-30: Maximum possible moment at top of column M121505 due to eccentricity of the applied forces from the center of the loading block as shown for absolute values and normalized to the base moment ....................................................................................................................... 81 Figure 3-31: Variation of axial load during the test for both columns ......................................... 82 Figure 3-32: Force-displacement response of test columns (a) M1230, (b) M1215 and (c) M1015C ....................................................................................................................................................... 84 Figure 3-33: Experimental P-Δ moment for column M123005.................................................... 85 Figure 3-34: Lateral displacement profiles for column M1230 .................................................... 86 Figure 3-35: Lateral displacement profiles for column M1215 .................................................... 87 Figure 3-36: Deformation profiles along test columns ................................................................. 88 Figure 3-37: Average curvature profiles after yield for column M1230 ...................................... 90 Figure 3-38: Average curvature profiles after yield for column M1215 ...................................... 91 Figure 3-39: Average curvature profiles for test columns after correction at the base level ........ 93 Figure 3-40: Experimental profiles of nonlinear moment on test columns .................................. 95 Figure 3-41: Moment-curvature response of test columns from analyses and experiments......... 96 Figure 3-42: Contribution of shear deformations to top displacement ......................................... 97 Figure 3-43: Decomposition of top displacement into sources of deformation for (a) column M123005 and (b) M121505 .......................................................................................................... 98 Figure 3-44: Loss of lateral strength due to P-Δ effect in columns (a) M1230, (b) M1215 and (c) M1015C ...................................................................................................................................... 100 Figure 3-45: Lateral displacement profiles for column M1215 depicting parameters of Δ and δ ..................................................................................................................................................... 101 xvii Figure 3-46: Decomposed moment profiles for column M1230 at (a) μΔ = 1 and (b) μΔ = 3 .... 103 Figure 3-47: Decomposed moment profiles for column M1215 at (a) μΔ = 1 and (b) μΔ = 3.5 . 103 Figure 3-48: Decomposed moment profiles for column M1015C at (a) μΔ = 1 and (b) μΔ = 4.. 104 Figure 3-49: Length of the plastic region (Lpr) on columns M1230, M1215, and M1015 ......... 106 Figure 3-50: Extent of the plastic region (Lpr) extracted from the experiments ......................... 108 Figure 3-51: Extents of the plastic region (Lpr) and spalling zone (Ls) depicted along the test columns at multiple displacement ductility levels ...................................................................... 110 Figure 3-52: Major contributors to Lpr ........................................................................................ 111 Figure 3-53: Predicted crack angle and the associated tension shift effect on test columns ...... 112 Figure 3-54: Effect of nonlinear moment profile and P-δ on the extent of the plastic region (Lpr) ..................................................................................................................................................... 114 Figure 3-55: Increase in spread of plasticity due to P-δ effect ................................................... 115 Figure 3-56: Profiles of curvature along test columns M1015C and M1015D .......................... 116 Figure 3-57: Asymmetric profiles of curvature along column M1015D .................................... 117 Figure 3-58: Spread of plasticity (Lpr) along the height of columns M1015C and M1015D ..... 118 Figure 3-59: Distribution of plastic curvatures (ϕp) over the plastic region of the test units at the highest displacement ductility (μΔ) to which they were tested ................................................... 119 Figure 4-1: Schematic representation of concrete stress-strain response with compressive and tensile strengths ........................................................................................................................... 127 Figure 4-2: Schematic representation of reinforcement steel stress-strain response .................. 128 Figure 4-3: Parts of finite element model for column SD0630 showing (a) entire model and (b) cross-section ................................................................................................................................ 130 Figure 4-4: Schematic of 8-node linear brick element with reduced integration ....................... 130 Figure 4-5: Radial mesh of the cross-section .............................................................................. 131 Figure 4-6: Loading and boundary conditions for test columns in (a) experimental setup and (b) finite element models (same as Figure 3-7) ................................................................................ 132 xviii Figure 4-7: Strength and damage properties of column SD0630 for (a) confined concrete and (b) unconfined concrete .................................................................................................................... 135 Figure 4-8: Stress-strain curve used in numerical modeling of longitudinal bar from column M1215 compared against experimental data ........................................................................................... 136 Figure 4-9: Optimum time-rate of loading for test unit M1215 as depicted for (a) forcedisplacement response; (b) kinetic energy content ..................................................................... 139 Figure 4-10: Force-displacement response of test column M1015 ............................................. 141 Figure 4-11: Total energy for column M1215 as shown by (a) the envelopes of force-displacement response and (b) plots of energy against top displacement......................................................... 142 Figure 4-12: Total energy of column M1215 considering effective and net lateral forces ........ 143 Figure 4-13: Complete net force cycles at displacement ductility levels of 2 and 3 for column M1230 ......................................................................................................................................... 144 Figure 4-14: Plastic energy dissipated in FE simulations compared to experimental data ........ 145 Figure 4-15: Curvature profiles for column M1215 at different displacement ductility levels .. 147 Figure 4-16: Spread of plasticity along the height of column M1215 ........................................ 148 Figure 4-17: Rotation profiles for column M1230 at different displacement ductility levels .... 150 Figure 4-18: Force-displacement response for all columns ........................................................ 152 Figure 4-19: Curvature profiles for four test columns at their maximum tested displacement ductility ....................................................................................................................................... 154 Figure 4-20: Curvature profiles for test column SD0630 at its maximum tested displacement ductility ....................................................................................................................................... 155 Figure 4-21: Flexural curvature values for column SD0630 plotted over the grid of location and displacement ductility level as shown for (a) FE analyses results and (b) test data ................... 157 Figure 4-22: Overlay of experimental space-time grid on that of FE analysis for curvature profile data of column SD0630 .............................................................................................................. 158 Figure 4-23: Scatter plot of experimentally measured lateral forces versus the FE results ........ 159 Figure 4-24: Scatter plot of experimentally measured curvatures versus the FE results ............ 159 xix Figure 4-25: Compressive strain profiles in cover concrete for different displacement ductility levels ........................................................................................................................................... 166 Figure 4-26: Predicted spalling region for the test units at their highest displacement ductility level ..................................................................................................................................................... 168 Figure 4-27: Comparative evaluations of cover concrete spalling between FE model and experiment................................................................................................................................... 170 Figure 4-28: Growth of the spalled region over the height of test columns versus displacement ductility ....................................................................................................................................... 172 Figure 4-29: Growth of the spalled region over the height of test columns versus drift ratio .... 173 Figure 5-1: Deformed column and forces acting on it with respect to: (a) vertical fixed reference axis and (b) rotated element chord axis ...................................................................................... 176 Figure 5-2: Equilibrium of column top segment in its deformed shape ..................................... 178 Figure 5-3: Bilinear idealization of moment-curvature response ............................................... 181 Figure 5-4: Moment-curvature response for the selected test columns with bilinear idealization options ......................................................................................................................................... 188 Figure 5-5: Equivalent flexural section stiffness of column M1215 before yield (EIel) ............ 190 Figure 5-6: Moment profiles for column M1215 at the base moment corresponding to μΔ = 3.5 ..................................................................................................................................................... 192 Figure 5-7: Curvature profiles for column M1215 at the base curvature corresponding to μΔ = 3.5 ..................................................................................................................................................... 193 Figure 5-8: Displacement profiles for column M1215 at μΔ = 3.5 ............................................. 194 Figure 5-9: Theil’s inequality coefficient for error for the analytical solution predictions ........ 196 Figure 5-10: P-δ moment profiles for column M1230 at μΔ = 1, 2, and 3 .................................. 198 Figure 5-11: Maximum P-δ moment for column M1230 at different displacement ductility levels ..................................................................................................................................................... 199 Figure 5-12: Maximum P-δ moment for all test columns - Ratio of experimental values to results from the solution ......................................................................................................................... 200 Figure 5-13: Effect of P-δ on spread of the plastic region along column height ........................ 201 xx Figure 5-14: Spread of plastic region (Lpr) for column M1230 plotted vs. base moment to yield ratio (MBase/My) ........................................................................................................................... 202 Figure 5-15: Spread of plastic region (Lpr) for column M1230 plotted vs. displacement ductility (μΔ = Δ/Δy) .................................................................................................................................. 202 Figure 5-16: Effect of P-δ on spread of plasticity for column M1215 ....................................... 204 Figure 5-17: Effect of P-δ on spread of plasticity for all test columns - Ratio of experimental values to results from solution ............................................................................................................... 205 Figure 5-18: Effect of aspect ratio (L/D) on the spread of the plastic region due to P-δ moments ..................................................................................................................................................... 206 Figure 5-19: Effect of axial load ratio (P/Agf ′c) on the spread of the plastic region due to P-δ moments ...................................................................................................................................... 206 Figure 5-20: Effect of longitudinal reinforcement ratio (ρsl) on the spread of the plastic region due to P-δ moments ........................................................................................................................... 207 Figure 5-21: Schematics depicting: (a) bilinear moment-curvature response, (b) deformation of a cantilever column, (c) nonlinear moment gradient, and (d) profile of flexural stiffness. ........... 208 Figure 5-22: Cantilever and effective forces and moments in: (a) reference; (b) rotated coordinate systems; and within: (c) elastic; and (d) inelastic segments. ...................................................... 210 Figure 5-23: Moment-curvature response of test columns from analyses and experiments....... 218 Figure 5-24: Profiles of flexural stiffness (EI) for test columns at: (a) μΔ = 1; and (b) μΔ = 3. .. 220 Figure 5-25: Profiles of: (a) moment, (b) curvature, and (c) displacement for test column M123005 at μΔ = 1 and μΔ = 3. .................................................................................................................... 221 Figure 5-26: Comparing accuracy of the linear, nonlinear elastic, and nonlinear inelastic solutions to: (a) moment and (b) curvature profiles along the test columns’ height; and (c) top displacement. ..................................................................................................................................................... 223 Figure 5-27: Force-displacement response of the test columns from analyses and experiments. ..................................................................................................................................................... 225 Figure 5-28: Profiles of second-order P-δ moments along the height of test columns at μΔ = 1.5 ..................................................................................................................................................... 226 Figure 5-29: Profiles of second-order P-δ moments along the height of test columns at μΔ = 3 227 Figure 5-30: Maximum P-δ moment from nonlinear analytical solutions versus test data ........ 228 xxi Figure 5-31: Extent of the plastic region (Lpr) on the test columns from analytical solutions ... 230 Figure 5-32: Effect of P-δ moments on Lpr in terms of percent increase due to nonlinear P-δ moments ...................................................................................................................................... 231 Figure 5-33: Comparing accuracy of nonlinear analytical solutions for predicting P-δ effect on Lpr ..................................................................................................................................................... 232 Figure 6-1: Effect of parameters 1, 2, 5, and 6 on the sign of post-yield flexural stiffness (EIin) ..................................................................................................................................................... 240 Figure 6-2: Effect of parameters 3, 4, 7, and 8 on the sign of post-yield flexural stiffness (EIin) ..................................................................................................................................................... 241 Figure 6-3: (a) Probabilistic distribution of sign of post-yield flexural stiffness (EIin); (b) Distribution of test columns reported in the PEER’s database ................................................... 243 Figure 6-4: Effect of parameters 1, 2, 5, and 6 on the moment overstrength ratio (Mu/My) ...... 245 Figure 6-5: Effect of parameters 3, 4, 7, and 8 on the moment overstrength ratio (Mu/My) ...... 246 Figure 6-6: Distribution of Mu/My ratio with respect to P/Agf ′c and fusl/fysl for: (a) the normal strength concrete (f ′c = 45 MPa); and (b) the high-strength concrete (f ′c = 90 MPa) ............... 248 Figure 6-7: Effect of parameters 1, 2, 5, and 6 on the elastic slenderness factor (κel) ................ 250 Figure 6-8: Effect of parameters 3, 4, 8, and 9 on the elastic slenderness factor (κel) ................ 251 Figure 6-9: Distribution of (a) κel/(L/D) and (b) L/D limit for considering P-δ effects with respect to P/Agf ′c and ρsl ......................................................................................................................... 252 Figure 6-10: Effect of P-δ moments on Lpr in terms of L/D and P/Agf ′c for RC sections with (a) ρsl = 1% and (b) ρsl = 4% ................................................................................................................. 254 Figure 6-11: Approximate versus exact values for (a) Mu/My, (b) adjusted Mu/My, and (c) adjusted κel/(L/D) ...................................................................................................................................... 255 Figure 6-12: Relative frequency of main parameters in Equation (5-45) assessed for RC bridge columns ....................................................................................................................................... 259 Figure 6-13: Valid (empty) and invalid spaces (filled) for the key parameters of Equation (5-45) ..................................................................................................................................................... 261 Figure 6-14: Cumulative probability for instability of the nonlinear inelastic solution in terms of drift ratio ..................................................................................................................................... 262 xxii Figure 7-1: Strength loss due to P–Δ effect on the inelastic force–deformation response of RC columns ....................................................................................................................................... 264 Figure 7-2: Effect of increasing strength on the force–deformation response of test column M121505 ..................................................................................................................................... 265 Figure 7-3: Effect of P–Δ on the force–displacement response of RC columns in terms of reduced cyclic stiffness ............................................................................................................................. 266 Figure 7-4: Ratio of reloading to initial lateral stiffness in cyclic tests. ..................................... 267 Figure 7-5: Comparing the accuracy of various seismic design codes for predicting the length of the plastic region (Lpr) at different displacement ductility levels in terms of mean absolute percent error (MAPE) .............................................................................................................................. 279 Figure 7-6: Comparing the accuracy of various seismic design guidelines in predicting the maximum length of the plastic region (Max Lpr) at maximum displacement ductility in terms of absolute percent error (APE) ...................................................................................................... 279 Figure 7-7: Comparison of the proposed formula for Lpr against test data and the model by Hines et al. [35] ..................................................................................................................................... 281 Figure 7-8: Cumulative probability of Lpr/L at: (a) μΔ = 3, and (b) μΔ = 4. ................................ 283 xxiii KEY TO ABBREVIATIONS AASHTO = American association for state highways and transportation officials ACI = American concrete institute APE = Absolute percent error DT = Displacement transducer MAPE = Mean absolute percent error PE Percent error = PEER = Pacific earthquake engineering research center FIB = International Federation for Structural Concrete Ag = Gross area of RC sections Asl = Area of longitudinal steel reinforcement D = Diameter of the column’s section EIel = Pre-yield flexural stiffness of a cracked RC section before yield EIin = Post-yield flexural stiffness of a RC section after yield L = Shear span of the column Lp = Plastic hinge length in a lumped plasticity model Lpm = Moment-gradient of the plastic hinge length Lpr = Length of the plastic region Lpr,L = Length of the plastic region as predicted by linear solution Lpr,NL = Length of the plastic region as predicted by nonlinear solution Lpr,P-δ = P-δ component of Lpr Ls = Extent of the spalling region on cover concrete Lsp = Strain penetration component of Lp xxiv Lts = Tension shift component of Lpr MBase = Bending moment at the cantilevered column base Mel = Internal moments along the elastic segment of a column Min = Internal moments along the inelastic segment of a column ML = Linear bending moment gradient as the sum of primary and P-Δ moments MNL = Nonlinear bending moment gradient as the sum of primary, P-Δ, and P-δ moments Mp = Primary moment MP-Δ = Linear second-order moment caused by P-Δ effect MP-δ = Nonlinear second-order moment caused by P-δ effect Mt = Total bending moment as the sum of primary and secondary moments Mu = Ultimate moment capacity of RC section My = Yield moment of RC section N = Number of data points used for error assessment P = Axial force applied at the top of the column V = Shear force with respect to vertical axis V′el = Internal shear forces along the elastic segment of column V′in = Internal shear forces along the inelastic segment of column U = Theil’s inequality coefficient dsh = Diameter of transverse reinforcing steel dsl = Diameter of longitudinal reinforcing steel f'c = Peak compressive strength of unconfined concrete f't = Peak tensile strength of concrete ful = Ultimate strength of longitudinal reinforcing steel xxv fyl = Yield strength of longitudinal reinforcing steel fyt = Yield strength of transverse reinforcing steel n = Ratio of the inelastic to elastic slenderness parameters s = Spacing of transverse reinforcing steel, spiral pitch x′max = Location of maximum member deformation (δmax) yi = Value of a response parameter at a point from experimental measurements yˆ i = Value of a response parameter at a point from numerical model y′ = Column deformation away from the chord line y′el = Bending deformations within the elastic segment y′in = Bending deformations within the inelastic segment ΔTop = Lateral displacement at column top y = Yield displacement of cantilever column at top ΩM = Moment overstrength α = Dimensionless factor as defined by x′ / L β = Normalized length of the plastic region defined by β=Lpr/L δ = Deformation of the column away from its chord-line δmax = Maximum column deformation away from the chord line εshl = Strain at the beginning of strain-hardening of longitudinal reinforcing steel εul = Ultimate strain of longitudinal reinforcing steel θ = Angle between the column chord line and the vertical axis κel = Elastic slenderness parameter κin = Inelastic slenderness parameter μ = Displacement ductility xxvi μϕ = Curvature ductility ρsh = Transverse volumetric reinforcement ratio ρsl = Longitudinal reinforcement ratio ϕBase = Flexural curvature at the base level of cantilever column ϕu = Ultimate curvature capacity of RC section ϕy = Yield curvature of RC section xxvii CHAPTER 1 1. INTRODUCTION 1.1. General Seismic design of modern structures involves provisions for allowing inelastic deformations to develop in pre-defined locations within the structure in order to provide an overall system inelastic response to strong ground motions. This inelastic response is desired since the elastic response lacks the capacity to dissipate seismic energy through loading and unloading cycles. The reason is that the energy within the elastic cycles is by definition conserved. Therefore, modern structures are designed to dissipate seismic energy by undergoing plastic deformations that are often accompanied with some levels of damage to the critical inelastic regions. In this regard, current practices for seismic design and retrofit of reinforced concrete bridges permit ductile inelastic deformations in pre-defined locations. These inelastic deformations are neither desired nor feasible to take place in the superstructure of bridges, leaving the substructure, mainly the bridge columns, as the principal elements to undergo inelastic deformations in the event of an earthquake. Reinforced concrete (RC) bridge columns are thus expected to exhibit stable ductile inelastic 1 response when subjected to medium to strong earthquakes. As a result, the study of the inelastic response of RC bridge columns has been the topic of research for many years. Decades of research on the inelastic response of RC bridge columns have led to current seismic design specifications. Despite extensive experimental, numerical, and analytical studies that have been carried out to characterize the inelastic response of RC bridge columns, the slenderness effects on the inelastic response of RC bridge columns have not been directly addressed. This is particularly an issue for slender bridge columns built with aspect ratios, the length to width ratios, greater than the test columns in the previous studies. This research aims at contributing to the stateof-the-art of seismic analysis and design of RC bridge columns by implementing slenderness effects into the inelastic response of RC bridge columns. 1.2. Knowledge Gap and Research Motivation 1.2.1. Scarcity of Experimental Evidence for Inelastic Response of Slender RC Columns Current analysis and design procedures for slender reinforced concrete (RC) bridge columns with aspect ratios (shear span length to section width/diameter or L/D) greater than 10 require further investigation and reevaluation. This follows from the relative lack of experimental evidence on their performance as depicted in Figure 1-1, in which the distribution of the aspect ratio is shown for the test columns reported in the latest PEER’s structural performance database [1] and the most recent FIB state of the art report [2]. It can be seen that the majority of the test columns upon which current seismic design specifications were developed have aspect ratios less than 7. Despite the relatively substantial number of research projects on RC columns, the inelastic response of slender columns with aspect ratios greater than 7 has barely been studied. This is in contrast to the fact that slender RC bridge columns are commonly seen in existing bridge structures. Slender bridge columns are commonly found in elevated highway interchanges. In addition, the ever-increasing 2 use of high strength concrete in construction sites along with transverse confinement reinforcement eliminated the need for large cross-sections, which leads to more slender columns. Also, the soilstructure interaction in the event of strong ground motions at the column base can potentially extend the shear span length of the column, which increases the effective column slenderness. These cases are discussed in detail as follows. Figure 1-1: Existing experimental data on seismic performance of RC columns 1.2.2. Prevalence of Slender RC Bridge Columns in Structures 1.2.2.1. Slender RC Bridge Columns as Vital Elements in Modern Infrastructure Slender RC bridge columns are prevalent in modern infrastructure due to the need for elevated highway interchanges. Architecture and geometry of the interchange often dictate the external dimensions of bridge columns. This can lead to slender columns with very large aspect ratios. The inelastic response of such columns plays a crucial role in earthquake-resistant design of elevated highway intersections located at seismically active regions. The Judge Harry Pregerson 3 interchange in Los Angeles, CA, for instance, is one of many bridges with slender RC columns located in seismically active regions close to active fault lines. Arial and side-view pictures of the Judge Harry Pregerson interchange are shown in Figure 1-2. It can be seen from Figure 1-2 that slender columns are inevitable in elevated interchanges. Also, a map that depicts active faults around Judge Harry Pregerson interchange is provided in Figure 1-3, where the noted interchange is marked by a blue star at the junction of Interstates I-105 and I-110. It is evident from Figure 1-3 that slender columns in the Judge Harry Pregerson interchange are expected to withstand probable earthquakes caused by the nearby faults. Figure 1-2: Slender RC bridge columns of Judge Harry Pregerson interchange in Los Angeles, CA 4 Figure 1-3: Map of active faults near judge Harry Pregerson interchange shown by the blue star 1.2.2.2. Increased Slenderness due to Soil-Structure Interaction Bridge columns supported on pile groups can experience an unexpected increase in their effective shear span length due to soil-structure interaction effects during earthquakes, despite having nonslender dimensions. That is, the interaction of the column foundation with the soil beneath it can affect the distribution of moments along the column height in a way that the point of contraflexure is shifted towards one end of the column. Considering the fact that column length is measured from the point of maximum moment to the point of contraflexure, the effective column aspect ratio can significantly increase due to soil-structure interaction. Recent research on dynamic response of RC bridge columns supported on pile groups showed that considering soil-structure interaction increases the effective column aspect ratio [3-5]. 5 To demonstrate the effect of foundation flexibility on increasing the effective length of bridge columns, three cases of double-curvature and single-curvature bridge columns with various levels of rotational fixity at the top are discussed next. In particular, a multi-bent double-curvature frame column with a fully fixed top, a multi-bent double-curvature column with a partially fixed top, and a cantilever single-curvature column with a free top are respectively shown in Figure 1-4, Figure 1-5, and Figure 1-6 to depict the effects of base rotational fixity on the effective column length and slenderness. It is evident from the figures that the effective length (Le), defined as the length between the point of contraflexure and maximum fixed-end moment, and the effective aspect ratio (ae), as per effective length over the cross-sectional dimension or Le/D for circular columns, increases in all three cases as the base rotational fixity decreases due to foundation flexibility and soil-structure interaction. The numeric values for the effective length of columns with different levels of rotational fixity at two ends, as presented in Figure 1-4, Figure 1-5, and Figure 1-6, were obtained using the method presented by Hellesland [6] for the effective length (Le) of unbraced columns according to Equation (1-1), in which β is factor that can be found by solving Equation (1-2). Le   H (1-1)    2         ktop kbot  tan   ktop  kbot            (1-2) In Equation (1-2), ktop and kbot are the non-dimensional rotational restraint stiffness which are defined according to ktop = Ktop/Kcol and kbot = Kbot/Kcol, where Kcol is the column bending stiffness (EI/H), Ktop and Kbot are the rotational restraint stiffness (M/θ) at top and bottom of the column respectively. 6 The fixity of the column end constraints is presented by the rotational degree of fixity (R), which is calculated according to Equation (1-3). It is worth noting that R = 1 represents a fully fixed end whereas, R = 0 represents a pinned end or zero fixity. Partially fixed restraints with R = 0.5 institutes a restraint condition in which the rotational stiffness of the end point is 2.4 times the bending stiffness of the column. Rtop  1 2.4 1 ktop Rbot  and 1 2.4 1 kbot (1-3) Once the effective lengths of the column at each end, i.e., Le,top and Le,bot are obtained according to Equation (1-4), the length of the column for calculating the design aspect ratio is equal to L = max( Le,top and Le,bot).  Rtop   Le,top  Le  R R  bot   top  Rbot   Le,bot  Le  R R  bot   top and (1-4) In case of double-curvature framed columns with a fixed top, the point of contraflexure moves downward from the middle, as base rotational fixity decrease, as shown in Figure 1-4. Accordingly, the effective length and the associated aspect ratio increases with the increased base flexibility. At ultimate, the base provides zero rotational fixity and the column acts as an inverted cantilever, where the effective length is equal to the total height. 7 Figure 1-4: Effect of base flexibility on the effective length of double-curvature bridge columns with fully fixed top In the case of a double-curvature column with a partially fixed top, which is represented by a rotational degree of fixity R = 0.5 in Figure 1-5, the point of contraflexure moves towards the base as the base rotational fixity decreases. At the same time, the location of maximum fixed-end moment moves below the base and the effective length of the column extends into the foundation and soil beneath it. The extension of the effective column length below surface has been extensively documented for integral pile shaft columns [7]. However, the principle applies to all columns in which foundation flexibility is significant. In case of a double-curvature column with a partially fixed top, the extension of the column effective length into the foundation peaks at Rbot = 0.5 and decreases afterward, as depicted in Figure 1-5. 8 Figure 1-5: Effect of base flexibility on the effective length of double-curvature bridge columns with partially fixed top In the case of a single-curvature cantilever columns, in which the rotational fixity provided by the super structure is negligible, the point of contraflexure remains at the top for all levels of rotational fixity at the base; as depicted in Figure 1-6. Yet, the location of the maximum fixed-end moment extends downward into the foundation and soil beneath the column, as the base fails to provide a fully-fixed restraint due to foundation flexibility and soil-structure interaction. It is evident from Figure 1-6 that the effective length of the column significantly increases as base fixity decreases. 9 Figure 1-6: Effect of base flexibility on the effective length of single-curvature bridge columns To highlight the effect of base fixity on the slenderness of bridge columns, the effective aspect ratio (ae) was calculated at different levels of base fixity for three columns with nominal aspect ratios a = 4, 6, and 8, where a = L/D and L is the shear span length, i.e., the length between point of contraflexure and maximum fixed-end moment, assuming a fully-fixed base for the column. The results for double-curvature framed columns with a fully-fixed top (Rtop = 1) and a partially fixed top (Rtop = 0.5) are respectively shown in Figure 1-7(a) and (b). It can be seen that the effective aspect ratio of columns can increase by more than 100% as the base rotational degree of fixity decreases. 10 (b) (a) Figure 1-7: Effect of base rotational fixity on the effective aspect ratio of double-curvature columns with: (a) fully fixed top (Rtop = 1), and (b) partially fixed top (Rtop = 0.5) The effective aspect ratio (ae) for cantilever columns with a = 4, 6, and 8 at different levels of base fixity was calculated and plotted in Figure 1-8. It can be seen that the effective aspect ratio and slenderness of cantilever columns dramatically increases with decreasing base fixity. It is worth mentioning that reaching zero fixity at the base of cantilever columns is not possible since the column becomes unstable. Figure 1-8: Effect of base rotational fixity on the effective aspect ratio of single-curvature cantilever columns 11 It is worth noting that the effect of soil-structure interaction on increasing the effective length of columns was presented to demonstrate practical cases for RC columns with large slenderness. Therefore, the length of the column, which is defined as the distance between the maximum moment and the point of contraflexure, is equal to the greatest effective length when soil-structure interaction is properly considered. However, herein this research, all columns are considered as cantilevers with fully fixed bases. Therefore, the length of the columns (L) is equal to the entire height of the column (H), i.e., L = H. 1.2.2.3. Slender RC Columns due to Use of High-Strength Concrete High strength concrete (HSC), which is generally referred to concrete with compressive strength (f ′c) greater than 50 MPa (7.25 ksi) [8], is nowadays commonly used thanks to recent advances in technology and reduction in costs [9-13]. Ultra-high strength concrete (UHSC), which typically has compressive strength (f ′c) greater than 100 MPa (14.5 ksi) [14], has also been recently introduced to construction sites, and high-strength materials are now more frequently used in load carrying structures [14]. Through the ever-increasing use of high-strength and ultra-high-strength concrete in practice, a substantial reduction in the cross-section of columns is obtained. This allows smaller sections for RC columns built using HSC and UHSC than ones built using normal-strength concrete. Thus, the recent advancements in concrete technology contribute to the prevalence of RC bridge columns with aspect ratios large enough to consider them as slender [9, 11]. In addition to the increased use of high-strength concrete in construction, the confinement effect provided by the closely spaced transverse reinforcement and steel tubes can increase the strength of the concrete core in columns [8, 15]. This is particularly the case for bridge columns that are 12 often provided with sufficient transverse reinforcement. As a result of this confinement, the core concrete can be designed for higher strengths. The additional strength gained by the effect of transverse confinement also contributes to smaller sections for bridge columns, which in turn increases their slenderness. Since the geometry and the stiffness of the column jointly contribute to the second order effects, the reduction of the cross-section increases the slenderness and amplifies the associated second order effects. 1.3. Slenderness Effects on Inelastic Response of RC Bridge Columns 1.3.1. Amplified Second-Order Effects The response of slender columns at the global and local levels is expected to be significantly different from shorter and stockier columns due to their greater flexibility and larger bending deformations. Second-order moments caused by the global displacements (P-Δ moments) and member bending deformations (P-δ moments) can potentially dominate the inelastic response and seismic performance of slender RC bridge columns [16]. Lateral displacement (Δ) and member deformation (δ) are shown in Figure 1-9, in which the gravitational forces applied to the top of the column in its deformed configuration are represented by P. 13 Figure 1-9: Global lateral displacement (Δ) and member deformation (δ) in deformed configuration of a cantilever column 1.3.1.1. Destabilizing Effect of Second-order P-Δ Moments Lateral displacement (Δ) of bridge superstructure induces an additional moment (P-Δ) to the column due to eccentricity of the gravitational loads. The P-Δ moment is applied to the base, independent from the deformed shape of the column. The second-order moment at the base level is accommodated by the RC section’s capacity and thus the effect reduces the actual shear force resisted by the column, leading to instability of the element [16, 17]. The effect of P-Δ moments on reducing the lateral load carrying capacity of RC columns is schematically depicted in Figure 1-10, in which a typical force-displacement response of well-confined ductile RC column is shown. It can be seen that the P-Δ moments reduce the actual load-carrying capacity of RC columns in proportion to the lateral displacement and axial load. The reduced strength and the 14 softening effect due to the second-order P-Δ moment can cause instability in slender columns with large axial load levels. Figure 1-10: Effect of P-Δ moment on force-displacement response of typical well-confined ductile RC columns The destabilizing effect of P-Δ moments at the global level can significantly reduce the load bearing capacity of slender columns and lead to sudden failure of the column before significant material deterioration and damage occur. Slender columns are expected to experience significant P-Δ moments due to their additional flexibility. The destabilizing effects of the P-Δ along with the reduction in strength of slender RC columns requires further study, and this research is aimed to provide knowledge on second-order effects in the response of slender RC columns. 1.3.1.2. Nonlinear Second-order Moment Profiles due to P-δ Effect Along with the columns’ lateral displacement at the global level, RC columns bend and deform as illustrated in Figure 1-11. The axial component of the gravitational forces or P·cos(Δtop/L) combined with the deformation of the column perpendicular to the element’s chord line (termed as δ) creates member second-order moments, referred to as P-δ effects. The P-δ moments reshape 15 the profile of bending moments along the column height resulting in a nonlinear distribution with larger effects at mid-height. Localization of inelastic deformations over the plastic region generates more flexibility and deformation, which makes P-δ effects even more significant within this region of the column. P-δ moments can thus increase the spread of inelastic deformations to elevations higher than predicted by a linear distribution of moments, as schematically shown in Figure 1-11. Although the base moment is not directly affected by the nonlinear distribution of the moments, extension of the inelastic region along the height of the column increases the flexibility of the column, which in turn leads to a larger lateral displacement at the global level. Amplified secondorder effects on the seismic design of slender RC bridge columns have received inadequate attention in the literature. Thus, this research aims at evaluating the slenderness effects on the inelastic response of RC columns. Figure 1-11: P–δ effects on the spread of plasticity along cantilever columns 16 1.3.2. Damage States for Performance-Based Seismic Design Reinforced concrete (RC) bridge columns are typically designed to endure moderate to strong earthquake events by experiencing significant inelastic deformations that are usually accompanied by some level of damage. Unlike conventional seismic design philosophies, in which collapse prevention under the maximum probable earthquake is the only design target [18, 19], modern seismic design guidelines based on performance-based seismic design (PBSD) philosophy aim at limiting damage for moderate, yet more probable, levels of earthquake risk [20-22]. As a result, it is critical for PBSD to properly identify intermediate damage limit states (IDLS) that can occur in RC bridge columns under moderate seismic demand levels. Successful implementation of PBSD thus requires a satisfactory level of confidence in predicting damage states for the design of RC bridge columns [23]. The main framework for PBSD according to Moehle and Deierlein [24] is shown in Figure 1-12, in which one of the main steps along the path from the facility information, e.g. location and site parameters, towards the final design decision is damage analysis (Step 3). Damage analysis is needed to obtain damage measures for different levels and probabilities of seismic hazard. The identified damage levels are often related to engineering design parameters such as displacement ductility and drift [23]. Determining the link between damage measures and design parameters has been the focus of the studies on damage states of RC columns. 