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The Shaking Table Tests Column Database and evaluation of drift capacity models for non-ductile columns Li, Yaqiong 2012

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THE SHAKING TABLE TESTS COLUMN DATABASE AND EVALUATION OF DRIFT CAPACITY MODELS FOR NON-DUCTILE COLUMNS  by  YAQIONG LI  B.Eng., Tongji University, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  January 2012 © Yaqiong Li, 2012  Abstract  A Shaking Table Tests Column Database comprised of 59 reinforced concrete columns subjected to shaking table tests is compiled in this research. Key geometrical information, material properties and test data of the columns are provided and used to improve the seismic assessment of old reinforced concrete structures. Several structural models estimating the column effective stiffness and drift capacities at shear failure and axial-load failure are evaluated. Models based on the design parameters, such as transverse reinforcement ratio and axial load ratio, generally underestimate the column drift at shear failure (loss of 20% of lateral resistant force) in the dynamic test. Empirical plastic drift capacity models for columns subjected to subduction earthquakes and general types of ground motions are proposed. For column drift at axial-load failure, models based on shear-friction mechanism could capture the drift capacity fairly well. The provisions of the current seismic rehabilitation standard ASCE/SEI 41 Supplement 1 regarding the concrete columns are also evaluated with this dynamic database. It is found that this standard generally overestimates the column effective stiffness and could provide the mean value of the column shear strength. The modeling parameters specified in the standard provide fairly conservative estimate of the column drift capacities and are consistent with the targeted probability of failure. Refinements of the shear strength model and the criteria for column classifications are suggested in this research.  ii  Table of Contents  Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables .......................................................................................................................... ix List of Figures ......................................................................................................................... xi Acknowledgements ............................................................................................................ xviii Dedication ............................................................................................................................. xix Chapter 1: Introduction ........................................................................................................ 1 1.1  Background ................................................................................................... 1  1.1.1  Inelastic Seismic Analysis ................................................................................ 1  1.1.2  Shaking Table Tests .......................................................................................... 2  1.1.3  Column Classifications by Failure Type........................................................... 3  1.2  Summary of ASCE/SEI 41 Provisions for Concrete Columns ..................... 4  1.2.1  Introduction ....................................................................................................... 4  1.2.2  Generalized Load-Deformation Relation .......................................................... 4  1.2.3  Shear Strength Model ....................................................................................... 6  1.2.4  ASCE/SEI 41 Column Classification ............................................................... 8  1.3  Objective and Scope ..................................................................................... 9  1.4  Organization ................................................................................................ 11  Chapter 2: Shaking Table Tests Column Database .......................................................... 13 2.1  Introduction ................................................................................................. 13  2.2  Static Cyclic Test Column Databases ......................................................... 13  iii  2.2.1  Introduction to Static Cyclic Test Column Database ...................................... 13  2.2.2  PEER Structural Performance Database ......................................................... 14  2.2.3  Sezen (2002) Column Database ...................................................................... 15  2.3  Properties of Shaking Table Tests Column Database ................................. 16  2.4  Information of Shaking Table Tests Programs ........................................... 20  2.4.1  Introduction ..................................................................................................... 20  2.4.2  NCREE 2009 Tests ......................................................................................... 20  2.4.3  NCREE 2007 Tests ......................................................................................... 26  2.4.4  NCREE 2005 Tests ......................................................................................... 31  2.4.5  NCREE 2004 Tests ......................................................................................... 35  2.4.6  Ghannoum 2006 Tests .................................................................................... 38  2.4.7  Shin 2005 Tests ............................................................................................... 40  2.4.8  Elwood 2002 Tests .......................................................................................... 45  2.5  Test Results Interpretation .......................................................................... 47  2.5.1  P-Delta Correction .......................................................................................... 47  2.5.2  Column Section Analysis ................................................................................ 49  2.6  The Number of Cycles at Key Points ......................................................... 50  2.6.1  Procedure to Count the Number of Cycles at Key Points ............................... 50  2.6.2  Number of Cycles for Columns in Shin 2005 Tests ....................................... 53  2.7  Experimental Envelope of Lateral Force-Drift Relations ........................... 55  Chapter 3: Column Effective Stiffness............................................................................... 57 3.1  Introduction ................................................................................................. 57  3.2  Measured Column Effective Stiffness ........................................................ 57  iv  3.2.1  Introduction ..................................................................................................... 57  3.2.2  Effect of Column Axial Load ......................................................................... 59  3.2.3  PEER Method ................................................................................................. 60  3.2.4  TEST Method.................................................................................................. 62  3.2.5  Measured Column Effective Stiffness ............................................................ 63  3.3  Effect of Key Parameters on Measured Column Effective Stiffness .......... 66  3.4  Evaluation of ASCE/SEI 41 Procedure ...................................................... 69  3.5  Evaluation of Three-component Model ...................................................... 70  3.6  Comparison with the Static Cyclic Tests Data ........................................... 75  3.7  Effects of Number of Cycles....................................................................... 77  Chapter 4: Column Drift at Shear Failure ........................................................................ 80 4.1  Definition of Column Drift at Shear Failure ............................................... 80  4.2  Effect of key Parameters on Measured Drift at Shear Failure .................... 82  4.3  Evaluation of Pujol et al. (1999) Model...................................................... 88  4.3.1  Procedure ........................................................................................................ 88  4.3.2  Evaluation with The Database ........................................................................ 89  4.3.3  Comparison with the Evaluation with Sezen (2002) Database ....................... 93  4.4  Evaluation of Kato and Ohnishi (2002) Model.......................................... 94  4.4.1  Procedure ........................................................................................................ 94  4.4.2  Evaluation with The Database ........................................................................ 96  4.4.3  Comparison with the Evaluation with Sezen (2002) Database ....................... 98  4.5 4.5.1  Evaluation of Zhu et al. (2006) Model....................................................... 99 Procedure ........................................................................................................ 99  v  4.5.2  Evaluation with The Database ...................................................................... 100  4.5.3  Comparison with the Evaluation with Cyclic Test Data ............................... 102  4.6  Evaluation of Elwood and Moehle (2003, 2005) Model ......................... 103  4.6.1  Procedure ...................................................................................................... 103  4.6.2  Evaluation with The Database ...................................................................... 104  4.6.3  Comparison with the Evaluation with Sezen (2002) Database ..................... 107  4.6.4  Column Plastic Drift at Shear Failure ........................................................... 109  4.7  Effect of Number of Cycles ...................................................................... 113  Chapter 5: Column Drift Capacity at Axial-Load Failure ............................................ 115 5.1  Definition of Axial-Load Failure .............................................................. 115  5.1.1  Definition Based on Hysteretic Response of Axial Load ............................. 115  5.1.2  Definition Based on Degradation of Lateral Resistance ............................... 118  5.1.3  Definition Based on Column Shortening/Lengthening................................. 119  5.1.4  Measured Column Drift at Axial-Load Failure ............................................ 120  5.2  Effect of Key Parameters on Measured Drift at Axial-Load Failure ........ 121  5.3  Evaluation of Shear-Friction Model ........................................................ 126  5.3.1  Procedure ...................................................................................................... 126  5.3.2  Evaluation of Shear-Friction Model with The Database .............................. 127  5.3.3  Comparison with the Cyclic Test Data ......................................................... 129  5.4  Evaluation of Kato and Ohnishi (2002) Model......................................... 131  5.4.1  Procedure ...................................................................................................... 131  5.4.2  Evaluation with The Database ...................................................................... 132  5.5  Evaluation of Zhu et al. (2006) Model..................................................... 134  vi  5.5.1  Procedure ...................................................................................................... 134  5.5.2  Evaluation of Zhu et al. (2006) Model.......................................................... 134  5.5.3  Comparison with Static Cyclic Test Data ..................................................... 137  5.6  Effect of Number of Cycles ...................................................................... 138  Chapter 6: Evaluation of ACSE/SEI 41-06 Supplement ................................................ 140 6.1  Introduction ............................................................................................... 140  6.2  Column Shear Strength ............................................................................. 140  6.2.1  Evaluation of Shear Strength Ratio VMax/Vn ................................................ 140  6.2.2  Evaluation of Shear Strength Ratio VMax/V0 ................................................ 142  6.2.3  Evaluation of Shear Strength Ratio VMax/V0‟ ............................................... 143  6.2.4  Influence of Axial Load Ratio ...................................................................... 146  6.3 6.3.1 6.4  Column Classification............................................................................... 147 ASCE/SEI 41 Column Classification ........................................................... 147 Modeling Parameters of Generalized Load-Deformation Relation .......... 150  6.4.1  Measured Plastic Rotations ........................................................................... 150  6.4.2  Influence of Varying Column Axial Load .................................................... 152  6.4.3  Assessment of ASCE/SEI 41 Modeling Parameter a ................................... 155  6.4.4  Assessment of ASCE/SEI 41 Modeling Parameter b ................................... 158  6.4.5  Modeling Parameters for Columns in Condition i ........................................ 163  6.5  Column Lateral Load-Deformation Backbones ........................................ 165  6.5.1  Experimental Envelope of Effective Lateral Force ...................................... 165  6.5.2  ASCE/SEI Backbone Curve ......................................................................... 176  Chapter 7: Conclusions and Future Work ...................................................................... 188  vii  7.1  Summaries................................................................................................. 188  7.2  Conclusions ............................................................................................... 189  7.2.1  Column Effective Stiffness ........................................................................... 189  7.2.2  Drift Capacities at Shear Failure ................................................................... 190  7.2.3  Drift Capacities at Axial-Load Failure ......................................................... 192  7.2.4  Evaluation of ASCE/SEI 41.......................................................................... 193  7.3  Future Research ........................................................................................ 194  References ............................................................................................................................ 196 Appendices ........................................................................................................................... 200  viii  List of Tables  Table 2.1  Properties of PEER database ............................................................................... 14  Table 2.2  Properties of Sezen (2002) database .................................................................... 15  Table 2.3 Characteristics of specimens in seven shaking table test programs..................... 16 Table 2.4  Information of the shaking table test programs ................................................... 18  Table 2.5 Properties of columns in The Database ............................................................... 19 Table 2.6 Results of the section analysis for columns in Elwood 2002 Tests ..................... 50 Table 2.7  Number of cycles for columns in Shin 2005 Tests .............................................. 53  Table 3.1 Statistical results of column effective stiffness ratio for columns in The Database ................................................................................................................................................. 75 Table 3.2  Properties of the dataset in Elwood and Eberhard (2009) ................................... 76  Table 3.3  Evaluation of column effective stiffness models with dynamic and static cyclic  test data ................................................................................................................................... 76 Table 4.1  Properties of column database in Pujol et al. (1999) ........................................... 89  Table 4.2 Statistical results of Pujol et al. (1999) model .................................................... 93 Table 4.3 Statistical results of Kato and Ohnishi (2002) model .......................................... 98 Table 4.4 Properties of the column database used in Zhu et al. (2006) ............................. 100 Table 4.5 Statistical results of Zhu et al. (2006) model ..................................................... 102 Table 4.6 Statistical results of Elwood and Moehle (2003, 2005) models ........................ 107 Table 4.7  Statistical results of proposed plastic drift models ............................................ 112  Table 5.1 Statistical results of shear-friction model .......................................................... 128 Table 5.2 Statistical results of Kato and Ohnishi (2002) model ........................................ 134  ix  Table 5.3 Statistical results of Zhu et al. (2006) model ..................................................... 137 Table 6.1  Evaluation of ASCE/SEI 41 shear strength model ............................................ 147  Table 6.2  Evaluation of ASCE/SEI 41 column classification method .............................. 148  Table 6.3  Evaluation of variation of ASCE/SEI 41 classification method ........................ 149  Table 6.4  Evaluation of modified ASCE/SEI 41 classification method ............................ 150  x  List of Figures  Figure 1.1  Generalized force-deformation relations in ASCE/SEI 41 .................................. 5  Figure 1.2  Modifying factor k for shear strength model ........................................................ 7  Figure 1.3  ASCE/SEI 41 column classification ..................................................................... 8  Figure 2.1  Pictures for Specimen HCFS in NCREE 2009 Tests ......................................... 22  Figure 2.2  Definition of chord rotations; (a) chord rotations at the top and base of the  column; (b) comparison of chord rotation with the total rotation ........................................... 22 Figure 2.3  Information of specimens of NCREE 2009 Tests .............................................. 23  Figure 2.4  Response spectra for NCREE 2009 Tests .......................................................... 23  Figure 2.5  Hysteretic response of non-ductile columns in NCREE 2009 Tests .................. 25  Figure 2.6  Response spectra for NCREE 2007 Tests .......................................................... 27  Figure 2.7  Information of specimens of NCREE 2007 Tests .............................................. 28  Figure 2.8  Hysteretic response of columns in Specimen S1 of NCREE 2007 Tests ........... 29  Figure 2.9  Hysteretic response for columns in Specimen S3 and S4 of NCREE 2007 Tests  ................................................................................................................................................. 30 Figure 2.10  Response spectra for NCREE 2005 Tests ........................................................ 32  Figure 2.11  Information of specimens of NCREE 2005 Tests ............................................ 33  Figure 2.12  Hysteretic response of non-ductile columns in NCREE 2005 Tests ................ 34  Figure 2.13  Information of Specimen S2 and S3 of NCREE 2004 tests ............................. 36  Figure 2.14  Response spectra for NCREE 2004 Tests ........................................................ 36  Figure 2.15  Hysteretic response of non-ductile columns in NCREE 2004 Tests ................ 37  Figure 2.16  Information of specimen of Ghannoum 2006 Tests ......................................... 