17 Figure 1-12: Main framework for performance-based seismic design (PBSD) of structures Post-earthquake damage evaluations and investigations, as reported in EERI’s learning from earthquakes (LFE) reconnaissance archive [25], and large-scale laboratory experiments (as reported in PEER’s structural performance database [1],) have been traditionally the main sources of studying damage states in RC columns. Among others, first yield of the longitudinal reinforcement, initial spalling of the cover concrete and significant growth of the spalled region have been identified as IDLS for RC bridge columns [16]. Despite the relatively large amount of research on RC bridge columns, the occurrence of all damage states including IDLS has not been consistently reported [26]. In addition, columns with aspect ratios greater than 6 have not been investigated enough to establish damage states for slender RC columns [27]. Therefore, reassessment of damage limit states for slenderness effects is required. 1.3.3. Lengths of the Plastic Region (Lpr) and Plastic Hinge (Lp) Reinforced concrete bridge columns, designed according to current seismic design guidelines, are expected to dissipate seismic energy by experiencing significant plastic deformations. These plastic deformations, which are often associated with damage to the column, concentrate at the end regions where moment demands are maximum [28]. The portion of the columns over which 18 inelastic deformations take place is commonly known as the critical plastic region, and a ductile inelastic flexural response is necessary along this region to dissipate the seismic energy without losing the load carrying capacity. Design guidelines [20, 29-34] require special detailing to ensure such ductile response by providing adequate confinement steel reinforcement over the plastic region. Also, seismic retrofitting and strengthening strategies typically involve providing additional transverse confinement over the critical plastic region of earthquake-resistant RC columns with inadequate transverse reinforcement. Having an accurate account of the length of the plastic region (Lpr) is thus essential for seismic design, retrofitting, and strengthening plans. In slender columns, the additional flexibility due to slenderness generates excessive member deformations away from the chord-line (δ). These member deformations, in conjunction with the applied axial load (P), cause additional second-order P-δ moments that reshape the distribution of moments along the columns and contribute to the extent of the plastic region (Lpr). This is schematically depicted in Figure 1-13, in which the nonlinear moment profile leads to a significantly greater length of plastic region than the one from a linear moment profile. It should be noted that P-δ moments are in addition to the second-order P-Δ moments, which are the product of the axial load (P) and the global displacement of the column at top (Δtop) and are considered in the linear moment profile. Therefore, considering second-order effects of P-Δ on the length of the plastic region is not enough to fully capture the spread of plasticity along slender RC columns. Considering P-δ effects on Lpr is also crucial since underestimating Lpr for seismic design of RC columns leads to insufficient detailing and inadequate confinement over the critical region of the column to produce a ductile response. This adversely affects the reliability of the structure for sustaining significant inelastic deformations before failure. 19 The plastic region length (Lpr) should be distinguished from the plastic hinge length (Lp) since the former represents the physical region over which plastic deformations actually spread along the height of RC columns; whereas the latter is an analytical term used in lumped plasticity analysis models to combine all sources of inelastic deformations to determine the column’s post-yield displacement [35, 36]. Yet, the plastic hinge length (Lp) is clearly related to the length of the plastic region (Lpr). This relationship is based on the fact that Lpr captures the moment gradient component of Lp, which is commonly associated with the portion of lateral displacements that is caused by inelastic flexural deformations that start from the base level and spread along the column height [14-16]. The moment gradient component of Lp, which is referred to Lpm in this document, is defined by the equivalent length along the column height that has a constant curvature and is capable of generating plastic rotations identical to the sum of plastic rotations along the plastic region. The derivation of Lpm from the plastic curvatures along Lpr is schematically depicted in Figure 1-13. It can be seen that Lpm = 0.5Lpr if we assume a linear distribution for the inelastic curvatures along the plastic region. It is also evident from the figure that including P-δ moments in the analysis, which increases the length of the plastic region (Lpr) and the distribution of inelastic curvatures along Lpr, affects the moment gradient component of the analytical plastic region length (Lpm) as well. Considering that more flexible slender columns bend further and exhibit larger deformations under the applied lateral loads than stiffer shorter columns, current standards and guidelines for calculating the moment gradient component of Lp (or Lpm) require further evaluation to determine the effects of slenderness. 20 Figure 1-13: Effect of P-δ moments on the length of critical plastic region (Lpr) and the moment-gradient component of plastic hinge length (Lpm) 1.4. Research Significance and Contributions 1.4.1. Significance Current methods of seismic analysis and design RC bridge columns do not fully consider the effects of geometrical nonlinearities and second-order moments on the inelastic response of slender columns. Notably, the spread of the plastic region over the height of slender RC columns and the extent of their damage are currently predicted without consideration of second-order P-δ effects. In addition, predicting damage on slender RC columns when subjected to moderate ground motions is currently done according to experiments on shorter columns. Yet, it is not clear that to what extent the performance of slender columns will be affected by member deformations and Pδ effects. This research aims at understanding the possible effects of member deformation on the inelastic response of slender RC columns. Implications of slenderness on seismic design of RC bridge columns are also studied and recommendations are provided for the design of slender columns. As 21 a result of this research, slender RC bridge columns will be designed with higher reliability by accounting for slenderness effects. 1.4.2. Main Contributions The research presented here contributes to seismic analysis, design, and retrofit of RC columns through the contributions discussed below. 1.4.2.1. Expanded Structural Performance Database of RC Bridge Columns This research expands current understanding about the structural performance of RC bridge columns by testing four large-scale slender RC bridge columns with an unprecedented aspect ratio, which makes them more slender than any RC bridge column tested and reported before. Inelastic response of RC columns with extreme slenderness to lateral loads, equivalent to forces induced by earthquakes, is documented and reported for the first time. As a result of this experimental investigation, slenderness effects on the seismic performance of RC columns are better understood. 1.4.2.2. Evaluated Slenderness Effects on Inelastic Response of RC Columns Although the stability of slender columns has been extensively studied in the literature, the effects of second-order moments on the inelastic response of RC columns in the case of earthquake have not received enough attention. Data obtained from the test units studied here are used to measure the effect of column slenderness on the second-order moments. This research uses a novel approach to extract second-order effects directly from the test data by combining measurements from different instruments. It was thus possible to experimentally evaluate slenderness effects from the test data. 22 1.4.2.3. Finite Element Models to Capture Seismic Damage in RC Columns The research presented here led to the development of reliable FE models to simulate the inelastic response of slender RC columns and predict the damage that they undergo under seismic demands. The methods that are proposed for developing and verifying finite element models can enable researchers to study the inelastic response of RC columns via numerical simulations. Further, the reliance on experiments to study damage on slender RC members can be reduced by the use of high-resolution three-dimensional (3D) FE models. 1.4.2.4. Derived Analytical Expressions for Second-order Effects Analytical expressions and closed-form solutions were derived to accurately describe second-order effects of slenderness on the inelastic response of RC columns. Using this method, second-order moments are elegantly expressed in terms of mechanical properties and geometry of RC columns. This helps researchers to precisely evaluate the effects of different structural parameters, including concrete strength, axial load, and column aspect ratio, on the inelastic response of slender RC columns. In addition, the proposed analytical expressions were simplified to derive approximate formulas that can be used as design formulas. 1.4.2.5. Evaluated Sensitivity of Second-order Effects to Design Parameters The sensitivity of second-order effects on RC columns to different design parameters were evaluated and presented. This can help engineers adjust the design parameters in way that secondorder effects are minimized. Also, the most effective design parameters on the second-order moments and inelastic response of RC columns are identified. In addition, limits for each design parameter were established beyond which second-order effects on the seismic performance of RC columns are significant enough to be included in the analysis and design process. 23 1.4.2.6. Proposed Guidelines for Seismic Design of Slender RC Columns As a part of this research, design guidelines and recommendations were made to improve safety and reliability of structures with slender RC columns. Second-order effects that are more prominent in slender columns were fully considered in the proposed seismic design guidelines. 1.5. Research Hypothesis The response of slender RC bridge columns is expected to differ considerably from their shorter counterpart due to the greater influence of member deformations and P-δ effects. It is hypothesized that slenderness increases the error associated with predicted the inelastic response of RC columns if current analysis models and design practices are used. Slender RC columns exhibit substantial flexibility that makes them more susceptible to second-order effects. The P-δ effects reshape the distribution of internal bending moments along the column’s height. This leads to the spread of the critical plastic region and damaged zone to a larger zone than in shorter columns, where second-order effects are less pronounced. 1.6. Research Methods and Dissertation Outline The main objective of this research is to test the abovementioned hypothesis through experimental, numerical, analytical, and parametric studies. These four different approaches were undertaken to provide a comprehensive understanding of the effect of slenderness on the inelastic response of RC bridge columns. The present dissertation is organized in a way that each chapter discusses one of the research methods, starting from Chapter 3. Chapter 1, the current chapter, presents the introduction and motivation behind the topic. Chapters 3 to 6 discuss experimental, numerical, analytical, and parametric studies, respectively. The implications of this research in seismic analysis and design of slender RC columns are presented in Chapter 7, in which recommendations and guidelines for including the slenderness effects in design are provided. Chapter 8 is dedicated 24 to the conclusions from the study and suggestions for further research. The methods used in the presented research are introduced and briefly discussed in the following. 1.6.1. Experimental Studies Large-scale test units resembling slender RC bridge columns were constructed, instrumented and tested to create an experimental basis for evaluating the research hypothesis. The focus of these experimental studies was to evaluate second-order effects of slenderness on the inelastic response of four large-scale test columns with extreme aspect ratios that have not been experimentally studied before. This was achieved by extracting second-order P-Δ and P-δ moments from the test data and evaluating their impact on the inelastic response and seismic performance of RC columns. Structural response parameters, essential to seismic analysis and design of bridge columns, were also studied. Specifically, strength, deformation capacity, onset of yield, spread of plasticity, magnitude of second-order effects, and damage limit states of slender columns were experimentally investigated. The effects of several geometrical and structural parameters were studied by a set of test columns that were similar to each other except for the parameter of interest. Parameters considered in the experimental studies include the longitudinal reinforcement ratio, the axial load ratio, the aspect ratio and the displacement history. The experimental results were analyzed to extract the effect of geometrical nonlinearities, which are magnified for slender columns, on global and local performance parameters. Findings of this study demonstrated how inelastic response parameters, such as the spread of plasticity and ultimate displacement ductility, of slender columns is affected by their slenderness. The experimental studies are presented in Chapter 3 of this dissertation. The following objectives were pursued via conducting experiments on RC columns: 25 1. Obtaining inelastic responses of slender columns subjected to quasi-static cyclic and pseudo-dynamic lateral loading patterns. 2. Recording the onset of different damage states that occur during the loading cycle of the test units for use in performance-based seismic design of slender RC bridge columns. 3. Developing a reliable procedure to extract the effects of slenderness, namely P-Δ and P-δ effects, from test data. 4. Investigating the sensitivity of the second-order effects on the inelastic response of slender RC columns due to variations in different structural and design parameters. The following structural parameters were investigated. a. Ratio of longitudinal steel reinforcement, which affects the strength of RC columns. b. Aspect ratio, which represents the geometrical aspect of column slenderness. c. Axial load level, which is crucial in intensifying the slenderness effects in columns d. Displacement history, which was studied by comparing the experimental results from cyclic and dynamic tests 1.6.2. Numerical Studies Numerical simulations are important complementary tools for studying the inelastic response of RC columns, considering the high cost of large-scale experimental programs. Recent advances in finite element (FE) analysis methods along with the enhanced computing power available to researchers have promoted an increasing interest towards using FE simulations for predicting inelastic response in RC structures. Three-dimensional (3D) continuum-based FE simulation of RC columns can provide a unique opportunity for accurately predicting the structural response of columns as well as simulating seismic damage. The reason is that 3D continuum-based models offer a level of accuracy and detail that cannot be matched by 1D or 2D models. 26 The focus of the numerical studies in this research was on the development, validation, and use of 3D FE models to enhance the understanding of the interaction between slenderness of columns and their inelastic performance. Three-dimensional FE models were also utilized to simulate the probable damage in slender RC bridge columns under seismic demands. Use of 3D FE models to simulate the effects of slenderness on the inelastic response of RC columns was thus pursued. In addition, a comprehensive procedure is proposed for verification and validation of FE models that are developed to simulate the inelastic response of RC columns. This is done by quantitatively evaluating the errors associated with predicted responses of the columns at the global and local levels. The numerical studies are discussed in Chapter 4 of this dissertation. The following objectives were sought through the numerical studies: 1. Accurately predicting the global response of RC columns, e.g., the force-displacement response and dissipated energy through plastic deformations. 2. Obtaining the local responses of RC columns, e.g., curvature profiles and local strain demands. 3. Simulating seismic damage states of RC columns, e.g., the onset and the height of spalling region over the cover concrete. 4. Establishing a quantitative validation and verification process to evaluate 3D FE models used to simulate the inelastic response of RC columns. 1.6.3. Analytical Studies Analytical expressions and closed-form solutions, which are based on the mechanics of the structure, give a deeper insight into the effective factors and their impact on second-order effects and the inelastic response of slender RC columns. The goal of the analytical studies was to derive 27 mathematical expressions for the inelastic response of slender RC columns, which is primarily affected by their slenderness. Closed-form solutions were derived to study the effect of slenderness, P-δ effects in particular, on the inelastic response of RC columns using nonlinear beam-column theory. Also, analytical expressions for calculating the length of the plastic region considering P-δ effects was sought in this research. The analytical studies are presented in Chapter 5 of this document. The following objectives were pursued by conducting the analytical studies: 1. Developing closed-form solutions to the length of the plastic region (Lpr), which accounts for the second-order effects in slender columns. 2. Deriving analytical expressions for the magnitude of P-δ moments in slender RC columns, considering their inelastic response. 3. Providing simple and approximate, yet sufficiently accurate, design formulas, to be used in bridge design practice, to estimate the extent of P-δ effects in slender RC columns. 1.6.4. Parametric Studies In this part of the research, the analytical expressions derived in the previous section were utilized to determine the magnitude by which the inelastic response of RC columns is affected due to their slenderness. Also, a slenderness limit for RC bridge columns is defined in terms of design variables beyond which second-order effects on the inelastic response of RC columns cannot be ignored. In addition, the influence of different structural, geometrical, and material properties of RC columns on their vulnerability to second-order effects was evaluated. Finally, simple expressions are proposed to predict the effects of second-order P-δ moments on the length of the plastic region 28 (Lpr). The parametric studies are presented and discussed in Chapter 6 of this dissertation. The following objectives were pursued by conducting the parametric studies: 1. Defining parameters to distinguish a slender RC column from a non-slender one. 2. Establishing a criterion to decide whether considering the slenderness effects in seismic design of RC columns is necessary. 29 CHAPTER 2 2. LITERATURE REVIEW 2.1. Previous Experimental Studies on the Inelastic Response of RC Columns Experimental studies have been a great source for acquiring knowledge about the structural response of RC bridges for many years [3, 37-39]. With the introduction of new structural materials and design techniques, further experimental studies were needed to investigate the response of RC bridges [40-45]. In this chapter, previous studies that are closely related to the subject of the presented research are briefly discussed. 2.1.1. Tests on Slender Columns Early experimental studies on slender RC bridge columns have been reported by Dodd [46]. One sixth-scale column with an aspect ratio of 10 and an axial load ratio of 0.4f ′cAg was dynamically tested on a shake-table. Results from this study qualitatively demonstrated the effect of secondorder effects on the spread of the plastic region through visual observations of damage on cover concrete. Dodd [46] also discussed the global instability resulting from the slenderness of the columns by studying their force-displacement response. However, enhanced understanding of 30 second-order effects on the seismic response of slender RC columns needs a quantitative evaluation at global and local levels with consideration of current design practice, which generally adopts lower axial load ratios. Lehman et al. [37, 47] reported more recent experimental studies on half-scale slender RC columns designed according to modern codes. The effect of increasing slenderness (up to 10) on the reduction of the strain limit state for spalling of the cover concrete was reported. However, the effects of P-Δ and P-δ were not directly addressed. 2.1.2. Tests on Shear-Dominated Columns Hines et al. [35, 48] studied local effects of shear cracking on the spread of the plastic region in RC columns with aspect ratios up to 6, where shear has a more significant role than second-order effects. Yet, similar approaches for determining the spread of plasticity in slender bridge columns for which second-order effects govern remain underexplored. 2.1.3. Tests on High Axial Load Columns Second-order effects on RC building columns have been also studied. Bae and Bayrak [49] conducted experimental studies on the effect of aspect ratio on ultimate drift ratio of building columns. They reported significant loss of drift capacity when the aspect ratio increased from 5 to 7, the largest aspect ratio tested experimentally. This was contrary to the fact that columns with high aspect ratios are considered ductile members with adequate capacity to deform before ultimate failure at the material level [50]. Although, Bae and Bayrak [51] extended the results of their findings to columns with aspect ratios up to 10 by proposing a new plastic hinge model that considers slenderness effect, the derivation and calibration of the model were based on 31 experimental data from test columns with lower aspect ratios. Experimental data has not been available to evaluate the proposed model for columns with aspect ratios beyond the testing range. Barrera et al. [27, 52] tested RC building columns with aspect ratios up to 15 and documented instability as the ultimate failure mode for slender columns. It was concluded that displacement ductility, as obtained from the instability limit of slender columns, decreases with the longitudinal reinforcement ratio but does not necessarily decrease with aspect ratio. They noted that further comment on the influence of slenderness on deformation capacity of columns was not possible due to limited experimental data on slender columns. 2.2. Previous Experimental Studies on Column Slenderness Effects The effects of slenderness and second-order moments on the response of RC columns has been primarily investigated through experimental tests. Although experiments have offered a valuable basis to study the response of slender columns, overreliance on the empirical results has limited our understanding about the interaction between column slenderness effects and different design parameters. The limitation is caused by restrictions on the range of test units and the way test data is interpreted. This could be a factor on the lack of agreement among researchers about the effects of slenderness on the inelastic response of RC columns. Some researchers stipulate that the plastic hinge region is independent of the column slenderness and the axial load level [17, 53]. This conclusion was based on empirical results from tests conducted on non-slender RC columns with aspect ratios less than 6, which made these test units less susceptible to P-δ effects [27]. More recent experimental studies, such as the one by Bae and Bayrak [49], on test units with aspect ratios greater than 6, found a strong effect of slenderness on the plastic region length. Accordingly, new plastic hinge models were proposed to include the combined effect of axial load and column aspect ratio [51]. The different positions regarding slenderness effects, e.g., member flexibility and 32 P-δ moments, on the plastic hinge region length can be attributed to two main reasons: (1) largescale experimental studies usually involve a small number of test units that makes it hard to infer the sensitivity of the results to different parameters; and (2) the slenderness of RC columns has been traditionally defined by their aspect ratio. Current lumped-plasticity (Lp) models and expressions for the length of the critical plastic region (Lpr) either ignore the element’s slenderness [26, 53-57] or define it in terms of aspect ratio (L/D) [51, 58, 59]. Therefore, a better measure of slenderness needs to be implemented in lumped-plasticity models, consistent with nonlinear mechanics of axially-loaded beam-columns. 2.3. Previous Numerical Studies on Predicting Seismic Damage in RC Columns Finite element (FE) analyses have been a reliable method to numerically simulate the seismic response of RC bridges in lieu of experimental data. Models developed using one-dimensional beam-column elements have been mainly used to capture the global response of bridges. However, 1D models are not capable of simulating local responses of RC structures, such as spread of plasticity along columns’ height and seismic damage. As a consequence, two-dimensional (2D) and three-dimensional (3D) finite element models have recently attracted attention among researchers who focus on the local inelastic response of RC structures [60-63]. In this section, pertinent cases of previous research on FE analysis of RC structures are discussed. 2.3.1. 1D and 2D Finite Element Simulations Recent advances in finite element (FE) analysis methods along with the increasing computing power available to researchers have promoted an increasing interest towards the use of FE simulations for predicting damage limit states in RC structures. Nonlinear 1D frame models, as proposed by Neuenhofer and Filippou [64, 65], in conjunction with distributed plasticity formulations [66-70] can partially capture the effect of geometrical nonlinearities on the inelastic 33 response of RC columns. However, lumped-plasticity (Lp) models are widely used to analyze RC columns mainly because they offer a more economical approach towards simulating the inelastic behavior of RC columns than distributed plasticity models [26]. The accuracy of lumped plasticity models is comparable to the distributed models since the inelastic deformations are expected to concentrate at the ends of the elements [28]. Iranmanesh and Ansari [71] evaluated damage limit states for RC bridge columns using distributed plasticity fiber elements. An energy-based damage index, determined for different lateral displacement levels, was used to evaluate damage states. A similar approach was used by Kim et al. [72, 73] to study damage limit states of RC columns, where two-dimensional (2D) plane-stress FE analyses were used to compute damage indices. The predicted damage could only be categorized under general levels, such as minor, moderate and severe. Thus, identifying the exact mechanism of the damage state, similar to experimental observations of yielding and spalling, was not feasible. Similarly, Zhao et al. [54] used 2D plane-stress elements to determine the onset of different damage states such as rebar yielding and concrete crushing. Yet, the 2D model is not capable of visualizing the damage region. 2.3.2. 3D Continuum-Based Finite Element Simulations Damage indices work well for evaluating general condition states but provide limited detail regarding the type of damage. Strain limit states at the material level can be alternatively used for identifying element damage limit states since intermediate damage limit states can be directly linked to strain demands in the concrete and reinforcement [16]. Recent research [74, 75] has shown that use of beam-type finite element models in conjunction with strain limits at the material level are not reliable for predicting intermediate damage states of RC columns. This can be attributed to the details lost by modeling the complex material behavior of concrete and steel when 34 simplifying the three-dimensional (3D) response as unidirectional. Contrary to one or 2D models, 3D continuum-based finite element analyses provide a high level of detail for simulating material and structural response for evaluating damage states in RC columns. Continuum-based 3D FE models have long been used to perform structural analyses on RC members and structures. A great deal of results at the local and element levels, which can be utilized to determine limit states, are typically generated. Yet, the potential of using continuumbased 3D FE simulations for identifying and establishing damage limit states remains untapped. The work by Alemdar et al. [76] is a recent demonstration of using 3D FE models for studying the response of RC columns at a local level. Their work exemplifies the potential for using FE simulations in predicting the displacement and rotation profiles along the column’s height. However, interpreting the state of damage and identifying the associated damage limit states based on the extracted local responses have not been extensively reported in the literature. 2.4. Previous Analytical Studies on Second-order Effects 2.4.1. Large-Deflection Solutions to Elastic Columns The effect of P-δ on beam-column members, in general, has been rigorously studied in largedeflection theories of nonlinear mechanics and exact closed-form solutions for cantilevers subjected to different loading and boundary conditions have been presented. Wang [77] proposed a solution to the large-deflection problem of inclined cantilever columns under the effect of end loads using linear elastic material properties. Lee [78] extended the solution to large deflections of cantilevers into a nonlinear elastic constitutive equation by defining a power function for stressstrain response. Aristizábal-Ochoa [79] studied the effect of semi-rigid boundary conditions for slender beam-column elements on the closed-form solution. 35 Semi-exact solutions to cantilever elements with consideration of large-deflections of cantilever have also been provided [80, 81]. Despite the abundance of closed-form solutions to the problem of a cantilever column, their application for studying the inelastic response of RC columns has been hindered due to complexity of the solutions. Furthermore, the issues specific to common practices of seismic analysis and design of RC bridge columns has not been directly addressed in the existing closed-form solutions. 2.4.2. Nonlinear Beam-Column Theory Despite the abundance of closed-form solutions to the problem of a cantilever column, their application for studying the inelastic response of RC columns has been hindered due to the solutions’ complexity. Simpler derivations based on nonlinear beam-column theory, which were mainly developed to address the stability of steel columns and to magnify moment demands obtained from first-order analyses, provide solutions to the nonlinear response of beam-column elements [82-84]. Yet, these solutions do not address the issues specific to the seismic analysis and design of RC columns. 2.5. Previous Models for Lpr and Lp Design guidelines such as ACI [33], AASHTO [20], and EN [31] require special detailing and additional reinforcement for a ductile inelastic response over the plastic region. The plastic region length (Lpr) is thus an essential parameter in the seismic design of RC bridge columns [36]. The length of the plastic hinge (Lp) is another important design parameter that characterizes inelastic response of RC column. In this section, previous models and specifications for Lpr and Lp are reviewed. 36 2.5.1. Current Seismic Design Specifications for Lpr Seismic design guidelines require special detailing and enhanced confinement steel reinforcement in the plastic regions of RC columns in order to ensure stable ductile response. The ACI code (ACI-318-14) [33] and Eurocode 8 for buildings (EN 1998-1:2004) [30] specify the length of the critical region (lo and lcr as referred to by ACI and EN-1, respectively) in columns of ductile moment frames to be the larger of: (a) maximum dimension of the cross section, (b) one-sixth of the clear span, or (c) 450 mm. The length of the special detailing zone (lo) according to the Canadian standard (CAN/CSA-A23.304) [29] for axial load ratios (P/fc′Ag) less than 0.5 is the greater of: (a) 1.5 times the largest cross-section dimension, or (b) One-sixth of the clear span length. The AASHTO seismic design guidelines [20] define a plastic region length (Lpr as referred to by AASHTO) in bridge piers as the largest value of: (a) 1.5 times the dimension of the cross section in the bending direction, (b) length of the region where the moment demand exceeds 75% of the maximum plastic moment, or (c) the analytical plastic hinge length (Lp). Caltrans [34] mandates a ductile response for the plastic hinge region (Lpr) in a way similar to the AASHTO recommendations with the exception of the last criterion, which is switched by the one- 37 fourth of the distance from the location of maximum moment to the point of contra-flexure. Therefore, the length of the plastic region according to Caltrans is greater value of: (a) 1.5 times the dimension of the cross section in the bending direction, (b) length of the region where the moment demand exceeds 75% of the maximum plastic moment, or (c) one-fourth of the distance from the location of maximum moment to the point of contra-flexure. The New Zealand standard (NZS 3101) [32] and Eurocode 8 for bridges (EN 1998-2:2005) [31] require the ductile detailing length (ly and lh as they are respectively referred to by NZS and EN2) to be the greater of: (a) dimension of the cross section in the direction of the resisting moment, or (b) the length of the region over which the design moment exceeds 80% of the end moment for P/fc′Ag < 0.25. It is worth noting that P/f ′cAg < 0.25 is often the maximum axial load ratio in current bridge columns. The noted design recommendations for the length of the plastic region need to be reassessed for slenderness effects in light of the test data reported here. 2.5.2. Previous Lpr and Lp Models Most researchers have dealt with Lpr as the moment gradient component for plastic hinge length (Lp) models [26, 51, 53]. As a consequence, explicit expressions for Lpr, similar to the models proposed by Hines et al. [35], and Dowel and Hines [85], are rare in the literature. The length of the plastic region according to Hines et al. is provided in Equation (2-1), where T and Ty,avg are 38 flexural tensile force resultants after yield and at yield, respectively. jd is the distance between flexural tension and compression centroids; and V is the horizontal force applied to the columns. At, fyst, and s respectively are the area, yield stress, and pitch length of transverse reinforcement steel; f1 is the principal tensile stress; and w is the effective section width. V  jd  L pr  (T  Ty ,avg )  A f  V  2 t yst  f1w   s  (2-1) Although, current Lp models have distinct terms for the moment gradient component, the relationship between the moment gradient term and Lpr is not well-defined due to uncertainty about the distribution of inelastic curvatures over the plastic region. If a linear distribution for the inelastic curvatures is assumed, which is reasonable for most RC columns [48], a factor of 2 is used to relate the lumped-plasticity models to the extent of the plastic region according to Equation (2-2): L pr  2L p  Lsp  (2-2) where Lsp represents the terms of Lp model that accounts for the strain-penetration (or bond-slip) effect. While Lp models do not characterize the physical length of the plastic region, it is of importance to reassess the moment gradient components of Lp models with respect to experimental results from slender test units. 2.6. Slenderness Effects in Previous Lpr and Lp Models Dowell and Hines [85] and Hines et al. [35] proposed expressions to predict the extent of the plastic region (Lpr). However, the expressions are expected to underestimate Lpr for slender columns since the effects of member deformation (δ) and the associated second-order P-δ moments on Lpr are neglected. The same holds for the moment gradient component of all commonly used plastic hinge 39 (Lp) models, such as the models by Priestley and Park [53], Priestley et al. [16], Bae and Bayrak [51], Berry et al. [26], and Goodnight et al. [57], where the combined effects of column slenderness and axial force (P-δ effects) on Lpr are essentially neglected. This explains, to the author’s knowledge, the reason that most of the available Lpr expressions lack any term to include the effect of slenderness. Therefore, developing an expression for Lpr that includes the effects of slenderness and second-order nonlinear moment gradient on the extent of the plastic region is essential for the proper design of slender RC columns. 40 CHAPTER 3 3. EXPERIMENTAL STUDY 3.1. General In this chapter, experimental tests on four large-scale reinforced concrete bridge columns with aspect ratios 10 and 12 are reported. The test units were constructed and tested at Michigan State University’s Civil Infrastructure Lab (CIL) to study column slenderness effects, namely P-Δ and P-δ moments, on the inelastic structural response and seismic performance of RC bridge columns. The results from these experiments significantly enhanced the current understanding of seismic response of slender columns and contributed to the existing database for the performance of RC columns by adding data from columns with large aspect ratios. The destabilizing effect by P-Δ moments was studied using stability indices proposed in the literature. The effect of nonlinear distribution of bending moment profiles and P-δ moment on the length of the plastic region (Lpr) along column height was experimentally evaluated. The results of this experimental investigation were utilized to evaluate seismic response of slender RC bridge columns. 41 3.2. Test Units Four test columns were designed, constructed, and tested as one-half-scale prototypes of actual RC bridge columns to experimentally evaluate slenderness effects on their inelastic structural response. 3.2.1. Geometry All test columns had a circular cross-section with a diameter of 610 mm (24 in.) and were detailed to behave in a ductile manner with a flexure-dominated response. All columns were designed to have larger than average aspect ratio. The test units were similar in overall design, yet differed in one or more test parameters to study their isolated effect. 3.2.2. Test Parameters Three main design parameters that play a significant role in structural response of RC bridge columns were studied. These parameters are discussed in detail in the following section. 3.2.2.1. Aspect Ratio Slenderness of RC columns is generally represented by their aspect ratio (L/D), which is defined for circular columns by the ratio of the length (shear span length or L) to section diameter (D). Test columns were designed with aspect ratios of 10 and 12 to capture the effect of L/D. All columns had the same diameter. Therefore, the length (height) of test columns, as measured from the base level to the center of the top loading block, was 6.096 m (240 in.) and 7.315 m (288 in.) for aspect ratios 10 and 12, respectively. 3.2.2.2. Longitudinal Reinforcement Ratio Test columns were designed with two different longitudinal reinforcement ratios (ρsl), which is defined by the ratio of the total area of longitudinal reinforcement (Asl) to the section gross area 42 (Ag). One test column was designed with ρsl of 3% to represent columns with relatively heavy reinforcement, whereas the three remaining columns were designed to have lower ρsl at 1.5%. Considering the 1% and 4% limits on ρsl, imposed by design specification codes as discussed later in this chapter, the test matrix reasonably covered the variations in longitudinal reinforcement ratio. 3.2.2.3. Axial Load Ratio The test columns differed in their axial load ratio (P/f ′cAg). Considering the limitations imposed by the capacity of test equipment available at the CIL, a maximum of 1334 kN (300 kips) was applied to the test columns. Two columns were tested with a nominal axial load of 712 kN (160 kips) and the other two were subjected to a nominal axial load of 1334 kN (300 kips). 3.2.3. Naming Convention For ease of reference, the four test units were identified according to the following designation: (1) the first letter denotes the university where the column was tested (M = Michigan State); (2) the next two digits define the slenderness ratio (L/D), and (3) the last two digits define the longitudinal reinforcement ratio in percent. The letters “C” and “D” at the end of the tag indicates the condition of loading, where “C” represents reversed cyclic and “D” denotes pseudo-dynamic loading conditions. The geometry and reinforcement layout of the test columns are shown in Figure 3-1. Also shown is the effective height of the columns (L) that was measured from the columnfooting interface to the height of lateral loading, which is at the mid-height of a loading block at the top of the column unit. Therefore, columns with aspect ratio of 12 and 10 had effective heights of 7315 mm (288 in.) and 6096 mm (240 in.), respectively. 43 3.2.4. Design Specifications 3.2.4.1. Longitudinal Reinforcement Longitudinal reinforcement ratio was one of the structural and design parameters studied in the experiments. The minimum and maximum reinforcement ratio for the longitudinal steel are 1% and 4%, respectively, according to the Caltrans seismic design guidelines. These limits were adjusted according to the AASHTO specifications when necessary. Two design configurations were used for the longitudinal reinforcement of the test columns. The first configuration had 1.5% reinforcement ratio, which was provided by 16 bars with a nominal diameter of 19 mm (6/8 in.). The other configuration had 3% reinforcement by using 18 bars with a diameter of 25 mm (1 in.). 3.2.4.2. Transverse Reinforcement Maximum spacing for lateral reinforcement was obtained based on Caltrans guidelines that require the spacing to be smaller than: • one-fifth the least dimension of the cross-section, • six times the nominal diameter of the longitudinal reinforcement bar, or • eight inches (~ 200 mm). In this project, the spacing was selected as 76 mm (3 in.) for all test columns along their entire length. The minimum shear reinforcement area for the critical plastic region was calculated based on the specifications by Caltrans and AASHTO. It was found that using a spiral with a nominal diameter of 13 mm (0.5 in.) spaced uniformly at 76 mm (3 in.) was sufficient for all test columns. 