39  xi  Figure 2.17  Hysteretic response of non-ductile columns in Ghannoum 2006 Tests ........... 39  Figure 2.18  Information of specimens of Shin 2005 Tests .................................................. 41  Figure 2.19  Input ground motions of Shin 2005 Tests ........................................................ 42  Figure 2.20  Response spectra for Shin 2005 Tests .............................................................. 42  Figure 2.21  Hysteretic response of non-ductile columns in Shin 2005 Tests ...................... 44  Figure 2.22  Response spectra for Elwood 2002 Tests ......................................................... 45  Figure 2.23  Information of specimens of Elwood 2002 Tests ............................................. 46  Figure 2.24  Hysteretic response of non-ductile columns in Elwood 2002 Tests................. 46  Figure 2.25  Illustration for correcting P-Delta effect .......................................................... 47  Figure 2.26  P-Delta correction for columns in The Database ............................................. 48  Figure 2.27  Semi-cycle points for column Shin 2005-II-Test 1-Chile-0.1-WestC.............. 51  Figure 2.28  The maximum lateral force and drift points in each cycle ............................... 51  Figure 2.29  Number of cycles with column axial load ratio................................................ 52  Figure 2.30  Number of cycles for columns in Shin 2005 Tests........................................... 55  Figure 2.31  Experimental envelope for column Shin 2005-II-Test 1-Chile-0.1-WestC ..... 56  Figure 3.1  Column lateral stiffness keff from lateral load-deformation relations ................. 58  Figure 3.2  Influence of varying column axial load on column effective stiffness .............. 59  Figure 3.3  Interpretation of column effective stiffness by PEER method ........................... 61  Figure 3.4  Column effective stiffness ratio of PEER method ............................................. 61  Figure 3.5  Interpretation of column effective stiffness by TEST method ........................... 62  Figure 3.6  Column effective stiffness ratio from TEST method ......................................... 63  Figure 3.7  Comparison of column effective stiffness from PEER method and TEST method  ................................................................................................................................................. 63  xii  Figure 3.8  Examples of column effective stiffness from PEER method and TEST method 64  Figure 3.9  Large discrepancies in effective stiffness from PEER method and TEST method  ................................................................................................................................................. 65 Figure 3.10  The measured effective stiffness with key parameters ..................................... 68  Figure 3.11  Column effective stiffness ratio estimated by ASCE/SEI 41 procedure .......... 69  Figure 3.12  Evaluation of column effective stiffness estimated by ASCE/SEI 41 procedure  ................................................................................................................................................. 70 Figure 3.13  Deformation components from three-component model ................................. 72  Figure 3.14  Evaluation of column effective stiffness estimated by three-component model  ................................................................................................................................................. 73 Figure 3.15  Evaluation of column effective stiffness estimated by simplified three-  component model .................................................................................................................... 74 Figure 3.16  Comparison of effective stiffness from three-component model and simplified  model....................................................................................................................................... 74 Figure 3.17  Number of cycles with column effective stiffness keff .................................... 77  Figure 3.18  Number of cycles with effective stiffness ratio EI eff EI g .............................. 79  Figure 4.1  Axial-load failure occurred before 20% loss of its lateral load .......................... 81  Figure 4.2  Measure column drift at Max Veff with key parameters ..................................... 84  Figure 4.3  Measured column drift at 80% Max Veff with key parameters........................... 85  Figure 4.4  Measured column plastic drift at 80% Max Veff with key parameters ............... 86  Figure 4.5  Measured column drift with key dynamic parameters ....................................... 87  Figure 4.6  Calculated drift by Pujol et al. (1999) model with measured drift at 80% Max  Veff ........................................................................................................................................... 90  xiii  Figure 4.7  Calculated drift by Pujol et al. (1999) with measured drift at Max Veff ............. 91  Figure 4.8  Evaluation of Pujol et al. (1999) model.............................................................. 92  Figure 4.9  Evaluation of Pujol et al. (1999) model with columns in The Database and Sezen  (2002) database ....................................................................................................................... 93 Figure 4.10  Equivalent axial load ratio from Kato and Ohnishi (2002) Model ................... 95  Figure 4.11  Calculated column drift at 80% Max Veff by Kato and Ohnishi (2002) model  ................................................................................................................................................. 96 Figure 4.12  Evaluation of Kato and Ohnishi (2002) model................................................. 97  Figure 4.13  Evaluation of Kato and Ohnishi (2002) model with dynamic and cyclic test  data .......................................................................................................................................... 98 Figure 4.14  Calculated column drift by Zhu et al. (2006) model ...................................... 101  Figure 4.15  Evaluation of Zhu et al. (2006) model............................................................ 101  Figure 4.16  Evaluation of Zhu et al. (2006) model with dynamic and cyclic tests data.... 102  Figure 4.17  Calculated column drift by Elwood and Moehle (2003, 2005) models ......... 105  Figure 4.18  Evaluation of Elwood and Moehle (2003, 2005) models ............................... 106  Figure 4.19  Evaluation of Elwood and Moehle (2005) model with The Database and Sezen  (2002) database ..................................................................................................................... 107 Figure 4.20  Measure shear strength versus measured drift normalized by the calculated  shear strength and calculated column drift ........................................................................... 108 Figure 4.21  Plastic drift calculated by Elwood and Moehle (2003, 2005) model ............. 109  Figure 4.22  Proposed empirical plastic drift model for flexure-shear-critical columns .... 112  Figure 4.23  Number of cycles with normalized column drift at shear failure ................... 114  xiv  Figure 5.1  Axial-load failure based on axial load response for column in Shin 2005 Tests  ............................................................................................................................................... 116 Figure 5.2  Axial-load failure based on axial load response for column in NCREE 2007  Tests ...................................................................................................................................... 116 Figure 5.3  Axial-load failure definitions for columns without sudden change in axial load  response................................................................................................................................. 117 Figure 5.4  Axial-load failure based on shear degradation ................................................. 118  Figure 5.5  Inappropriate to define the column axial-load failure based on shear degradation  ............................................................................................................................................... 119 Figure 5.6  Axial-load failure based on column lengthening/shortening............................ 120  Figure 5.7  Axial-load failure at low lateral drift for column in NCREE 2009 Tests ........ 121  Figure 5.8  Measured drift at axial-load failure with key parameters ................................. 123  Figure 5.9  Measured plastic drift at axial-load failure with key parameters ..................... 124  Figure 5.10  Measured drift at axial-load failure with dynamic parameters ....................... 125  Figure 5.11  Calculated column drift at axial-load failure of shear-friction model ............ 127  Figure 5.12  Evaluation of shear-friction model ................................................................. 128  Figure 5.13  Relation between effective coefficient of friction  and column drift at axial-  load failure ............................................................................................................................ 129 Figure 5.14  Evaluation of shear-friction model by dynamic and cyclic test data.............. 130  Figure 5.15  Evaluation of shear-friction model by dynamic and cyclic tests data ............ 131  Figure 5.16  Calculated column plastic drift at axial-load failure by Kato and Ohnishi (2002)  model..................................................................................................................................... 132  xv  Figure 5.17  Calculated column drift at axial-load failure by Kato and Ohnishi (2002) model  ............................................................................................................................................... 133 Figure 5.18  Evaluation of Kato and Ohnishi (2002) model............................................... 133  Figure 5.19  Calculated column drift at axial-load failure by Zhu et al. (2006) model ...... 