44 3.2.5. Components and Reinforcement Layouts of Test Units As shown in Figure 3-1, the column components included a 1676 mm (66 in.) square footing to provide fixity at the base. In addition, there was a 1066×711×457 mm (42×28×18 in.) loading block, which ensured an evenly applied load to the top of the column. Figure 3-1: Test columns’ geometry and reinforcement layout 3.2.5.1. Column Element The column reinforcement was designed to provide the column with sufficient strength and ductility, and to meet specifications outlined in the AASHTO Seismic Design Specifications [86] and Caltrans Seismic Design Criteria [34]. Longitudinal reinforcement was provided with a 25 mm (1 in.) concrete cover. The reinforcement was extended into the footing and the loading block with the longitudinal bars anchored into the footing with 90-degree hooks. Confinement was 45 provided with transverse steel reinforcement in the form of a continuous #4 spiral at 76 mm (3 in.) pitch spacing. This resulted in a transverse reinforcement ratio (ρst) of 1.2% as defined by Equation (3-1). It is worth noting that the provided transverse reinforcement satisfied the minimum shear reinforcement ratio, which is 0.005 for columns designed for high-seismicity regions.  st  4 Asp Ds (3-1) where Asp is the cross-sectional area of the transverse steel bar, D′ is the outer diameter of the spiral and s is the spacing of the spiral pitch. Detailed geometrical properties of all test units are presented in Table 3-1. Details on the reinforcement steel geometry are also given in Table 3-1, where dsl is the diameter of longitudinal steel reinforcement, dst is the diameter of transverse reinforcement (spiral), Ag is the gross cross-sectional area of columns, and P is the axial load. 3.2.5.2. Footing The column footing was designed to resist maximum moment and shear forces at the base of the column due to lateral loading applied at the top loading block. The footing geometry was a 1676 mm (66 in.) square with a 483 mm (19 in.) depth. The width and base dimensions were chosen to have sufficient space for the column longitudinal anchor bars, as well as include enough points for post-tensioning the entire test unit to the ground. The footing reinforcement consisted of two mats of #6 reinforcing bars in the direction of loading with 90-degree anchor hooks. Transverse reinforcement was provided by #5 straight reinforcing bars, and #3 stirrups crossed nearly every intersection of the main reinforcement to resist shear stresses. The detailed footing design is shown in Figure 3-2. 46 Table 3-1: Geometry and reinforcement details for test units Test Unit L/D M1230 M1215 M1015C M1015D (M123005) (M121505) (M101510) 12 12 10 10 D mm (in.) L mm (in.) dsl mm (in.) 610 (24) 610 (24) 610 (24) 610 (24) 7315 (288) 7315 (288) 6096 (240) 6096 (240) 25 (1) 19 (0.75) 19 (0.75) 19 (0.75) ρsl 3.1% 1.6% 1.6% 1.6% dst mm (in.) 12.7 (0.5) 12.7 (0.5) 12.7 (0.5) 12.7 (0.5) ρst 1.2% 1.2% 1.2% 1.2% P/Agf ′c 5.3% 5.7% 8.8% 9.6% 712 (160) 712 (160) 1334 (300) 1334 (300) 25 (1) 25 (1) 25 (1) 25 (1) 76 (3) 76 (3) 76 (3) 76 (3) P kN (kips) cover mm (in.) s mm (in.) 3.2.5.3. Loading Block The reinforced concrete block at the top of the column was designed as a loading block to ensure no damage would occur in the column when axial and lateral loads were applied. The block’s function was to provide a flat surface to apply axial and lateral loading to the column at its effective height. The actuator was attached to the block using high-strength rods passing through these pipes. 47 15 #5 @ 4.5 in. Top and bottom mats straight bars 66 in. 16 #6 C bars @ 4.5 in. Top and bottom mats 2.5 in. PVC 24 in. 66 in. 9 in. 9 in. (a) Layer of #5 straight bars #3 stirrups at every crossing #6 C-bars 19 in. 1.5 in. cover 66 in. (b) Figure 3-2: Footing Design as shown in (a) plan and (b) elevation 3.3. Test Setup The columns were tested as free-standing cantilevers with a footing affixed to the laboratory reaction floor and lateral load applied with a servo-controlled hydraulic actuator at the column top through a loading block. The test setup is schematically shown in Figure 3-3. Also shown are the steel frame and support blocks to react against the lateral forces from the hydraulic actuator, and 48 the fixtures used for applying loads. Pictures of the actual test setup for columns M1230 and M1215 during the tests are shown in Figure 3-4. The deformation of the test columns due to lateral loads is also evident from the pictures. (b) Figure 3-3: Typical test setup and fixtures to apply axial and lateral loads as shown in (a) 3D and (b) side views 3.3.1. Steel Loading Frame The hydraulic actuator used for lateral loading was supported by a structural steel reaction A-frame atop three reinforced concrete blocks to bring the frame to the test columns’ height. The A-frame was designed to attach the actuator at several locations for columns with different aspect ratios. 49 Loading Frame Hydraulic Actuator Hydraulic Loading Actuator Frame Reaction Wall Reaction Wall Figure 3-4: Test setup for columns (a) M1230 and (b) M1215 as shown at their deformed configuration 3.3.2. Axial Loading Setup Two levels of constant axial loads, namely, 712 kN (160 kips) and 1334 kN (300 kips) approximately corresponding to 0.05 f ′cAg and 0.1 f ′cAg were applied to the test columns throughout the tests. The axial load was applied using hydraulic jacks and two external highstrength rods reacting against the top block and anchored to the laboratory floor at the lower end. A spreader beam was placed above the loading block in the transverse direction to evenly apply the axial load. Above the beam, the rods passed through two hydraulic jacks that continuously applied half the desired load each during testing, as shown in Figure 3-5. At the base of the column, the rods pass through the footing and in to the laboratory floor anchors. Tapered slots in the footing provided space for the rods to sway as the column deformed. 50 Figure 3-5: Axial load application setup 3.3.3. Lateral Load Application Lateral loading was applied to the top of the column by means of a hydraulic actuator connected through the loading block, as shown in Figure 3-6. Two distinct load protocols, namely, reversed cyclic and pseudo-dynamic patterns were used for lateral loading of the test columns. The reversed cyclic loading pattern, which was characterized with symmetric responses, was determined according to the results from static analyses of columns under monotonic loading, as discussed later in this chapter. The pseudo-dynamic loading pattern was defined using the results from a nonlinear time-history dynamic analysis of a column subjected to earthquake records. 51 Figure 3-6: Lateral load application setup 3.3.4. Resultant Applied and Reaction Forces The applied and constraint forces on the test columns in a typical experimental setup, which is shown in Figure 3-7(a), were represented by three forces, as shown in Figure 3-7(b). Dictated by the direction of the high-strength rods used for applying the axial load, the direction of the axial load (P) follows the rotation of the chord line of the columns as illustrated in Figure 3-7(b), in which the axial load is directed towards an anchorage point rather than being vertical. Lateral loads were applied to the column by imposing horizontal displacements to the end side of the loading block. The force applied by the actuator is called Feff and the horizontal reaction force at the base of the column is called Fnet. These forces are related by Equation (3-2),      , Fnet  Feff  P sin  tan1  L  L bot    (3-2) where Lbot is the height of footing as measured from the column-footing interface to the anchorage point of the post-tensioning rods. 52 Figure 3-7: Schematic diagram of (a) loading and boundary conditions for test columns; (b) free body diagram of the applied and reaction forces on test columns 3.4. Instrumentation The test columns were instrumented to measure various global and local responses during the tests. Flexural curvatures were monitored at several sections along the height of the columns using sets of vertically oriented displacement transducers (DT) that measured extension and contraction at the extreme sides of columns. The distribution of the DTs along the height is illustrated in Figure 3-8 for column M1215. Besides the instruments that were placed to capture local responses of the test columns, horizontal displacement transducers, a clinometer (rotation-meter) and multiple load cells were utilized to trace the deflected shape, top displacement, rotation angle and the force transferred from the actuator to the column. Average shear deformations were also measured on three segments located at the bottom 1219 mm (48 in.) of the columns’ length. Foil-gage electric 53 strain gauges were attached to the steel reinforcement at key locations on extreme rebars to evaluate the local strain demands and detect reinforcement yielding. Figure 3-8: Instrumentation layout on column M1215 54 3.4.1. Displacement and Rotation Measurements 3.4.1.1. Lateral Displacement In order to measure the horizontal displacement of the column at intermediate heights, horizontal DTs were placed at seven elevations along the column length, as shown in Figure 3-9. The instruments were secured to a reference frame and extended to a target near the loading face of the column as shown in figure. Figure 3-9: Horizontal Displacement Transducers on Reference Frame 3.4.1.2. Top Rotation The rotation angle of the top loading block was measured at its center using a rotation meter. The data from the rotation meter was used to estimate the angle at which the axial and lateral loads were applied to the columns. 55 3.4.2. Deformation and Strain Measurements 3.4.2.1. Flexural Curvatures Section flexural curvatures of the column were measured by placing displacement transducers (DT) at the heights shown in Figure 3-8 at identical locations on two sides of the column in the direction of loading. To attach all instruments to the column, two rods were placed during construction, each protruding from the column concrete and connected to an L-shaped aluminum plate. The DT was attached to the face of the plate parallel to the column, while the perpendicular face was used as a target for the instrument. The described setup can be seen in Figure 3-10. Figure 3-10: Instrumentation for curvature measurement Average flexural curvatures associated with sections of the column at different elevations were experimentally determined for the segment of the column length by using the relative vertical displacements measured between two sections on two sides of the column. The average curvature for the segment wass calculated using Equation (3-3), 56  ht  hc , g  D  dt  d c  (3-3) where ϕ is the experimental curvature of the section, the gauge length g is the length of the segment under consideration, and D is the diameter of the column section. The variables dt and dc are the clear distances of the instruments from the surface of the column, where the subscripts t and c denote the tension and compression sides, respectively. The relative vertical displacements over the gauge length were measured by linear displacement transducers on the tension and compression sides of the column and are denoted by Δht and Δhc, respectively. The last two variables are related to the deformed gauge lengths gt and gc according to Equations (3-4) and (3-5). ht  gt  g (3-4) hc  gc  g (3-5) A schematic segment of a column instrumented by two vertical DTs to measure the average flexural curvature of the segment is depicted in Figure 3-11. Also shown in the figure are the deformed shape of the segment due to bending and the associated measurements in tension and compression. 3.4.2.2. Shear Strains Shear in the column was measured by installing three rectangular panels of sliding DTs on the face of the column perpendicular to the direction of lateral loading. These panels included a diagonal component to measure the shear deformation upon loading, as shown in Figure 3-12. Aluminum rods protruding from the column concrete were once again used to attach frictionless swiveling rod connectors to the aluminum panels. 57 Figure 3-11: Instrumentation setup for measurement of average curvatures on a segment of columns Figure 3-12: Shear instrumentation 58 3.4.2.3. Reinforcement Strains Strains were measured with electrical resistance quarter-bridge foil strain gages with 120-ohm resistance and a 5-milimeter gauge length. The strain gages were powered by a 10-volt external power supply during testing. Strain gages were placed on the two outermost longitudinal reinforcing bars in the direction of loading as well as the spiral on both follower and generator sides. 3.4.3. Force Measurements Load cells were used to measure the applied forces from the axial loading setup and the hydraulic horizontal actuator. Real-time load measurement from the actuator was also used to control the progress of the tests before reaching yield in the extreme longitudinal reinforcement. 3.5. Material Properties Concrete strength properties were evaluated through standard testing and the compressive (f ′c) and tensile (f ′t) strengths for test columns at the day of testing were obtained. Material properties for the steel reinforcement, namely, yield strength (fy), ultimate strength (fu), onset of strain-hardening (εsh), and ultimate strain (εsu) for the longitudinal (denoted by l) and transverse reinforcement (denoted by t) were determined based on data from three tensile tests for each of the reinforcing bars types. 3.5.1. Compressive Strength of Concrete Three cylinders 102 × 203 mm (4 × 8 in.) were tested in compression to obtain compressive properties of unconfined concrete for each test column. Results at the day of test are provided in Table 3-2 for all test columns. The stress-strain graphs obtained from compressive tests on concrete samples of column M1230 are shown in Figure 3-13. 59 Table 3-2: Concrete Compressive Strengths Test Unit f ′c MPa (ksi) Avg. f ′t MPa (ksi) Avg. S.D. S.D. M1230 M1215 46.2 (6.7) 2.06 (0.298) 3.37 (0.489) 0.110 (0.016) 42.7 (6.2) 3.99 (0.579) 3.16 (0.462) 0.117 (0.017) M1015C M1015D 51.8 (7.5) 2.36 (0.343) 4.34 (0.630) 0.469 (0.068) 47.6 (6.9) 2.63 (0.382) 3.67 (0.533) 0.578 (0.084) 3.5.2. Split Tensile Strength of Concrete Three cylinders were tested to obtain the split-tensile strength of concrete. The resulting strengths from these tests are shown in Table 3-2. Figure 3-13: Compressive response of concrete specimens from M1230 60 3.5.3. Uniaxial Stress-Strain Response of Reinforcement Steel 3.5.3.1. Longitudinal Reinforcement No. 8 Bars Coupons were made of the #8 reinforcing bars in order to characterize their tensile constitutive response. This figure also shows an image of a failed specimen. The three test samples gave very similar stress-strain data as shown in Figure 3-15. Based on these curves, the key parameters characterizing the uniaxial response of the reinforcement steel were obtained and are presented in Table 3-3. Figure 3-14: Stress-strain response of no. 8 reinforcing steel 61 fu Stress fy εsy εsu εsh εmax Strain Figure 3-15: Schematic stress-strain curve of reinforcing steel with key parameters Table 3-3: Reinforcing steel properties Reinforcement Type fyl or fyt MPa (ksi) ful or fut MPa (ksi) εsu mm/mm εmax mm/mm εsh mm/mm Longitudinal Longitudinal Transverse Rebar #8 Rebar #6 Rebar #4 486 455 469 (70.5) (66) (68) 738 689 689 (107) (100) (100) 0.08 0.11 0.12 0.12 0.11 0.12 0.01 0.008 0.005 3.5.3.2. Longitudinal Reinforcement No. 6 Bars Material properties of the #6 reinforcing bars were determined by testing three samples of the rebar. The resulting stress-strain curves are shown in Figure 3-16 for three specimens. The key parameters defining the uniaxial response of the #6 rebar are presented in Table 3-3. 62 Figure 3-16: Stress-strain response of no. 6 reinforcing steel 3.5.3.3. Transverse Reinforcement No. 4 Spirals Tension tests on the steel spiral were performed on three samples. The results of these tests are shown in Figure 3-17 in terms of stress-strain curves. It is seen that the samples failed at different ultimate strains; however, the curves up to 0.12 are very similar. Based on this data, a strain of 0.12 was selected as the ultimate strain for the #4 rebar. Other material properties of transverse reinforcement are provided in Table 3-3. 63 Figure 3-17: Stress-strain response of no. 4 transverse steel 3.6. Experiment Design 3.6.1. Preliminary Analyses 3.6.1.1. Section Moment-Curvature Analysis Sectional analyses were conducted to predict the moment-curvature (M-ϕ) response of the test columns. Fiber-discretized RC sections with different nonlinear constitutive models for confined concrete, cover concrete and steel reinforcement were used to capture the combined effect of axial load and bending moment. A force-based formulation, as per Spacone et al. [68], was used to obtain a stable M-ϕ response in case of loss of strength due to softening. The effect of confinement by the transverse reinforcement on the compressive response of the core concrete was considered by following the methods from Mander et al. [87] and Chai [88]. The uniaxial constitutive relation proposed by Mander [89] for steel was implemented to define the post-yield hardening response. Standard material tests were carried out on concrete and steel samples and the results were used to define the parameters for the constitutive models. 64 Moment-curvature responses for the test columns are shown in Figure 3-18. Also shown on the plots is the yield point (ϕy, My) that was used to define the location of the ideal yield. The yield curvature, ϕy, was calculated according to the method by Priestley et al. [7]; yet, My was adjusted to lay on the actual M-ϕ curve. The adjusted My allowed a coherent treatment of curvature and moment profiles that produced consistent results. A detailed discussion about the selection of the yield point properties and the merits of the proposed method is presented in the following section. Figure 3-18: Moment-curvature response of test columns from analysis 65 3.6.1.2. Yield and Ultimate Points Yield Moment (My) and Yield Curvature (ϕy) The yield point on the moment-curvature (M-ϕ) response of an RC section can be defined at (1) the first onset of yield, (2) the nominal yield, or (3) the ideal yield, as schematically depicted in Figure 3-19. First yield is associated with the yielding of the extreme longitudinal rebar in tension. The moment and curvature values at first yield are denoted as M′y and ϕ′y, respectively. The nominal yield point corresponds to a moment (Mn) that generates compressive strains larger than 0.004 in the extreme concrete fiber or tensile strains greater than 0.015 in the extreme rebar, whichever occurs first. The section curvature at nominal yield (ϕy) is given by ϕy = (Mn∙ϕ′y) / M′y. This method of defining the yield point of a section leads to a point located outside of the actual M-ϕ response with no physical correspondence between the yield moment and the yield curvature. To overcome this issue, Hines et al. [35] proposed an alternative method to define the yield point that requires tracking the flexure-induced tensile force resultant, which is not available from common moment-curvature analyses. Here, it is proposed to have an adjustment to the yield point from Priestley et al. [7] and the ideal yield point is defined as that which corresponds to an actual section moment (My) at the ideal yield curvature (ϕy). Figure 3-19: Schematic representation of the first, nominal, and ideal points 66 Bilinear idealizations of the M-ϕ curve according to different methods of defining the yield point are denoted as cases (1), (2) and (3) for the first, the nominal, and the ideal yield points, respectively; as shown in Figure 3-20 for test column M1215. The actual M-ϕ response of the RC section was also shown in this figure. Figure 3-20: Moment-curvature response for column M1215 with bilinear idealization options Ultimate Moment (Mu) and Ultimate Curvature (ϕu) The ultimate point on the M-ϕ curve was consistently defined for all four cases of bilinear and curvilinear M-ϕ responses as the point at which the strain in the confined concrete or steel rebar exceeds the ultimate crushing or fracture limits, respectively, whichever happened first. The ultimate crushing limit for concrete was defined per the energy balance method by Mander et al. [87]. The ultimate strain limit for the reinforcing steel was governed by low-cycle fatigue as proposed by Dodd and Restrepo-Posada [90]. Accordingly, the values of Mu and ϕu were taken as 67 the moment and curvature corresponding to the ultimate response of the RC section as listed in Table 3-4. Table 3-4: Key parameters of moment-curvature response for the test columns Parameter ϕy -3 (10 m-1) My (kN∙m) ϕu -3 (10 m-1) Mu (kN∙m) Mu / My M1230 M1215 M1015 9.62 8.59 8.49 958 552 668 125 123 126 1395 779 891 1.455 1.411 1.335 3.6.1.3. Anticipated Force-Displacement Response The moment-curvature response of the test columns was extended over the length of the columns to obtain an estimate of the force-displacement response of the test columns. This was achieved by using a force-formulation fiber-section element that implements the cross-section response. A force-formulation element was used over typical displacement-formulation model to be able to capture force and moment softening while ensuring equilibrium is satisfied. Considering the fact that the model consisted of only one element, which is the column, a large number of monitoring sections, 1000 sections, along the height was used to achieve accurate results. Having accurate preliminary results was needed before conducting the tests since the yield force and displacement were used to define the loading pattern. The results are shown in Figure 3-21, where the first and ideal yield points are highlighted. 68 Figure 3-21: Expected force-displacement response of test columns obtained from nonlinear structural analysis 3.6.2. Loading Protocol 3.6.2.1. Reversed Cyclic Loading Protocol Testing was done quasi-statically for three test units (M1230, M1215, and M1015C) following a fully reversed cyclic loading pattern. Four cycles were first applied in force control until the 69 theoretical first yield force, Fy', defined as the force corresponding to the theoretical yield moment (My′) was reached as given by Equation (3-6). Fy  M y L (3-6) The theoretical first yield force corresponds to a moment at base level that causes the onset of yielding on the extreme longitudinal reinforcing rebar in tension, as obtained from a momentcurvature analysis. The top displacement at the theoretical first yield, Δ′y, determined experimentally, was used to define the ideal yield displacement, Δy, corresponding to displacement ductility one (μΔ = 1). The ideal yield displacement Δy was determined using Equation (3-7), My y   M  y  y ,   (3-7) where My is the moment at which either the extreme concrete fibers reached εc = 0.004 in compression or the extreme steel fiber in tension reached εs = 0.015, whichever occurred first. The remainder of the test was conducted in displacement control with three cycles at each different displacement ductility level, as defined by Equation (3-8).    y (3-8) One full cycle was applied at each of the pre-yield load levels of F = 0.25F′y, 0.5F′y, 0.75F′y, and F′y. Three cycles were applied at each post-yield displacement levels of μΔ = 1, 1.5, 2, 3, 3.5 and 4. The columns were not subjected to larger deformation levels due to limitations on the actuator stroke (1016 mm (40 in.)). The lateral loading patterns for the columns tested according to the reversed cyclic loading protocol (M1230, M1215 and M1015C) are depicted in Figure 3-22, Figure 3-23, and Figure 3-24. 70 Figure 3-22: Reversed cyclic lateral loading pattern for column M1230 Figure 3-23: Reversed cyclic lateral loading pattern for column M1215 71 Figure 3-24: Reversed cyclic lateral loading pattern for column M1015C The lateral loading steps for columns tested using reversed cyclic protocol are listed in Table 3-5. It is worth noting that the pre-yield steps of the tests were conducted in force-control mode in which the target was a fraction of the yield force Fy'. Whereas, the post-yield steps were governed by the displacement targets defined as multiples of the yield displacement Δy. 3.6.2.2. Pseudo-Dynamic Loading Protocol One column (M1015D) was tested pseudo-dynamically according to the seismic response obtained from nonlinear time-history analyses. The column was analyzed as it was subjected to a ground acceleration. The displacement at the top of the column due to ground motion was recorded during the analysis and was imposed on the actual test column. The displacement history from the dynamic analysis had a random regime without any type of symmetry; however, it met the displacement limitations of the test setup since it did not exceed 508 mm (20 in.) in any direction. In addition to limits on displacements, the hydraulic actuator had limitations on the range of 72 velocities and accelerations that can be applied to the test unit. Unlike the quasi-static testing procedure, which is only restricted by displacement limits, i.e. the length of actuator stroke, the pseudo-dynamic tests were restricted by velocity and acceleration limits. Consequently, the test protocol was adjusted based on the actuator limitations. The process of defining the pseudodynamic loading protocol is discussed later in this document. The displacement history used as the lateral loading protocol for pseudo-dynamic testing of column M1015D is shown in Figure 3-25. Table 3-5: Target load or displacement levels for lateral loading using reversed cyclic protocol Test Column Step 0.25F′y 0.50F′y 0.75F′y F′y M1230 Force kN (kips) 27.8 (6.25) 55.5 (12.5) 83.3 (18.75) 111 (25) M1215 Displacement mm (in.) N.R. N.R. N.R. N.R. 173 (6.8) 259 (10.2) 345 (13.6) 518 (20.4) μΔ = 1.0 N.R. μΔ = 1.5 N.R. μΔ = 2.0 N.R. μΔ = 3.0 N.R. μΔ = 3.5 N.T. N.T. μΔ = 4.0 N.T. N.T. M1015C Force Force Displacement Displacement kN kN mm mm (in.) (kips) (kips) (in.) 16.7 25.5 N.R. N.R. (3.75) (5.75) 33.3 51.1 N.R. N.R. (7.5) (11.5) 50.0 76.6 N.R. N.R. (11.25) (17.25) 66.6 102.1 N.R. N.R. (15) (23) 137 112 N.R. N.R. (5.4) (4.4) 206 168 N.R. N.R. (8.1) (6.6) 274 224 N.R. N.R. (10.8) (8.8) 411 335 N.R. N.R. (16.2) (13.2) 480 N.R. N.T. N.T. (18.9) 447 N.T. N.T. N.R. (17.6) Note: 1- Cells that contain “N.R.” represent fields that are not relevant. Displacement targets are not relevant to pre-yield steps whereas force targets are not applicable to post-yield levels. 2- Cells that contain “N.T.” represent levels beyond test limits (508 mm (20 in.)). 73 Figure 3-25: Displacement history of pseudo-dynamic loading protocol for testing M1015D The displacement ductility history of test column M1015D is shown in Figure 3-26. Also shown is the effective lateral force applied by the horizontal actuator to the top of the column, normalized to the yield force (F′y). Figure 3-26: Pseudo-dynamic lateral loading pattern for column M1015D 74 3.7. Test Observations and Damage States Different damage limit states were observed during the cyclic tests. The onset of each damage state was linked to engineering design parameters of displacement ductility (μΔ) and drift ratio. The following describes the observed states of damage in the order of occurrence. 3.7.1. Tensile Cracking Tensile cracking initiated at a load level approximately equal to half that of the yielding limit (F = 0.5F'y), matching the predictions from sectional analyses. The extent of such damage along the height of test columns at the onset of cracking is presented in Table 3-6. Upon reaching the ideal yield condition (μΔ = 1) the crack height was documented. Also, the highest section to which the tensile cracks reached by the end of the tests is listed in Table 3-6. Governed by the spiral pitch, which was 76 mm (3 in.), cracks were generally spaced at multiples of 76 mm (3 in.). The spacing between cracks started at 152 mm (6 in.) but decreased to 76 mm (3 in.) by the end of the test. The cracking pattern of column M1230 at the end of the test is shown in Figure 3-27. Table 3-6: Extent of tensile cracking along the height of test columns Test Column Onset of Cracking (F = 0.5F′y) Ideal Yield (µ = 1) End of Test (µ varies) M1230 M1215 M1015C 3658 mm (12 ft) 3658 mm (12 ft) 1524 mm (5 ft) 5486 mm (18 ft) 6401 mm (21 ft) 5486 mm (18 ft) 6401 mm (21 ft) 3962 mm (13 ft) 4572 mm (15 ft) 75 Figure 3-27: Tensile cracking pattern of column M1230 at µΔ=3 3.7.2. First Yielding The onset of first yield is defined as the first time when the strain in the extreme longitudinal steel reinforcing bar exceeds its yield strain in tension. Experimentally, yielding of the extreme longitudinal reinforcement was determined from strain gage readings at the base of the column (critical section). Analytically, a sectional analysis using a fiber-based approach was used to predict the curvature and the moment associated with the first yield limit state. Experimental test data showed that the columns reached the first yield limit at the force-level close to that predicted from the moment-curvature analyses. Yet, the columns’ slenderness led to high drift ratios at first yield: 2%, 1.6% and 1.5% for columns M1230, M1215 and M1015C, respectively. 76 3.7.3. Initiation of Spalling Using the definition by Hose et al. [91, 92], the initiation of spalling occurs when the size of the spalled region of the cover concrete reaches 1/10 of the section diameter. Spalling of test columns initiated at μΔ values presented in Table 3-7. In all cases, spalling first appeared at the base level (next to the footing block). The drift ratios at initial spalling are also listed in Table 3-7. In general, the column with higher longitudinal reinforcement (M1230) experienced spalling at lower levels of displacement ductility. Table 3-7: Onset of initial spalling of test columns Test Column Displacement Ductility (μΔ) Drift Ratio M1230 1.5 3.6% M1215 2 3.8% M1015C 2 3.7% 3.7.4. Significant Spalling The onset of significant spalling was defined, after Hose et al. [13], at the moment when the spalling region’s size on the cover concrete exceeded half of the section diameter. Significant spalling was observed at the displacement ductility levels listed in Table 3-8. Also presented in Table 3-8 are the drift ratios associated with the onset of significant spalling of the test columns. The height of the spalled region at the maximum imposed displacement ductility μΔ = 3 for M1230, μΔ = 3.5 for M1215 and μΔ = 4 for M1015C) is provided in Table 3-8. The maximum observed damage in terms of significant spalling at the end of the tests is shown in Figure 3-28. 77 Table 3-8: Onset of significant spalling and height of spalled region at the end of tests Test Column Displacement Ductility (μΔ) Drift Ratio Maximum Spalling Height M1230 2 4.7% 660 mm (26 in.) M1215 2 3.8% 533 mm (21 in.) M1015C 3 5.5% 635 mm (25 in.) Figure 3-28: Maximum observed extent of spalling region on columns (a) M1230, (b) M1215 and (c) M1015C 3.8. Test Results The experimental results are presented in this section. The results are either directly extracted from the instruments or minimally post-processed. In later sections, these measurements were combined and processed to obtain the results that can be used to experimentally evaluate the effects of slenderness on the inelastic response of RC bridge columns. 78 3.8.1. Direct Measurement Results In this section, the results from direct sensor measurements are presented. The results include force and displacement measurements from the load cells, displacement transducers, and rotation meters. 3.8.1.1. Top Rotation Response The rotation angle of the top of the column, where the external lateral and axial loads are applied to the column, was measured and plotted against the top displacement in Figure 3-29. It can be seen that the rotation angle of the top of the column is linearly related to the top displacement, which is expected from a small rotation angle. Also, there is no difference between the loading and unloading cycles, which can be seen from the flat response. This can be justified by the fact that the relationship between the top rotation and the top displacement is mainly governed by the geometry of the column. In addition, it is of importance to calculate the deviation of the applied loads from the assumed direction to estimate the error associated with the assumptions of the analytical models. The calculations are presented in the following section. Figure 3-29: Rotation measured at the top of the test columns 79 3.8.1.2. Accidental Moment at Column Top The test columns were assumed to have zero moment and no constraint at the top. Ideally, this can be achieved by applying the external loads, axial and lateral, exactly at the center of the top loading block. In practice, the loads are applied to the external sides of the loading block at a distance from the center. When the loading block is perfectly aligned with the horizontal and vertical axes, there are no accidental moments applied to the top of the column. However, as the top loading block rotates, there is a slight additional moment applied to the column top due to the non-zero lever arms that extend from the center of the loading block to the faces where the external loads are applied. Using the geometry of the top loading block and the location of the applied forces, the length of the lever arms associated with the axial and lateral loads was calculated as presented in Error! R eference source not found.. The results are also plotted in Figure 3-30, where the absolute moment value and the ratio of the top moment to base moment are shown against the top displacement. It can be seen that the accidental moments at the top are a fraction (less than 1%) of the base moment. Therefore, it is safe to neglect the top moment and assume that the top of the column is free. 80 Figure 3-30: Maximum possible moment at top of column M121505 due to eccentricity of the applied forces from the center of the loading block as shown for absolute values and normalized to the base moment Table 3-9: Maximum possible error associated with the non-zero moment at top of the test column M121505 Test Step Δ (in.) Feff (kips) P (kips) θTop deg θChrd-Ln deg L.A.ax in. L.A.lat in. MTop,ax kips-in. MTop,lat kips-in. MTop,tot kips-in. MTop,tot / MBase % 0.25 F'y 0.5 F'y 0.75 F'y F'y Δy 1.5 Δy 2.0 Δy 3.0 Δy 3.5 Δy 0.45 1.22 2.70 4.55 5.40 8.10 10.67 16.09 18.80 3.8 7.5 11.3 15.1 16.1 18.3 18.8 19.5 19.5 159.3 160.3 162.9 163.7 163.5 162.8 162.9 163.2 163.7 0.110 0.292 0.645 1.101 1.313 1.911 2.423 3.500 4.032 0.083 0.228 0.504 0.848 1.008 1.511 1.991 3.000 3.504 0.004 0.010 0.022 0.040 0.048 0.063 0.068 0.079 0.083 0.040 0.107 0.236 0.404 0.481 0.700 0.888 1.282 1.477 0.7 1.6 3.6 6.5 7.9 10.2 11.0 12.8 13.6 0.2 0.8 2.7 6.1 7.7 12.8 16.7 25.0 28.8 0.8 2.4 6.3 12.6 15.6 23.1 27.7 37.8 42.4 0.08 0.11 0.19 0.28 0.32 0.41 0.47 0.59 0.65 81 3.8.1.3. Axial Force Response Axial load, which was applied by post-tensioning two high-strength rods, was measured continuously during the tests and is plotted in Figure 3-31. Since the axial loads were applied prior to the lateral loading, the post-tensioning was carried out on the test columns in their vertical configuration. Ideally, the top of the test columns moves on a circle with radius equal to the column height and the origin at the column base. This would keep the length of the post-tensioning rods unchanged during lateral deformations. However, it can be seen from the variation of the axial load in Figure 3-31 that there is a slight deviation from the circle that causes additional axial load as the lateral displacement increases. Also, the lateral load applied by the hydraulic actuator generates a small axial component as the top of the column as it rotates. These effects contributed to the axial load value measured by the load cells as shown in Figure 3-31. Figure 3-31: Variation of axial load during the test for both columns 82 3.8.2. Lateral Force-Displacement Response 3.8.2.1. Cyclic Response The lateral force-displacement response of the column at the global level was obtained by measuring the displacements at the middle of top loading block, and the force applied by the actuator or the effective force (Feff). This should be distinguished from the net shear force (Fnet), which is resisted by the columns and transmitted to the base, since a portion of the force applied by the actuator is used to counteract the horizontal component of the axial force (see Figure 3-7(b)). This horizontal force is introduced to the column as the lateral displacement increases and the angle of rotation for the post-tensioning rods becomes larger. The actuator force was modified accordingly to obtain the actual shear force acting on the column. The force-displacement response of both columns in terms of effective and net lateral forces is shown in Figure 3-32. The response is indicative of a stable energy dissipating response. However, P-Δ effects decrease the columns’ lateral load resisting capacity as the top displacement increases. By comparing the strength loss due to P-Δ effect in both columns, it can be seen that the column with lower longitudinal reinforcement ratio and greater axial load (M1015C) is more affected. 83 (a) (b) (c) Figure 3-32: Force-displacement response of test columns (a) M1230, (b) M1215 and (c) M1015C 84 3.8.2.2. P-Δ Moments The P-Δ moment was calculated at the column base level by multiplying the applied axial force by the top displacement. The P-Δ moment is plotted against the lateral displacement in Figure 3-33. Since the axial load is relatively constant during the test, the P-Δ moment shows a linear relation with respect to the top displacement. The average axial force to calculate P-Δ moments was obtained by using a linear regression to the plot. This is slightly higher than the nominal axial force due to the reasons discussed in the previous section. Figure 3-33: Experimental P-Δ moment for column M123005 3.8.3. Response Profiles 3.8.3.1. Lateral Displacement Profile The eccentricity of the column’s deformed shape with respect to the chord line, which connects the top to the bottom of the member, was obtained by determining the column lateral displacement profiles. This eccentricity, in conjunction with the applied axial load (P), generates extra moments 85 referred as P-δ moments. As a result of secondary moments, the distribution of internal moments along the height is not linear. The profile of the deformed column was determined by tracking the results from horizontal DTs and plotting the location of each measurement section at different displacement ductility levels. Lateral displacement profiles of columns M1230 and M1215 at μΔ levels of 1 and 3 are given in Figure 3-34 and Figure 3-35, respectively. Figure 3-34: Lateral displacement profiles for column M1230 86 Figure 3-35: Lateral displacement profiles for column M1215 3.8.3.2. Deformation Profile Lateral displacement of the test columns, which was monitored at 7 sections along the height, was used to measure the bending deformation of the columns away from their chord-line (δ). Bending deformation of the test columns is illustrated in Figure 3-36, from which the magnitude of P-δ moment was estimated. Nonlinear moment profiles were consequently constructed by adding the P-δ moment to the linear distributions. 87 Figure 3-36: Deformation profiles along test columns 3.8.3.3. P-δ Moment Profile The P-δ moment along the columns’ height was calculated from the experimental data by multiplying the axial load by the deformation of the column away from its chord line (δ). Using 88 this method, the nonlinear second-order moment profile along the column height was obtained from the experiments. 3.8.3.4. Curvature Profile Flexural actions on the test columns were assessed by determining section curvature demands along their height. The flexural curvature of each section was calculated from the relative displacements measured between two sections at the extreme tensile and compressive sides of the column and divided by the respective gage length to obtain average strains. The calculated strains at a given elevation were then used to calculate the average flexural curvature for the section. The flexural curvature profiles along the height of test columns M1230 and M1215 are shown in Figure 3-37 and Figure 3-38, respectively. It can be seen that the measurements at the base are not consistent with the rest since the fixed-end rotation of the column caused by the strain-penetration effect is included in the measurements for the curvatures. Therefore, the curvature values at base level were eliminated from the test data for further analyses. 89 Strain Penetration Figure 3-37: Average curvature profiles after yield for column M1230 90 Strain Penetration Figure 3-38: Average curvature profiles after yield for column M1215 Curvature profiles were calculated from the relative vertical extensions and contractions, measured by the vertical DTs, between multiple sections along the height. The average (from push and pull directions) curvature profiles along the test columns are shown in Figure 3-39. The plastic region length, Lpr, was found from the length of the region over which the curvature values exceeded the yield curvature (ϕy), indicated by a vertical dashed line in Figure 3-39. In this research, the use of the curvature profiles for assessing the extent of the plastic region is denoted as Experimental Method (1). A drawback of this method is the difficulty of determining the contribution from 91 various sources to the spread of plasticity, e.g., primary moment gradient, P-Δ effect, nonlinear Pδ moment, and the tension shift effect, since the curvature profiles cannot be broken down into their components. 3.8.3.5. Total Nonlinear Moment Profile Experimental distribution of the bending moments along the height of the test units was calculated using the deformed geometry of the columns in conjunction with the applied forces measured at the top of the cantilever columns. The linear primary moment (Mp) without second-order effects was obtained according to Mp = V(L-h) where V is the lateral force applied to the top of the column by the horizontal actuator, L is the total height of the column from the base to the center of the top loading block, and h is the height of the monitored section from the base. Additional moment demands due to P-Δ and P-δ were respectively estimated by MP-Δ = PΔ and MP-δ = Pδ. Following this method, experimental profiles of the linear moment (ML=Mp+MP-Δ) and the nonlinear moment (MNL=Mp+MP-Δ+MP-δ) along the columns’ height were constructed from the test data at different displacement ductility levels. 92 Figure 3-39: Average curvature profiles for test columns after correction at the base level The nonlinear moment gradients, which include the additional moments due to P-δ effect, are depicted in Figure 3-40, in which the yield moment (My) is shown by a vertical dashed line. The 93 extent of the plastic region due to nonlinear moment gradients (Lpr,NL) with P-δ moments included was estimated by the length of the region over which the MNL exceeded the yield moment (My). Similarly, the length of the plastic region due to linear moment gradients (Lpr,L), i.e., with P-δ effects ignored, was the length of the region over which ML exceeded My. In this work, experimental values of Lpr obtained from the moment profiles were denoted as method (2). Depending on the type of the moment profiles from which Lpr values were derived, the results were marked as ‘w/ P-δ’ and ‘w/o P-δ’ for nonlinear and linear moment gradients, respectively. Method (2) offered the possibility of decomposing Lpr into its components to find the contribution of P-δ moments as well as the tension shift effect. 3.8.3.6. Base Moment-Curvature (M-ϕ) Response Moment-curvature responses for the test columns was experimentally evaluated using the moment and curvature values obtained at the monitoring section closest to the base, which was 102 mm (4 in.) above the base. The results are shown in Figure 3-41, where the overlaid test data verifies the accuracy of the sectional analyses performed prior to the tests. Also shown on the plots is the yield point (ϕy, My) that was used to define the regime change point from elastic to plastic. The coordinates of the yield point are later used to estimate Lpr from experimental curvature and moment profiles. 