135  Figure 5.20  Adjusted calculated column drift at axial-load failure by Zhu et al. (2006)  model..................................................................................................................................... 136 Figure 5.21  Evaluation of Zhu et al. (2006) model............................................................ 136  Figure 5.22  Evaluation of Zhu et al. (2006) model with the dynamic and cyclic test data 138  Figure 5.23  Number of cycles with column drift capacity at axial-load failure ................ 139  Figure 6.1  Evaluation of shear strength ratio (VMax/Vn) .................................................... 141  Figure 6.2  Evaluation of shear strength ratio (VMax/V0) .................................................... 142  Figure 6.3  Evaluation of shear strength ratio (VMax/V0‟)................................................... 143  Figure 6.4  Sudden increase in lateral resistance for columns in NCREE 2007 Tests ....... 145  Figure 6.5  Shear strength ratio calculated with different axial load ratio .......................... 146  Figure 6.6  Evaluation of modeling parameter a for non-ductile columns ......................... 153  Figure 6.7  Evaluation of modeling parameter b for non-ductile columns ......................... 154  Figure 6.8  Plastic rotation ratio at significant loss of lateral resistant force ...................... 156  Figure 6.9  Lognomal fragility curve for plastic rotation at significant loss of lateral  resistance ............................................................................................................................... 157 Figure 6.10  Plastic rotation ratio at loss of axial load support........................................... 158  Figure 6.11  Lognomal fragility curve for plastic rotation at loss of axial-load support .... 160  Figure 6.12  Comparison of plastic rotation at axial-load failure and shear failure ........... 162  Figure 6.13  Evaluation of modeling parameter b for columns in Condition i ................... 163  xvi  Figure 6.14  Plastic rotation ratio for columns in Condition i ............................................ 164  Figure 6.15  Normalized experimental envelopes for columns in NCREE 2009 Tests ..... 166  Figure 6.16  Normalized experimental envelope for Elwood 2002 Tests .......................... 166  Figure 6.17  Normalized envelopes for columns subjected to Chile Earthquake in Shin 2005  Tests ...................................................................................................................................... 167 Figure 6.18  Normalized envelopes for columns subjected to Kobe Earthquake in Shin 2005  Tests ...................................................................................................................................... 167 Figure 6.19  Normalized envelopes for shear-critical columns in NCREE 2007 Tests ..... 168  Figure 6.20  Normalized envelopes for non-ductile columns with proposed simplified  envelopes............................................................................................................................... 171 Figure 6.21  Envelope for non-ductile columns with normalized force and column  horizontal drift ...................................................................................................................... 173 Figure 6.22  Envelope with normalized force and column plastic drift.............................. 175  Figure 6.23  ASCE backbone cures for column Shin 2005-II-Test 1-Chile-0.1-WestC .... 176  Figure 6.24  ASCE/SEI 41 backbone curves for columns in NCREE 2009 Tests ............. 179  Figure 6.25  ASCE/SEI 41 backbone curves for columns in NCREE 2005 Tests ............. 180  Figure 6.26  ASCE/SEI 41 backbone curves for columns in Ghannoum 2006 Tests ........ 181  Figure 6.27  ASCE/SEI 41 backbone curves for columns in Elwood 2002 Tests .............. 181  Figure 6.28  ASCE/SEI 41 backbone curves for columns in NCREE 2004 Tests ............. 182  Figure 6.29  ASCE/SEI 41 backbone curves for columns in Shin 2005 Tests ................... 184  Figure 6.30  ASCE/SEI 41 backbone curves for columns in NCREE 2007 Tests ............. 185  xvii  Acknowledgements  I would like to express my sincerest appreciation for all the guidance, support and help from my supervisor Dr. Kenneth Elwood. I offer my special thanks to Professor Shyh-Jiann Hwang and Professor Chiun-lin Wu, who gave me numerous valuable suggestions and helped me greatly when I was visiting Taipei. I also greatly appreciated the support of Civil Engineering Department of UBC and the generous help from my colleagues, Dr. Yavari and Nazli.  xviii  Dedication  To my parents  xix  Chapter 1: Introduction 1.1 1.1.1  Background Inelastic Seismic Analysis In seismic regions of Canada and all over the world, many existing structures are old  reinforced concrete buildings with structural deficiencies that do not satisfy the requirements specified in current seismic codes. The major vertical supporting components and the lateral resistant systems of these old structures are usually non-ductile columns with inadequate transverse reinforcement detailing, such as large spacing of transverse reinforcing bars and 90-degree hooks. Equation Chapter 1 Section 1 During a strong earthquake, the lateral demand from the ground shaking may exceed the capacities of these non-ductile columns, causing severe shear degradation and column damage. With reduced lateral strength, the columns may fail to support the vertical load imposed by the gravity load of the structures and additional axial load from the overturning moment. Thus these old structures are very vulnerable to catastrophic collapse and will endanger human lives. To retrofit these structures economically and efficiently, the capacities of the column components to resistant earthquake load need to be identified. Columns can develop inelastic deformations when subjected to a strong earthquake and tend to suffer shear failure and severe shear degradation after reaching a certain inter-story drift level. With increased horizontal drift, columns may lose the ability to support for gravity load, leading to the collapse of the whole structure. Unlike the traditional linear structural analysis and design, the inelastic behaviour and drift capacities of the column can be more important than the column strength in seismic  1  assessment of existing buildings. Thus the focus of research on seismic rehabilitation for old reinforced concrete structures has shifted from strength-based, linear elastic evaluation to deformation-based, non-linear inelastic analysis.  1.1.2  Shaking Table Tests In current seismic analysis for reinforced concrete structures, column response is often  modeled by a force-deformation backbone curve, which is constructed by using several key parameters, such as column effective stiffness, shear strength and the deformation capacity at significant loss of lateral force and at loss of axial load. Most available models for estimating these key parameters for non-ductile columns in reinforced concrete moment frames are developed based on the data of monotonic static tests or reversed-cyclic tests. These tests were mostly conducted on single columns and often terminated before the axial-load failure of the columns. More and more shaking table tests have been performed on specimens of simple reinforced concrete moment frames. Compared with other testing methods, shaking table tests can directly simulate earthquake ground motions and better reflect the dynamic effect on structures. Since some of those tests were conducted on frames with multiple columns, axial load redistribution and column interactions within the frame are captured, making the test results for column components more realistic. Since most shaking table tests are conducted on the specimens scaled from the real frames, the input motions need to be compressed and adjusted in the time-domain. Thus the strain rate might be higher than that experienced by real structures in earthquakes.  2  1.1.3  Column Classifications by Failure Type Generally speaking, reinforced concrete columns could be classified into three types  roughly by their failure modes: flexure-critical columns, flexure-shear-critical columns and shear-critical columns. Flexure-critical columns usually have well-detailed transverse reinforcement and are expected to demonstrate extended deformation beyond the column yielding when subjected to seismic loads. With adequate shear strength and ductility capacities, shear failure is not likely to occur and these ductile columns often experience failure associated with flexure damage, such as the concrete spalling, buckling of longitudinal reinforcement, crushing of concrete blocks, formation of plastic hinges at column ends, etc. Lateral load-deformation relations for flexure-critical columns are characterized by a plastic plateau after the initial linear response. Non-ductile columns with inadequate transverse reinforcement are vulnerable to the damage associated with shear failure when subjected to seismic loads, such as diagonal cracks and fracture of transverse reinforcements. If shear failure occurs after the column yielding, these non-ductile columns are often referred to as flexure-shear-critical columns. After the initial linear response in lateral load-deformation relations, flexure-shear-critical columns tend to develop limited inelastic deformation and suffer severe degradation in lateral strength. As shear-damaged plane develops, columns may lose their capacity to sustain gravity loads, eventually leading to the collapse of the whole structural systems. The non-ductile columns that experience shear failure before column yielding are often classified as shear-critical columns, for example, the short columns with low aspect ratio ( a / d ) and inadequate transverse reinforcement. Shear-critical columns subjected to seismic  3  loads tend to develop very limited inelastic deformations and suffer severe lateral force degradation due to the shear distress or diagonal cracking. Those non-ductile columns tend to lose their capacity to support gravity loads shortly after the shear failure and experience very sudden and brittle failure.  1.2  Summary of ASCE/SEI 41 Provisions for Concrete Columns  1.2.1  Introduction Seismic Rehabilitation of Existing Buildings, known as ASCE/SEI 41-06 supplement 1  (ASCE, 2008), is a national consensus standard in the United States of America and used internationally for seismic assessment and rehabilitation of existing concrete buildings. To better reflect the observed performance of concrete components from laboratory tests (Elwood et al., 2007), the latest edition of ASCE/SEI 41/06 supplement 1 updated the concrete provisions from its predecessor Seismic Rehabilitation of Existing Buildings (41/06) (ASCE, 2007). These updates include significant changes to the modeling parameters and acceptance criteria for concrete columns. For brevity, ASCE/SEI 41/06 supplement 1 will be referred to as ASCE/SEI 41 herein.  1.2.2  Generalized Load-Deformation Relation ASCE/SEI 41 recommends modeling the column behaviors by a generalized load-  deformation relation shown in Figure 1.1. The ordinate Q / Qy represents the normalized lateral force with Qy = lateral force at the time of flexural yielding. The abscissa is usually taken as the lateral drift ratio calculated by dividing the lateral deformation with column clear height.  4  Source: ASCE/SEI 41 Figure 1.1  Generalized force-deformation relations in ASCE/SEI 41  Before the column yielding, the column response is represented by a linear line from unloaded point A to effective yield point B, slope of which equals to lateral stiffness keff . Then, column behavior is described as linear response at reduced stiffness from point B to point C. The ordinate of point C equals to the column shear strength and the corresponding abscissa shall be taken as the deformation at the time of significant loss of lateral force. It is generally believed that column lateral strength degrades rapidly beyond the point that column loses 20% of its peak lateral resistant force and the corresponding deformation shall be used as the abscissa of point C. The slope from point B to C is usually between zero and 10% of the initial stiffness. Beyond point C, the lateral strength suddenly degrades to point D and then column deforms at the residual shear resistant force until axial load failure point E, at which column fails to sustain gravity load. The modeling parameter a referring to the portion of deformation from the yield point B to point C, which could be considered as the plastic rotation at shear failure. Similarly, the modeling parameter b is used to describe the plastic rotation at the time of axial-load failure. The modeling parameter c represents the percentage of the residual force in terms of the maximum lateral force at the stage between point D and E.  5  1.2.3  Shear Strength Model Several models and methods (Priestley et al., 1994; Caltrans, 1995; Kato, Ohnishi,  2002; ACI 318-02) have been developed to estimate the column shear strength. For columns with inadequate transverse reinforcement, ASCE/SEI 41 adopted the shear strength model proposed by Sezen (2004), which was developed based on an assembled database consisting of 51 flexure-shear-critical columns subjected to cyclic lateral loads. According to Sezen (2004), column shear strength Vn could be calculated by summing the contribution from concrete compression block Vc and transverse reinforcement Vs .  Vn  kV0  k Vc  Vt   0.5 f ' Nu c Vc    1  M / Vd 0.5 f c' Ag  A f d Vs  st yt s    0.8 Ag    (MPa)  Equation 1.1  where, Ast = cross-sectional area of one layer of transverse reinforcement parallel to the direction of lateral load; f yt = yield stress for transverse reinforcement; s = spacing of transverse reinforcement;   1.0 for normal weight concrete while   0.75 for light weight concrete; f c' = the concrete compressive stress on test day; N u = axial compression force (=0 for tension force); M / Vd is the largest ratio of moment to shear times column effective depth under design loadings for column, but shall not be taken greater than 4 or less than 2; Ag = gross cross-sectional area of column; d = the effective depth of column section and is  permitted to use 80% of column section height.  6  For columns, the ratio of M/V could be taken as shear span a and M Vd equals to the aspect ratio a / d . For columns in The Database that had fixed-fixed restrained at both ends,  a equals to half of the column clear height L . k is the modifying factor related to the displacement ductility demand  , which is the  ratio of ultimate displacement (ultimate drift) over yield displacement (yield drift). It is generally believed that columns experience shear failure at the point that column lost 20% of its peak lateral load and the corresponding displacement (drift) is considered to be the ultimate displacement (ultimate drift ratio). Yield displacement is taken as the measured yield displacement obtained by TEST Method in section 3.2.4. Research shows that the column shear strength will decrease as the ductility demand increases. As shown in Figure 1.2, when ductility demand  is greater or equal than 6, its modifying factor k  0.7 ; when ductility demand is less than or equal to 2, then k  1 ; when ductility demand is in between these two extremes, k is calculated by linear interpolation.  Source: Sezen (2004) Figure 1.2 Modifying factor k for shear strength model  According to truss models, column shear strength contributed by transverse reinforcement could be defined as Vs   Ast f yt d / s . Sezen (2004) compared the quantity  VTest  Vc with  Ast f yt d / s for all the columns included in the assembled database and found  7  that   1 achieved the best match. Therefore, the transverse reinforcement contribution is expressed as Vs  Ast f yt d / s . According to ASCE/SEI 41, if spacing of column transverse reinforcement is larger than half of column section effective depth ( s  d 2 ) in the direction of shear force, the transverse reinforcement should be taken as no more than 50% effective when resisting shear or torsion and thus Vs  0.5 Av f y d / s .However, if the spacing is larger than the effective depth of the column section ( s  d ), the transverse reinforcement shall be considered ineffective and set Vs  0 .  1.2.4  ASCE/SEI 41 Column Classification According to ASCE/SEI 41, columns are classified into three conditions based on shear  strength Vn , plastic shear capacity V p and transverse reinforcement detailing. Detailed classification criteria can be found in Figure 1.3.  Source: ASCE/SEI 41 Figure 1.3  ASCE/SEI 41 column classification  The plastic shear capacity VP is the shear demand corresponding to the flexural yielding of plastic hinges and could be calculated by dividing the maximum moment strength obtained from section analysis by shear span a. The shear strength Vn is calculated by the  8  shear strength model ( Vn  kV0  k Vc  Vs  ) mentioned in section 6.2.3 and the term Vn k is the shear strength excluding the reduction factor related to ductility demand. As shown in the first column of Figure 1.3, by comparing plastic shear capacity VP with V0  Vn / k , columns with transverse reinforcement including 135-degree hooks are classified into three conditions: condition i, ii and iii, which approximately correspond to the failure modes, namely flexure failure, flexure-shear failure and shear failure respectively. Past researches have shown that poor transverse reinforcement detailing reduces the column lateral deformation capacity. Therefore, columns with transverse reinforcement including 90-degree hooks or having lap splices are adjusted from condition i to condition ii if Vp / V0  0.6 ; columns with lap-spliced transverse reinforcement are classified into condition iii instead of condition ii for the case of 0.6  Vp / V0  1.0 . By classifying columns into the aforementioned three conditions roughly by failure modes, ASCE/SEI 41 then provides recommendations on allowed plastic rotations in generalized load-deformation relation for each condition.  1.3  Objective and Scope The overall objective of this research is to compile a comprehensive database providing  the key properties and test data for columns that have been subjected to shaking table tests and use this database to improve the accuracy of the seismic assessments of older concrete buildings with non-ductile columns. Available structural models and provisions of current seismic rehabilitation standards ASCE/SEI 41 regarding concrete column are evaluated with this database. Influence of varying axial load and the dynamic effects will be investigated. Only the large-scale columns in relatively simple moment-frames are intend to be included in 9  this Shaking Table Tests Column Database. A total of 59 columns from seven shaking table tests programs are selected, of which 36 are non-ductile columns with inadequate transverse reinforcement. The effective lateral force will be obtained from the recorded shear data by correcting the P-delta effect and section analysis will be conducted based on column properties to get several key points, such as yield point and plastic shear demand. The available models of estimating column effective stiffness, shear strength, drift capacities at shear failure and axial-load failure are evaluated with the test data from the dynamic column database in this research and the results are compared with the evaluation with static cyclic test data. This research will focus on the provisions of ASCE/SEI 41 regarding the reinforced concrete columns. The test data will be used to assess the accuracy of estimated column stiffness and column shear strength recommended by ASCE/SEI 41. Columns included in the compiled database are classified into three conditions and the level of conservatism of the modeling parameters specified in ASCE/SEI 41 for each condition are being evaluated. Columns are subjected to varying axial load during the shaking table tests. Especially for outside columns, the difference between axial load in the positive and negative direction could be very significant due to the overturning moment. When evaluating the models for column shear strength and drift capacities, this research will compare the effects of different column axial load on the calculated results.  