94 Figure 3-40: Experimental profiles of nonlinear moment on test columns 95 Figure 3-41: Moment-curvature response of test columns from analyses and experiments 3.8.4. Shear Deformations Panels of displacement transducers (DT) were used to measure shear deformations over the critical region of the column from the base level up to 1219 mm (48 in.) along the column height. The geometry-based method proposed by Lehman et al. [47] was used to extract the average shear strains from each panel, which consisted of two vertical and two horizontal DT instruments arranged around a rectangular panel with one extra DT measuring the displacements along the diagonal. The contribution of the shear deformations to the top displacement was calculated by 96 integrating the shear strains along the length, as shown in Figure 3-42 for column M1230. It can be seen that the shear deformations have a small effect on the total displacement of the test columns. This is to be expected since the test units are extremely slender and the response of slender RC columns is mainly governed by flexural deformations and base rotation due to strain penetration. Therefore, it is valid to neglect the contribution of shear deformations, which constitutes less than 5% of the total deformations, and assume that displacements and deformations measured during the tests were mainly caused by flexural deformations (elastic or plastic) and strain-penetration. The contribution of different deformation mechanisms to the top displacement is depicted in Figure 3-43 for test columns M1230 and M1215. It is evident from the plots that shear deformations have a negligible effect on the total displacement. Figure 3-42: Contribution of shear deformations to top displacement 97 (a) (b) Figure 3-43: Decomposition of top displacement into sources of deformation for (a) column M123005 and (b) M121505 98 3.9. Analysis of Test Results In this section, the results from the tests were further analyzed to isolate the effects of slenderness on the inelastic response of the columns. In addition, measurements from different instruments were combined to calculate the responses that cannot be directly measured from the test units. Furthermore, the effects of the phenomena that are closely related to the column slenderness, such as P-Δ and P-δ moments, on other responses of the test columns were studied. 3.9.1. Effect of P-Δ Moments Plotting the bending moment demand at the column base, where the internal moment is at its maximum, allows evaluating the effect of secondary moments (P-Δ) on structural stability. The contribution of primary and secondary moments to the total moment experienced at different levels of loading is illustrated in Figure 3-44 for the nearest section to the base, for which experimental data was available (102 mm (4 in.) above the base). The results show that the P-Δ moment accounts for a significant portion of the internal moment, particularly after yielding. This simply indicates the necessity of considering the secondary effect in order to satisfy general equilibrium of the forces and moments acting on the structure, otherwise the base moment would be underestimated. However, moments caused by member deformation (P-δ) are very small at sections near the base. The loss of moment capacity for resisting lateral loads is thus caused mostly by the increased magnitude of the P-Δ moments. 99 (a) (b) (c) Figure 3-44: Loss of lateral strength due to P-Δ effect in columns (a) M1230, (b) M1215 and (c) M1015C 100 3.9.2. Decomposition of Moment Profile Lateral displacement profiles of column M1215 at μΔ levels of 1 and 3 are given in Figure 3-45, where the parameters of Δ and δ are depicted. Throughout this document, δ is the normal distance between the actual location of each section and the chord line of the column; whereas, Δ is the horizontal distance of the chord line from the top of the column at each elevation. Δ δ Figure 3-45: Lateral displacement profiles for column M1215 depicting parameters of Δ and δ Using the deformations profile of test columns, the moment profiles were decomposed into their main contributors using Equation (3-9), M t  M p  M P   M P  , 101 (3-9) where Mt is the total moment at a section, equal to the moment associated with the experimental curvature of the section; and Mp is the primary moment at a section, estimated by assuming a linear distribution of moments from Equation (3-10), M p  Fnet  h , (3-10) where h is the vertical distance of the section of interest from the top of the column. MP-Δ is the secondary moment caused by P-Δ effects and calculated using Equation (3-11). Also, MP-δ is the internal moment developed at each section due to the effect of P-δ, given by Equation (3-12). M P Δ  P   (3-11) M P δ  P  δ (3-12) The resulting profiles from the decomposition of the total moments at yield (μΔ = 1) for test columns M1230, M1215 and M1015C are shown in Figure 3-46(a), Figure 3-47(a) and Figure 3-48(a), respectively. Similar results for decomposed bending moment profiles at the end of the test (maximum displacement ductility to which columns tested) are shown in Figure 3-46(b), Figure 3-47(b), and Figure 3-48(b) for test columns M1230, M1215, and M1015C, respectively. It can be seen that at lower ductility demand the existing moment at a section of a column mainly comes from the primary moment. Higher stiffness, which limits the column’s deformation, is the reason that secondary effects show little impact on the distribution of internal moments early on. In contrast, the effect of secondary actions caused by P-Δ and P-δ is more profound at the maximum displacement ductility levels. It can be observed that at this later stage of inelastic deformation a main portion of the moment demand comes from the second-order effects. 102 Figure 3-46: Decomposed moment profiles for column M1230 at (a) μΔ = 1 and (b) μΔ = 3 Figure 3-47: Decomposed moment profiles for column M1215 at (a) μΔ = 1 and (b) μΔ = 3.5 103 Figure 3-48: Decomposed moment profiles for column M1015C at (a) μΔ = 1 and (b) μΔ = 4 3.9.3. Plastic Region Length (Lpr) The height of plastic region at different displacement ductility levels was obtained by determining the height at which the flexural curvatures exceeded the ideal yield curvature (ϕy). The experimental results for spread of plasticity as a function of displacement ductility are shown in Figure 3-49 for all test columns. The plotted traces show the growth of the plastic region during testing. Also shown in Figure 3-49 are traces for the spread of plasticity obtained from the total moment profiles. Comparisons of the traces that correspond to cases with and without P-δ show the substantial second-order effects (P-δ) in increasing the plastic region height. Thus, while the additional moment caused by P-δ effects is only a small increase to the internal bending moment it can lead to a significant shift in the associated curvatures. It is worth noting that, the primary moment by itself is not large enough to develop inelastic demands. Therefore, no plastic region would be developed in the column if the P-Δ and P-δ effects were neglected. 104 The contribution of P-δ on the spread of plasticity can be better appreciated by highlighting the amount of increased plastic region length due to this effect as depicted in Figure 3-50, which shows the percentage increase in the plastic region caused solely by consideration of the P-δ effect. The traces start with a spike at displacement ductility around 1 because the consideration of P-δ moments expedites the growth of plastic region such that spread of plasticity starts with a small delay when P-δ moments are not taken into account. Neglecting the spike at the beginning of the diagram, which occurs over small range of μΔ about 1, the data shows a steady 20%, 25% and 23% increase in the plastic region height of columns M1230, M1215 and M1015C, respectively. It is worth noting that the column with lower reinforcement ratio, M121505, exhibits more sensitivity to the effect of P-δ on the spread of plasticity. 105 Figure 3-49: Length of the plastic region (Lpr) on columns M1230, M1215, and M1015 106 3.9.3.1. Methods of Evaluating Lpr Previous experimental research on determining the length of the critical plastic region used either visual evaluation of the damaged region [13, 49] or curvature profiles along the columns’ height [35, 36, 93]. In addition to the aforementioned methods, moment profiles are utilized in this research to assess Lpr and its components. Method (1): Using Curvature Profiles Spread of plasticity along the columns’ height was assessed by using the distribution of curvatures. Accordingly, the extent of the plastic region at the end of each loading cycle (μΔ = 1.5, 2, 3, 3.5, and 4) was determined from the corresponding curvature profiles. The evolution of Lpr obtained from curvature profiles, Method (1), is plotted in Figure 3-50 against the deformation progress of the tests. It can be seen that the plastic region approximately begins at μΔ=1 and extended to higher elevations as the lateral displacement increased. Method (2): Using Moment Profiles In this method Lpr was determined from the length of the portion of the column over which the nonlinear moment profile exceeded the yield moment (My). Evolution of Lpr by this method, with P-δ effects included, with respect to μΔ is depicted in Figure 3-50. Also shown is the extent of the plastic region from method (2) when P-δ moments are ignored. It can be seen that Lpr from nonlinear moment profiles (w/ P-δ) closely follow the results from curvature profiles (method (1)). The small difference between Lpr values from methods (1) and (2) with P-δ is mainly caused by non-flexure mechanisms, such as tension shift, that contribute to the spread of plasticity. 107 Figure 3-50: Extent of the plastic region (Lpr) extracted from the experiments Results from linear moment gradients (w/o P-δ) significantly underestimated the Lpr values obtained from method (1). This emphasizes the importance of considering nonlinear moment gradients and P-δ effects for estimating the extent of the plastic region. Method (3): Using Visual Observation Visual evaluation of cover spalling was also used to estimate the damage zone on the test columns. The length of the spalling region (Ls) was measured and recorded during tests at different displacement ductility levels. Values for Lpr were then estimated assuming that the observable 108 damage zone matched the critical plastic region, as depicted by Figure 3-51 in which Lpr values from method (1) are marked at different displacement ductility levels along with the extent of the observed spalling zone (Ls). It can be seen that the plastic region extended beyond the region with observable damage. The ineffectiveness of using the damage zone to estimate Lpr from the experiments is also illustrated by Figure 3-50, where results from the visual evaluation of spalling damage are depicted as method (3). It can be seen that method (3) dramatically underestimates the extent of the plastic region. While previous research has found correlations between the extent of the spalling region (Ls) and Lpr in non-slender columns [36], such correlation was not found for the slender columns reported here. Thus, visual inspection of the cover damage cannot always serve as a method for experimental evaluation of the extent of the plastic region, i.e., Ls ≠ Lpr. Due to the discrepancy between Lpr and Ls, results obtained from methods (1) and (2) were chosen for further analysis. 3.9.3.2. Components of Lpr It is essential to identify the components of Lpr and their contribution to the extent of plasticity on RC columns in order to assess the effects of slenderness. The main contributors to Lpr are the linear moment gradient (defined as the sum of primary and P-Δ moments), the tension shift effect and the P-δ moment due to column bending away from the chord line. It is worth mentioning that strain-penetration, which is a main component in plastic hinge (Lp) models, mainly contributes to the extent of the plastic region into the footing (or adjacent RC member). Therefore, the extent of plasticity along the columns height is not directly affected by strain-penetration. 109 Figure 3-51: Extents of the plastic region (Lpr) and spalling zone (Ls) depicted along the test columns at multiple displacement ductility levels The influence of major contributors to Lpr is shown in Figure 3-52, in which the darkest area represents the extent of the plastic region due to linear moment gradient. The section in the middle is the contribution of P-δ moments to Lpr and the region with the lightest shade shows the spread of plasticity caused by non-flexural mechanisms including shear effects. It is worth noting that the primary moment (Mp) does not reach the yield limit if P-Δ moments are not considered. Therefore, the contribution of Mp and P-Δ moments could not be determined separately. 110 Figure 3-52: Major contributors to Lpr 3.9.3.3. Effect of Tension Shift on Lpr The extent of the plastic region due to tension shift effect (Lts) was predicted for the test columns according to the method by Hines et al. [35]. The crack angle was calculated by estimating the direction of the principal tensile stress according to Collins and Mitchell [94]. The crack angle predicted for the test columns is plotted against μΔ in Figure 3-53(a), in which the plots begin at 111 the displacement ductility corresponding to the start of cracking. It can be seen that crack angles are small, i.e., cracks are primarily perpendicular to the columns’ longitudinal axis. Unlike the inclined cracks in RC members with strong shear effects, the nearly perpendicular cracks in flexure-dominated slender columns contribute marginally to the extent of the plastic region due to tension shift (Lts), which is plotted against displacement ductility in Figure 3-53(b). Small crack angles were experimentally verified by inspecting the crack patterns on the test columns. The contribution of tension shift on extending the plastic region was experimentally measured by the difference between the experimental Lpr from methods (1) and (2) with P-δ moments included. The difference provided the extent of the plastic region due to mechanisms other than moment gradients, such as tension shift. Figure 3-53: Predicted crack angle and the associated tension shift effect on test columns 3.9.3.4. Effect of P-δ Moment on Lpr Lateral displacement (Δ) of the bridge column at the top induces an additional moment demand (P-Δ) due to eccentricity of the axial load. The P-Δ moment (Pcosθ in the case of a non-vertical 112 P) is applied to the base independent from the deformed shape of the column. Therefore, the effect of P-Δ is typically considered by linearly increasing the primary bending moment profile caused by the shear force, as illustrated in Figure 3-54(a) and (b), in which the axial load (P) is shown inclined to be consistent with the test setup. The second-order moment at the base level is accommodated by the RC section’s capacity and thus the effect reduces the actual shear force resisted by the column leading to instability of the element. Besides the lateral displacement at the top, slender columns also bend as illustrated in Figure 3-54(a); where δ is the normal distance between the actual location of each section and the chord line of the column, and Δ is the horizontal distance of the chord line from top of the column. The axial load combined with the deformation of the column perpendicular to its chord line (δ) creates member second-order moments, referred to as P-δ effects. The P-δ moments (P·δ) reshape the bending moment gradient and lead to a nonlinear distribution with larger effects at mid-span. Nonlinear P-δ moments are expected to increase the spread of inelastic region to elevations higher than predicted by a linear moment distribution as shown in Figure 3-54(c). Although the base moment demand is not directly affected by the P-δ moment, extension of the inelastic region along the height of the column increases the flexibility of the column, which in turn leads to a larger lateral displacement at the top. 113 Figure 3-54: Effect of nonlinear moment profile and P-δ on the extent of the plastic region (Lpr) The effects of slenderness and the associated member flexibility on Lpr were assessed through the contribution of P-δ moments (Lpr,P-δ). Comparison of the results obtained according to method (2) from linear (ML) and nonlinear (MNL) moment profiles demonstrated the increase of the plastic region due to P-δ. Therefore, Lpr,P-δ was experimentally determined according to (Lpr,NL – Lpr,L). To illustrate the contribution of moment nonlinearity to the spread of plasticity, the ratio Lpr,P-δ / Lpr is plotted in Figure 3-55. It can be seen that, on average, P-δ moments account for 20% of the total Lpr. 114 Figure 3-55: Increase in spread of plasticity due to P-δ effect 3.9.3.5. Effects of Pseudo-Dynamic Loading on Lpr Experimental results from testing column M1015D using a pseudo-dynamic loading protocol demonstrated the effect of dynamic loading on the inelastic response of RC columns. It was found that pseudo-dynamic loading disturbs the smooth distribution of curvatures along the height of columns. Pseudo-dynamic loads contribute to the random distribution of tensile cracks along the column height, which leads to curvature profiles that are less smooth than the ones from quasistatic tests. This phenomenon is demonstrated by comparing the profiles of curvatures along the height of columns M1015C (quasi-static) and M1015D (pseudo-dynamic) in Figure. It can be seen from the distribution of curvatures along the height of these columns that column M1015D, which was tested using a dynamic loading protocol, exhibits less smooth profiles than column M1015C, which was tested according to cyclic loading protocol. 115 Figure 3-56: Profiles of curvature along test columns M1015C and M1015D Another feature that is unique to results from pseudo-dynamic tests is that curvature profiles are not symmetric with respect to two loading directions. The reason is that the columns subjected to dynamic loading may experience different levels of displacement ductility at different loading directions. Therefore, profiles of flexural curvatures along the columns height are not symmetric in two directions. The asymmetric distribution of flexural curvatures in two loading directions is depicted in Figure in which the profiles of curvatures along column M1015D are plotted in two loading directions. It can be seen from the smaller magnitude of negative curvatures in comparison with the positive curvatures that the test column M1015D experienced smaller deformations in the “pull” direction than the “push” direction. 116 Figure 3-57: Asymmetric profiles of curvature along column M1015D Although the distribution of curvatures along the columns’ height is significantly affected by the loading history and dynamic effects, the spread of the plastic region along column height is less sensitive to the loading history and rate. The evolution of the length of the plastic region (Lpr) with respect to increasing displacement ductility is plotted in Figure for test columns M1015C and M1015D to demonstrate the effects of dynamic loading and loading history. It can be seen that the Lpr curves from columns M1015C and M1015D have a similar trend despite the fact that results from the static test are smoother than the ones from the dynamic test. Moreover, the ultimate length of the plastic region at μΔ=4 is approximately the same for both columns. Thus, it can be concluded that the length of the plastic region is not significantly affected by the loading history and rate. 117 Therefore, the results from quasi-static cyclic tests on the spread of plasticity along the height of RC columns is also applicable to columns that are subjected to dynamic time-history loadings. Figure 3-58: Spread of plasticity (Lpr) along the height of columns M1015C and M1015D 3.9.4. Plastic Hinge Length (Lp) The plastic hinge length (Lp) is indeed related to the extent of the plastic region (Lpr). In particular, the moment gradient term in Lp models is proportional to Lpr. One can relate the plastic moment gradient portion of the plastic hinge length (Lp,MG) to the amount of plastic rotations (θp) that occur along the plastic region. The calculation is done by integrating plastic curvatures, ϕ(x) - ϕy, along the plastic region (Lpr) and diving them by the plastic demand at the base level (ϕBase - ϕy) using Equation (3-13). L pr L p , MG  p  p , Base    x dx  L 0 Base   y 118 pr y (3-13) Assuming a linear distribution of curvatures along the plastic region yields Lp,MG = Lpr / 2. However, the distribution of curvatures was not linear in the columns reported here. To better illustrate the nonlinear distribution of plastic curvatures (ϕp) over the plastic region, they were plotted against the length of columns at the highest displacement ductility (μΔ) to which they were tested. These plots are provided in Figure 3-59. Also shown are linear and quadratic regressions for the plotted experimental data. It can be seen that the linear distribution does not correlate well with the experimental data. However, a nonlinear distribution, quadratic in these plots, closely follows the experimental pattern. Figure 3-59: Distribution of plastic curvatures (ϕp) over the plastic region of the test units at the highest displacement ductility (μΔ) to which they were tested 119 In addition to visual assessment of the plastic curvatures’ distribution on the test columns, Equation (3-13) was numerically evaluated and results are shown in Table 3-10. Table 3-10: Equivalent plastic hinge length (Lp,MG) for the test columns μΔ Test Column M123005 M121505 M101510 Average θp (rad) 0.0047 0.0055 0.0064 1.5 Lp,MG (mm) 552 510 442 Lp,MG / Lpr 46% 53% 40% 46% θp (rad) 0.0151 0.0132 0.0146 2 Lp,MG (mm) 495 520 492 Lp,MG / Lpr 35% 35% 33% 35% θp (rad) 0.0338 0.0311 0.0349 3 Lp,MG (mm) 690 609 607 Lp,MG / Lpr 43% 33% 34% 37% θp (rad) 0.0409 3.5 Lp,MG (mm) 614 Lp,MG / Lpr 32% 32% θp (rad) 0.0518 4 Lp,MG (mm) 614 Lp,MG / Lpr 33% 33% Average Lp,MG / Lpr 41% 38% 35% 37% It can be seen from the results that the equivalent moment gradient portion of the plastic hinge length (Lp,MG) is less than 50% of the actual plastic region (Lpr). The reason is the nonlinear distribution of curvatures over the plastic region. Two trends are evident from the results presented in Table 3-10. First, the Lp,MG / Lpr ratio decreases with displacement ductility (μΔ). This implies that the distribution of curvatures along the plastic region is closer to a linear one at small inelastic deformation and becomes more nonlinear as the inelastic demand increases. Second, the Lp,MG/Lpr 120 ratio decreases with the slenderness ratio (κel), which justifies the observation that the distribution of curvatures in non-slender columns is close to linear. Establishing the proper relationship between Lp and Lpr is thus a complicated task due to the issues demonstrated above. 121 CHAPTER 4 4. NUMERICAL STUDY 4.1. General This chapter presents the development and use of three-dimensional (3D) continuum-based finite element (FE) models for simulating intermediate damage limit states in slender RC bridge columns. Results from the 3D FE simulations are compared and validated against experimental data from five large-scale test columns, three of which were reported in the previous chapter. Statistical error measures and correlation metrics were utilized to quantitatively evaluate the accuracy of the models. The validated models were employed to determine intermediate damage states of RC columns in case of earthquakes. 4.2. Methodology and Background 4.2.1. Intermediate Damage Limit States The focus of this chapter is on demonstrating and evaluating the use of 3D continuum-based FE models for predicting intermediate damage limit states (IDLS) in RC bridge columns that are 122 subjected to moderate levels of damage. IDLS represents conditions that introduce considerable damage to a structural element and that define key levels of performance and repair requirements. The AASHTO seismic specifications [20] considers cracking, reinforcement yielding, and major spalling of concrete as target limit states for damage control in RC bridges. Tensile cracking is not a significant damage limit state for RC bridge columns since it is likely to be exceeded at early stages of inelastic deformation [16]. Reinforcement yielding is defined as the first yielding of the extreme reinforcement in tension. Since this condition is the starting point for the significant inelastic response, it is commonly considered as the first IDLS. Spalling of the cover concrete refers to the observation of damage on the column surface and has been associated with the formation of a local damage mechanism. Repair at this IDLS can be costly and may require closure of the bridge [95]. A limit of 1/10 of the section diameter for the size of damaged region has been proposed by Hose and Seible [96] to indicate the onset of spalling. Significant spalling is associated with the growth of the spalled region to a significant level. It is considered an important IDLS since expensive and time-consuming measures are required to repair the damage [37]. One half of the section diameter was used in this work as the minimum size of the spalled region to identify significant spalling after Hose and Seible [96]. 4.2.2. Methods for Numerical Studies The procedure that was followed in the numerical studies consisted of three steps: (1) model development, (2) experimental validation and (3) application of the FE models for predicting IDLS in RC bridge columns. The models emulated, as accurately as possible, the conditions of the test columns by (1) defining the material properties of the model based on test results of concrete and steel samples, and (2) creating the loading and boundary conditions of the model in accordance with the laboratory setup. An optimum value for the dynamic analysis parameter, namely the time123 rate of loading, was found such that the quasi-static loading condition of the columns during the tests was recreated in the analysis without sacrificing the computational efficiency of the dynamic explicit solution algorithm. The FE models were then evaluated and validated against experimental data at global and local levels. Flexural curvatures at different sections along the columns’ height were used to validate the FE models at a local level. Following validation and evaluation, the FE models were used to determine the onset of intermediate damage limit states. Reinforcement yielding, initial spalling and significant spalling limit states were determined using the FE simulation results and compared to experimental data and observations. 4.3. Experimental Database Experimental results from five half-scale RC columns tested as free-standing cantilevers under constant axial load followed by quasi-static reversed cyclic loading were selected as case studies. All the test units were detailed to behave in a ductile manner and had a flexure-dominated response. 4.3.1. Geometry The tested columns had slenderness ratios (effective length over diameter or L/D) of 6, 8, 10 and 12. These aspect ratios generally represent columns with higher slenderness than the average. Therefore, slenderness effects could be studied by considering the selected test columns. All five columns had circular cross sections with a diameter (D) of 610 mm (24 in.). The column with the aspect ratio L/D = 6 was reported by Hose and Seible [96] as SRPH1. The column with L/D = 8 is the test unit defined as 815 from the experimental program reported by Lehman and Moehle [47]. The columns with L/D ratios of 10 and 12 were tested as a part of the present research and are reported in the previous chapter. 124 In addition to different aspect ratios, test columns differed in their longitudinal reinforcement ratio (ρsl), transverse reinforcement ratio (ρst) and axial load ratio (P/f ′cAg). For ease of reference, the five test units were identified according to the following designation: (1) the first letter denotes the university where the column was tested (SD = UC San Diego, B = UC Berkeley and M = Michigan State); (2) the next two digits define the slenderness ratio (L/D), and (3) the last two digits define the longitudinal reinforcement ratio in percent. The geometry and reinforcement details for the test units are presented in Table 4-1 in which s is the spiral pitch (the spacing between the transverse reinforcement loops), dsl and dst are the diameter of the reinforcement bar in the longitudinal and transverse directions, respectively. 4.3.2. Material Properties 4.3.2.1. Concrete Concrete strength properties were evaluated through standard testing of cylindrical specimens. The compressive (f ′c) and tensile (f ′t) strengths for the test columns at the day of testing were used to define the material properties for the concrete parts in the FE models. A schematic stress-strain diagram for unconfined concrete is shown in Figure 4-1, in which compressive and tensile strengths are depicted. Concrete material properties for all test columns are presented in Table 4-1. 125 Table 4-1: Geometry, reinforcement details and material properties for test units Test Column SD0630 B0815 M1015 M1215 M1230 L/D 6 8 10 12 12 L mm (in.) dsl mm (in.) 3658 (144) 22 (0.875) 4877 (192) 16 (0.625) 6096 (240) 19 (0.75) 7315 (288) 19 (0.75) 7315 (288) 25 (1) ρsl 2.7% 1.5% 1.6% 1.6% 3.1% dst mm (in.) 10 (0.375) 6 (0.25) 13 (0.5) 13 (0.5) 13 (0.5) ρst 0.9% 0.7% 1.2% 1.2% 1.2% P/Agf 'c 14.8% 7.2% 8.8% 5.7% 5.3% P kN (kips) f 'c MPa (ksi) f 't MPa (ksi) Cover mm (in.) fyl MPa (ksi) ful MPa (ksi) 1780 (400) 41.1 (6.0) 4.0 (0.58) 28 (1.1) 455 (66) 746 (108) 654 (147) 31 (4.5) 3.5 (0.51) 22 (0.88) 462 (67) 630 (91) 1334 (300) 51.8 (7.5) 4.3 (0.63) 25 (1) 455 (66) 689 (100) 712 (160) 42.7 (6.2) 3.2 (0.46) 25 (1) 455 (66) 689 (100) 712 (160) 46.2 (6.7) 3.4 (0.49) 25 (1) 486 (70.5) 738 (107) εul 0.12 0.12 0.11 0.11 0.08 εshl 0.008 0.008 0.008 0.008 0.01 fyt MPa (ksi) s mm (in.) 414 (60) 57 (2.25) 607 (88) 32 (1.25) 469 (68) 76 (3) 469 (68) 76 (3) 469 (68) 76 (3) 126 σ ft ′ εsp εco εct ε - fc ′ Figure 4-1: Schematic representation of concrete stress-strain response with compressive and tensile strengths 4.3.2.2. Reinforcement Steel Material properties for the reinforcement steel were experimentally obtained through tensile testing. A typical stress-strain response for reinforcement steel is schematically illustrated in Figure 4-2. Indicated in the plot are the yield strength (fy), ultimate strength (fu), onset of strainhardening (εsh), and ultimate strain (εsu). These key parameters for defining the material response of reinforcement steel are presented in Table 4-1 for all test columns as obtained from tensile tests on three specimens for each of the reinforcement bar types. It is worth noting that the properties for the longitudinal and transverse reinforcement are denoted by l and t subscripts, respectively. 127 fu Stress fy εsy εsu εsh εmax Strain Figure 4-2: Schematic representation of reinforcement steel stress-strain response 4.3.3. Instrumentation The response of all test columns was monitored by means of various instruments to measure local and global responses. Among all the measurements, three sets of data were specifically used for validation of the FE models: (1) the applied lateral load by the actuator to the top of the columns, as measured by the actuator load cell, (2) the horizontal displacement at the top of the columns measured by external deformation transducers, and (3) sectional curvatures calculated from displacement transducers mounted on the two loading sides of the columns. 4.3.4. Test Unit Components The test units studied in this chapter consisted of three parts: (1) the base block, which acts as the footing for the column and provides fixity to the bottom of the column, (2) the top block, which is used to uniformly distribute and apply the axial and lateral loads to the column without creating local damage to the column, and (3) the column, which is located between the top and bottom blocks. The focus of this study is to simulate the inelastic damage response of the column. However, the other two components, namely, the base block and the top block were also modeled 128 to create the same type of interactions between the components of the test columns in the simulations. 4.4. Finite Element Model Description 4.4.1. Parts 4.4.1.1. Concrete Parts Continuum-based 3D finite element models were assembled using the standard tools available in the commercial FE platform Abaqus [97]. The column, base block and the top loading block concrete sections were modeled using 8-node brick elements to match the geometry of the test units, as shown in Figure 4-3(a). To avoid shear locking due to full integration, reduced integration elements with hourglass control (C3D8R) (see Figure 4-4) were used and preferred because of their computational efficiency. 4.4.1.2. Steel Reinforcement Parts Reinforcement steel in the longitudinal and transverse directions was modeled by threedimensional 2-node beam elements (B31). The beam formulation was used to model the longitudinal reinforcement because straight beam elements, unlike truss elements, are numerically stable under axial and lateral loading. The spiral reinforcement was modeled by individual hoops spaced along the height of the column according to the spiral pitch. A cross-sectional cutoff from the FE model for column SD0630 is shown in Figure 4-3(b). The colored regions represent different material model assignments. Perfect bond was assumed between the reinforcement elements and the concrete part. Rotations at the column base due to axial deformation of the embedded portion of longitudinal reinforcement into the footing block was expected to be partially captured in the model through the inelastic material definitions for both concrete and steel parts and by modeling the extension of longitudinal rebar into the footing. 129 Figure 4-3: Parts of finite element model for column SD0630 showing (a) entire model and (b) cross-section 8 7 5 6 Integration Point 4 3 1 2 Figure 4-4: Schematic of 8-node linear brick element with reduced integration 130 4.4.2. Mesh A radial mesh configuration was adopted for the cross-section of the model in order to preserve the rotational symmetry of the circular section of the columns. Mesh sensitivity of the concrete elements was minimized by proper selection of the mesh size such that as many of the elements as possible included a rebar. Therefore, the size of the elements in the axial direction of the column was governed by the spacing of the transverse reinforcement, whereas the distance between the longitudinal rebars defined the mesh size along the column perimeter. An adaptive mesh size was used in the radial direction of the column to ensure that the shape of the elements was close to a cube. Mesh configuration of the element cross a section of the column is depicted in Figure 4-5. Figure 4-5: Radial mesh of the cross-section 4.4.3. Boundary Conditions Constraints and loading conditions were set to simulate the boundary conditions of the test columns in a typical experimental setup (shown in Figure 4-6(a)). Tie-downs, used to anchor the base block to the ground, were replaced with a fixed boundary condition at the bottom of the footing. This may not be realistic for the footing, but it provides appropriate boundary conditions 131 for the column, which is the focus of the study. Rigid surfaces were provided to the side and top surfaces of the loading block atop the column to distribute the applied concentrated load and imposed displacement, and avoid element distortions due to localized forces. The direction of the axial load was set to follow the rotation of the loading block to simulate the experimental test setup as shown in Figure 4-6(b), in which the axial load is directed towards an anchorage point rather than being vertical. Lateral displacements were imposed on the columns by defining a monotonically increasing horizontal displacement to the end side surface of the loading block at its mid-height via a rigid surface. Figure 4-6: Loading and boundary conditions for test columns in (a) experimental setup and (b) finite element models (same as Figure 3-7) 132 The force applied by the actuator is called the effective lateral force (Feff) and the net horizontal reaction force at the base of the column is called Fnet. These forces are related through Equation (4-1),      , Fnet  Feff  P sin  tan1   L  Lbot   (4-1) where, with reference to Figure 4-6(b), Lbot is the height of footing that is measured from the column-footing interface to the anchorage point of the post-tensioning rods, P is the axial load applied to the column’s top, and Δ is the lateral displacement of the column at the top. 4.4.4. Material Models 4.4.4.1. Elastic Properties The initial elastic modulus for concrete was defined according to the equation suggested by ACI [33] for normal weight concrete, as given by Equation (4-2)a, using the compressive strengths from material tests. The concrete Poisson’s ratio was taken as 0.2. For the reinforcing steel, an elastic modulus of 200 GPa (29000 ksi) and a Poisson’s ratio of 0.3 were used. Ec  4700 f c Ec  57 1000  f c MPa  ksi  (4-2)a (4-2)b 4.4.4.2. Concrete Plasticity Properties Abaqus’ concrete damage plasticity model [97] was used to define the properties of concrete elements. The model is based on a yield function developed by Lubliner et al. [98] and modified by Lee and Fenves [99] with an isotropic scalar damage parameter. The concrete part of the column 133 was divided into two regions such that one corresponds to the core concrete confined by the transverse reinforcement spiral and the other represents the unconfined cover concrete. Two different sets of uniaxial compressive stress-strain data were used to define the concrete material model for the core and cover sections. The compressive strength of unconfined concrete was obtained from material tests, while the core concrete strength was adjusted for the effect of confinement following the model by Mander et al. [87]. Results from split tensile tests on concrete cylinders were used to define the tensile strength of concrete for both confined and unconfined regions. Following the recommendations by Caltrans [34], the ultimate compressive strain for unconfined concrete (εsp), which defines the spalling limit for cover concrete, was taken as -0.005. The ultimate strain for confined concrete was computed using the energy balance method. 4.4.4.3. Concrete Damage Properties Compressive damage parameters for the unconfined concrete (dco) were calculated based on the loss of strength after peak response following the method suggested by Jankowiak and Lodygowski [100], as given by Equation (4-3), d co  1  fc f c  c   co , (4-3) where fc and εc are the stress and strain values upon unloading, and εco is the strain corresponding to the compressive strength of unconfined concrete (f ′c). For confined concrete, the ratio of the unloading stiffness (Eun) to the initial undamaged stiffness (Ec) was used to determine the compressive damage parameters (dcc) with Equation (4-4). Stiffness degradation due to compressive damage was defined to start with plastic deformations. The 134 damaged stiffness upon unloading was calculated using the difference between the residual plastic strain (εpl) and the strain at the unloading point (εc) according to Equation (4-5). d cc  1  Eun Ec Eun   pl  0 fc  c   pl (4-4) (4-5) The plastic strain (εpl) was obtained from the method proposed by Mander et al. [87] for unloading behavior of confined concrete in compression. The loading and unloading behavior of confined and unconfined concrete for test unit SD0630 is illustrated in Figure 4-7. The decreasing slope of the unloading branches with respect to the initial stiffness shows the damage incurred onto the material during loading. 1 Figure 4-7: Strength and damage properties of column SD0630 for (a) confined concrete and (b) unconfined concrete Values for the remaining parameters required to complete definition of the concrete inelastic material model were based on the recommendations from the literature for unconfined concrete. 135 The dilation angle was taken to be 38°, according to Jankowiak and Lodygowski [100]; and the eccentricity, the ratio of equibiaxial yield stress to uniaxial yield stress, the ratio of the second stress invariant on the tensile meridian to that on compressive meridian, and the viscosity parameter were taken at the default values suggested by Abaqus [97] as 0.1, 1.16, 0.667, and 0, respectively. 4.4.4.4. Steel Plasticity Properties The inelastic material model for the reinforcing steel parts was based on the uniaxial stress-strain response by Mander [89]. Experimental data from bar samples were used to define stress and strain values at yield and ultimate as well as the strain at the onset of strain-hardening. Classic J2 plasticity with an isotropic hardening rule was used. A comparison between the uniaxial stressstrain curve from the test data and the numerical model for the longitudinal rebar of column M1215 is illustrated in Figure 4-8. Figure 4-8: Stress-strain curve used in numerical modeling of longitudinal bar from column M1215 compared against experimental data 136 4.4.5. Solution Algorithms A monotonically increasing displacement-controlled dynamic explicit algorithm was used with an optimum time-rate of loading that was slow enough to resemble quasi-static loading. A cyclic analysis was not necessary to study IDLS on RC columns since it has been reported [101, 102] that IDLS, i.e., spalling and yielding, are not dependent on the history of loading but they are rather closely related to the maximum displacement that the structure has experienced. Therefore, a monotonic analysis was sufficient for predicting IDLS. 4.4.6. Identification of Optimum Loading Rate The developed FE models were intended to simulate quasi-static testing protocols in which the external work by the applied forces is either converted to strain energy of the system or dissipated through plastic deformations. In a dynamic analysis, a portion of the external work needs to provide the change in kinetic energy of the system. Therefore, the force responses from a dynamic analysis are different from those obtained from quasi-static loading. The error in force values is proportional to the kinetic energy content of the system and the time-rate of loading. Therefore, higher accuracy of the FE simulation results can be achieved by minimizing the loading rate. It is thus desired to conduct the dynamic analyses at the lowest possible time-rate. However, the computational cost associated with such analyses is extremely high; whereas, analyses at faster loading rates are computationally more efficient. An optimum loading rate thus needs to be found such that the dynamic analyses become accurate and efficient. This was done by developing a computer program to conduct preliminary monotonic pushover analyses on the test columns. Fiber-based beam elements with distributed plasticity and force-formulation were utilized to conduct the preliminary monotonic pushover analyses. Results from these analyses were used as 137 the benchmark for accuracy of the 3D FE simulations conducted at different loading rates. The lateral force-displacement response at the column top was set as the response target to which results from the 3D FE analyses were compared to, as shown in Figure 4-9(a) for column M1215. By comparing the force-displacement responses from FE analyses conducted at different loading rates, it was found that the accuracy of the simulation generally improves by decreasing the loading rate. This can be also seen from Figure 4-9(b) in which the kinetic energy content of the FE model for column M1215 as a ratio of the external work is significantly smaller for the analysis with lower loading rate. A downside of analyses with lower loading rate is that the computational cost increases proportionally with the rate decrease. An optimum value for the loading rate was found when further decrease in the time-rate of loading did not enhance the accuracy of the force-displacement response, despite increasing the analysis’ computational cost. For the models studied in this work, loading rates of about 16 mm/s (0.6 in./s) provided satisfactory agreement with quasi-static analyses and reasonable computational efficiency. 