10  1.4  Organization Chapter 2, “Shaking Table Tests Column Database”, introduces the dynamic column  database compiled in this research. Detailed information and literature review of each shaking table test are presented. Two important static cyclic test column databases are briefly introduced. This chapter also provides the main properties of this dynamic column database, adjustments of the test data (i.e. P-Delta correction) and other key parameters (e.g. the yielding point from section analysis). The number of the cycles that each column went through during the dynamic tests and experimental envelope of the column response are also investigated in this chapter. Chapter 3, “Column Effective Stiffness”, discusses two methods to interpret the measured column effective stiffness EI eff , namely the “PEER Method” and “TEST Method”. With the measured effective stiffness, performance of the method proposed by ASCE/SEI 41 and the Three-Component model is evaluated in this chapter. Chapter 4, “Column Drift at Shear Failure”, presents three methods to obtain the measured column drift at shear failure. Four available drift capacity models for non-ductile columns, namely Pujol (1999) model, Kato and Ohnishi (2006), Zhu et al. (2006) and Elwood and Moehle (2005), are evaluated in this chapter. The procedure of each model is introduced and the evaluation with the test data of Shaking Table Tests Column Database is presented along with the comparison of the evaluation with static cyclic test data in past studies to gain better understanding of the dynamic effect on drift capacities. Chapter 5, “Column Drift at Axial-Load Failure”, discusses different ways to define column axial-load failure based on measured column axial load, shear degradation or the column lengthening/shortening. Measured column drifts at axial-load failure are compared  11  with calculated drifts from Shear-Friction models, Kato and Ohnishi (2006) and Zhu et al. (2006) in this chapter. Chapter 6, “Evaluation of ASCE/SEI 41-06 Supplement 1”, evaluates the performance of current rehabilitation standard (ASCE, 2008) in terms of column classification and estimations of modeling parameters a and b . This chapter also suggests some refinements and changes of this standard based on the test data of Shaking Table Tests Column Database. Finally, the Chapter 7, “Conclusions and future work”, summarizes the main conclusions from each chapter and provides some suggestions for future research.  12  Chapter 2: Shaking Table Tests Column Database 2.1  Introduction This chapter introduces the compiled Shaking Table Tests Column Database and  detailed information of each shaking table test program is presented. Two important static cyclic tests column databases are briefly introduced. This chapter also provides interpretations of the test data, such as the effective force obtained after P-Delta correction, key parameters (e.g. the yielding point from section analysis) from section analysis. The methods to count the number of the cycles that each column went through and develop experimental envelopes of the column lateral response are also presented in this chapter. Equation Chapter 2 Section 1 2.2 2.2.1  Static Cyclic Test Column Databases Introduction to Static Cyclic Test Column Database Many available models estimating the column shear strength and drift capacities were  developed based on the static test data. Two important static test column databases will be introduced in this section, namely the PEER Structural Performance Database and Sezen (2002) column database. By comparing the evaluation of the available models with both cyclic test data and the shaking table test data, an improved understanding of the column behaviour, dynamic effects, as well as the performance of the column drift capacity models could be gained.  13  2.2.2  PEER Structural Performance Database The PEER Structural Performance Database is a comprehensive database compiled by  numerous researches for columns tested by static reversed-cyclic lateral load. It includes 253 rectangular reinforced concrete columns and 163 spiral columns. Column geometry, material properties and digitalized data of lateral force-deformation histories are provided for each column in the database. Herein, the PEER Structural Performance Database will be referred to as PEER database for brevity. A number of studies have been carried out using the column data in PEER database to investigate the column behaviour, evaluate the available structural models and develop new methods. For example Camarillo (2003) evaluated four existing shear-strength models using the PEER database. Since all the columns included in the compiled dynamic database had rectangular cross section, this research will focus on the rectangular columns included in the PEER database. The main properties of the rectangular columns included in the PEER database are summarized in Table 2.1. Table 2.1  Properties of PEER database  Parameters  Symbol  Aspect Ratio Concrete compressive strength  a/d  f c'  Long. Rein. Yield Stress  f yl  Long. Rein. Ratio  l  Trans. Rein. Yield Stress  f yt  Trans. Rein. Ratio Axial Load Ratio  Unit  Max  Min  Mean  STD  COV  3.7  0.5  3.58  1.46  41%  MPa  118  16  51.7  29.2  56%  MPa  587  318  429  75  18%  0.0400  0.0600  0.0240  0.0010  40%  1424  249  487  221  45%  t  0.0600  0.0002  0.0020  0.0010  61%  Pini f c' Ag  0.90  0.00  0.27  0.20  74%  MPa  14  2.2.3  Sezen (2002) Column Database Sezen (2002) compiled a database consisting of 50 flexure-shear-critical columns with  light transverse reinforcement ( t  0.007 ) that are subjected to static cyclic lateral load . Most of the columns included in Sezen (2002) database were large scale specimens representative of the columns commonly used in older reinforced concrete buildings, while around 20% of the columns were one-third scale. These columns were subjected to unidirectional cyclic lateral load and observed to suffer severe degradation of lateral strength and shear failure. The properties of the Sezen (2002) column database are given in Table 2.2. Table 2.2  Properties of Sezen (2002) database  Parameters  Symbol  Aspect Ratio  a/d  Unit  Max  Min  Mean  4  2  3  Concrete compressive strength  f  ' c  MPa  44.8  13.1  24.8  Long. Rein. Yield Stress  f yl  MPa  524.0  324.1  406.8  Long. Rein. Ratio  l  0.04  0.01  0.023  Trans. Rein. Yield Stress  f yt  648.1  317.2  427.5  Trans. Rein. Ratio  t  0.0065  0.0010  0.003  0.6  0.0  0.2  0.71  0.23  0.46  Initial Axial Load Ratio Maximum shear stress  MPa  ' c  Pini f Ag  v  f c'  MPa  Most of the column drift models at shear failure and axial-load failure studied in this thesis had been developed from or evaluated with the Sezen (2002) database.  15  2.3  Properties of Shaking Table Tests Column Database Similar to PEER database, a Shaking Table Tests Column Database is assembled to  provide the geometrical information, material properties and the test data for columns belonging to reinforced concrete frames subjected to shaking table tests. For brevity, The Shaking Table Tests Column Database will be referred to as “The Database” herein. The Database intends to include only relatively large scale columns in simple twodimensional frames. As a result, seven shaking table tests programs are included in The Database, three of which were conducted by scholars of UC Berkeley (Ghannoum, 2007, Elwood, 2003, Shin, 2007) and the remaining four were carried out at NCREE (National Center for Research on Earthquake Engineering) of Taiwan. (Yavari, 2011, Su, 2007, Kuo, 2008, Wu et al., 2006) The information of these seven test programs is summarized in Table 2.3 and details of each program will be introduced in section 2.4. Most shaking table tests are conducted up to collapse of the frame specimens. Table 2.3  Characteristics of specimens in seven shaking table test programs  Test programs 1  NCREE 2009  2  NCREE 2007  3 NCREE 2005 4 NCREE 2004 5 Ghannoum 2006 6  Shin 2005  7  Elwood 2002 Total  No. of No. of Specimens Stories 4 1 1 2 4 3 1 4 4 4 2 29  2 1 1 1 1 1 3 1 1 1 1 N.A.  Per Specimen No. of No. of No. of No. of Ductile |Non-Ductile Bays Columns Columns Columns 2 6 0 6 2 3 2 1 1 2 2 0 1 2 0 2 3 4 2 2 1 2 0 2 3 12 6 6 1 2 2 0 1 2 1 1 1 2 0 2 1 3 2 1 N.A. 97 34 63  16  Table 2.4 summarizes the information about the specimens and columns included in The Database. Only first-story columns out of the total 97 columns in these dynamic tests are included in The Database, since the first-storey columns were subjected to the greatest axial load and were considered to be the most vulnerable and critical columns in multiple-story moment-resisting frames Also the test data was more complete for first-story columns, while it was relatively difficult to measure the axial load and shear force for upper-story columns. Some first-story columns were excluded for various reasons. For example, the circular outside columns in Elwood 2002 Tests were neglected as The Database focused on square columns; the ductile columns in specimen S2 of NCREE 2007 were omitted because the test data file is damaged and unreliable. Among the 59 columns included in the Database, 36 columns are non-ductile columns with light transverse reinforcement ( t  Av / bs  0.002 ) and poor configurations that were typically used in older reinforced concrete structures and do not satisfy the current seismic code. These columns are expected to demonstrate limited inelastic response during earthquake shaking and experience pure shear or flexure-shear failures. There are 23 ductile columns with adequate transverse reinforcement (i.e. closely spaced transverse reinforcement with 135 degree hooks) in The Database and the column responses are expected to be dominated by flexural behavior. Appendix A presents the column geometry, material properties and reinforcement details for each column. Some columns were subjected to more than one test series and Appendix B provides the key information for each test series, such as initial axial load ratio and PGA.  17  Test data including time step, table acceleration, horizontal inter-story drift ratio, shear, axial load and column shortening/lengthening are also available for most of the dynamic tests series. Table 2.4  1 2 3 4 5 6 7  Information of the shaking table test programs  Per Specimen No. of No. of No. of No. of Test programs Ductile Non-Ductile Specimens Columns Columns Columns 4 3 3 0 NCREE 2009 1 3 2 1 NCREE 2007 2 2 0 2 2 4 2 2 NCREE 2005 2 2 0 2 NCREE 2004 1 4 2 2 Ghannoum 2006 4 2 2 0 4 2 1 1 Shin 2005 3 2 0 2 2 2 0 2 Elwood 2002 25 59 23 36 Total  Per Column No. of Test series * 2 2 1 2 1 4 1 1 1 1 88  * : “No. of test series” = the number of tests to which each column is subjected.  