138 (a) (b) (b) Figure 4-9: Optimum time-rate of loading for test unit M1215 as depicted for (a) force-displacement response; (b) kinetic energy content 4.5. Model Evaluation and Validation 4.5.1. Evaluation Levels Validation of any numerical model needs to be performed according to the intended purpose of the modeling [103, 104]. In this work, it was of interest to use element strain results from the FE analyses to determine IDLS. These results differ from global responses (i.e., force and 139 displacement values) since their errors may cancel out during integration over the model’s domain. Therefore, limit states determined from local results of FE analyses will not be reliable unless they are validated against experimental data at the same level. The validity and accuracy of the FE models developed for this study were evaluated at two levels: structural/global and local/sectional. 4.5.1.1. Global Responses Structural responses at a global level are essential pieces of information for analysis and design of RC columns. It is of interest to predict the forces and deformations that develop in columns under different loading conditions. The following results from FE simulations represent responses at a global level. Lateral Force-Displacement Lateral force-displacement response of the columns is an important response of the structure since it provides valuable information about parameters such as stiffness, strength, and ductility; which are used extensively in seismic design of bridges. Experimental data from RC columns subjected to quasi-static cyclic loading is the source for studying the force-displacement response of the columns. The force can be either represented by the effective force applied by the hydraulic actuator to the column (Feff), or by the net shear force resisted by the column (Fnet). In Figure 4-10, the predicted force-displacement response from a quasi-static monotonic pushover analysis is plotted over the experimental data from the reversed cyclic testing of column M1015. The results are shown for the effective force as well as the net shear force. The monotonic response is expected to provide an envelope to the cyclic response of the columns as illustrated in Figure 4-10. 140 Figure 4-10: Force-displacement response of test column M1015 Total Energy The total energy of the structure is a quantity that represents column behavior at a global level. The total energy is the sum of all potentially stored, dissipated, and kinetic energies of the system; and should be equal to the work done by the external forces on the structural system. External work on the system can be experimentally obtained by integrating the envelope of lateral force response of the system over the displacement domain as shown in Figure 4-11(a) for column M1215. This can be seen as the area under the force-displacement response of the column as illustrated in Figure 4-11(b) for column M1215. The difference between the energy obtained from the effective and net forces is mainly due to the work done by the axial load on the column. 141 (a) (b) Figure 4-11: Total energy for column M1215 as shown by (a) the envelopes of force-displacement response and (b) plots of energy against top displacement The total energy of the system was also obtained from the FE simulations by adding the energy content of the elements for the entire model. Close agreement between the total energy predicted by the FE simulations and the external work measured from the experiments provides insight into the ability of the simulations for predicting global responses of the columns. The total energy of 142 the model is expected to match the external work obtained from the experimental effective forcedisplacement response; whereas the energy associated with the net force should equal the total energy minus the work from the axial load. Depicted in Figure 4-12 is the total energy of the system as predicted by the FE simulations and compared against experimental data. Figure 4-12: Total energy of column M1215 considering effective and net lateral forces Dissipated Energy through Plastic Deformations The energy dissipated through plastic deformations during reversed cyclic loading is a key design parameter in displacement-based seismic design of RC columns. Equivalent inelastic damping is one such parameter and is associated with the energy dissipated in each loading cycle. The area enclosed in each complete loading cycle of Figure 4-13 represents the energy dissipated through plastic deformations during cycles at displacement ductility levels of 2 and 3 of column M1230. It can be seen from Figure 4-13 that the area enclosed by the two plots are different, such that at larger displacement levels a greater amount of energy is dissipated. 143 Figure 4-13: Complete net force cycles at displacement ductility levels of 2 and 3 for column M1230 Similar to the total energy discussed in the previous section, the dissipated energy through plastic deformations can also be obtained from the FE simulations. The dissipated energy was calculated at the material level by integrating plastic strains over the entire model. Plastic dissipation of the column M1215 as predicted by FE analyses is shown in Figure 4-14. Experimental data is also presented at different displacement ductility levels for comparison. 144 Figure 4-14: Plastic energy dissipated in FE simulations compared to experimental data 4.5.1.2. Local Responses Local response parameters of RC columns as obtained from FE simulations are considered in this section. The accuracy of the models in predicting local responses is of interest since damage on RC columns tends to be closely linked with local strain demands rather than global structural responses. Different responses of the columns at the local level are studied in this section. Flexural Curvature Profile Profiles of flexural curvatures associated with sections of the column at different elevations at any given displacement level are considered in this section. Average flexural curvatures are experimentally determined for a segment of the column length by using the relative vertical displacements measured between two sections on two sides of the column. The curvature at each element height was also calculated from the FE simulations. Axial strain outputs (SE11) from the extreme longitudinal rebars at two loading sides were used. Logarithmic 145 strains in the member direction were extracted for each node on the extreme rebar elements on two sides of the model. Sections were defined based on the planes passing through two nodes at an equal height in the undeformed configuration. The predicted curvature associated with each section was calculated from the simulation results according to Equation (4-6),  t  c , dc (4-6) where εc and εt are the strain components of the longitudinal rebars at the extreme compression and tension sides of the model, respectively. Also, dc is the diameter of the circle going through the centers of the longitudinal rebars. Figure 4-15 illustrates the curvature profiles along the height of column M1215 obtained from the FE analyses plotted against experimental data at different displacement ductility levels. Spread of Plasticity Spread of plasticity, which is defined as the length of the plastic region, was estimated from the profiles of flexural curvature along the height of the columns. The extent of plasticity along the height of the column at different displacement ductility levels was obtained by finding the intersection of the curvature profile with the ideal yield curvature value (ϕy). Figure 4-16 shows the spread of the plastic region obtained from the FE analyses as well as the experimental test data for column M1215. 146 Figure 4-15: Curvature profiles for column M1215 at different displacement ductility levels 147 Figure 4-16: Spread of plasticity along the height of column M1215 It can be seen that the FE model performed adequately in predicting the trends of the plastic region growth; however, it underestimated the height of the plastic region at large displacement levels. This can be due to the fact that the developed FE models used continuum solid elements. Thus, the formation and propagation of cracks were not directly considered. The existence of these cracks is linked to the extent of the plastic region by the effect known as tension shift, which is not well captured by the continuum-type FE models. Inclined cracks on test columns propagate through the depth of the concrete section as the top displacement increases; and as a result of the tension shift effect the plastic region grows higher than what would be predicted by a continuum-type FE model. The spread of plastic region above the base level is also governed by the primary moment gradient and P-δ effects. The effect of moment gradient on spread of the plastic region was captured reasonably well by the FE models. This was achieved by defining realistic inelastic material behavior for the concrete and reinforcing steel. In addition, full consideration of the geometric nonlinearities in the FE simulations allowed capturing contributions to the spread of the plastic region by P-δ effects, which become important for columns with large aspect ratios. 148 Flexural Rotation Profile Integration of the flexural curvatures over the height of the columns results in flexural rotation profiles. Figure 4-17 illustrates the rotation profiles for column M1230 at different displacement ductility levels. It can be seen that the errors in curvatures canceled during the integration. This leads to a smooth rotation profile that matches the experimental data closely. It is worth noting that the flexural rotations start from the base of the footing block. This leads to a rotation at the column base. Therefore, the rotation has a non-zero value at height = 0. It is evident from the plots that the base rotation of the columns was precisely captured by the FE models. This proves that the effect of the footing on the response of the test column was fully captured by the models. 4.5.1.3. Selected Responses for Comprehensive Evaluations The lateral force-displacement response at the column’s top was chosen as a global level response while sectional flexural curvatures along the column’s height were considered as local responses. The lateral force-displacement response was chosen because it is a key design feature that is commonly used for evaluating FE models. The analyses were conducted using a displacementcontrolled method and thus displacements cannot be used to evaluate the models’ performance. Therefore, the lateral force applied by the hydraulic actuator to the top of the column was used as the basis for evaluating the FE analysis results at the global level. 149 Figure 4-17: Rotation profiles for column M1230 at different displacement ductility levels 150 Flexural curvature of the sections along the height of the column was used to validate the FE models at the local response level. Curvature profiles were preferred over other local responses of the columns, such as strain measurements, since they can be measured from the experiments with reasonable accuracy and reliability using external displacement transducers mounted on opposing sides of the test column. Curvature profiles for the FE simulations were extracted from the longitudinal strain outputs at nodes of extreme rebar elements on two sides of the model. Use of reinforcement strains for extracting curvatures from the FE simulations is valid because of straincompatibility of the continuum-based models. Experimental curvatures, in contrast to those from the simulations, must be measured by external instruments and cannot be reliably captured by the strain gages on the rebars, where discontinuities due to concrete cracking violate the straincompatibility across the section. 4.5.2. Evaluation Methods 4.5.2.1. Visual Qualitative Evaluation Qualitative evaluation of the accuracy of FE models by visual comparison of the overlaid plots of simulation results and experimental data is presented in this section. The evaluation was conducted on two sets of structural responses: (1) force-displacement response of the columns, and (2) flexural curvature. Accuracy of the models was qualitatively evaluated by visual evaluation of the graphs for the responses. Force-Displacement Response The experimental cyclic force-displacement results of the test columns are compared to the predicted backbone responses from the monotonic FE simulations in Figure 4-18. It can be seen that the result from the FE analyses follow the envelope to the cyclic test results very closely. This was characteristic for all test columns. 151 M1230 M1215 M1015 B0815 SD0630 Figure 4-18: Force-displacement response for all columns 152 Flexural Curvature Profiles Figure 4-20 illustrates the experimental and simulated (FEA) curvature profiles along the height for four test columns (M1230, M1215, M1015, and B0815) at the maximum displacement ductility level to which the columns were tested. Flexural curvature profile along the height of test column SD0630 at its highest tested displacement ductility level is presented in Figure 4-20. Visual comparison of the overlaid curvature plots shows acceptable agreement between the FE analyses and the experimental data for all the sections except for the curvatures measured at the base level. The discrepancy follows from the effect of strain penetration (or bond-slip) into the footing block, which was not explicitly considered in the FE models since a perfect bond between the rebar and concrete was assumed. Therefore, the experimental measurements at the base level were omitted from the data and the quantitative comparisons with the FE results. Nonetheless; the fixed-end rotation caused by strain penetration was partially captured in the simulations by defining inelastic materials inside footing, which resulted in the accumulation of plastic curvatures and rotations below the base level and consequently a rotation at the columns’ bottom. It is worth noting that the good agreement between the FE results and test data was expected since the RC columns studied here are well-confined and have a flexure-dominated ductile response, which is easier to capture than brittle response of sections with inadequate reinforcement. Therefore, it is expected from 3D FE models, which perfectly resemble the geometry and boundary conditions of test columns and benefit from nonlinear inelastic constitutive models specifically defined for the confined and unconfined concrete, to be capable of predicting the response of flexure-dominated RC columns. 153 M1230 M1215 M1015 B0815 Figure 4-19: Curvature profiles for four test columns at their maximum tested displacement ductility 154 Figure 4-20: Curvature profiles for test column SD0630 at its maximum tested displacement ductility Conversely, accurate predicting of the inelastic response for brittle RC sections with sub-sufficient transverse reinforcement, in which concrete brittleness, concrete-reinforcement bond behavior, and shear-flexure cracking dominate the response of the section, is more challenging and requires sophisticated constitutive models to capture concrete cracking and bond-slip. 4.5.2.2. Quantitative Evaluation Visual evaluation of simulated responses against experimental data, despite its popularity, is inadequate for validating complex 3D FE models and particularly for the purpose of this study. Quantitative evaluation of the accuracy for the simulated results is required to demonstrate the validity of a model. However, quantitative approaches need appropriate data preparation in order 155 to create a one-to-one correspondence of the location and time between analysis results and experimental data. Flexural curvature data from the FE simulations and experiments of column SD0630 are plotted over a two-dimensional grid of location and displacement ductility in Figure 4-21(a) and (b), respectively. Comparison of the data points between the FE results (Figure 4-21(a)) and experimental data (Figure 4-21(b)) illustrates the lack of correspondence between the grids. Results from FE analyses can thus only be compared to experimental measurements for the same location and time. Data Preparation for Quantitative Evaluation To establish a one-to-one correspondence, a 2D linear interpolation over the independent variables of time and space was used in lieu of a perfect match between the data points. The overlaid spacetime grids for the FE results and experimental data for column SD0630 are shown in Figure 4-22, in which the coarser and thicker grid represents the experimental data points. Results from the FE simulations were thus interpolated to obtain projected values at the experimental data locations on the time-space grid. 156 (a) (b) Figure 4-21: Flexural curvature values for column SD0630 plotted over the grid of location and displacement ductility level as shown for (a) FE analyses results and (b) test data 157 Figure 4-22: Overlay of experimental space-time grid on that of FE analysis for curvature profile data of column SD0630 Scatter Plots Once a one-to-one correspondence between the experimental and simulated data is established, a comparison can be made by means of a scatter plot of observed versus predicted data as depicted in Figure 4-23 and Figure 4-24 for lateral force and flexural curvature responses, respectively. The slope of the fitted line, which is used as a measure of the accuracy for the FE results, represents the bias in the predictions. The closer the slope is to a value of one, the less bias the prediction has in simulating the experimental results. It can be seen from Figure 4-23 and Figure 4-24 that the predictions’ bias is small and that the mean of the experimental data and FE results are close to each other. It can also be observed that data dispersion in the case of curvatures is greater than for 158 the lateral force responses. Evaluation of the variance and dispersion of the data needs goodnessof-fit measures as discussed in the following section. Figure 4-23: Scatter plot of experimentally measured lateral forces versus the FE results Figure 4-24: Scatter plot of experimentally measured curvatures versus the FE results 159 Relative Accuracy Measure The relative accuracy of the results obtained from a numerical model was evaluated by the mean absolute percentage error (MAPE) as defined by Equation (4-7), MAPE  1 N N  i 1 yˆi  yi , yi (4-7) where ŷi and yi are the response parameters obtained from the FE model and the experiment, respectively. N is the total number of data points. Relative error measures are sensitive to the magnitude of the experimental data; therefore, they are less useful when the experimental data tend to zero [105]. The MAPE tends to infinity as the experimental data goes to zero, which can show an unrealistic high error value, despite reasonable accuracy of the model. For instance, sectional curvatures in elastic regions are susceptible for large relative error since their values are very small. As a result, absolute error measures can be more reliable tools to evaluate the accuracy of numerical models when all responses are not in the same level of magnitude and small or even zero values are expected to occur. Absolute Accuracy Measure Absolute error measures, unlike relative error measures, are independent from the magnitude of experimental data and lead to a better evaluation of a model’s accuracy. The root mean square error (RMSE), as defined by Equation (4-8), has been shown [105] to be the most suitable statistical parameter to evaluate the goodness-of-fit for numerical models when the magnitude of data can be very small. 160 RMSE  1 N N   yˆ i 1 i  yi  2 (4-8) The magnitude of the RMSE indicates dispersion of the numerical results from the experimental data. The RMSE is known to penalize large errors with higher values compared to other error measures since the differences are squared. The RMSE has the units of the measured or predicted response parameters and this is an issue only when it is utilized to evaluate response parameters with different units. Normalized Accuracy Measure One way to remove the units from the RMSE is to divide it by the mean value of experimental data (ȳ). The normalized root mean square error (NRMSE) is defined by Equation (4-9). NRMSE  1 N 2  yˆi  yi   N i 1 y (4-9) The NRMSE, despite being dimensionless, is not bounded to a higher value. It is often desirable to use a measure of accuracy that is bounded between two values, where one bound corresponds to a perfect match between the numerical data and the experiment and the other bound corresponds to the worst prediction with no correspondence to the experimental data. The normalization method proposed by Theil [106] was found as the most applicable correlation metric to be utilized in multi-level evaluation of the local and global results from the FE simulations. Bounded Inequality Coefficients Theil’s inequality coefficient (U) is a bounded error measure, i.e., its value is bounded between two extremes, where one bound corresponds to a perfect match between the numerical data and the experiment and the other bound corresponds to the worst prediction without any proportionality 161 to the experiment. The U value thus provides a statistical tool to measure the dispersion between numerical results and experimental measurements. Defined by Equation (4-10), the U value is bounded between 0 and 1, where zero implies a perfect model and one is for the worst model. U 1 N N 1 N   yˆ  y  N N i 1  yˆ i 1 i 2 i  2 i 1 N RMSE  1 N y i 1 2 i N  yˆ i 1 2 i  1 N (4-10) N y i 1 2 i 4.5.3. Results of Evaluation and Validation Theil’s inequality coefficient was used to evaluate the accuracy of the FE models at local and global levels. The results of quantitative evaluations are presented here. 4.5.3.1. Global Level The simulation results in terms of the effective and net forces were compared to the corresponding data from the tests. Table 4-2 summarizes the error measures MAPE, NRSME and U calculated for the force responses of the FE simulations. Table 4-2: Accuracy of FE models in predicting the lateral force-displacement response Error Measure Response M1230 M1215 M1015 B0815 SD0630 MAPE NRMSE U Net Force Effective Net Force Effective Net Force Effective (Fnet) Force (Feff) (Fnet) Force (Feff) (Fnet) Force (Feff) 11% 14% 13% 16% 0.057 0.070 10% 6% 12% 8% 0.053 0.034 7% 10% 7% 14% 0.033 0.060 6% 10% 7% 7% 0.035 0.032 4% 3% 5% 4% 0.023 0.019 162 4.5.3.2. Local Level The accuracy of the FE simulations at the local level was evaluated by determining different error measures for the flexural curvature. The error parameters, as presented in Table 4-3, yield a quantitative evaluation of the model’s accuracy at the local level. Table 4-3: Accuracy of FE models in predicting the flexural curvature profiles Error Measure M1230 M1215 M1015 B0815 SD0630 MAPE NRMSE U 14% 17% 21% 36% 31% 34% 37% 43% 44% 28% 0.101 0.103 0.100 0.108 0.078 It can be seen that the errors at a local level were generally greater than those at the global level. This difference was not evident from the visual comparison of the overlaid plots. The accuracy levels of the model in predicting curvatures are anticipated to transfer to the predicted IDLS since the identification of these conditions depend on the local response parameters. In general, the FE models in this study predicted curvatures with an inequality level of about 0.1. 4.6. Determining Intermediate Damage Limit States 4.6.1. First Yield of Longitudinal Reinforcement Bars The first intermediate damage limit state determined during the test of a RC column is the yielding onset of the longitudinal bars in tension. Typically, strain gages attached to the extreme bars are used to measure the demand level. Taking a similar approach for the FE analyses, the longitudinal strains were extracted from the extreme reinforcing bar in tension and compared to the yield strain 163 of the material. By doing so, the onset of first yield was associated with the instance when the strains on the rebar exceeded the yield strain for the first time. Table 4-4 and Table 4-5 present the design parameters of displacement ductility and drift ratio at the IDLS studied here, including first yield. Results from the FE simulations, along with the experimental observations of damage states in terms of displacement ductility and drift ratio are listed in Table 4-4 and Table 4-5, respectively. The correlation of the FE results with experimental data shows that the first yield limit state was predicted well for the test columns studied here. The general trend for all test columns, except for M1015, is that the FE simulations predicted the onset of yielding later than the experimental measurements. However, the predictions were satisfactorily close to the test data. Table 4-4: Displacement ductility at intermediate damage limit states IDLS Exp. / FE SD0630 B0815 M1015 M1215 M1230 First Yield Initial Spalling Significant Spalling Exp. FE Exp. FE Exp. FE 0.98 1.03 2.0 1.2 2.0 2.7 0.64 0.83 1.5 1.2 2.0 2.0 0.80 0.76 2.0 1.2 3.0 1.6 0.84 0.94 2.0 1.6 2.0 2.1 0.89 0.91 1.5 1.2 2.0 1.6 Table 4-5: Drift ratio at intermediate damage limit states IDLS Exp. / FE First Yield Initial Spalling Significant Spalling Exp. FE Exp. FE Exp. FE SD0630 B0815 M1015 M1215 M1230 (%) (%) (%) (%) (%) 1.1 1.2 1.5 1.6 2.1 1.1 1.5 1.4 1.8 2.2 2.2 2.7 3.7 3.8 3.5 1.3 2.1 2.2 3.0 2.8 2.2 3.6 5.5 3.8 4.7 2.9 3.6 2.9 3.9 3.8 164 4.6.2. Tracking the Spalling of Cover Concrete Spalling of the cover concrete is related to the strains in the compression side of the column. The strains are extractable from continuum-based 3D FE analyses of the columns. In this work, the profile of the compressive strains on the extreme surface elements of the models was used to evaluate the damage limit states involving spalling of the cover concrete. Plotting the distribution of compressive strains along the column height at each displacement ductility level provides the profile of strains. Similar plots for test columns M1215, M1015, B0815 and SD0630 can be found in Figure 4-25. Estimating the extent of the spalling region and determining the initial and significant spalling damage limit states requires a spalling limit strain to be defined for the unconfined concrete. This spalling limit strain differs from the average spalling strain estimated from the experimental curvatures since the former is related to the minimum strain that causes the cover concrete to spall, whereas the latter is the average compressive strain over the spalled region of the cover concrete. The spalling limit needs to be exceeded in order for a spalling region on the cover to develop. Therefore, the average strains over the spalled region are usually larger than the initial spalling limit strain and may not be directly used for determining the spalling limit strain. A limit of -0.005 was chosen for this work in accordance to recommendations by the AASHTO [20] and Caltarns [34] design guides. It should be noted that the same spalling strain limit was used in defining the material model for unconfined concrete. 165 M1215 M1015 B0815 SD0630 Figure 4-25: Compressive strain profiles in cover concrete for different displacement ductility levels Use of continuum-type solid elements in the FE simulations make it possible to visualize the potential spalling region on the cover concrete in addition to predicting the onset of the limit states. The longitudinal compressive strain outputs from the FE analyses at the nodes located on the surface of the model were used to graphically show the predicted spalled region for different 166 displacement ductility levels. This process resulted in strain contour plots representing spalling zones similar to the damage observed on the actual tests. The distribution of longitudinal strains (LE22) on the compression side of the FE model is shown in Figure 4-26(a) for columns M1230, M1215 and M1015. It is worth noting that the height of elements was 76 mm (3 in.). The strain contours represent the state of the column at the highest displacement ductility level reached during testing. In Figure 4-26(b) the region of the model’s surface where the nodal longitudinal strains exceeded the spalling limit is highlighted. The highlighted region shows the potential spalling zone of the cover at the specified displacement ductility demand. The actual state of the damage on the cover concrete of the test units is shown in Figure 4-26(c). It can be seen that the extent of spalling damage was well captured in size and shape by the FE simulations. 4.6.1. Onset of Spalling of the Cover Concrete Spalling of cover concrete is evaluated experimentally from visual observations of the damage on the surface of the column. Following the definition of the initial spalling previously presented in this document, the onset of spalling limit states was taken as the point when the size of the spalled region reached a tenth of the section width, after Hose and Seible [96]. The onset of spalling obtained from the simulations in terms of displacement ductility and drift are presented in Table 4-4 and Table 4-5, respectively. The experimental observations are also included for comparison. 167 M1230 M1215 M1015 (b) (a) (c) Figure 4-26: Predicted spalling region for the test units at their highest displacement ductility level 168 In experimental studies, the state of spalling damage is typically evaluated and reported at the peak displacement ductility level associated with the first observation of the damage. Thus, the experimental displacement ductility levels and the related drift ratios presented in Table 4-4 and Table 4-5 correspond to peak values. It can be seen that the FE simulations predict the onset of spalling to occur sooner than the experimental observations for all cases. However, it can be argued that the actual damage level is reached at some point along the loading cycle, i.e., between the previous and the reporting peaks, during a typical cyclic loading protocol. Therefore, it is reasonable to consider that the continuous damage prediction from FE simulations leads to limit states smaller than those reported experimentally. 4.6.2. Significant Spalling of the Cover Concrete The onset of significant spalling was related to a state where the length of the damaged region exceeded half the section width, according to Hose and Seible [96]. As for the onset of spalling, determining the exact occurrence of significant spalling from testing is challenging, since damage is typically evaluated and reported at the loading cycle peaks. In contrast, the FE models provide continuous results for the initiation and growth of spalling of the cover concrete; which makes it possible to determine the exact instance when significant spalling develops. Figure 4-27 depicts the initiation and spread of the spalling region on the cover concrete as well as the measurements made during testing. 169 M1230 M1215 M1015 B0815 SD0630 Figure 4-27: Comparative evaluations of cover concrete spalling between FE model and experiment 170 Comparison between the experimental data and the FEA results reveal that the spalling damage state was predicted fairly well by the FE models, considering the fact that the growth of the spalled region on actual test units is relatively random and sensitive to local variations of the quality of the cover concrete. The predicted limit state for significant spalling obtained from the FE analyses is compared to test measurements in Table 4-4 and Table 4-5. It can be seen from these tables that, in general, the FE simulations predicted the occurrence of significant spalling closely to the observed level from tests. 4.6.3. Engineering Design Limits States It is often useful to link damage limit states occurring at a local level on a bridge column to global structural design parameters, such as displacement ductility and drift ratio. The global design parameters are selected from responses of the column that can be reliably and easily predicted by most of structural analysis methods, including models that are based on beam-column formulations. This helps engineers to determine the onset of damage states without conducting sophisticated 3D FE analyses. In this section, displacement ductility and drift ratio are presented for damage limit states. 4.6.3.1. Displacement Ductility Displacement ductility, as defined by the ratio of the lateral displacement over the ideal yield displacement, is a common design limit state for bridge columns. Figure 4-28 presents the growth of the spalling region on the cover concrete as a function of displacement ductility. Two horizontal lines passing the vertical axis at 0.1 and 0.5 represent the limit states for initial and significant spalling limit states, respectively. It can be seen that for all columns spalling started at a common point of displacement ductility level 1, but it grew more rapidly in slender columns. 171 Figure 4-28: Growth of the spalled region over the height of test columns versus displacement ductility 4.6.3.2. Drift Ratio Drift ratio, as defined by the ratio of the lateral displacement of the column at the top to the length of the column, is another common engineering design parameter to be considered. Figure 4-29 illustrates the growth of the spalling region on the cover concrete for all test columns as it increases with drift ratio. It can be seen that the initiation of spalling region on columns with different geometric configurations varied in terms of drift ratio. That is, spalling starts later in a slender column when drift ratio is taken as the engineering limit state. 172 Figure 4-29: Growth of the spalled region over the height of test columns versus drift ratio 173 CHAPTER 5 5. ANALYTICAL STUDY 5.1. General This chapter presents the development of analytical expressions and closed-form solutions to the effect of member deformations (δ) on the inelastic response of RC bridge columns are presented. Two dimensionless slenderness parameters that measure the susceptibility of RC columns to second-order effects before and after yield are identified. Simplified formulas for the spread of the plastic region (Lpr) and the magnitude of P-δ moment are proposed for use in seismic design. The design formulas, derived from the mechanics-based solution, are shown to be able to predict experimental data on the effects of P-δ with satisfactory accuracy. Two distinct sets of analytical expressions for the effects of member deformations and P-δ moments on the inelastic response of slender RC columns were developed. The first set of the expressions, referred to as the elastic approach, assumed an elastic structure with a constant flexural stiffness for the cracked RC section. The assumption of constant stiffness allowed deriving closed-form solutions and simple design formulas to calculate the effects of second-order 174 moments. The second set of expressions, labeled as the inelastic approach, were developed to include the effect of material inelastic response on the P-δ moments. The latter approach led to more accurate solution to the P-δ moments than the elastic approach. Yet, the simplicity of the elastic approach was compromised. 5.2. Linear Elastic Solution to Lpr The simplest way to derive an expression for the extent of the plastic region due to moment gradient along the height of RC columns is to assume a linear moment distribution. Accordingly, the length of the plastic region is obtained by Lpr,L = L (1 - My/MBase), in which Lpr,L is the length of the plastic region due to linear moment profile, L is the length of the column as defined by the distance from the point of contra-flexure to the point with maximum moment, and My/MBase is the ratio of the yield moment (My) to the maximum moment at the base of the column (MBase). 5.3. Nonlinear Elastic Solution to Moment Profile A closed-form solution to the nonlinear moment profile along a cantilever that includes the effect of P-δ was derived. This was achieved by a coordinate transformation that substitutes the cantilever member with a simply-supported beam-column that has two concentrated moments at the ends and a constant axial load. The solution to the substitute structure, as presented by [107], was extended to the case of a cantilever through a coordinate transformation as discussed in the following. 5.3.1. Coordinate Transformation The deformed configuration of a cantilever column, as illustrated in Figure 5-1(a), was substituted by a simply-supported element (Figure 5-1(b)) that shares the same chord line with the cantilever. The chord line, which connects the top to the base of the cantilever column, was selected as the reference axis of the element. The continuously rotating chord line allows for the formulation of 175 P-δ effects that is independent from the lateral displacement of the column, Δ. Yet, the effect of PΔ is implicitly considered in the transformed substitute structure by introducing the rotation angle of the chord line θ. The applied and constraint forces in the original cantilever and the transformed substitute structure are shown in Figure 5-1(a) and (b), respectively. Figure 5-1: Deformed column and forces acting on it with respect to: (a) vertical fixed reference axis and (b) rotated element chord axis The applied and constraint forces/moments from one configuration can be converted to the other configuration using a planar rotation transformation by the angle θ, according to Equation (5-1). The forces and moments in the rotated configuration are distinguished from their counterparts by use of prime notation. 176  P  cos     V     sin   M   0     sin  cos 0 0  P    0  V   1  M  (5-1) It is worth noting that the moments in the substitute structure are the same as the moments in the actual cantilever, i.e., M = M′. As a result, the end moments of the substitute simply-supported structure are identical to those of the actual cantilever column, where the moment at the top is zero and the moment at the base level is given by MBase = VL + PΔ = V′L. In this transformation, it was assumed that the top of the column is moving on a circle with a radius equal to the height of the column and an origin located at the column base. 5.3.2. Member Deformation Profile (δ) The closed-form solution to the member deformations (δ) was derived for a deformed column in its rotated coordinate system. The free body diagram of such a column under the applied and constraint forces and moments is shown in Figure 5-2. Using the nomenclature for the transformed coordinate system in Figure 5-2, which is indicated by the primed notation, Equation (5-2) was obtained as the governing differential equation for the member deformation. d2y' P' V'  y'   x' 2 dx ' EI el EI el (5-2) In Equation (5-2), EIel is the flexural stiffness of the cracked RC section before yielding of the steel reinforcement and is assumed to be constant; which means that the degradation of flexural stiffness due to yielding of the longitudinal reinforcement is not taken into account. 177 Figure 5-2: Equilibrium of column top segment in its deformed shape The general and particular solutions for the governing differential equation are provided in Equation (5-3), in which C1 and C2 are constants.  P'   P'  V ' y '  C1 sin  x '   C2 cos  x '  x ' EI EI el el     P' (5-3) Applying the boundary conditions at the column’s top and bottom in which the member deformations (δ) are zero and solving for the constants yield Equation (5-4) for the deformations of the column away from the chord line. y '     M Base  sin  el    P '  sin  el   178 (5-4) In Equation (5-4), α represents the ratio of x′/L and the dimensionless factor κel is defined according to Equation (5-5). This factor is termed the ‘elastic slenderness parameter’ for reasons that will be discussed later in this document.  el  L P' EI el (5-5) 5.3.3. Bending Moment Profile (Mt) The total bending moment Mt is the sum of the primary moment Mp and the secondary moment Ms, where the primary moment (Mp) is a linear distribution of the end moments over the length of the element according to Equation (5-6). The secondary moment (Ms) is the product of the axial load P by the transverse member deformation y′, as given by Equation (5-7). Therefore, the total moment is given by Equation (5-8).  x  M p  M Base   L (5-6) M s  P  y (5-7)  x  M t  M Base    P  y  L (5-8) Differentiating Equation (5-8) twice leads to Equation (5-9), which can then be solved for the noted moment distribution. d 2 M t P '  M t  0 dx '2 EI el 179 (5-9) General solution for the differential equation for the total moments (Equation (5-9)) is shown in Equation (5-10) in which A and B are constants to be found by the application of boundary conditions.  P'   P'  M t  A sin  x '   B cos  x '  EI el   EI el  (5-10) Boundary conditions at the top of the column require the moment to be zero. At the base level, the total moment is equal to MBase. Imposing the aforementioned boundary conditions yields the solution to the distribution of total moments as expressed by Equation (5-11). M t    sin  el  M Base sin  el  (5-11) 5.4. Nonlinear Elastic Solution to Displacement Profile The moment profile derived for RC columns in the transformed configuration of the substitute structure is also valid for a cantilever column because the moments are identical in both configurations (M′t = Mt). Therefore, the proposed solution in Equation (5-11) also describes the bending moment diagram along a cantilever column. The complete response of the column in terms of curvatures and displacements can be obtained by combining the moment-curvature response of the RC section with the solution for internal moments. 5.4.1. Cross-Section Moment-Curvature Response The cross-section response of RC sections in flexure, commonly known as the moment-curvature (M-ϕ) response, is needed to extend the proposed solution to the full response of RC columns. In this document, the M-ϕ response of RC sections was implemented in two forms: (1) a “curve” that represents the actual response of the RC section, and (2) an idealized ‘bilinear’ response. 180 5.4.1.1. M-ϕ as a Curvilinear Response The moment-curvature (M-ϕ) response of RC sections was numerically calculated through a sectional analysis on a fiber-discretized section with inelastic material properties for the concrete and reinforcement. The M-ϕ response obtained from a section analysis provides curvature values corresponding to different levels of bending moment. The term ‘curvilinear’ is used throughout this chapter whenever the full curve of M-ϕ response from the section analysis was directly employed to extract the results. 5.4.1.2. M-ϕ as an Idealized Bilinear Response An Idealized response of an RC section in terms of bending moment versus flexural curvature was assumed to follow a bilinear response as depicted in Figure 5-3, where My and ϕy are the moment and curvature of the sections at yield, and Mu and ϕu are the ultimate moment and curvature of the section. Figure 5-3: Bilinear idealization of moment-curvature response 181 Using the bilinear assumption for the moment-curvature response, the corresponding curvature ϕ for a given moment at any section along the column is obtained from Equation (5-12) and Equation (5-13) for the pre-yield and post-yield regimes, respectively.   Mt  M y Mu  M y Mt y My  u 0  Mt  M y for  y   y for (5-12) M y  Mt  Mu (5-13) 5.4.2. Solution to Curvature Profiles (ϕ) The curvature profile along the column height was extracted from the moment profile using the curvilinear and bilinear M-ϕ responses. When the actual M-ϕ curve is used the moment of sections along the height of the column is replaced by the corresponding curvature from the M-ϕ response, as obtained from the sectional analysis. On the other hand, a bilinear approximation for the M-ϕ response of the section yields closed-form solutions for the curvature profile of the column as presented in Equation (5-14) and Equation (5-15) for situations before and after yield. Lpr is the length of the plastic region over which the moment demand exceeds the yield moment capacity.         M Base sin  el  y M y sin  el  for M Base sin  el   M y sin  el  EI in sin  el   y 182 0    1 Lpr 1 Lpr for (5-14) L L  1 (5-15) 5.4.3. Solution to Rotation Profile (θ) The closed-form solutions to elastic and inelastic rotations of the column are given in Equation (5-16) and Equation (5-17), respectively, where β = Lpr/L. Elastic rotations θel represent the accumulation of elastic curvatures (ϕ < ϕy) over the region outside of the plastic region. Rotations due to inelastic curvatures within the plastic region are given by the inelastic rotations θin. el     in     y   L y M Base cos  el   cos  el   el     el sin  el  M y  My  M Base L cos  el   cos  el   1    L   EI in  EI in el sin  el   (5-16) (5-17) 5.4.4. Solution to Lateral Displacements (Δ) The displacement profile for the cantilever column was extracted from the proposed solution by integrating the rotations along the height. Column rotations due to flexural curvatures were found by numerical integration based on trapezoidal quadrature rule (for the case of the M-ϕ curve) and by closed-form integration of the curvatures (for the bilinear M-ϕ case) over the column height. The resulting solutions to the elastic and inelastic rotations are given in Equation (5-18) and Equation (5-19), respectively. The elastic deformations outside the plastic region and the inelastic deformations within the plastic region were found by integrating the rotations along the column height.  el    L2 y M Base sin  el   el    sin  el    el 1      cos  el   el     el2 sin  el M y  183 (5-18)  in    L2 2   My   M Base L2 2  y  EI  1      EI  2 sin   sin  el   sin  el    el 1    cos  el     in  el    in el (5-19) Using the equations for the elastic and inelastic displacements, the top lateral displacement of the column is obtained by the sum of three components: (1) the elastic deformations over the length of the column that is still in elastic region, (2) the inelastic deformations of the column within the plastic region, and (3) the lateral displacement of the elastic region due to plastic rotations of the plastic region. The expression for the top displacement of a cantilever column with a nonlinear distribution of moments and a bilinear assumption for the M-ϕ response of the section is provided in Equation (5-20). Top   L2 y M Base sin  el   el     el 1    cos  el   el      el2 sin  el M y  L2  2 2  My    M Base L2     y EI   EI  2 sin   sin  el   sin  el   el     el  cos  el    in    in el  el   (5-20) M   M Base L   y  y   L  cos  el   el    cos  el   EI in  EI in el sin  el    5.4.5. Solution to Length of the Plastic Region (Lpr) In order to obtain the length of the plastic region, the total moment was set equal to the yield moment (My). Solving for α yields a closed-form formula for the length of the plastic region Lpr, which is given in Equation (5-21).   M  1 Lpr  L 1  sin 1  y sin  el     M Base    el 184 (5-21) 5.5. Verification of the Nonlinear Elastic Solutions The proposed solutions for moment, curvature, and displacement profiles were verified against experimental data from large-scale RC columns. The distribution of bending moments along the columns’ height was obtained from Equation (5-11), which is independent from the sectional response since a constant flexural stiffness was assumed. Unlike the moment profile, the distributions of curvatures and displacements along the column depend on the M-ϕ response of the section. Therefore, the results of the proposed solution were evaluated for the two noted assumptions of the M-ϕ response. 5.5.1. Experimental Database for Evaluating the ‘Elastic Solution’ Experimental results from four half-scale RC columns were selected for verification. The columns were tested as free-standing cantilevers under quasi-static reversed cyclic loading and constant axial load. All test columns were detailed to behave in a ductile manner and had a flexuredominated response. The tested columns had aspect ratios (effective length over diameter or L/D) of 8, 10 and 12. All four columns had circular cross sections with a diameter of 610 mm (24 in.). The column with L/D = 8 (B0815) is the test unit known as column 815 from the experimental program reported by Lehman and Moehle [47]. The columns with L/D ratios of 10 and 12 were tested by the authors. The designated name tag, geometry, reinforcement details and key material properties for the test columns are presented in Table 5-1. 185 Table 5-1: Properties of RC test columns for evaluation Property B0815 M1015 M1215 M1230 L/D L (mm) Lft (mm) dsl (mm) ρsl dsh (mm) ρsh 8 4877 450 15.9 1.5% 6.4 0.7% 10 6096 482 19.1 1.6% 12.7 1.2% 12 7315 482 19.1 1.6% 12.7 1.2% 12 7315 482 25.4 3.0% 12.7 1.2% P/Agf ′c P (kN) f 'c f (MPa) (MPa) 7.2% 654 8.8% 1334 5.4% 712 5.6% 712 31 462 630 0.12 32 51.8 455 689 0.12 76 42.2 455 689 0.12 76 46.1 486 724 0.12 76 yl ful (MPa) εul s (mm) The response of all columns was monitored by means of various instruments. Average curvatures over small segments along the columns height were calculated from vertical displacement transducers (DTs) mounted on the two loading sides of the columns. The lateral displacements and the member deformations of the columns were measured at different elevations above their base and at their top by horizontal DTs. Experimental results extracted from the measurements by the vertical and horizontal DTs, which were used to verify the closed-form solutions, are labeled as “Experimental (1)” and “Experimental (2)”, respectively. 5.5.2. Solution Parameters for Test Columns The solutions presented in this document require defining the equivalent flexural stiffness (EIel) for the cracked RC section prior to yield as well as the moment and curvature values at yield and ultimate. Discussion on the definitions for these parameters follows. 186 5.5.2.1. Yield Moment (My) and Yield Curvature (ϕy) The yield point of an RC section can be defined at (1) the first onset of yield, (2) the nominal yield and (3) the ideal yield. First yield of an RC section is associated with the yielding onset of the extreme longitudinal rebar in tension. The moment and curvature values at first yield are denoted as M′y and ϕ′y, respectively. The nominal yield point corresponds to a moment (Mn) that generates compressive strains larger than 0.004 in the extreme concrete fiber or tensile strains greater than 0.015 in the extreme longitudinal rebar, whichever occurs first. The section curvature at nominal yield (ϕy) is given by ϕy = (Mn∙ϕ′y) / M′y. This method of defining the yield point of a section leads to a point located outside of the actual M-ϕ response of the section. Consequently, there is no actual physical correspondence between the yield moment and the yield curvature. To overcome this issue, it was proposed to use an ideal yield point that corresponds to the actual moment (My) of the section at the ideal yield curvature (ϕy), which was taken equal to the nominal yield curvature. Bilinear idealizations of the M-ϕ curve according to different methods of defining the yield point were used for the closed-form formulation for displacement and curvature profiles. These bilinear responses are denoted as cases (1), (2) and (3) based on the first, the nominal and the ideal yield points, respectively, as shown in Figure 5-4. The actual M-ϕ response of the RC section was also implemented into the solution by numerically integrating the moment profile to achieve displacement and curvature profiles. Yield moments and yield curvatures as determined according to the different methods for defining the yield point are given in Table 5-2 for all test columns. 187 M1230 M1215 M1015 B0815 Figure 5-4: Moment-curvature response for the selected test columns with bilinear idealization options 5.5.2.2. Ultimate Moment (Mu) and Ultimate Curvature (ϕu) The ultimate point was consistently defined for all four cases of bilinear and curvilinear M-ϕ responses as the point at which the strain in the extreme fiber of the confined concrete core or extreme tensile rebar exceeds the ultimate crushing or rupture limits, respectively, whichever happens first. The ultimate crushing limit for concrete was defined according to the energy balance method. The ultimate strain limit for the reinforcing steel was governed by low-cycle fatigue as proposed in the literature [89]. Accordingly, the values of Mu and ϕu were taken as the moment 188 and curvature corresponding to the ultimate response of the section. Moment and curvature values at ultimate for the test columns are given in Table 5-2. Table 5-2: Analysis parameters for the test columns Ultimate (3) Ideal Yield (2) Nominal Yield (1) First Yield State Property ϕ′y (μ 1/mm) M′y (kN∙m) EIel,1 3 (10 kN∙m2) κel,1 ϕy (μ 1/mm) Mn (kN∙m) EIel,2 3 (10 kN∙m2) κel,2 ϕy (μ 1/mm) My (kN∙m) EIel,3 (103 kN∙m2) κel,3 ϕu (μ 1/mm) Mu (kN∙m) B0815 M1015 M1215 M1230 6.39 6.49 6.18 6.97 453 601 482 821 70.9 92.4 77.9 117.9 0.511 0.790 0.745 0.606 8.50 8.33 8.38 9.53 603 770 653 1123 70.9 92.4 77.9 117.9 0.511 0.790 0.745 0.606 8.50 8.33 8.38 9.53 513 664 548 955 60.3 79.7 65.5 100.2 0.555 0.851 0.813 0.657 139.9 140.8 136.0 139.3 665 861 750 1287 5.5.2.3. Flexural Stiffness of Cracked RC Section (EIel) The proposed solution is based on a constant section flexural stiffness (EIel) given by EIel = My/ϕy, where the yield properties of the section are obtained according to one of the three previously 189 described approaches for defining the yield point. The value of EIel, which is the slope of the elastic branch of the bilinear M-ϕ response, is given for column M1215 in Figure 5-5. It can be seen that cases (1) and (2) yield the same value for EIel and that it is greater than the average stiffness of the actual curvilinear response before yield. On the other hand, using case (3) to define the yield point leads to a flexural stiffness that is more consistent with the actual stiffness of the cracked RC section. Figure 5-5: Equivalent flexural section stiffness of column M1215 before yield (EIel) The slenderness parameter κel was calculated for the case study columns according to Equation (5-5). The values of κel for the three definitions of yield point on the RC sectional response are provided in Table 5-2 for all test columns. In calculating κel, P′ was taken equal to P (axial load) because it leads to a condition that is more consistent with typical laboratory test setups for applying axial load via post-tensioning steel rods. Moreover, the height of the test columns was adjusted to include the height of the columns’ base, or foundation, block (Lft), This was done to 190 simulate the constraint condition, which is typically provided to the base block rather than the column. 5.5.3. Experimental Data vs. Closed-Form Solution Profiles for the bending moment, curvature and displacement over the height of the test columns were obtained from the proposed analytical solution (through the respectively derived expressions) and compared against test measurements. In implementing the closed-from solution, different options for defining the section response, namely three bilinear cases and a curvilinear case, were attempted. 5.5.3.1. Bending Moment Profiles Bending moment profiles were extracted from measurements of applied forces and lateral displacements at several sections along the columns’ height. The experimental moment profile for column M1215 is plotted against the results from Equation (5-11) at the same base moment in Figure 5-6. Satisfactory agreement is seen between the results from the analytical solution and the experiment. The experimental results slightly diverged from the proposed solution over the plastic region because the proposed solution does not include the reduction in flexural stiffness after the yield point. 191 Figure 5-6: Moment profiles for column M1215 at the base moment corresponding to μΔ = 3.5 5.5.3.2. Curvature Profiles The moment profiles were extended to curvature profiles by using the M-ϕ responses of the RC section. Bilinear idealization of the M-ϕ response and the actual curvilinear response were considered. The results for column M1215 are plotted against the experimental curvatures with the same curvature at the base level in Figure 5-7. It can be seen that the different methods of defining the bilinear M-ϕ response significantly affect the solution’s accuracy, with the curvilinear M-ϕ response providing a closer prediction to the experimental results. 192 Figure 5-7: Curvature profiles for column M1215 at the base curvature corresponding to μΔ = 3.5 5.5.3.3. Displacement Profiles The extracted displacement profiles from the solution only include deformations due to flexure. Thus, end-rotation at the column base due to strain-penetration, which is a significant contributor to lateral displacement of the actual columns, was not considered in the proposed solution. This effect was added to the solutions’ results (bilinear and curvilinear) according to Equation (5-22).  sp    L1      y Lsp (5-22) where Lsp is the extent of the plastic region into the footing and below the base level. The method proposed by Priestley and Park [53] for the length of strain-penetration (Lsp) was used according to Equation (5-23) and Equation (5-24); where dsl and fyl are the diameter and yield strength of the longitudinal rebar, respectively. 193 Lsp  0.022 d sl  f yl Lsp  0.15d sl  f yl for MPa (5-23) for ksi (5-24) Experimental measurements of the test columns’ deformation profiles were compared to the displacement profiles extracted from the analytical solution at the same top displacement, as it is illustrated for column M1215 in Figure 5-8. Satisfactory agreement is seen between the test data and the results from the analytical solutions. Figure 5-8: Displacement profiles for column M1215 at μΔ = 3.5 5.5.4. Quantitative Evaluation of the Proposed Solution The root mean square error (RMSE), normalized according to the method by Theil [106], was used to test the goodness-of-fit between the observed and predicted quantities. The normalized 194 correlation measure, known as Theil’s inequality coefficient (U), is an error measure whose value is bounded between 0 and 1, where zero implies a perfect model and one is for the worst model. Theil’s inequality coefficient (U) was used to evaluate the accuracy of the solution in predicting the experimentally measured profiles for moment, curvature and displacement. The evaluations were done on the basis of an equal base moment (MBase), base curvature (ϕBase), and top displacement (ΔTop). The error (U) was calculated separately for each profile as the average of errors along the profile. Moreover, different methods for defining the RC section M-ϕ response, i.e. bilinear and curvilinear, were considered. The summary of error measures for the profiles of moment, curvature and displacement is presented in Figure 5-9. Also shown is the average of the errors for predicting the structural response of the test columns. It can be seen that the proposed solution is most accurate for the moment profiles. In contrast, the accuracy of the solution declines as it extends to curvature and displacement profiles, where the M-ϕ response needs to be added to the model. Comparison of the error measures for different definitions of the bilinear M-ϕ response of RC sections, as shown in Figure 5-9, reveals that the proposed method of defining the yield point (case (3)) is superior to the first and nominal yield points. Among all the studied cases, the use of first yield (case (1)) exhibited the poorest performance, whereas the common method of defining yield at the nominal moment (case (2)) had a medium performance. The proposed bilinear (case (3)) and the curvilinear M-ϕ responses were selected for further analysis of the results. 5.6. Application of the Nonlinear Elastic Solution The closed-from solutions presented in this document for member deformations and total moments were used to derive design formulas to estimate the extent of secondary effects on the inelastic 195 response of RC columns. These effects were studied in terms of the following parameters: (1) maximum P-δ moment, (2) length of the plastic region and (3) spread of plasticity due to P-δ along the column height. The aforementioned parameters are affected the most by slenderness and are of importance in the seismic design of slender RC columns. Figure 5-9: Theil’s inequality coefficient for error for the analytical solution predictions 5.6.1. Maximum P-δ Moment The proposed closed-form solution for the deformed shape of columns was used to obtain the maximum member deformation δmax and the corresponding maximum P-δ moment expected to develop along the height of a column. By differentiating Equation (5-4) and equating the derivative 196 to zero (dy′/dx = 0), the location of the maximum member deformation (x′max) along the height of the column is found as expressed in Equation (5-25).  sin  el   xmax 1  cos 1   L  el   el  (5-25) By substituting the location of the maximum member deformation into the solution for the deformed shape, the maximum displacement of the column away from the chord line (δmax) and the associated P-δ moment can be found as given by Equation (5-26). P  max 2   M Base    el  1  sin  el     1  cos   el   sin  el   el      (5-26) The solution presented above was simplified using series expansions to replace the trigonometric functions with polynomials. The approximate formula for the maximum ratio of P-δ moment to the base moment is given in Equation (5-27). It can be seen from Equations (5-26) and (5-27) that the magnitude of the P-δ moments is only related to κel, which is the measure of slenderness effects on RC columns. P  max  0.064 el2  0.011 el4 M Base (5-27) The distribution of P-δ moments along the height of column M1230 normalized to the base moment at displacement ductility levels (μΔ = Δ/Δy) of 1, 2 and 3 are shown in Figure 5-10. Results from the analytical solution are plotted along with experimental data and the prediction by the simple formula in Equation (5-27). It can be seen that the proposed formula provides a satisfactory estimate of the magnitude of P-δ moment at μΔ = 1. 197 The maximum P-δ values at different displacement ductility levels of column M1230 are shown in Figure 5-11. It is evident that the analytical solutions are capable of predicting the experimental values up to μΔ = 2.0. As the inelastic deformations of the column increases (μΔ > 2.0), the proposed solution underestimates the experimental measurements. This can be explained by the fact that the solution does not include degradation of the column’s flexural stiffness after yield. As a result, greater P-δ effects were measured during the test than estimated by Equation (5-27) when the column undergoes large inelastic deformations. Figure 5-10: P-δ moment profiles for column M1230 at μΔ = 1, 2, and 3 The ratio of the experimental P-δ moment to values obtained from Equation (5-27) is presented in Figure 5-12 for all test columns at different levels of displacement ductility. The efficiency of the design formula, as depicted in Figure 5-12, can be studied in three different regimes of response, 198 e.g., pre-cracking/pre-yield, intermediate inelastic deformations and large inelastic deformations. Before major cracking or yielding the flexural stiffness of the column is greater than EIel of the formula. Therefore, the effect of P-δ is overestimated by the formula. The second regime includes post-cracking response up to intermediate inelastic deformations (0.5 ≤ μΔ ≤ 2.5). During this response span, the columns’ flexural stiffness is in the range of EIel and as a result the maximum P-δ moments were satisfactorily estimated. At the end of the spectrum the columns undergo large plastic deformations and their flexural stiffness declines below EIel. This leads to a greater effect from secondary moments than predicted by the analytical solution. Yet, these discrepancies occur at demand levels likely affected by global instability due to P-Δ. Figure 5-11: Maximum P-δ moment for column M1230 at different displacement ductility levels 199 Solution Figure 5-12: Maximum P-δ moment for all test columns - Ratio of experimental values to results from the solution 5.6.2. Nonlinear Solution to the Length of the Plastic Region (Lpr, NL) The proposed equation for the length of the plastic region along the columns’ height (Equation (5-21)) was based on a geometrically nonlinear model that includes the effects of member deformations on the spread of the plastic region. This is contrary to the plastic region or plastic hinge equations found in the literature, where P-δ effects are generally ignored (Figure 5-13(a)) and the bending moment demand is assumed to be linearly distributed along the height as depicted in Figure 5-13(b). Yet, the member deformations (Figure 5-13(c)) and the associated P-δ moments tend to extend the plastic region to higher sections along the column height as shown in Figure 5-13(d). 200 Figure 5-13: Effect of P-δ on spread of the plastic region along column height The plastic region length along the height of column M1230 is depicted in Figure 5-14 and Figure 5-15 as a function of the ratio of base moment to yield moment (MBase / My) and displacement ductility (μΔ = Δ/Δy), respectively. The formula in Equation (5-21) was directly applied to obtain the results presented in Figure 5-14. However, the solutions for the top column displacement in conjunction with Equation (5-21) were necessary to obtain the results shown in Figure 5-15 for the case of a bilinear section response. It can be seen that the proposed solution used with the curvilinear M-ϕ response yields a close estimate of the experimental data; whereas, the bilinear Mϕ response exhibits poor agreement. 201 Figure 5-14: Spread of plastic region (Lpr) for column M1230 plotted vs. base moment to yield ratio (MBase/My) Figure 5-15: Spread of plastic region (Lpr) for column M1230 plotted vs. displacement ductility (μΔ = Δ/Δy) It is of interest to estimate the error associated with the use of conventional linear-based plastic region length (Lpr,L) models in slender columns and identify an appropriate adjustment factor to 202 account for the effect of geometrical nonlinearities. To achieve this goal, the closed-form equation for the plastic region was simplified using series expansions and an approximate formula (Equation (5-28)) was obtained for the length of plastic region with P-δ effects included (Lpr,NL). M y    el2   Lpr , NL  L  1   1 M Base   3   (5-28) A basic plastic region length (Lpr,L) can also be derived assuming linear distribution of the flexural moments along the RC column height according to Equation (5-29). My    Lpr , L  L1  M Base   (5-29) By substituting this definition for the linear plastic hinge length (Lpr,L) in Equation (5-28), the effect of slenderness and geometrical nonlinearities on the spread of the plastic region was obtained as given by Equation (5-30).  2  Lpr , NL  Lpr , L  1  el  3   (5-30) It is evident that the spread of the plastic region due to P-δ effects is only related to the dimensionless slenderness factor κel. Therefore, the expression (κel2/3) can be used to adjust current plastic hinge and plastic region models, in which geometrical nonlinearities are typically ignored, to account for the contribution of P-δ effects on the spread of inelastic actions along the height of RC columns. The adjustment factor per Equation (5-30) for column M1215 is illustrated in Figure 5-16. This figure also shows experimental measurements on the effect of P-δ on the spread of the plastic region, which represents the difference in plastic hinge lengths with and without P-δ, i.e., Lpr,NL 203 and Lpr,L respectively, and is normalized to Lpr,L. The experimental trace displays a spike around μΔ = 1, which occurs because the spread of plasticity does not start simultaneously for cases with and without P-δ consideration. Neglecting the spike at the beginning of the diagram, since it occurs over a small range, the test data shows a steady effect of P-δ over a wide range of μΔ. This is consistent with the proposed adjustment factor, which remains constant for μΔ greater than one (post-yield). Figure 5-16: Effect of P-δ on spread of plasticity for column M1215 Ratios of experimental data to the predictions from Equation (5-30), i.e. κel2/3, are shown in Figure 5-17. Neglecting the spike at the beginning of yield (μΔ = 1) for the reasons discussed before, it can be seen that the proposed formula provides a lower-bound for the effect of P-δ on the length of the plastic region. That is, member deformations cause the plastic region to extend to higher elevations than the values predicted by the proposed formula. Therefore, Equation (5-30) appears to provide a lower-bound estimate for the effect of P-δ on the extent of the plastic region. 204 (Lpr,NL – Lpr,L) / Lpr,L Experiment / Solution Figure 5-17: Effect of P-δ on spread of plasticity for all test columns - Ratio of experimental values to results from solution 5.6.3. Slenderness Parameter The slenderness of RC bridge columns has been traditionally represented by their aspect ratio (L/D). Although aspect ratio reflects how slender a column appears it fails to address the effect of slenderness on its inelastic response. This relates to the results obtained by researchers who have studied the effect of aspect ratio on the length of plastic region and reported lack of a meaningful relationship between them. The reason is that slenderness effects are influenced by the axial load (P) and the area of the longitudinal steel reinforcement (Asl). Yet, these two structural parameters are typically considered separately as the axial load ratio (P/Agf ′c) and the longitudinal reinforcement ratio (ρsl). Similar to the aspect ratio, previous studies have not consistently found a meaningful relationship between the length of the plastic region and P/Agf ′c or ρsl. Yet, the combination of the aspect ratio, axial load ratio, and reinforcement ratio influence the degree to which member deformations and P-δ moments affect the inelastic response of RC columns as illustrated in Figure 5-18, Figure 5-19, and Figure 5-20, respectively. It is evident from the results 205 that the square of the slenderness parameter κel2 = PL2/EIel is an appropriate measure for the severity of slenderness effects. The aforementioned parameter can thus be used to obtain design limits on slenderness, develop stability indices for ultimate response, and provide magnification factors for the inelastic response of RC bridge columns. Figure 5-18: Effect of aspect ratio (L/D) on the spread of the plastic region due to P-δ moments Figure 5-19: Effect of axial load ratio (P/Agf ′c) on the spread of the plastic region due to P-δ moments 206 Figure 5-20: Effect of longitudinal reinforcement ratio (ρsl) on the spread of the plastic region due to P-δ moments 5.7. Nonlinear Inelastic Solution to Moment Profiles In this section, the analytical solution to the nonlinear distribution of the moments along RC columns height is enhanced by implementing the effect of the reduced stiffness over the plastic region. 5.7.1. Methods and Assumptions for Deriving the Nonlinear Inelastic Solution An analytical expression was developed to determine the distribution of nonlinear second-order bending moments and Lpr for a cantilever column with an idealized bilinear moment-curvature response (Figure 5-21(a)) and variable stiffness along its height that is subjected to axial and lateral loads at its free end, as schematically depicted in Figure 5-21(b). 207 Figure 5-21: Schematics depicting: (a) bilinear moment-curvature response, (b) deformation of a cantilever column, (c) nonlinear moment gradient, and (d) profile of flexural stiffness. 5.7.1.1. Nonlinear Beam-Column Theory Nonlinear beam-column theory was used to derive the governing equations since it captures the effects of second-order P-Δ and P-δ moments. Lpr was obtained by determining the extent of the region over which moments exceeded the yield moment (My) as shown in Figure 5-21(c). 5.7.1.2. Column Partition In order to implement variable flexibility along the RC column, which is caused by the reduced post-yield flexural stiffness over the plastic region, the column was divided into two parts: elastic and inelastic. Variable flexural stiffness along the height was considered by assigning relevant EI values from the section M-ϕ response to each part, as depicted in Figure 5-21(d). The slope of the pre-yield and post-yield branches of the M-ϕ response respectively define the effective elastic (EIel) and inelastic (EIin) flexural stiffness values. 208 5.7.1.3. Coordinate Transformation A coordinate transformation was performed to isolate the slenderness effects and P-δ moments from P-Δ moments. Column bending deformations away from its chord-line (δ) and column lateral displacement with respect to a reference line (Δ) are depicted in Figure 5-21(b). The coordinate transformation, as illustrated in Figure 5-22(a) and (b), associated the cantilever column with end loads (P and V) at its original reference coordinate system (Figure 5-22(a)) to an equivalent simplysupported beam-column with an end moment (MB) at its locally rotated coordinate system (Figure 5-22(b)). A primed notation is used to describe parameters in the rotated coordinate system. Using the noted coordinate transformation, the equations for nonlinear moment gradient and P-δ effects can be derived independently from the position of the column top (ΔTop) and the associated P-Δ moment. Yet, the effect of P-Δ is implicitly considered inside the term MB (= VL + PΔ = V′L) despite being absent from the formulation, where MB is the moment at the base level, with V and V′ being the shear forces in the reference and rotated configurations, respectively. In this coordinate transformation the bending moment values remain unaffected, i.e., M′=M. Thus, the solution to the nonlinear moment gradient along the column height in the rotated coordinate system is also applicable to the column in its original reference coordinate system. 209 Figure 5-22: Cantilever and effective forces and moments in: (a) reference; (b) rotated coordinate systems; and within: (c) elastic; and (d) inelastic segments. 5.7.2. Derivation of the Nonlinear Inelastic Solution for Moment Profiles 5.7.2.1. Equilibrium of the Elastic and Inelastic Segments The free body diagrams of the elastic and inelastic segments of the column are illustrated in Figure 5-22(c) and (d). Following the nomenclature of Figure 5-22, the governing equations for nonlinear second-order moments along the elastic (Mel) and inelastic segments (Min) of the column are respectively given in Equations (5-31) and (5-32). M el  V x  Pyel M in  M B  Pyin  V x 0  x  L  L pr for for 210 0  x  Lpr (5-31) (5-32) In the equations above, Mel and Min are the nonlinear moments along the elastic and inelastic parts of the column. Also, y′el and y′in respectively represent the deformation of the column away from the chord-line in the rotated configuration over the elastic and inelastic segments. The parameters x′ and x″ determine the location of the sections along the elastic and inelastic parts of the column in the rotated coordinate system. It is worth noting that x′ and x″ have opposite directions and are bounded by the following inequalities: 0 ≤ x′ ≤ L – Lpr and 0 ≤ x″ ≤ Lpr. 5.7.2.2. Section Moment-Curvature (M-ϕ) Response The M-ϕ response of RC sections is commonly idealized by a bilinear relationship with a welldefined yield point as shown in Figure 5-21(a). Using such a bilinear assumption, the M-ϕ response of the elastic and inelastic segments of the column was respectively characterized by Equations (5-33) and (5-34). In this section, the EIin was assumed to be positive. This assumption was justified by the fact that RC bridge columns are designed to exhibit a stable inelastic response and undergo large inelastic rotations without a significant loss of strength. M el   EI el d 2 yel dx 2  d 2 yin  M in  M y     y  EI in 2  dx  for for d 2 yel  y dx 2 d 2 yin y   u dx 2 (5-33) (5-34) 5.7.2.3. Elastic Segment Differentiating Equation (5-31) twice and substituting the moment-curvature relationships from Equation (5-33), the governing differential equation for the nonlinear moment profile along the elastic part of the column is derived, as given by Equation (5-35). The method presented by Wang 211 et al. [107], in which the bending moment profiles are directly obtained without the need to solve for the deformations (δ), was adopted in this formulation. d 2 M el PM el  0 dx2 EI el for 0  x  L  L pr (5-35) For simplicity and consistency with the elastic solution, Equation (5-35) was rewritten in terms of normalized parameters, as provided in Equation (5-36); where κel is the dimensionless elastic slenderness parameter (as defined by κel2 = P′L2/EIel). Parameter α′ is the normalized length parameter that is defined by α′ = x′/L, and β is the normalized length of the plastic region (β = Lpr/L). d 2 M el   el2 M el  0 d 2 for 0    1  (5-36) The differential equation in Equation (5-36) was solved for the elastic segment of the column. Applying the boundary condition at top of the column (α′ = 0), which is a moment-free end, yields the solution to Mel as given in Equation (5-37). This solution has an undefined parameter C1el that is to be determined later by imposing continuity conditions at the transition point. M el  C1el sin  el  (5-37) 5.7.2.4. Inelastic Segment Similar to the elastic part of the column, the governing differential equation for the nonlinear moment profile along the plastic region (0 ≤ α″ ≤ β), as given in Equation (5-38), was found by differentiating Equation (5-32) twice and substituting the moment-curvature response from Equation (5-34). 212  1 d 2 M in P 1    M in  PM y   2 dx EI in  EI in EI el  for 0  x  Lpr (5-38) Equation (5-38) was rewritten using normalized dimensionless parameters as provided in Equation (5-39), where κin is the inelastic slenderness parameter defined according to κin2 = P′L2/EIin and α″ is the normalized length quantity along the plastic region that is obtained from α″ = x″/L.  d 2 M in   in2 M in  M y  in2   el2 2 d   for 0      (5-39) By solving the governing differential equation in Equation (5-39) and applying the boundary condition at the base level (α″ = 0), in which the moment is equal to MB, the moment profile for the inelastic segment (Min) was obtained as expressed in Equation (5-40), where C1in is a constant yet to be determined by applying the continuity conditions at the transition point.    2   2  M in  C1in sin  in    M B  M y 1  el2  cos in   M y 1  el2    in    in   (5-40) 5.7.2.5. Transition Point Continuity of Moments Equilibrium of moments at the transition point between elastic and inelastic segments imposes a continuity condition for the moment profile. To assure continuity, the elastic moment (Mel) at α′ = 1−β was set equal to the inelastic counterpart (Min) at α″ = β. Continuity of moment profiles at the transition point leads to an expression for C1el in terms of C1in as given by Equation (5-41).   el2  cos in    sin  in       C1el  C1in   M  M  y 1    B 2   sin  el   el      in  sin  el   el   My    el2   1  2   sin  el   el    in  213 (5-41) Continuity of Shear Forces The shear force within the elastic and inelastic segments of the column was determined by differentiating the moment profiles with respect to x′ and x″, respectively. The resulting shear forces at the transition point were set equal to each other with consideration of sign convention, i.e., V′el = −V′in. The shear continuity condition, which also guarantees smoothness of the moment profile, leads to an expression for C1el in terms of C1in, as given by Equation (5-42).   in cos in       el2   in sin  in     C1el  C1in   M  M  B y 1   2    el cos el   el      in   el cos el   el   (5-42) Combining Equation (5-41) with (5-42), and solving for C1el and C1in yields the expressions given in Equations (5-43) and (5-44), respectively.    2    in sin  in    C1el   M B  M y 1  el2      in    el cos el   el     M y in cos in     el2   sin      1   2  el el in        el sin  in   cos el   el    in el cos in   cos el   el         sin      el cos el   el   el el      el2    in sin  in   in cos in   cos in   in cos in     M  M   B y 1   2      cos     sin  el   el    el el el in          el sin  in   cos el   el    in el cos in   cos el   el      sin  el   el    el cos el   el     214 (5-43) C1in    el2    in sin  in   cos in      M B  M y 1  2      in    el cos el   el   sin  el   el        sin  in    in cos in        sin      el cos el   el   el el  (5-44) My    el2   1  2    sin     el el    in    sin  in    in cos in        sin  el   el    el cos el   el   Yield Condition The transition between the elastic and inelastic segments in a column occurs at a section in which the moment (or, equivalently, curvature) is equal to the yield moment, My (or yield curvature, ϕy). Implementing the yield requirement (Min = Mel = My) at the transition point led to Equation (5-45), in which n is the ratio of the inelastic slenderness parameter (κin) to its elastic counterpart (κel). The location of the transition point (β) along with the extent of the plastic region due to nonlinear moment gradient was thus determined by solving Equation (5-45) for β, provided that the base moment (MB) is greater than the yield moment (My) and less than the ultimate moment (Mu). M B 1  cos in   sin  in   cos el   el     1  0 My n2 n sin  el   el   (5-45) The expression for β, (Equation (5-45)) is a simple closed-form expression that can be easily solved using numerical root-finding techniques. Finding a solution to Equation (5-45) is particularly simple since the boundaries of β are well-defined (0 ≤ β ≤1). It is worth mentioning that Equation (5-45) determines the extent of the plastic region caused only by the moment gradient effect as defined in the literature [16, 17, 54, 59, 108]. Therefore, other effects that spread the plasticity 215 along the column height, e.g. tension-shift effect, are not included. However, other effects are insignificant in flexure-dominated slender columns [35], which are the focus of this research. 5.8. Verification of the Nonlinear Inelastic Solution 5.8.1. Experimental Database Experimental results from three RC columns, reported in Chapter 3 of this dissertation, were used to verify the proposed analytical solution. The test columns had aspect ratios of 10 (column M101510) and 12 (columns M123005 and M121505). The designated nametags follow the same convention used in Chapter 3, where further details about the geometrical configuration, reinforcement layout, and key material properties of the test columns can be found. The progress of the tests during the lateral loading stage was measured and monitored using the ratio of the top displacement (ΔTop) to the yield displacement (Δy). This ratio is herein referred to as displacement ductility (μΔ) and defined as per Δ/Δy. It is worth noting that μΔ= 1 represents the state at which the most critical section of the column reaches the ideal yield moment (or curvature). Larger values of μΔ indicate greater amount of inelastic deformation. In this chapter, experimental results from the test columns, which were reported in Chapter 3, are used to verify the proposed expression at multiples of displacement ductility, i.e., μΔ=1, 1.5, 2, and 3. 5.8.2. Moment-Curvature Response Moment-curvature analyses were conducted on the test columns to extract the parameters of Equation (5-45). Different nonlinear constitutive models for the core concrete, cover concrete, and reinforcement steel were used to closely capture the inelastic response of the RC sections. The effect of confinement on the core concrete was considered using the method by Mander et al. [87]. The post-yield hardening response of reinforcement steel was modeled based on the constitutive relation proposed by Mander [89]. The results from standard material tests conducted on steel and 216 concrete specimens were used to define the key material properties. Force-based element formulations, as per Spacone et al. [68], and fiber-discretized RC sections, as per Spacone et al. [69], were utilized to consider the combined effect of axial load and bending moment. The moment-curvature (M-ϕ) response of the RC sections was idealized as bilinear to obtain the parameters needed in Equation (5-45). This was done by defining a yield point that indicated the transition from the elastic regime to the inelastic one and an ultimate point that defined the end of the inelastic segment. The ideal yield and ultimate properties, i.e., ϕy, ϕu, My, and Mu were obtained based on the methods presented earlier in this chapter. Moment-curvature (M-ϕ) responses of the test columns are shown in Figure 5-23 with overlaid test data to verify the accuracy of the sectional analyses. Also shown is the idealized bilinear M-ϕ response that was constructed from the M-ϕ curve using the aforementioned definitions for the ideal yield and ultimate points. Lastly, the flexural stiffness parameters EIel and EIin were obtained by calculating the slope of the elastic and inelastic branches of the bilinear M-ϕ response, respectively. Results of the moment-curvature analyses on test column along with the key parameters required in Equation (5-45) are provided in Table 5-3. 217 Figure 5-23: Moment-curvature response of test columns from analyses and experiments. 5.8.3. Flexural Stiffness (EI) Profile Flexural stiffness (EI) profiles along the test columns were obtained from the experiments at two displacement ductility levels: μΔ = 1 and μΔ = 3, as respectively shown in Figure 5-24(a) and Figure 5-24(b). The two-piece step function that was used in the analytical derivations to model the flexural stiffness profiles is also depicted in Figure 5-24. It can be seen from Figure 5-24(a) that at μΔ = 1, which marks the onset of yielding at the base of the column, the entire length of the column is in the elastic regime. Despite the uneven distribution of the experimental data, which is mainly 218 caused by the non-uniform crack pattern along the columns height, the average flexural stiffness is reasonably close to EIel. Moreover, it can be seen from Figure 5-24(b) that at μΔ = 3, which corresponds to a level of large inelastic deformation, the plastic region follows the post-yield branch of the section M-ϕ response and has a much lower flexural stiffness EIin. Table 5-3: Key parameters of the proposed solution determined for the test columns Inelastic Inelastic Segment / Elastic Elastic Segment State Property ϕy (μ mm-1) My (kN∙m) EIel (MN∙m2) κel ϕu (μ 1/mm) Mu (kN∙m) EIin (MN∙m2) κin M123005 M121505 M101510 9.53 8.38 8.33 955 548 664 100.2 65.5 79.7 0.618 0.769 0.794 139.3 136.0 140.8 1287 750 861 3.79 1.99 1.91 3.17 4.37 5.10 n 5.13 5.68 6.42 Mu / My 1.455 1.411 1.335 219 (a) (b) Figure 5-24: Profiles of flexural stiffness (EI) for test columns at: (a) μΔ = 1; and (b) μΔ = 3. It is evident from the plots in Figure 5-24 that the assumed flexural stiffness profiles matched the experimental results with satisfactory accuracy. It should be noted that the location of the transition from the plastic to elastic regions of the column was obtained using the expression for Lpr as given in Equation (5-45). 220 5.8.4. Moment, Curvature, and Displacement Profiles Experimental moment profiles over the test columns were used to verify the proposed analytical solution of the nonlinear moment gradient, i.e., Equations (5-37) and (5-40). Details of extracting moment profiles from test data can be found in Chapter 3 of this document. Results from the nonlinear inelastic solution are plotted against test data from column M123005 in Figure 5-25(a) at μΔ levels of 1 and 3. Results from the linear and nonlinear elastic solutions are also provided. It can be seen from the overlaid plots that the nonlinear solutions, elastic and inelastic, perfectly match the test data, whereas the linear solution fails to capture the nonlinearity of the moment profiles. It is worth noting that the nonlinear elastic and inelastic solutions yield identical results at μΔ = 1. However, as the inelastic deformations increase (μΔ > 1), the inelastic solution starts to diverge from the elastic one. The superior performance of the inelastic solution over the elastic one may not be easily appreciated from the moment profiles since the difference between the two solutions is too small relative to the moment values. This issue is clarified later in this chapter by removing the linear portion of the moment profiles to obtain pure P-δ moments. (c) (b) (a) Figure 5-25: Profiles of: (a) moment, (b) curvature, and (c) displacement for test column M123005 at μΔ = 1 and μΔ = 3. 