Key properties of the Database are summarized in Table 2.5. The “initial axial load  Pini ” refers to the column axial load obtained before column was subjected to the input motions and could be considered as the gravity load imposed on the column. Initial axial load could be measured by the load cell at the column base, however for most columns the measured gravity load was not reliable and thus was obtained by computed structural model (e.g. the OpenSees model) instead. The “Max Axial Load” Pmax refers to the maximum axial load that column was subjected to during the shaking table tests. Notice that due to the limit of shaking table capacities, most of the columns are subjected to axial load ratios smaller than 0.2.  18  Table 2.5  Properties of columns in The Database  Parameters  Symbol  Unit  Max  Min  Mean  Standard Deviation  COV  4.3  1.5  3.2  0.7  20.9%  (a) Non-Ductile Columns Aspect Ratio Concrete compressive strength  a/d  f c'  MPa  36.5  23.9  29.9  4.2  13.9%  Long. Rein. Yield Stress  f yl  MPa  479  386  442  24  5.5%  Long. Rein. Ratio  l  Asl Ag  0.0324 0.0143  0.0248  0.0046  18.5%  Trans. Rein. Yield Stress  f yt  689.7  385.9  550.6  78  14%  Trans. Rein. Ratio  t  Ast bs  0.0018 0.0007  0.0014  0.0004  25.4%  Initial Axial Load Ratio Max. Axial Load Ratio Normalized shear Stress  MPa  Pini Ag f  ' c  Pmax Ag f  ' c  f c'  v  MPa  0.36  0.03  0.13  0.07  53.4%  0.43  0.04  0.19  0.068  42.5%  0.59  0.17  0.35  0.09  25.0%  3.5  1.5  3.2  0.4  12.1%  (b) Ductile Columns Aspect Ratio Concrete compressive strength  a/d  f c'  MPa  35.8  24.6  28.5  4.2  14.6%  Long. Rein. Yield Stress  f yl  MPa  483  232  413  86  20.8%  Long. Rein. Ratio  l  Asl Ag  0.0209  0.0055  26.2%  Trans. Rein. Yield Stress  f yt  568  59  10.4%  Trans. Rein. Ratio  t  Ast bs  0.0087  0.0027  31.1%  Initial Axial Load Ratio  Pini Ag fc'  Max. Axial Load Ratio Normalized shear Stress  MPa  Pmax Ag f  v  0.0253 0.0109  f c'  662  475  0.0119 0.0030  ' c  MPa  0.20  0.05  0.12  0.05  43.7%  0.43  0.12  0.23  0.10  43.9%  0.44  0.20  0.34  0.07  19.8%  19  2.4 2.4.1  Information of Shaking Table Tests Programs Introduction This section provides important information about each shaking table test program  included in The Database. Hysteretic response of each non-ductile column is presented with the key points marked. The “1st yield” point is defined by following the method discussed in section 3.2.4 (TEST Method). The “Max Veff Veff” point is the point that column achieve maximum effective force and the “80% Max Veff” point is the point that column degrades to 80% of its maximum effective force. The “Axial-load failure” point could be found in section 5.1.3. The “Max achieved drift” point refers to the point that achieves maximum lateral column drift during the tests.  2.4.2  NCREE 2009 Tests NCREE 2009 Tests include four two-storey, two-bay planar frame specimens subjected  to unidirectional seismic loads scaled from the ground motion recorded during 1999 Chi-Chi earthquake. These tests are carried out by scholars from University of British Columbia in Canada and National Center for Research on Earthquake Engineering (NCREE) in Taiwan. The elevation view of test specimens is shown in Figure 2.3. Specimen names summarized the main characteristics of each setup as follows: HCFS : High axial load ratio Confined joints Flexure-Shear columns MCFS : Moderate axial load ratio Confined joints Flexure-Shear columns MUFS : Moderate axial load ratio Unconfined joints Flexure-Shear columns MUF : Moderate axial load ratio Unconfined joints Flexure columns  20  Specimens MCFS and HCFS were identical except that the columns of HCFS is designed to have higher column axial load by placing hydraulic jacks that added prestressing force (designed axial load ratio P Ag fc'  0.2 for middle column in HCFS and 0.1 for that in MCFS). Test data show that the behavior of frame specimen, damage patterns and column failure modes were closely related to the axial load that the columns are subjected to. The gap of inter-story drift ratio between shear failure and axial-load failure for middle in MCFS was larger than that of HCFS. After axial-load failure, column B1 in HCFS also experienced vertical shortening and lateral force degradation more rapidly than MCFS. While in HCFS damage was observed in both stories, MCFS suffered damage concentrated in the bottom of first-story columns. Specimen MUF consisting of three ductile columns and could be used to observe the beam-column joint behavior when subjected to lateral seismic loads. By comparing the test results of MUF and MUFS, one can observe the interactions among beam, column and beamcolumn joints and check the influence of column strength on joint demand. Those two specimens are included in this column database, since the data is helpful to investigate the possible effects of joint dealing on column stiffness and drift capacities. Yavari (2011) observed that there was apparent flexure cracking in first-story beams of HCFS, because the beam to column strength ratio was relatively low for specimen HCFS of which columns were subjected to high axial load ratio (designed axial load ratio equal to 0.24). Therefore the total inter-story drift was larger than the actual column deformation due to the additional deformation contributed by the beam flexibility and the rotation of beamcolumn joints.  21  Figure 2.1  Pictures for Specimen HCFS in NCREE 2009 Tests  Yavari (2011) suggest using the chord rotation instead to describe the column behaviour and defined the chord rotation as “the angle between the chord connecting the column ends and the tangent at the member end” (Page 76) as shown in Figure 2.2.  Figure 2.2  Definition of chord rotations; (a) chord rotations at the top and base of the column;  (b) comparison of chord rotation with the total rotation  Where Total = the total rotation and equal to the inter-story drift ratio;  t = chord rotation at the top of the column;  b = chord rotation at the base of the column;  j = joint rotation. 22  (a) Elevation view of specimens  (b) Column section for MCFS, HCFS and MUFS  (c) Column section for MUF Source: (Yavari, 2011)  Information of specimens of NCREE 2009 Tests  NCREE 2009 Tests -Test 1  9 8  Specimens  7  HCFS  T=0.29  6  MCFS  5  MUFS  4  MUF  3 2 1 0 0  0.2  0.4  0.6 0.8 1 1.2 Time (Sec)  Figure 2.4  1.4  NCREE 2009 Tests - Test 2  9  Pseudo-Acceleration S(a) (g)  Pseudo-Acceleration S(a) (g)  Figure 2.3  1.6  1.8  2  Specimen MCFS/HCFS Specimen MUFS/MUF HCFS MCFS MUFS MUF  8 7 6 5 4 3 2 1  T=0.36  0 0  T=0.46  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec)  2  Response spectra for NCREE 2009 Tests 23  (a) NCREE 2009-HCFS-A1  120  1st Yield Max Veff Max Achieved Drift Test Data 1 Test Data 2  90  1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  90 60  Shear (kN)  Shear (kN)  60  (b) NCREE 2009-HCFS-B1  120  30 0  30 0  -30  -30  -60  -60  -90  -90  -120  -120 -4  -2  0  2  4  -4  -2  0  Hori Drift (%)  (c) NCREE 2009-HCFS-C1  120  90  90  60  60  Shear (kN)  Shear (kN)  120  30 0  (d) NCREE 2009-MCFS-A1  0 -30  -60  -60  -90  -90  -120  -120 -4  -2  0  2  4  -4  -2  0  Hori Drift (%)  2  4  6  8  Hori Drift (%)  (e) NCREE 2009-MCFS-B1  (f) NCREE 2009-MCFS-C1  120  90  90  60  60  Lateral Force (kN)  Shear (kN)  4  30  -30  120  2  Hori Drift (%)  30 0 -30  30 0 -30  -60  -60  -90  -90  -120  -120 -4  -2  0  Hori Drift (%)  2  4  -4  -2  0  2  4  Hori Drift (%)  24  (g) NCREE 2009-MUFS-A1  100  100  (h) NCREE 2009-MUFS-B1  1st Yield  80  Max Veff  80  60  Max Achieved Drift  60  Test Data 1  40  Test Data 2  Shear (kN)  Shear (kN)  40 20 0  20 0  -20  -20  -40  -40  -60  -60  -80  -80 -100  -100 -4  UFS-B1  -2  0  Hori Drift (%)  2  -4  4  -2  0  2  4  Hori Drift (%)  (i) NCREE 2009-MUFS-C1  100 80 60  Shear (kN)  40 20 0 -20 -40 -60 -80 -100 2  4  -4  -2  0  2  4  Hori Drift (%)  Note: The column is named by the form “Test name-Specimen name-column name”. For example, the column “NCREE 2009-MUFS-B1” refers to the column B1 in specimen MUFS in NCREE 2009 Tests. Column B1 is the first-story column in axis B. Figure 2.5  Hysteretic response of non-ductile columns in NCREE 2009 Tests  25  2.4.3  NCREE 2007 Tests Su (2007) performed shaking table tests on four half-scale, single-story reinforced  concrete frames. Specimen S1, as shown in Figure 2.7 (a), had one non-ductile column in the middle and two ductile columns outside. Specimen S2, S3 and S4 were portal frames with two columns connected by a rigid beam, as illustrated in Figure 2.7 (b). The test data of specimen S2 consisting of two ductile columns are not available due to the damage of instruments during the tests. Specimen S3 and S4 included two non-ductile columns with column clear height equal to 750mm and 1000mm respectively. Since the ratio of column clear height to column section width ( L / D ) fell between 3 to 4, these columns are generally regarded as short columns or shear critical columns, which are expected to experience pure shear failures without flexural yielding. Figure 2.7 (c) illustrated the cross-section of the ductile and non-ductile columns. The dynamic tests were conducted up to collapse of the test frames. It is observed from the tests that shear-critical columns experienced in-cycle or cyclic shear strength degradation and exhibited negative slope in hysteretic response of lateral force soon after column shear failure. Backbone curves based on the key points estimated by various shear strength models and drift capacity models were also drawn for these short columns and it is found that most backbone curves could predict the column