221 In addition to moment profiles, profiles of curvature and lateral displacement along the length of the test columns, as respectively depicted in Figure 5-25(b) and Figure 5-25(c) for column M123005, were used to verify the proposed nonlinear inelastic solution for moment gradients in Equations (5-37) and (5-40). Curvature profiles were obtained by substituting moment values with the associated curvatures from the M-ϕ response of the test columns. Displacement profiles were calculated by numerically integrating the curvature profiles twice and adding the contribution from the footing to the deformation of the column, i.e., strain penetration according to Hines et al. [35]. Similar to the moment profiles, it can be seen from the overlaid profiles of curvature and lateral displacement in Figure 5-25 that the nonlinear inelastic solution provides an accurate description of the column’s global response. However, these profiles offer little insight into the better performance of the nonlinear inelastic solution and the effect of non-uniform flexural stiffness along the columns’ height on the extent of the plastic region, which are discussed later. Besides visual evaluation and verification of the proposed solution, the normalized root mean square error (RMSE), referred to as U in this dissertation, was utilized to quantitatively assess the accuracy of the analytical solutions for predicting moment and curvature profiles along the height of the test columns. The percent error (PE) values were also calculated for the analytical predictions of the top displacement (ΔTop). The results of the error analyses on moment profiles, curvature profiles, and ΔTop values are respectively shown in Figure 5-26(a), Figure 5-26(b), and Figure 5-26(c), where the average error is depicted by the columns’ height. Also included in the graphs as error bars are the 95% confidence intervals for the noted error measures. 222 (a) (b) (c) Figure 5-26: Comparing accuracy of the linear, nonlinear elastic, and nonlinear inelastic solutions to: (a) moment and (b) curvature profiles along the test columns’ height; and (c) top displacement. It is evident from the error values provided in Figure 5-26 that the results from the proposed nonlinear inelastic solution generally offer slightly more precise predictions of the moments, curvatures, and displacements than the linear and nonlinear elastic solutions. However, the improved accuracy of the predictions, which was achieved by the use of the proposed nonlinear 223 inelastic expression, was not statistically significant as it is shown in Figure 5-26 by the overlapping confidence intervals. This statistical insignificance can be attributed to the smallness of the test sample (only 3 test units) and relatively large margins of error from measurements in the large-scale structural tests. Nonetheless, it is clear from the average error values shown in Figure 5-26 that the nonlinear inelastic solution may not offer a practically significant improvement in terms of global structural responses. However, local responses, such as the maximum P-δ moment and the extent of the plastic region (Lpr), are expected to be more sensitive to the effects of geometrical nonlinearity, second-order actions, and varying flexural stiffness along the columns’ height as discussed later. 5.8.5. Force-Displacement Response The force-displacement response of the test columns was extracted from the analytical solutions and the results were compared against test data to verify the method used in here to estimate displacement values. The force-displacement responses are presented in Figure 5-27 in terms of shear force in the rotated configuration (V′ = MB/L) versus the lateral displacement at the top (ΔTop). V′ was preferred over the shear force in the original coordinate system (V) since V′ includes the additional demand due to P-Δ effect. This is evident from the relationship between V and V′, which is defined according to V′ = V+PΔTop/L for rotation angles smaller than 0.1 rad. Similar to the profiles of moment and curvature, an excellent agreement was found between the analytical results and test data for the force-displacement response of the test columns. In addition, it can be seen from the graphs in Figure 5-27 that the results from the nonlinear inelastic solution offer a slight improvement in the accuracy of the predictions over the ones from the linear and nonlinear elastic expressions. 224 Figure 5-27: Force-displacement response of the test columns from analyses and experiments. 5.9. Comparing Different Solutions (Linear, Nonlinear Elastic, and Nonlinear Inelastic) 5.9.1. Second-Order P-δ Moment Profile Profiles of second-order P-δ moments were obtained by subtracting the linear portion of the moment profile, which included the P-Δ and primary moments, to highlight the excellent performance of the nonlinear inelastic solution. The analytical results from the proposed nonlinear 225 inelastic solution along with the results from the nonlinear elastic solution are shown in Figure 5-28 and Figure 5-29 for μΔ = 1.5 and 3, respectively. Figure 5-28: Profiles of second-order P-δ moments along the height of test columns at μΔ = 1.5 226 Figure 5-29: Profiles of second-order P-δ moments along the height of test columns at μΔ = 3 It is worth mentioning that results from the linear solution are not shown in the figures since they yielded zero for P-δ moments, as expected from a linear model. It can be seen from Figure 5-28 and Figure 5-29 that the nonlinear inelastic and elastic solutions to P-δ moments provided similar 227 results at μΔ = 1.5 but, started to separate from each other with increasing μΔ. The nonlinear inelastic solution closely followed the test data while the P-δ moments from the elastic solution remained almost unaffected as μΔ increased. 5.9.2. Maximum P-δ Moment Maximum P-δ moment values were extracted from the analytical nonlinear moment profiles and plotted against experimental data at different levels of μΔ in Figure 5-30. It can be seen that the nonlinear inelastic solution matched the experimental data with satisfactory accuracy at all displacement ductility levels to which the columns were tested. However, the nonlinear elastic solution started to diverge from the test results as the inelastic deformations and μΔ increased. Figure 5-30: Maximum P-δ moment from nonlinear analytical solutions versus test data 228 The divergence of the analytical results from the elastic solution can be explained by the fact that the ratio of the maximum P-δ moment to the base moment (P-δ)max/MB is independent from the inelastic deformations and remains constant for all values of μΔ. Conversely, in the proposed nonlinear inelastic solution, the (P-δ)max/MB ratio is not constant and grows rapidly with increasing μΔ. Thus, the nonlinear moment profiles according to Equations (5-37) and (5-40) were found to be significantly more accurate in modeling second-order P-δ moments than the previous expressions. This is particularly true at large inelastic deformations (μΔ > 2). 5.9.3. Length of the Plastic Region (Lpr) The extent of the plastic region (Lpr) along the height of the test columns was evaluated according to the ‘linear elastic’ (Equation (5-29)), ‘nonlinear elastic’ (Equation (5-21)), and ‘nonlinear inelastic’ (Equation (5-45)) expressions. The results are presented in Figure 5-31 in which test data with and without P-δ effects on Lpr are also provided. It can be seen that the linear solution to Lpr matches the experimental results if the P-δ moments are ignored. Therefore, the linear solution underestimates the length of the critical plastic region by the amount that is increased due to nonlinear P-δ moments. In contrast, the nonlinear elastic and inelastic solutions closely fit the test data with P-δ moments. Comparison of the overlaid plots of Lpr at different displacement ductility levels demonstrates that the nonlinear inelastic solution provides a more accurate estimate of the extent of plasticity than the nonlinear elastic solution. 229 Figure 5-31: Extent of the plastic region (Lpr) on the test columns from analytical solutions 5.9.4. Effect of P-δ on Lpr Column bending due to flexural moments creates a second-order P-δ moment that changes the moment distribution from linear to nonlinear. The nonlinearity of the moment profile is proportional to column flexibility and thus its slenderness. Nonlinear moment profiles increase the inelastic deformation further along the column height and extend the length of the plastic region beyond the limits obtained from a linear moment distribution. The percent increase in the length of the plastic region solely due to the effect of P-δ moments was found from the nonlinear 230 analytical expressions (elastic and inelastic) and results are shown in Figure 5-32. This was achieved by comparing the results from the nonlinear expressions (Lpr,NL) to those from the linear expression (Lpr,L). Also shown in Figure 5-32 is the effect of P-δ on Lpr according to the simplified formula in Equation (5-30), which approximates the nonlinear elastic solution. Experimental data from the test column are also provided to have a basis for comparison. Figure 5-32: Effect of P-δ moments on Lpr in terms of percent increase due to nonlinear P-δ moments It can be seen from the graphs in Figure 5-32 that the nonlinear inelastic solution provides the closest prediction of the P-δ effects on Lpr. It is also clear from the figures that the simplified 231 formula in Equation (5-30), despite being approximate, offers a slightly better correlation with test data than the exact nonlinear elastic solution. The reason is that the results from the exact elastic solution generally exhibits a descending trend with respect to increasing μΔ, while the simplified formula remains constant and thus closer to the test data. To better demonstrate the effectiveness of the proposed expression, the ratio of the experimental data to analytical results in terms of (Lpr,NL – Lpr,L)/Lpr,L is plotted in Figure 5-33. Figure 5-33: Comparing accuracy of nonlinear analytical solutions for predicting P-δ effect on Lpr 232 It can be seen from the graphs that the results yielded by Equation (5-45) closely group around the line that defines a perfect correlation with the test data. It is also evident that the simplified formula to account for the effects of slenderness on the spread of plasticity along column height captures the test data with a reasonable accuracy. 233 CHAPTER 6 6. PARAMETRIC STUDY 6.1. General The analytical expressions and closed-form solutions, which were discussed in Chapter 5 of this dissertation to quantify the effects of second-order moments on Lpr, were used to conduct a parametric study on the effects of different design parameters. The motivation for conducting the parametric study is that the noted analytical expressions require a precise estimate of parameters that are not typically calculated during a typical design process and are thus less commonly known to engineers. Therefore, a full implementation of the nonlinear model in seismic design of RC columns needs expressions in terms of well-known design parameters. In addition, the effect of common structural/material parameters, such as aspect ratio, axial load ratio, steel reinforcement ratio, and concrete strength, on the susceptibility of RC columns to second-order effects remains unclear. In this chapter, a parametric study on the effects of second-order moments on the plastic region along slender RC columns is presented. This study led to the identification of design parameters that significantly affect the second-order response of RC columns. Furthermore, the 234 parameter ranges for which second-order effects on Lpr can be safely ignored were identified. Finally, in lieu of more sophisticated analyses, simple design formulas are proposed to estimate the effects of second-order moments on the plastic region of RC columns. 6.2. Parametric Study Based on the Nonlinear Elastic Solution 6.2.1. Key Variables of the Nonlinear Elastic Solution Expressions for the extent of the plastic region due to nonlinear moment profiles and the effect of second-order P-δ moments on Lpr, as respectively given by Equation (5-21) and Equation (5-30), are in terms of two key variables: (a) the moment overstrength ratio (Mu/My) and (b) the column’s elastic slenderness factor (κel). Accuracy of the analytical results from the nonlinear model directly depends on the precision to which these key variables are estimated. It is thus essential to evaluate the relationship between typical structural design parameters and the key variables of the nonlinear analytical solution, i.e., Mu/My and κel. 6.2.1.1. Slope Sign of Post-yield M-ϕ Branch (EIin) The closed-form solution presented here predicts the extent of the plastic region due to nonlinear moment profile, commonly known as the moment gradient effect. Therefore, it is only applicable to RC sections with positive post-yield flexural stiffness (EIin > 0). Moreover, depending on the post-yield response of RC sections, the extent of the plastic region is governed by different phenomena. For instance, moment gradient significantly contributes to the spread of plasticity in slender RC columns with positive post-yield flexural stiffness (Mu > My). Conversely, the effect of moment gradient on Lpr for columns with a softening branch after yield is negligible. Therefore, knowledge about the sign of the post-yield flexural stiffness (positive for hardening and negative for softening response) is essential for proper implementation of the nonlinear solution to the second-order effects on Lpr. 235 6.2.1.2. Moment Overstrength Ratio (Mu / My) The ratio of the ultimate moment (Mu) to the yield moment (My), also known as moment overstrength factor, is a key variable in determining the extent of the plastic region in RC columns according to the nonlinear elastic solution presented in Chapter 5. Thus, it is crucial to have an accurate estimate of the moment overstrength factor to implement second-order effects on the inelastic response of slender RC columns, in lieu of sophisticated section analyses. This becomes even more important considering the fact that most specification codes define Lpr in terms of the My/Mu ratio, which is the inverse of the moment overstrength factor. If the Mu/My ratio is underestimated, the actual Lpr can exceed the length required by design guidelines to have special detailing for ductile inelastic response. That is, the plastic demands can spread beyond the ductile zone and potentially cause brittle failures. 6.2.1.3. Elastic Slenderness Parameter (κel) The susceptibility of RC columns to second-order moments on their inelastic response can be fully captured by the slenderness parameter κel according to the nonlinear elastic approach, as defined in Chapter 5. Unlike the aspect ratio (L/D), which describes the slenderness of RC columns from a geometrical point of view, the slenderness parameter κel is capable of addressing the structural response of slender columns. The reason is that κel includes, in addition to the aspect ratio, the effects of parameters such as axial load (P), the compressive strength of concrete (f ′c), and the area of the longitudinal steel reinforcement (Asl). The combination of the noted parameters influences the degree to which second-order moments affect the inelastic response of RC columns. Yet, the contribution of these structural parameters on the susceptibility of RC columns to second-order effects has not been evaluated in prior studies. 236 6.2.2. Methods 6.2.2.1. Structural Parameters In this section, nine normalized dimensionless parameters that are sufficient to describe critical structural and material properties of circular RC bridge columns were considered. Section diameter (D), steel reinforcement ratios (ρsl, ρst), diameter of rebars (dsl, dst), spiral hoop spacing (s), concrete compressive strength (f ′c), steel yield and ultimate stress values (fysl, fusl), axial load ratio (P/Agf ′c) and the shear span length (L) were the considered geometric and material parameters. The subscripts of l and t denote longitudinal and transverse steel reinforcement layouts. To normalize the geometrical parameters with length dimensions they were divided by the section diameter D, which ranged from 305 mm to 1219 mm. Similarly, material properties with units of stress were normalized by the yield strength for the longitudinal steel reinforcement (fysl), which was set to 455 MPa. The normalized parameters used in this study are listed in Table 6-1. Also presented in Table 6-1 is the range of values for the noted parameters. Design specifications for reinforced concrete sections according to ACI-318 [33] along with seismic design guidelines from AASHTO [20] and Caltrans [34] were utilized to establish the limiting bounds for the structural parameters. Typical values of material properties for current practice were used to define the limits for material parameters. The average values for the parameters of this study were extracted from the PEER’s structural performance database [1] for 52 circular RC columns that met the conditions for ductile flexural inelastic response. These average values, which are presented in Table 6-1, are generally consistent with the mean value of the ranges except for the volumetric transverse reinforcement ratio (ρst) and concrete compressive to steel yield strength (f ′c / fysl). The reason is that current seismic design guidelines generally require higher ratios of transverse reinforcement to provide confinement than those in effect 237 decades ago when most of the columns reported in the PEER’s database were designed and tested. Moreover, recent advancements in concrete mixture design and technology have led to higher strength values than the mixes used in most columns reported in the PEER’s database. Table 6-1: Dimensionless parameters for the parametric study No 1 2 3 4 5 6 7 8 9 Dimensionless Lower-bound Parameter ρsl 1% dsl / D 0.029 ρst 1.20% dst / D 0.013 s/D 0.042 f ′c / fysl 0.1 fysl / fusl 1.3 P / Agf ′c 5% L/D 2 Upper-bound Average PEER Average 4% 0.083 3.90% 0.042 0.230 0.2 1.7 20% 12 2.5% 0.056 2.55% 0.0275 0.136 0.15 1.5 13% 7 2.4% 0.040 1.10% 0.018 0.12 0.1 1.5 15% 5.5 6.2.2.2. Parametric Study Method The effect of the structural and material parameters presented in Table 6-1 on the second-order response of RC bridge columns was studied. A total of 11,088 cases of possible design configurations that meet the requirements of the aforementioned code specifications were generated by taking at least three equally-spaced values from the ranges presented in Table 6-1 and omitting the combinations that were not allowed. Section analyses were conducted on all of the allowed cases to obtain the key parameters of the nonlinear closed-form solution to the secondorder effects of P-δ on the extent of the plastic region (Lpr). 6.2.2.3. Section Analyses Key parameters of the closed-form solution, namely, the slenderness parameter κel and the moment overstrength ratio Mu/My, were extracted from the M-ϕ response of the RC sections. Pre-yield 238 flexural stiffness for the cracked RC sections was obtained by the ratio of the yield moment My to the yield curvature ϕy per EIel = My / ϕy, and the slenderness parameter was calculated according to Equation (5-5). The moment overstrength ratio was also computed by the ratio of the moment at ultimate state (Mu) to the yield moment (My). 6.2.2.4. Regression Models Results from the study were used to identify the most effective parameters in defining the key variables of the nonlinear closed-form solution. Linear regression models were derived to serve as simplified design formulas with the least number of parameters required to provide reasonably accurate estimates for the second-order effect on the inelastic response of RC columns. To accomplish this, the ordinary least square estimator method [109] for multi-variable regression problems was utilized. Regression models that provided the highest accuracy with the least number of variables were selected. 6.2.2.5. Factor of Safety Regression models were adjusted using factors of safety to ensure conservative estimates of the key variables. Factors of safety were calculated based on 95% confidence intervals. That is, the regression models were adjusted to provide conservative predictions for at least 95% of the cases. For both key variables, i.e., Mu/My and κel, larger values and overestimates led to more conservative designs. Therefore, the regression models were magnified by a factor of safety to meet the aforementioned conditions. 239 6.2.3. Parametric Study Results for Nonlinear Elastic Solution 6.2.3.1. Sign of Post-yield Flexural Stiffness (EIin) A parametric study was conducted to evaluate the sign of the slope of the post-yield momentcurvature response (EIin). Eight out of nine parameters from Table 6-1 were used since the aspect ratio (L/D) has no effect on the sign of EIin, which is a property of the RC section. The results are illustrated in Figure 6-1 and Figure 6-2, in which the correlation between eight parameters and the sign of the post-yield flexural stiffness (EIin) is provided. Figure 6-1: Effect of parameters 1, 2, 5, and 6 on the sign of post-yield flexural stiffness (EIin) 240 Figure 6-2: Effect of parameters 3, 4, 7, and 8 on the sign of post-yield flexural stiffness (EIin) Also provided in Figure 6-1 and Figure 6-2 are the equations for linear regression lines obtained by the least square estimator method. The slope of the regression line multiplied by the average value of the parameter simply indicates the significance of the parameter in determining the sign of EIin. Using the adjusted partial correlation coefficients, it was found that the sign EIin is mostly 241 dominated by two parameters: (a) the axial load ratio (P/Agf ′c) and (b) the concrete compressive strength (f ′c). The probability of positive EIin is depicted in Figure 6-3(a) as a probabilistic distribution based on different values of the effective parameters. It is evident from Figure 6-3(a) that RC sections with lower axial load and lower concrete strength have higher probability of a positive post-yield flexural stiffness. That is, the moment-curvature response of the section has a positive slope after yield and the flexural stiffness of the section has a positive value, i.e., Mu > My. This probability was obtained by calculating the ratio between the number of design configurations that led to a positive EIin to the total number of permissible designs. Therefore, numbers depicted on the contours of Figure 6-3(a) show the probability of having a positive EIin. Statistical analysis of the results from the parametric study revealed that positive EIin occurred in 69% of all design configurations studied here. This percentage increased to 91% for sections with normal-strength concrete (f ′c = 45 MPa) and reduced to 52% for high-strength concrete (f ′c = 90 MPa). Therefore, the closed-form solution presented in this dissertation for the second-order effects is applicable to almost all RC columns with normal-strength concrete (NSC). It is also appropriate for RC sections with high-strength concrete (HSC) with axial load ratios of less than 10%. The valid ranges for the noted parameters are better highlighted in Figure 6-3(b), in which the shaded area represents the ranges that are most likely to cause negative post-yield stiffness. The shaded area covers the region in which the combination of higher axial load level and higher concrete compressive strength makes it less likely (less than 50%) to have a positive EIin. 242 Figure 6-3: (a) Probabilistic distribution of sign of post-yield flexural stiffness (EIin); (b) Distribution of test columns reported in the PEER’s database 243 Also shown in Figure 6-3(b) is the distribution of test columns from the PEER’s database that have axial load ratios less than or equal to 20%. It can be seen that all test columns are located in a region with higher probability for positive EIin. This shows that the closed-form solution in Equation (5-21) is applicable to most plausible design configurations. It is worth mentioning that the uniaxial constitutive models used in the present parametric study to predict the response of high-strength concrete (HSC) sections do not take into account the increased brittleness of HSC. That is, the constitutive models for HSC and NSC were similar and the features that are unique to HSC were not directly implemented in the models. However, the use of general constitutive relations, which are mainly developed for NSC, to model HSC sections is acceptable since the response slender columns with adequate transverse confinement is flexuredominated. The response of HSC sections in flexure-dominated columns with additional transverse reinforcement to compensate for the extra brittleness of HSC is not expected to be significantly affected by the constitutive model used for HSC. Therefore, it is not necessary for the purpose of this study to consider the unique brittleness of HSC in the model since it does not play a significant role in the inelastic response of flexure-dominated well-confined slender RC columns. 6.2.3.2. Moment Overstrength Ratio (Mu / My) A parametric study was carried out to identify parameters that significantly affect the ultimate to yield moment ratio (Mu/My) of RC sections that meet the precondition of Equation (5-21), which is a positive post-yield flexural stiffness (Mu/My > 1). Similar to the sign of EIin, only the first eight parameters of Table 6-1 were used for this study since aspect ratio does not affect Mu/My, which is governed only by the properties of RC sections. The correlation between the eight parameters used and Mu/My is depicted in Figure 6-4 and Figure 6-5. 244 Figure 6-4: Effect of parameters 1, 2, 5, and 6 on the moment overstrength ratio (Mu/My) 245 Figure 6-5: Effect of parameters 3, 4, 7, and 8 on the moment overstrength ratio (Mu/My) Equations for the best fitted lines are also provided in Figure 6-4 and Figure 6-5. The products of the line slopes and the average values of the parameters can be used to identify the most significant parameters. By computing the adjusted partial correlation coefficients, it was found that three parameters dominate the moment overstrength ratio (Mu/My) of RC sections. These parameters, in order of significance, are: (a) the axial load ratio (P/Agf ′c), (b) the concrete compressive strength (f ′c), and (c) the ratio of ultimate to yield strength for the longitudinal steel reinforcement (fusl/fysl). 246 The distribution of Mu/My ratio with respect to two main parameters, i.e., axial load ratio (P/Agf ′c) and ultimate to yield stress for longitudinal reinforcement steel (fusl/fysl), is plotted in Figure 6-6(a) and Figure 6-6(b) for NSC and HSC sections, respectively. It is evident from the plots that the Mu/My ratio is generally higher for greater values of the fusl/fysl ratio. The direct relationship between Mu/My and fusl/fysl is explained by the fact that strain-hardening of the steel reinforcement is the main factor that contributes to the moment overstrength in RC sections. However, the axial load ratio has an adverse effect on Mu/My. That is, the Mu/My ratio generally decreases with increasing P/Agf ′c up to 20%. The ratio of Mu/My is also of importance since most design codes describe the extent of the plastic region as that over which the moment demands exceeds a factor of the maximum end moment. The My/Mu ratio, which is the inverse of the overstrength factor, has been recommended by design guidelines and depends on the axial load ratio. The recommended values for My/Mu vary between 75% to 80% for columns with axial load ratios less than 25%. The parametric study presented in this dissertation was used to reevaluate the accuracy and reliability of the My/Mu ratios suggested by design guidelines. It was found that the average My/Mu ratio for RC section with NSC is 77% and 89% for axial load ratios of 10% and 20%, respectively. However, the average ratios may not be appropriate to use in design guidelines since the average values provide a non-conservative overestimate of My/Mu for about half of the design configurations. Therefore, the 95% confidence interval values were calculated as 71% and 81% for axial load ratios of 10% and 20%, respectively. Although current design guidelines for the extent of the plastic region successfully estimate the My/Mu ratio for columns with an axial load ratio of 20%, it is evident from the results of the parametric study that current design specifications lead to non-conservative overestimates for columns with an axial load ratio of 10%. 247 Figure 6-6: Distribution of Mu/My ratio with respect to P/Agf ′c and fusl/fysl for: (a) the normal strength concrete (f ′c = 45 MPa); and (b) the high-strength concrete (f ′c = 90 MPa) The ratio of My/Mu was found to be generally higher in RC sections with HSC. The 95% confidence interval for HSC sections was 78% and 86% for axial load ratios of 10% and 20%, respectively. 248 Increase in the My/Mu ratio due to higher concrete strength has been confirmed by Pam et al. [36] through experiments conducted on the extent of the plastic region along HSC columns. 6.2.3.3. Elastic Slenderness Parameter (κel) Effects of the nine parameters listed in Table 6-1 on the elastic slenderness factor κel were evaluated. The relationship between κel and the eight most effective parameters from Table 6-1, excluding the ultimate to yield stress ratio for reinforcement steel (fusl/fysl) is illustrated in Figure 6-7 and Figure 6-8. The plot that showed the relationship between κel and fusl/fysl was omitted from the results since no meaningful correlation was observed. This can be explained by the fact that κel is defined using properties of an RC section prior to yield, whereas the fusl/fysl ratio comes into play in the hardening response of RC sections after yield. A strong linear correlation between κel and aspect ratio (L/D) was observed, as depicted in Figure 6-8. Therefore, it was more convenient to conduct the parametric study on the slope of the best fitted line relating κel to L/D, which can be defined by κel/(L/D). The most significant parameters for predicting κel/(L/D) were identified by using adjusted partial correlation coefficients. These parameters, in order of significance, are: (a) the axial load ratio (P/Agf ′c), (b) the longitudinal steel reinforcement ratio (ρsl), and (c) the concrete compressive strength (f ′c). 249 Figure 6-7: Effect of parameters 1, 2, 5, and 6 on the elastic slenderness factor (κel) 250 Figure 6-8: Effect of parameters 3, 4, 8, and 9 on the elastic slenderness factor (κel) Distribution of κel/(L/D) with respect to its two most significant parameters, i.e., P/Agf ′c and ρsl, is shown in Figure 6-9(a), which can be used to estimate the slenderness parameter by multiplying the slope from the graph by the aspect ratio (L/D). It is evident from the plots that the axial load ratio contributes to the slenderness of RC columns as defined by their susceptibility to secondorder effects. By contrast, the longitudinal reinforcement ratio has an adverse effect on κel, which can be explained by the fact that RC sections with higher longitudinal reinforcement exhibit greater 251 flexural rigidity to bending and P-δ effects. The third significant parameter, concrete strength (f ′c), which is not shown in Figure 6-9(a), also contributes to the second-order effects on RC columns. That is, columns with higher concrete strength are more vulnerable to second-order effects. Figure 6-9: Distribution of (a) κel/(L/D) and (b) L/D limit for considering P-δ effects with respect to P/Agf ′c and ρsl 252 Results from Figure 6-9(a) were also used to find limits for the aspect ratio of RC columns beyond which second-order effects of P-δ moments on the extent of the plastic region (Lpr) cannot be ignored. This was accomplished by assuming a limit for the second-order effects and combining Equation (5-30) with the distribution of the slenderness parameter κel with respect to P/Agf ′c and ρsl. The aspect ratios beyond which second-order moments increase the extent of the plastic region by more than 10% are shown in Figure 6-9(b). Therefore, second-order effects cannot be ignored in RC columns with aspect ratios greater than the values depicted in Figure 6-9(b). That is, the aspect ratio beyond which second-order effects increase the length of the plastic region (Lpr) by more than 10% is plotted in Figure 6-9(b). Therefore, the numbers provided on the contour lines show the aspect ratio beyond which the slenderness effects on the spread of plasticity along column height is more than 10% for the given combination of the independent variable, axial load ratio and longitudinal reinforcement ratio. For instance, number 6 in the lower center of Figure 6-9(b) shows that for a column with longitudinal reinforcement ratio about 1.5% and axial load ratio of about 12.5%, if the aspect ratio of the column is greater than 6, second-order effects of slenderness on the inelastic response of the column is expected to be greater than 10%. 6.2.3.4. Effect of Second-order P-δ Moments on Lpr The effect of second-order P-δ moment on the extent of the plastic region (Lpr) was evaluated using Equation (5-30) for combinations of the parameters listed in Table 6-1 that provided positive postyield flexural stiffness. Statistical analyses were conducted to identify the most significant parameters. It was found that the aspect ratio (L/D) and the axial load ratio (P/Agf ′c) play the most significant roles in the second-order effects on Lpr. In addition, longitudinal reinforcement ratio (ρsl) and concrete strength (f ′c) were found to have a tangible effect on the susceptibility of RC columns to second-order moments. Results from this parametric study are illustrated in Figure 253 6-10(a) and Figure 6-10(b) for ρsl equal to 1% and 4%, respectively. The percentages shown in Figure 6-10 represent the increase in Lpr due to second-order P-δ moments. It can be seen that the length of the critical region can increase by more than 50% in columns with an aspect ratio of 12, axial load ratio of 20% and longitudinal reinforcement ratio of 1%. Figure 6-10: Effect of P-δ moments on Lpr in terms of L/D and P/Agf ′c for RC sections with (a) ρsl = 1% and (b) ρsl = 4% 254 6.2.4. Regression Models and Design Formulas A regression model was developed for the moment overstrength ratio Mu/My using the least square method. The correlation between exact values and predictions from the regression model is shown in Figure 6-11(a). Figure 6-11: Approximate versus exact values for (a) Mu/My, (b) adjusted Mu/My, and (c) adjusted κel/(L/D) 255 It can be seen that the regression model underestimates the Mu/My ratio for almost half of the design configurations. Thus, the regression model was adjusted by a factor of safety to ensure that the approximate values were greater than the exact ones for more than 95% of the cases. The adjusted regression model for Mu/My ratio, which was obtained by adjusting the best fitted linear model by a factor of 1.087, is given in Equation (6-1). M u M y  1.28  1.03 f c f P  0.254 usl  1.62 1 f ysl f ysl Ag f c (6-1) Similarly, the regression model for the slenderness parameter (κel) was obtained and adjusted by a factor of 1.074. The result is provided in Equation (6-2). The correlation between the exact values and predictions from the adjusted models for Mu/My and κel/(L/D) are depicted in Figure 6-11(b) and Figure 6-11(c), respectively. It can be seen that the linear model for the slenderness parameter, Equation (6-2), correlates very well with exact values. Conversely, the linear model for the moment overstrength ratio, Equation (6-1), provides a conservative overestimate of the Mu/My ratio with reasonable accuracy.  el  f L  P  0.043  0.742 sl  0.182 c  0.312 0 D  f ysl Ag f c  (6-2) 6.3. Parametric Study Based on the Nonlinear Inelastic Solution It is of interest to study the applicability of the proposed nonlinear inelastic solution to Lpr over a wide range of different RC bridge columns with various design configurations. A parametric study was thus conducted on a large number of RC bridge columns that meet current code specifications in order to: (a) obtain the domain of the key parameters of Equation (5-45), (b) evaluate the stability 256 and applicability of Equation (5-45), and (c) determine the extent to which the nonlinear inelastic solution increases Lpr beyond the linear solution. 6.3.1. Methods 6.3.1.1. Study Parameters Nine structural and material parameters, which are considered adequate to characterize the design of circular RC bridge columns, were investigated. Section diameter (D), longitudinal and transverse reinforcement ratios (ρsl, ρst), the concrete compressive strength (f ′c), the longitudinal steel yield and ultimate stress values (fysl, fusl), the transverse steel yield stress (fyst), the axial load ratio (P/Agf ′c), and the aspect ratio (L/D) were the investigated parameters. The ranges for the aforementioned parameters are provided in Error! Reference source not found.. Table 6-2: Range of parameters investigated in the parametric study based on the nonlinear inelastic solution No. Notation 1 D 2 ρsl 3 ρst 4 f ′c 5 fysl 6 fusl/fysl 7 fyst 8 9 P/Ag f ′c L/D Description Units Range investigated Section diameter mm 305, 610, 914 Longitudinal reinforcement ratio 0.01, 0.015, 0.02, 0.04 Transverse reinforcement ratio 0.01, 0.02 Concrete compressive strength MPa 28, 41, 55, 69 Longitudinal reinforcement steel yield MPa 414, 483 strength Longitudinal reinforcement steel ultimate to 1.3, 1.5, 1.7 yield strength ratio Transverse reinforcement steel yield MPa 414, 621 strength Axial load ratio 0.05, 0.1, 0.2, 0.3 Aspect ratio 3, 6, 9, 12 Design specifications for reinforced concrete columns according to ACI and the seismic design provisions by AASHTO and Caltrans were used to find the boundaries for the structural 257 parameters. The combinations of parameters listed in Error! Reference source not found. led to 9 ,216 viable design configurations among which 7,916 cases (about 86%) had a positive post-yield flexural stiffness (EIin > 0). The remaining cases (1,300 cases) were removed from the database since the expressions presented in this dissertation for Lpr are only valid for columns with a positive post-yield moment gradient, that is, a positive EIin. 6.3.1.2. Domain for Parameters κel, κin, and Mu/My It is evident from Equation (5-45) that κel, κin, and MB/My are the only parameters required to define the extent of the plastic region, Lpr, according to the nonlinear inelastic solution. Therefore, it is of importance to have a precise knowledge about their feasible range in the design of RC columns. The key parameters of Equation (5-45) were obtained from the moment-curvature analyses conducted on the parametric study cases. The relative frequency of κel, n (= κin/κel), and Mu/My is shown in Figure 6-12(a), Figure 6-12(b), and Figure 6-12(c), respectively. It is worth noting that MB varies between My and Mu, i.e., 1≤ MB/My ≤ Mu/My. It can be seen from the histograms that the typical ranges for the elastic slenderness parameter (κel), the ratio of the inelastic to elastic slenderness parameters (n), and the ratio of the ultimate to yield moments (Mu/My) are respectively (0.1-1.1), (5-13), and (1.1-1.6). 258 Figure 6-12: Relative frequency of main parameters in Equation (5-45) assessed for RC bridge columns 6.3.2. Parametric Study Results for the Nonlinear Inelastic Solution 6.3.2.1. Stability of the Inelastic Solution It was observed that the nonlinear inelastic expression to Lpr according to Equation (5-45) failed to yield an acceptable solution to β (= Lpr/L) for some combinations of the key parameters within their typical ranges. It was also found that as the inelastic deformations (equivalent to μΔ) increased, the probability of obtaining a solution from Equation (5-45) decreased. In particular, 259 out of the 7,916 cases studied here, the nonlinear inelastic expression in Equation (5-45) successfully provided a solution to Lpr for 99, 96, 90, and 83% of the cases at μΔ levels of 1.5, 2, 3, and 4, respectively. By comparison, the nonlinear elastic expression for Lpr, as per Equation (5-21), provided an acceptable solution to every combination of MB/My and κel. The valid space for the key parameters to yield a solution to Equation (5-45) was determined and shown in Figure 6-13, where the empty volume represents the valid space and the filled space illustrates the region for which there is no solution to Equation (5-45). Use of the nonlinear inelastic method is thus limited to the valid ranges of the noted key parameters within which the assumptions of the method are valid. Although it appears from Figure 6-13 that a significant portion of the space constructed by the typical ranges of the parameters falls into the invalid (filled) region, it should be noted that the invalid region covers only extreme values of the key parameters, which are less likely to occur in practice. Moreover, only extremely slender RC columns with aspect ratios greater than 10 that are subjected to high axial load levels can generate extreme values for the key parameters. In addition, instability of the nonlinear inelastic solution starts to appear at large inelastic deformation levels and drift ratios. This is illustrated in Figure 6-14 where the cumulative probability of instability of Equation (5-45) is plotted against drift ratios. It can be seen from the graphs that instability mainly occurred in slender columns subjected to high axial load levels and deformed beyond 10% drift ratios. This finding is in agreement with the assumptions made when deriving the analytical solution. That is, rotation angles greater than 0.1 rad, which is equivalent to a 10% drift ratio, invalidated the small angle assumption. 260 Figure 6-13: Valid (empty) and invalid spaces (filled) for the key parameters of Equation (5-45) 261 Figure 6-14: Cumulative probability for instability of the nonlinear inelastic solution in terms of drift ratio 262 CHAPTER 7 7. SEISMIC DESIGN IMPLICATIONS 7.1. General This chapter presents the implications of the research in seismic analysis and design of slender RC bridge columns. Current codes and standard practices for seismic analysis and design of slender RC columns were scrutinized. Experimental results from the tests conducted as a part of this research were used to reexamine the accuracy and safety of the current design guidelines in case of slender columns. Also, previous plastic region length (Lpr) and plastic hinge length (Lp) models, which are commonly used by engineers to determine the inelastic response of RC bridge columns under seismic demands, were evaluated for slenderness effects in light of the new experimental data presented in Chapter 3. Analytical expressions for the nonlinear effects of slenderness and second-order moments on the inelastic response of RC columns, as presented in Chapters 5 and 6, were used to derive simple and approximate, yet sufficiently accurate, design formulas. Lastly, recommendations and guidelines for including the slenderness effects in the design process of RC columns are provided. 263 7.2. Seismic Design Implications of Experimental Studies 7.2.1. Effect of Increased Strength on P-Δ Moments The destabilizing P-Δ effects on the force-deformation response of columns leads to a decrease in the lateral load bearing capacity of the element. A simple approach to compensate for the P-Δ effect is to increase the member’s strength. This concept is illustrated by means of a typical forcedeformation response in Figure 7-1. An increase of the base-moment capacity was recommended by Pettinga and Priestley [110] and Priestley et al. [16] for RC and steel structures by 50 and 100% of the P-Δ moment, respectively. The strength compensation approach, however, is based on assumed (from numerical modeling) yielding and degradation mechanisms, which need further experimental validation. Figure 7-1: Strength loss due to P–Δ effect on the inelastic force–deformation response of RC columns Increasing the strength of slender columns has been suggested as a practical way to counteract against P-Δ effects. The efficacy of this method was investigated by comparing the forcedisplacement response of columns M121505 and M123005, where the latter had higher lateral 264 strength (as much as 160 % of the maximum P-Δ effect.) Therefore, column M123005 was considered as a design modification to column M121505. Figure 7-2 illustrates the forcedisplacement response of both columns as well as the loss of strength due to P-Δ effects, shown by shaded areas. It can be seen that the strength modification succeeds in terms of providing extra strength to compensate for the strength loss due to P-Δ. However, providing extra strength had only a partial effect on enhancing the column’s stability since the ductility capacity increased only by 0.7 if the stability indices proposed by Priestley et al. [16] and the ACI [33] are used. Figure 7-2: Effect of increasing strength on the force–deformation response of test column M121505 The effectiveness of increased strength on the stability of slender columns can vary depending on the stability limit index being used. For instance, use of the stability limits proposed by Silva et al. [111] and Barrera et al. [27, 52] extend the displacement ductility capacity of the enhanced strength column (M123005) by 1 and 1.6 units, respectively, compared to column M121505. Therefore, the effect of increased strength on improving the stability of RC columns is more significant based on these recently developed criteria. 265 7.2.2. Stability of Cyclic Response In addition to the loss of strength, the destabilizing effect of P-Δ is also viewed as a degradation of stiffness. A typical cyclic response of a column with P-Δ effects is shown in Figure 7-3. The response is considered stable as long as the reloading stiffness (K s*) is positive and greater than 5% of the initial stiffness (Ki); that is, upon complete unloading there exists a minimum stiffness that resists further lateral displacements and generates a resisting load. In this research, the effects of P-Δ on the stability of RC columns in terms of loss of strength and stiffness were evaluated using experimental data. Figure 7-3: Effect of P–Δ on the force–displacement response of RC columns in terms of reduced cyclic stiffness Through numerical modelling of the cyclic response of RC columns, Priestley et al. [16] found that cyclic stability is insured (Ks* / Ki > 0.05) if the stability limit is below 0.3. To further evaluate this stability limit index the Ks* / Ki ratio was experimentally obtained for all loading cycles of test 266 columns M123005 and M121505, and results are plotted against displacement ductility in Figure 7-4. It can be seen that the reloading stiffness remained above 10 % of the initial stiffness. This would imply that for column M121505 the stability limit of 0.3 provided by Priestley et al. [16] can be exceeded without losing the cyclic stability. Comparing the results for both columns reveals that there is no significant difference between the rates of degrading stiffness at reloading cycles in spite of dramatic difference in lateral strengths. This is another evidence for the limited contribution of higher strength in slender columns to overall stability. Figure 7-4: Ratio of reloading to initial lateral stiffness in cyclic tests. 7.3. Seismic Design Implications of Analytical Studies 7.3.1. Comparison of Experimental Lpr with Previous Research Current knowledge about the extent of the plastic region due to moment gradient is reevaluated here for slender RC columns in light of the new experimental evidence from the tests reported in this dissertation (Chapter 3). 267 7.3.1.1. Comparison with Current Design Guidelines Design guidelines often provide recommendations for the largest extent of the plastic region at the ultimate limit state of the RC column. None of the columns tested and reported in Chapter 3 reached their ultimate failure, defined by the ultimate curvature ductility, since the lateral displacement was limited by the actuator stroke (508 mm). However, the test units reached the stability limits found in the literature [30, 33, 110, 111]. Experimental Lpr at displacement ductility levels (μΔ) that correspond to the onset of instability defined by ACI [33], Eurocode 8 for buildings [30], Pettinga and Priestley [110], Priestley et al. [16], and Silva et al. [111] were compared against the length of the critical plastic region required by the design guidelines with variable Lpr [20, 31, 32, 34] as given in Table 7-1. 268 M101510 M121505 M123005 Test Column Table 7-1: Comparison between experimental Lpr and design guidelines with variable Lpr MPE(7) MAPE(8) Notes: μΔ Exp. Lpr(5) (mm) 2.5(1) 3.0(2) 4.1(3) 2.9(4) 1.8(1) 2.3(2) 3.1(3) 3.5(4) 1.4(1) 1.7(2) 2.3(3) 4.0(4) - 1495 1627(9) 2105(9) 1611 1181 1530 1862 1900 1028 1234 1587 1870 - Lpr w/o P-δ (mm) 914 914 1218 914 914 914 977 1032 914 923 1099 1309 - AASHTO Lpr PE(6) w/ % P-δ (mm) -39 914 -44 1015 -42 1583 -43 975 -23 914 -40 1051 -48 1408 -46 1506 -11 1010 -25 1228 -31 1480 -30 1866 -35 35 - Design Guidelines with Variable Criteria NZS 3101 & EN 1998-2 Lpr Lpr Lpr PE w/o PE w/ PE w/o % P-δ % P-δ % P-δ (mm) (mm) (mm) -39 610 -59 610 -59 1829 -38 610 -63 610 -63 1829 -25 800 -62 1192 -43 1829 -39 610 -62 610 -62 1829 -23 610 -48 610 -48 1829 -31 610 -60 610 -60 1829 -24 610 -67 947 -49 1829 -21 613 -68 1057 -44 1829 -2 610 -41 618 -40 1524 0 610 -51 855 -31 1524 -7 765 -52 1135 -28 1524 0 990 -47 1553 -17 1524 -21 -57 -45 21 57 45 - Caltrans Lpr PE w/ % P-δ (mm) 22 1829 12 1829 -13 1829 14 1829 55 1829 19 1829 -2 1829 -4 1829 48 1524 24 1524 -4 1524 -19 1866 13 20 - (1) Stability limit according to ACI 318-14 [33]. (2) Stability limit according to Eurocode 8 for buildings [30], Priestley et al. [16], and Pettinga and Priestley [110]. (3) Stability limit according to Silva et al. [111]. (4) Maximum displacement ductility level to which the columns were tested. 269 PE % 22 12 -13 14 55 19 -2 -4 48 24 -4 0 14 18 Table 7-1 (cont’d) (5) Experimental Lpr from experimental curvature profiles (6) Percent error. (7) Mean percent error. (8) Mean absolute percent error. (9) Extrapolated from the available test data. 270 It is worth noting that the design guidelines that define Lpr according to the moment distribution along the column were implemented using both linear (w/o P-δ) and nonlinear (w/ P-δ) moment profiles and the results were compared separately against the experimental data. In addition, some design guidelines, such as AASHTO [86], Caltrans [34], NZS 3101 [32], and Eurocode [31], recommend a ratio of the maximum moment for predicting Lpr. The accuracy of this method is affected by the yield to ultimate moment ratio (My / Mu), with accuracy increasing as the moment ratio approaches the actual value for the column. Larger moment ratios underestimate Lpr, while smaller moment ratios provide a conservative overestimate. For the columns tested and reported in Chapter 3, the average ratio for My / Mu was 72%. Therefore, design guidelines that recommend a 75% threshold for determining Lpr gave more accurate estimates of the extent of the plastic region than the design guidelines based on a moment ratio of 80%. Yet, all underestimated Lpr. Similarly, a comparison of design guidelines that recommend a constant length for the critical region [5, 6, 9] is provided in Table 7-2. Percentage errors (PE) for the code-specified values of Lpr were calculated at four different displacement ductility levels, corresponding to the aforementioned stability limits. A review of the error values listed in Table 7-1 and Table 7-2 reveals that the extent of the plastic region on slender RC columns is generally underestimated by current design guidelines. The only exception is the seismic provisions by Caltrans, which yielded a conservative estimate of Lpr for all cases. Considering the fact that underestimating the length of the plastic region for special detailing potentially risks the ductility of RC columns that are designed to undergo large inelastic deformations, reassessment of the current design guidelines for the case of slender RC columns is vital for safe seismic design. 271 Table 7-2: Comparison of experimental Lpr against design guidelines with constant Lpr M101510 M121505 M123005 Test Column Design Guidelines with Constant Criteria Exp. Lpr ACI 318-14 CAN/CSA μΔ & EN 1998-1 A23.3-04 (mm) Lpr PE Lpr PE (mm) % (mm) % 2.5 1495 1219 -18 1219 -18 3.0 1627 1219 -25 1219 -25 4.1 2105 1219 -42 1219 -42 2.9 1611 1219 -24 1219 -24 1.8 1181 1219 3 1219 3 2.3 1530 1219 -20 1219 -20 3.1 1862 1219 -35 1219 -35 3.5 1900 1219 -36 1219 -36 1.4 1028 1016 -1 1016 -1 1.7 1234 1016 -18 1016 -18 2.3 1587 1016 -36 1016 -36 4.0 1870 1016 -46 1016 -46 MPE -25 -25 MAPE 25 25 It is worth noting that the ACI code [33] limits the maximum displacement ductility of slender columns through a stability limit defined by (Mp+MP-Δ)/Mp < 1.4. The noted limit restricts the displacement ductility of columns M123005, M121505, and M101510 to 2.5, 1.8, and 1.4, respectively. As seen from the results in Table 7-2, the ACI code successfully predicted Lpr on the test columns with a 7% average error at the maximum permitted μΔ. Therefore, it can be inferred the extent of the plastic region specified by the ACI code is consistent with the maximum permitted displacement ductility. The same can be said about the seismic provisions by Caltrans [34], which limits the displacement ductility of single column bents to 4. Comparing the experimental Lpr with the value prescribed by the Caltrans guidelines indicates that the code recommended Lpr is a conservative estimate for μΔ 272 levels less than 4. In contrast, some of the design guidelines studied here, such as, Eurocode 8 for buildings [30] and Eurocode 8 for bridges [31], were inconsistent between the maximum allowable displacement ductility and the extent of the recommended plastic region. Eurocode 8 for buildings allows RC columns to reach the stability limit defined by (P·Δ)/(V·L) = 0.3. The noted limit respectively corresponds to μΔ = 3, 2.3, and 1.7 for columns M123005, M121505, and M101510. However, results in Table 7-2 show that the length of the plastic region specified by Eurocode 8 for buildings significantly underestimates the test values. The inconsistency also exists in Eurocode 8 for bridges, which limits μΔ to 3.5 for RC columns with aspect ratios greater than 3. Results in Table 7-1 show that the length of the plastic region according to Eurocode 8 for bridges does not cover the entire Lpr from the experiments at displacement ductility levels less than 3.5. Inconsistency between the maximum μΔ and the recommended Lpr potentially imperils the reliability of RC columns by allowing large inelastic deformations while failing to impose the necessary transverse reinforcement required for a ductile inelastic response along the entire length of the critical plastic region. 7.3.1.2. Comparison with Current Expressions Expressions for Lpr and Lp from other researchers were compared against experimental Lpr values for the test columns reported here as shown in Table 7-3. The shear crack model by Hines et al. [12] was used to predict the extent of the plastic region on the test columns. Further, the moment gradient component from commonly used plastic hinge models [21-23] was extracted using Equation (2-2) and is compared against the experimental Lpr. Percent errors (PE) were computed for the models studied and the shear crack model has a superior performance, even though it significantly underestimates Lpr. The additional spread of plasticity not captured by the shear crack model was caused by the nonlinear moment gradient from P-δ effects. 273 Table 7-3: Comparison of experimental Lpr against current Lpr and Lp models Current Models Proposed for Lp Exp. Formula Priestley Bae & Berry Test Lpr Equation μΔ Hines et al. [35] & Park Bayrak et al. (7-9) Column [53] [51] [26] (mm) Lpr PE Lpr PE Lpr PE Lpr PE Lpr PE (mm) % (mm) % (mm) % (mm) % (mm) % 2.5 1495 1367 -9 1170 -22 362 -76 732 -51 1459 -2 3.0 1627 1530 -6 1170 -28 362 -78 732 -55 1660 2 4.1 2105 1797 -15 1170 -44 362 -83 732 -65 2011 -4 2.9 1611 1499 -7 1170 -27 362 -78 732 -55 1630 1 1.8 1181 799 -32 1170 -1 305 -74 732 -38 1056 -11 2.3 1530 1066 -30 1170 -24 305 -80 732 -52 1322 -14 3.1 1862 1379 -26 1170 -37 305 -84 732 -61 1644 -12 3.5 1900 1450 -24 1170 -38 305 -84 732 -61 1777 -6 1.4 1028 472 -54 975 -5 305 -70 610 -41 570 -45 1.7 1234 669 -46 975 -21 305 -75 610 -51 744 -40 2.3 1587 882 -44 975 -39 305 -81 610 -62 995 -37 4.0 1870 1227 -34 975 -48 305 -84 610 -67 1455 -22 MPE -27 -28 -79 -55 -16 MAPE 27 28 79 55 16 M101510 M121505 M123005 Current Models for Lpr 7.3.2. Proposed Design Expression for Lpr The development of an expression for the extent of the plastic region due to linear moment gradient (Lpr,L) is presented in this section. The Lpr,L expression can be adjusted for the effect of member deformation and nonlinear moment gradient due to P-δ effects using Equation (5-30). Consequently, a closed-form relationship between Lpr,NL and μΔ is also proposed. 7.3.2.1. Lpr due to Linear Moment Gradient (Lpr,L) Lateral displacement of a column after yield can be assumed to be equal to the sum of displacements from elastic deformations (Δy) along the entire length, as given by Equation (7-1), 274 and plastic deformations (Δp) from plastic rotations (about the base) over the plastic region (Lpr,L), as calculated according to Equation (7-2). y   y L2 (7-1) 3  p  L   y  Lpr , L 2 (7-2) Combining Equations (7-1) and (7-2), a relationship between displacement ductility (μΔ) and curvature ductility (μϕ = ϕ/ϕy) was obtained as given by Equation (7-3).     1 3Lpr , L 1 2L (7-3) A bilinear plot for the M-ϕ response of RC section is assumed such that the initial branch starts from the origin and ends at the yield point (ϕy, My) and the post-yield branch connects the yield point to the ultimate curvature (ϕu) and ultimate moment (Mu). The slope of the pre-yield branch (EIel), commonly known as the effective stiffness of the cracked section, is defined by EIel = My / ϕy. Similarly, the slope of the post-yield branch of the M-ϕ response is found according to EIin = (Mu - My)/(ϕu - ϕy). Accordingly, the relationship between curvature ductility (μϕ) and moment overstrength (ΩM = M/My) was found as given by Equation (7-4), where n2 (= EIel / EIin) is the ratio of the pre-yield effective flexural stiffness (EIel) to the post-yield slope of M-ϕ bilinear response (EIin).   n 2  M  1  1 (7-4) Assuming a linear moment gradient that starts at zero from the point of contra-flexure and reaches the maximum moment at a distance L, the extent of the plastic region (Lpr,L) along the column height is related to the moment overstrength (ΩM) according to Equation (7-5). 275  1 L pr ,L  L1   M    (7-5) Solving Equation (7-5) for ΩM results in Equation (7-6). M  L L  Lpr , L (7-6) By combining Equation (7-6) with Equation (7-4), and then substituting into Equation (7-3), the relationship between Lpr,L and μΔ was determined (Equation (7-7)).  Lpr , L  3Lpr , L  1  2L L  L pr , L     n 2  (7-7) Rearranging Equation (7-7) provides a binomial equation that always has a positive root for Lpr,L. Therefore, the solution to the extent of the plastic region due to a linear moment gradient (Lpr,L) as a function of displacement ductility (μΔ) was found as noted in Equation (7-8). Lpr , L     1 2    1     1   L    2   2  2     2   3n   3n    3n  (7-8) It is worth noting that the length of the plastic region is meaningful after yield. Therefore, Equation (7-8) is valid for displacement ductility levels greater than 1 (μΔ ≥ 1). 7.3.2.2. Lpr due to Nonlinear Moment Gradient (Lpr,NL) In Chapter 5 of this dissertation, it was demonstrated that the Lpr,L from a linear moment gradient can be adjusted to account for the effect of P-δ moment using an elastic slenderness parameter (κel2 = PL2/EIel) according to Lpr,NL = Lpr,L(1+ κel2/3). Therefore, the extent of the plastic region due to a nonlinear moment gradient (Lpr,NL) is given by Equation (7-9). 276 Lpr , NL 2     1 2    1     1     L    2   2  2     2 1  el  3   3n   3n    3n  (7-9) It is worth mentioning that Equation (7-9) provides the extent of the plastic region due to a nonlinear moment gradient. Therefore, the effect of tension shift on Lpr is not included in Equation (7-9); albeit moment gradient is the main contributor to Lpr in slender ductile RC columns. 7.3.2.3. Verification of the Proposed Formula The length of the plastic region for the test columns at different displacement ductility levels determined with Equation (7-9) are provided in Table 7-3. The parameters for Equation (7-9) were computed from the moment-curvature analysis results. As seen from the mean absolute percent errors (MAPE) in Table 7-3 the proposed formula is a simple and accurate way to estimate of the extent of the plastic region on flexure-dominated slender RC columns. The model by Hines et al. [35] predicted Lpr for the test columns with 27% error, and the moment gradient component from current Lp models underestimated Lpr by 54%. Conversely, the proposed formula predicted Lpr with 16% error. The mean absolute percent error (MAPE) values were calculated for Lpr from design guidelines and from Equation (7-9), and are compared in Table 7-4. Further, the error measures corresponding to the use of different design guidelines were calculated separately and presented in Figure 7-5. The numbers above the bar groups represent the average error for the three test columns (M123005, M121505, and M101510). It can be seen that the proposed formula offers a more accurate estimate of Lpr at different levels of displacement ductility than models from current design guidelines. 277 Table 7-4: Summary of the error in predicting Lpr in test columns MAPE (%) Current Current Proposed Test Codes Models Formula Column Equation w/o w/ L Lp (7-9) P-δ P-δ pr M123005 38 35 9 55 3 M121505 36 31 28 53 11 M101510 31 22 45 54 36 All Columns 35 29 27 54 16 The author acknowledges that the recommendations in seismic design codes are not meant to precisely predict Lpr. Nonetheless, it is essential for design guidelines to require the length of the potential plastic region for special detailing to be conservatively larger than the maximum Lpr from experiments to guarantee a ductile flexural inelastic response. Therefore, the highest elevation to which the plastic region on the test columns reached (maximum Lpr) was compared against the length of the plastic region recommended in seismic codes. The absolute percent error (APE) associated with the code-recommended plastic region is illustrated in Figure 7-6. Also included in Figure 7-6 is the error from the proposed formula (Equation (7-9)) calculated at the highest displacement ductility to which the columns were tested. It can be seen that Equation (7-9) predicts the maximum Lpr with reasonable accuracy. 278 Figure 7-5: Comparing the accuracy of various seismic design codes for predicting the length of the plastic region (Lpr) at different displacement ductility levels in terms of mean absolute percent error (MAPE) Figure 7-6: Comparing the accuracy of various seismic design guidelines in predicting the maximum length of the plastic region (Max Lpr) at maximum displacement ductility in terms of absolute percent error (APE) An exception to the superior performance of the proposed model is the case of column M101510, in which the experimental maximum Lpr showed better correlation with the Caltrans design recommendations than Equation (7-9). Better correlation with the Caltrans design guidelines can be explained by the fact that the code recommends a constant plastic region equal to 1/4 of the length (shear span, L), which is the highest ratio for Lpr/L among all the specification codes studied 279 here. While for most column configurations L/4 is a conservative overestimate of the actual extent of the plastic region (Lpr), it yields a better prediction for the case of column M101510. This is considered mainly a coincidence since assuming the extent of the plastic region as one-fourth of the shear span regardless of the aspect ratio, axial load ratio, longitudinal and transverse reinforcement ratio, etc., is a rough treatment of the issue. The proposed model, however, takes into account different geometrical and structural properties of the column and adjusts the extent of the plastic region accordingly. To avoid such non-conservative underestimates of Lpr from the proposed model, as was the case for column M101510, it must be increased by a factor of safety obtained from an exhaustive survey of previously tested RC columns before it can be used as a design guideline. It should be noted that not only is the proposed equation capable of predicting the extent of the plastic region at the ultimate state (maximum Lpr), but it also accurately follows the spread of the plastic region along the columns’ height due to increasing inelastic deformations (μΔ). The trends of experimental and predicted Lpr were compared to demonstrate this effect. The experimental extents of the plastic region (Lpr) with respect to displacement ductility (μΔ), with and without P-δ effect, are plotted in Figure 7-7. Equation (7-9) is also plotted to compare against experimental data. Results from the shear crack model by Hines et al. [12] are plotted as well. It can be seen that Equation (7-9) correlates well with the experimental data extracted from the nonlinear moment gradient in which P-δ effects were included. The shear crack model is shown to capture the extent of the plastic region due to linear moment gradient, i.e., if P-δ moments are ignored. Thus, the plots in Figure 7-7 show that Equation (7-9) accurately predicts Lpr for a wide range of μΔ levels. 280 Figure 7-7: Comparison of the proposed formula for Lpr against test data and the model by Hines et al. [35] The proposed model (Equation (7-9)) for the moment gradient part of Lpr is valid for all positive post-yield stiffness ratios (n > 0). Yet, the contribution of second-order effects to Lpr can be safely ignored for non-slender columns (κel ≤ 0.5) since for these cases P-δ moments contribute less than 10% to the moment gradient portion of Lpr and the effects of non-flexural phenomena on Lpr become significant. The derivations presented in this chapter were based on a positive post-yield M-ϕ response, i.e., EIin > 0. Almost all RC sections of normal-strength concrete with adequate confinement steel (i.e., 281 designed to modern seismic codes) exhibit a positive post-yield stiffness. However, post-yield softening (EIin ≤ 0) can occur in RC sections with high-strength concrete, with high axial load, or without proper confinement reinforcement for ductile response. For a non-positive post-yield stiffness, the moment gradient does not contribute to the growth of Lpr and the proposed expression becomes invalid. In the proposed expression for Lpr (Equation (7-9)), the effect of P-δ moments on extending the plastic region was derived from a closed-form solution to the deformation in cantilever columns using nonlinear elastic beam-column theory (Equation (5-30)). It was found that the response of RC columns at intermediate displacement ductility levels (μΔ ≤ 2.5) was effectively approximated by an equivalent elastic model. However, the response starts to deviate from elastic beam-column theory at large inelastic deformations (μΔ > 2.5). Therefore, the proposed formula introduces some error at large inelastic deformations, albeit the error remains small enough for design purposes. Lastly, the proposed formula tends to slightly underestimate Lpr. It is thus desirable to amplify the results with a factor to provide a conservative estimate of the potential plastic region for special detailing. However, this requires rigorous statistical analyses on a large array of RC columns. Therefore, a full implementation of the proposed expression for Lpr in design guidelines requires further research. 7.4. Seismic Design Implications of Parametric Studies Reinforced concrete design specifications typically recommend a fraction of the column length as the length of the critical plastic region that requires special detailing for ductile inelastic response. For instance, this fraction is one-sixth (≈0.17) and one-fourth (0.25) in ACI’s [33] and Caltrans’ [34] codes, respectively. However, defining a constant ratio for Lpr/L such that it is applicable to a 282 wide range of RC columns is not practical since the extent of the plastic region varies dramatically with column aspect ratio, axial load level, and concrete strength. As a consequence, any constant value for Lpr/L is going to be either overly conservative or unsafe. Nonetheless, cumulative probability graphs of Lpr/L can be used if a single value for Lpr/L is to be determined to serve as a design recommendation for the size of the critical plastic region. Cumulative probability graphs were thus developed for the linear (Equation (5-29)), nonlinear elastic (Equation (5-21)), and nonlinear inelastic (Equation (5-45)) expressions. The results are presented in Figure 7-8(a) and Figure 7-8(b) for μΔ levels 3 and 4, respectively. Figure 7-8: Cumulative probability of Lpr/L at: (a) μΔ = 3, and (b) μΔ = 4. The reason to choose μΔ levels of 3 and 4 was that most seismic design guidelines limit the inelastic deformation of RC columns to these displacement ductility levels. Considering the 90% probability line, which assumes 10% risk, Lpr/L from the linear solution is about 0.16 at μΔ=3. The 0.16 limit for Lpr/L correlates perfectly with the one-sixth criterion recommended by the ACI code. 283 Assuming that the length of the plastic region (Lpr) is equal to twice the size of the moment gradient component of Lp, the 0.16 criterion (Lpr = 0.16L) is in agreement with the plastic hinge length (Lp) model by Priestley and Park [53] in which the moment-gradient component of Lp is defined by Lpr,MG = 0.08L. Yet, the 0.16 (or one-sixth) limit is based on a linear moment gradient that underestimates the extent of the critical plastic region on more than 40% of possible design configurations. Therefore, the recommended value for Lpr/L should be increased to 0.2 to assure that more than 90% of RC columns deformed up to μΔ=3 have sufficient reinforcement over their critical plastic region. The Lpr/L ratio should be even higher, possibly 0.25, if displacement ductility levels up to 4 are permitted. It is worth mentioning that defining a more refined criterion for the length of the plastic region to be used in design guidelines requires further statistical and parametric analyses. 284 CHAPTER 8 8. SUMMARY AND CONCLUSIONS 8.1. Summary The destabilizing effects of P-Δ on reinforced concrete (RC) bridge columns was experimentally evaluated using test data from four half-scale slender columns with aspect ratios of 10 and 12. In addition, P-δ moments were found to significantly alter the distribution of internal moments, leading to an increased height of the plastic region. The extent of the critical plastic region in slender RC bridge columns was experimentally evaluated via test data from three large-scale columns with aspect ratios up to 12. Current design guidelines for the length of the potential plastic region that require special detailing were reassessed in light of the experimental results reported here [112, 113]. A framework for using 3D continuum-based FE models for predicting intermediate damage limit states (IDLS) was presented within the context of performance-based seismic design of RC bridge columns, and the promise of using 3D FE simulations for evaluating IDLS was demonstrated. The models were verified against results from tests conducted on flexure-dominated columns with 285 adequate transverse reinforcement to provide ductility. Thus, the models are only applicable to columns with aspect ratios of 6 or greater. Damage from non-flexure sources, such as shear cracks in columns with light transverse reinforcement and excessive bond-slip effects due to lack of sufficient anchorage for the longitudinal reinforcement, were not considered in this work. Due to monotonic nature of loading conditions applied to the FE models of this work, the findings may not be directly extended to the prediction of ultimate damage limit states that exhibit significant dependency on displacement history. Nonetheless, the presented methods and procedures for extracting global and local responses from 3D FE simulations are widely applicable and valid for quantitatively validating FE models against experimental data, and to identify IDLS in the damage analysis of RC structures [114]. Closed-form expressions for the effect of member deformations (δ) and P-δ moments on the inelastic response of RC columns, namely, the extent of the plastic region (Lpr) were attained [115, 116]. Formulas to estimate the maximum P-δ moments and their impact on Lpr were derived. A dimensionless slenderness parameter κel, which considers aspect ratio, axial load ratio and longitudinal reinforcement ratio, is recommended to predict the influence of P-δ on the plastic region along RC columns, as long as the assumptions to the problem solution hold. A parametric study was conducted to evaluate second-order effects on the extent of the plastic region on slender RC columns. Accordingly, the following conclusions were reached. 8.2. Conclusions 8.2.1. Experimental Studies 1. The height of the plastic region can be underestimated by up to 25% for slender columns if the effect of P-δ is ignored when obtaining the internal bending moment profiles. 286 Therefore, it is crucial to consider moment nonlinearity due to P-δ effects for predicting Lpr in slender columns. 2. Application of currently used stability limit indices to columns with an aspect ratio of 12 restricted the displacement ductility to values as low as 1.8, prior to the observation of significant damage, e.g., in the form of cover concrete spalling. However, the reloading stiffness from cyclic response showed satisfactory cyclic stability. 3. The experimental data considered here showed that for high slenderness ratios the approach of increasing strength to counteract P-Δ effects only leads to slight improvements on stability. 4. Use of curvature profiles is the most viable method to experimentally assess the extent of the plastic region (Lpr). Moment profiles can also be used to determine the moment gradient components of Lpr. However, the length of the damage zone on test columns may not necessarily correlate with the extent of the plastic region. Particularly in slender RC columns, the plastic region tends to spread significantly beyond the observable damage. 5. Tension shift effects on the spread of plasticity for the slender columns reported here were found to be minimal as the plastic region was extended by 6%, on average, due to this effect. This confirms the flexure-dominated response of slender RC columns. 6. Pseudo-dynamic loading with an irregular displacement history yielded similar results as the reversed cyclic testing regime in terms of observed damage, and local and global responses. Results from testing a column following a different displacement history showed no significant difference in the effects of P-Δ and P-δ on the loss of strength and spread of the plastic region, respectively. The only exception was that results from the 287 pseudo-dynamic testing exhibited more uneven responses compared to the smoother results from the quasi-static cyclic tests. 8.2.2. Numerical Studies 7. Accuracy of the FE simulations in predicting structural responses at different levels of global and local results varies. Therefore, model validations based on global responses cannot adequately reflect the performance of a model at a local level. In this work, global force displacement response of the columns was captured more accurately than flexural curvature profiles along the columns’ height. 8. Intermediate damage limit states in RC bridge columns can be adequately predicted from 3D continuum-based FE analyses results. Onset of yielding, onset of cover spalling and significant growth of the spalled region can be directly determined using the strain outputs from reinforcing steel and cover concrete elements of the model. 9. A strain limit of -0.005 for spalling of the cover concrete works well for tracking the size and shape of the spalling region when it is used as the minimum strain to cause the spalling rather than the average strain over the spalled region. 10. The onset of spalling in terms of drift increased from 1.3% to 3.0% as the columns’ aspect ratio increased from 6 to 12. The displacement ductility limit for the onset of spalling is almost the same for columns with different aspect ratios since the yield displacement changes accordingly with aspect ratio. 8.2.3. Analytical Studies 11. Experimental data from the post-yield response of slender RC columns confirmed the basic assumption of the proposed expression (Equation (5-45)), which approximates flexural 288 stiffness profile along column height as a bilinear distribution in which the plastic region has a lower stiffness. 12. Maximum P-δ moments that occur along the column height were accurately captured by the proposed nonlinear elastic expression (Equation (5-27)) for small inelastic deformations (~ μΔ < 2). The proposed nonlinear inelastic expression for maximum P-δ moments (Equations (5-37) and (5-40)), however, accurately predicted the experimental Pδ moments for all displacement ductility levels. 8.2.4. Parametric Studies 13. Ranges for the key dimension-free parameters (κel, Mu/My, and n) of the proposed expressions for Lpr (Equations (5-21) and (5-45)) were established through the parametric study. It was found that the elastic slenderness parameter (κel), the ratio of the inelastic to elastic slenderness parameters (n), and the ratio of the ultimate to yield moments (Mu/My) have variation ranges of (0.1-1.1), (5-13), and (1.1-1.6), respectively. 14. A majority of RC sections (91%) with normal-strength concrete (f ′c ≤ 45 MPa) that were designed according to the current seismic design guidelines exhibited a positive post-yield flexural stiffness. Therefore, analytical models based on a nonlinear moment gradient were applicable to almost all normal strength concrete (NSC) sections. For cases of sections with high-strength concrete (HSC), i.e., f ′c = 90 MPa, only half of the feasible design configurations led to a positive post-yield flexural stiffness. As a result, the use of Lpr models based on moment gradient is limited for RC columns with HSC. 15. The proposed nonlinear inelastic expression for calculating P-δ effects was shown to be stable over a wide range of the key parameters. The noted expression had a unique and valid solution for more than 90% of possible design configurations up to displacement 289 ductility 3. Instability of the solution occurred only in unlikely cases of extreme slenderness, axial load ratio, and drift ratios greater than 10%. 16. The ratio of yield moment to ultimate moment (My/Mu), which is commonly used by seismic design codes as a measure for the length of the critical plastic region (Lpr), was found to be non-conservatively overestimated by current design guidelines for NSC columns with axial load ratios less than 10%. The parametric study on My/Mu ratio showed that the limit needs to be modified to 70% to ensure a conservative estimate of Lpr along such columns. 17. The findings of this study helped to determine limits for the aspect ratio (L/D) of RC columns beyond which the effects of the second-order moments on Lpr can no longer be ignored. The limiting values for L/D were found to vary from 5 (for RC sections with axial load ratio of 20% and longitudinal reinforcement ratio of 1%) to 10 (for RC sections with axial load ratio of 5% and longitudinal reinforcement ratio of 4%). 18. The length of the critical region (Lpr) on slender columns can increase by up to 50% due to second-order P-δ moments. This can potentially extend inelastic deformation demands beyond the critical region over which special detailing is provided. 8.2.5. Seismic Design Implications 19. Out of seven different seismic design code recommendations for the length of the critical plastic region, only those by Caltrans [34] provided a conservative estimate of Lpr, whereas the other design codes underestimated Lpr by 34%. Therefore, slender columns designed according to current codes may not have sufficient transverse reinforcement along the entire length of the potential plastic region. 290 20. In some design guidelines, such as ACI [33] and Caltrans [34], recommendations for the length of the critical region are consistent with the maximum displacement ductility level permitted by the same code. That is, the experimental Lpr did not exceed the code-specified length except for displacement ductility levels that were beyond the code limit. In contrast, guidelines such as Eurocode 8 for buildings [30] and bridges [31], allow large inelastic ductility demands while the recommended length for ductile detailing is significantly smaller than the actual plastic region. 21. Current design guidelines for the length of the critical plastic region in RC columns that are based on one-sixth of the column length were found to be inadequate. Considering P-δ effects on Lpr, 0.2 and 0.25 of the column length are recommended to serve as design guidelines for the length of the critical region in columns expected to reach displacement ductility levels of 3 and 4, respectively. 22. The proposed expressions for Lpr (Equations (5-21), (5-45), and (7-9)), which consider a nonlinear moment profile along RC columns’ height, are capable of accurately predicting the extent of the plastic region on slender columns with adequate transverse reinforcement to ensure a ductile flexure-dominated response. 8.3. Suggestions for Future Research Research on the effects of slenderness and second-moments in RC columns needs to be continued and extended to provide solutions to current engineering problems concerning nonlinear inelastic response of slender columns. A list of suggested topics for future research along with a brief discussion on each topic is presented in this section. 291 8.3.1. Inelastic Response of Slender Concrete Columns Built with New Materials Recent advancements in concrete technology have introduced new materials and methods for design of concrete columns. High-strength concrete (HSC) sections provide additional strength to design more compact RC sections, which reduce the construction costs by minimizing the amount of construction material and weight of the structures, and maximizing the functional space in buildings. Similarly, ultra-high strength concrete-filled steel tubular (CFST) sections offer more compact alternatives to the conventional RC sections. Structural responses of RC sections that use new materials and construction methods require further research. Therefore, the inelastic response of slender HSC and CFST columns under extreme loading conditions, such as, earthquake, can be experimentally and numerically evaluated in future research. 8.3.2. Seismic Response of Slender RC columns with Sub-sufficient Reinforcement The research presented in this dissertation demonstrated that slender RC concrete columns designed according to current specification codes are likely to lack sufficient reinforcement over their entire plastic region. That is, the critical plastic region of slender RC column may not have enough confinement from transverse reinforcement to exhibit the intended ductile inelastic response. Therefore, it is of importance to evaluate the inelastic response and seismic performance of slender RC columns with sub-sufficient transverse reinforcement. Large-scale experimental programs are suggested to evaluate the nonlinear inelastic response of RC columns with sub-sufficient transverse reinforcement under extreme loading conditions. Further research is thus required to assess the ductility of the noted columns. It is also of interest to experimentally evaluate seismic damage in columns that are provided with sub-sufficient reinforcement. 292 8.3.3. Finite Element Models for Damage Analysis Development of numerical models that are capable of predicting the nonlinear inelastic response of RC elements and simulating damage due to extreme loading conditions is highly demanded. Despite recent advancements in FE analysis of structures, its full implementation in damage analysis requires further research. For instance, damage is typically modeled by simply reducing or removing elements from the model. Moreover, the progressive nature of damage is often simplified by considering the ultimate damage state. Instead, damage can be more accurately modeled to initiate and progress during the life-cycle of a structure through different performance levels before reaching the ultimate state. Thus, the current state of FE modeling for damage analysis can be extended by implementing damage mechanics into the structural models to predict the initiation and propagation of damage. This can be achieved by the use of fracture mechanicsbased. Particularly, cohesive zone models (CZM) can be implemented in finite element simulation of spalling and delamination in reinforced concrete columns. This can capture the initiation and progress of damage in case of extreme loading events, such as impact, explosion, and earthquake. 8.3.4. Effects of Slenderness on Dynamic Response of RC Columns Most of the research presented here was based on the quasi-static response of slender RC columns to lateral forces from ground motions. Therefore, dynamic performance of slender columns and the effects of slenderness on their time-history response requires further research. The effects of slenderness and second-order moments on dynamic response of RC columns can be experimentally evaluated using shaking-tables. Also, numerical models, which are capable of simulating the nonlinear inelastic response of RC sections, can be used for time-history analysis to evaluate the effects of slenderness on the seismic performance of RC columns. 293 8.3.5. Analytical Solution to the Plastic Hinge Length (Lp) Models for plastic hinge length (Lp) of RC beams and columns have been developed using empirical data from tests conducted on RC columns. Also, statistical and regression methods have been recently used to maximize the accuracy of the Lp models for predicting the inelastic response of RC elements. However, these Lp models heavily depend on the database from which the calibration data have been extracted. Therefore, previous Lp models cannot be used to design RC elements that have a feature different from the previous test columns. It is thus needed to derive an analytical expression for the plastic hinge length that is based on the structural mechanics of the elements and applicable to a wide range of RC elements. The presented research makes it possible to derive an analytical solution to the plastic hinge length in a way that the effects of slenderness and second-order moments are also considered. 8.3.6. Probabilistic FE Analysis of RC Structures One way of enhancing the performance of numerical models for predicting the nonlinear inelastic response of RC structures is to implement probability and uncertainty of the modeling parameters in to the FE model. In this regard, stochastic finite element method, which utilizes statistical analyses to intrinsically incorporate the probabilistic nature of the input parameters, is ideal. 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