behavior fairly well, yet generally underestimate the column shear strength. From the hysteretic response of these shear critical columns shown in Figure 2.8 and Figure 2.9, it is observed that the column lateral force was relatively small in the beginning  26  and then a sudden increase in the lateral force occurred in one cycle followed by the column shear failure and axial-load failure shortly afterwards. The dynamic test data are compared with the static cyclic test results of full size columns with similar properties. Wu et al. (2007) found that column shear strength in the dynamic test was slightly higher than that of static cyclic test, which probably could be explained by the higher strain rate in dynamic tests, the energy dissipation within reversed cycles in cyclic test and the size effect of column specimens in dynamic test. NCREE 2007 Tests - S1 T=0.14  3  T=0.46  Specimen S1-Test 1 Specimen S1-Test 2 Test 1 Test 2  2  1  0 0  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec)  Figure 2.6  NCREE 2007 Tests - S3/S4  4  Pseudo-Acceleration S(a) (g)  Pseudo-Acceleration S(a) (g)  4  2  T=0.1265  3  Specimen S3 Specimen S4 S3 S4  2  1 T=0.1316  0 0  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec)  2  Response spectra for NCREE 2007 Tests  27  C1  C2  C3  (a) Experimental setup for specimen S1  C3  C1 (b) Experimental setup for specimen S2, S3 and S4  90-degree hook  135-degree hook 8 #4  4mm @150mm  4mm @ 50mm  20mm clear cover  16 #4  25  90-degree hook  (c) Cross-section for ductile columns  20mm clear cover  25  (d) Cross-section for non-ductile columns Source: Wu et al. (2007)  Figure 2.7  Information of specimens of NCREE 2007 Tests  28  (a) NCREE 2007-S1-C1  150  150  100  100  50 0  1st Yield  -50  Max Veff Axial Load Failure  -100  (b) NCREE 2007-S1-C3  200  Lateral Force (kN)  Lateral Force (kN)  200  50 0 -50 -100  Max Achieved Drift -150  -150  Test Data 1 Test Data 2  -200  -200 -4  0  4  -4  8  Hori Drift (%)  007-S4-C1  0  2  4  6  Hori Drift (%)  (c) NCREE 2007-S1-C2  200  1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  150 100  Lateral force (kN)  -2  50 0 -50  -100 -150 -200  1  2  -5.0  -2.5  %)  0.0  2.5  5.0  Hori Drift (%)  Note: The column is named by the form “Test name-Specimen name-column name”. For example, the column “NCREE 2007-S1-C2” refers to the column C2 in Specimen S1 of NCREE 2007 Tests. Figure 2.8  Hysteretic response of columns in Specimen S1 of NCREE 2007 Tests  29  (a) NCREE 2007-S3-C1  250  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  Lateral Force (kN)  150 100  200 150  Lateral Force (kN)  200  (b) NCREE 2007-S3-C1  250  50 0 -50  100 50 0 -50  -100  -100  -150  -150  -200  -200  -250  -250 -4  -2  0  2  4  -4  -2  Hori Drift (%)  (c) NCREE 2007-S4-C1  2  4  (d) NCREE 2007-S4-C1  200  150  150  100  100  Lateral Force (kN)  Lateral Force (kN)  200  0  Hori Drift (%)  50 0 -50  50 0 -50  -100  -100  -150  -150  -200  -200  -2  -1  0  Hori Drift (%)  1  2  -2  -1  0  1  2  Hori Drift (%)  Note: The column is named by the form “Test name-Specimen name-column name”. For example, the column “NCREE 2007-S4-C1” refers to the column C1 in Specimen S4 of NCREE 2007 Tests. Figure 2.9  Hysteretic response for columns in Specimen S3 and S4 of NCREE 2007 Tests  30  2.4.4  NCREE 2005 Tests In 2005, Kuo (2008) conducted shaking table tests on four one-third-scale, three-bay by  one-story reinforced concrete frames in NCREE. As shown in Figure 2.11, specimens P1 and P2 had four columns, of which two non-ductile columns with inadequate transverse reinforcement were on the north side and two ductile columns were on the south side. Specimen L and W had similar setup as P1 and P2, except that the former had lap-spliced transverse reinforcement in the lower part of non-ductile columns and the latter had additional wing walls on the columns. The specimens were subjected to seismic load scaled from ground motions recorded in 1999 Chi-Chi earthquake. Specimen P1 was tested in a preliminary experiment to investigate the structural properties (for example, the frame period), check the instrumental system and validate the design methods and specimen W was used to observe the influence of additional wing walls. Thus, only columns from specimens P2 and L are included in the Database. Dynamic test results of specimen P2 showed that the flexural-shear-critical columns in reinforced concrete frames had fairly large drift capacities when subjected to low axial load ratio, thanks to the axial load redistribution mechanism and column interactions within frames. The non-ductile columns wouldn‟t experience lateral strength degradation until interstory drift ratio reached 3.5% to 4% and shear failure usually occurred when inter-story drift ratio reached 4.5% to 5.5%. For non-ductile columns in specimen L, large vertical splitting cracks were observed and bond splitting failure occurred due to lap splices. Comparing the hysteretic response of non-ductile middle column C2 in specimen P2 and L shown in Figure 2.12, columns with lap-spliced reinforcement experienced larger lateral displacement. However, in the other  31  direction, the maximum column drift was obviously less than columns without lap splices in P2. It was noticed that before non-ductile columns C1 and C2 in specimen L reached their maximum drift in the positive direction, large cracks and severe column damage had already been observed. Thus the maximum drift measured during the tests was larger than the actual column drift capacity. NCREE 2005 Tests - Specimen P2 Specimen P2  T=0.22  5  P2  4 3 2 1 0 0  0.2  0.4  0.6  0.8 1 1.2 1.4 Time (Sec)  Figure 2.10  1.6  1.8  NCREE 2005 Tests - Specimen L  6  Pseudo-Acceleration S(a) (g)  Pseudo-Acceleration S(a) (g)  6  2  Specimen L  5  L  4 3 2 1 T=0.24 0 0  0.2  0.4  0.6  0.8 1 1.2 1.4 Time (Sec)  1.6  1.8  2  Response spectra for NCREE 2005 Tests  32  C1 C2  C4  C3  (a) Experimental setup for specimens P2 and L  (b) Cross-section for ductile columns (C3, C4)  (c) Cross-section for non-ductile columns (C1, C2) Source: Wu et al. (2009)  Figure 2.11  Information of specimens of NCREE 2005 Tests  33  (a) NCREE 2005-P2-C1  40  (b) NCREE 2005-P2-C2  40  20  Shear (kN)  Shear (kN)  20  0  0  -20 Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  -40  -20  -60  -40 -6  -4  -2  0  2  4  6  -6  -4  -2  Hori Drift (%)  (c) NCREE 2005-L-C1  2  4  6  (d) NCREE 2005-L-C2  60  40  40  20  20  Shear (kN)  Shear (kN)  60  0  Hori Drift (%)  0  0  -20  -20  -40  -40  -60  -60 -10  -5  0  5  Hori Drift (%)  10  -10  -5  0  5  10  Hori Drift (%)  Note: The column is named by the form “Test name-Specimen name-column name”. For example, the column “NCREE 2005-L-C1” refers to the column C1 in Specimen L of NCREE 2005 Tests. Figure 2.12  Hysteretic response of non-ductile columns in NCREE 2005 Tests  34  2.4.5  NCREE 2004 Tests In NCREE 2004 Tests, Wu et al. (2006) conducted shaking table tests by on two half-  scale, single-story portal frames, specimen S2 and S3. As shown in Figure 2.13, specimens S2 and S3 had two identical non-ductile columns connected by a relatively rigid concrete beam. Since axial load could not be redistributed in two-column system, the system failed simultaneously with the failure of individual columns. Both Temposonic LDTs and image recorders were installed to record the column displacement history during the tests. Tests showed that image processing technique could interpret data more effectively and accurately, especially when large deformation and structural collapse are expected. Therefore for NCREE 2004 tests, the test data of column horizontal drift provided in the Database were obtained from processing the image data. However, column shortening/lengthening data could not be read reliably through this method and was not provided in the Database. The identical specimens S2 and S3 are subjected to different input motions with nearfault characteristics and the test results showed that post-peak behavior of columns was strongly dependent on the selected input ground motions. The input motion of S2 was scaled from the ground motion acceleration recorded by TCU076ns station during 1999 Chi-Chi earthquake with PGA as 1.29g and during the test, more flexural deformation was observed. Specimen S3 was subjected to input motions recorded by TCU082ew in Chi-Chi earthquake at two intensity levels with PGA as 0.63g and 1.16g respectively. In S3, shear deformation mainly contributed to the column failure.  35  C1  C2  (a) Experimental setup for specimens S2 and S3 D4 ties @ 125 mm c.c.  90-degree hook  8 #3 Bars  200 mm  (b) Cross-section for columns in S2 and S3 Source: Wu et al. (2006) Figure 2.13  Information of Specimen S2 and S3 of NCREE 2004 tests  NCREE 2004 Tests  Pseudo-Acceleration S(a) (g)  1.5  Specimen S2 Specimen S3 S2 S3  T=0.37  1  0.5  T=0.43  0 0  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec) Figure 2.14 Response spectra for NCREE 2004 Tests  2  36  (a) NCREE 2004-S2-C1  60  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  20  40  Shear (kN)  Shear (kN)  40  (b) NCREE 2004-S2-C2  60  0  20  0  -20  -20  -40  -40  -60  -60 -10  -5  0  5  -10  -5  Hori Drift (%)  (c) NCREE 2004-S3-C1  5  (d) NCREE 2004-S3-C2  60  40  40  20  20  Shear (kN)  Shear (kN)  60  0  Hori Drift (%)  0  0  -20  -20  -40  -40  -60  -60 -6  -4  -2  0  Hori Drift (%)  2  4  6  -6  -4  -2  0  2  4  6  Hori Drift (%)  Note: The column is named by the form “Test name-Specimen name-column name”. For example, the column “NCREE 2004-S3-C1” refers to the column C1 in Specimen S3 of NCREE 2004 Tests. Figure 2.15  Hysteretic response of non-ductile columns in NCREE 2004 Tests  37  2.4.6  Ghannoum 2006 Tests In 2006, Ghannoum (2007) in UC Berkeley conducted a series of dynamic tests on a  three-story, three-bay reinforced concrete frame. The specimen was subjected to unidirectional seismic loading until partial collapse of the frame. The experimental setup of the specimen is shown in Figure 2.16. The specimen had two non-ductile columns in axis A and B on east side and two ductile columns in axis C and D that were introduced to control the failure mode of the frame. The dynamic tests data included in the Database were a half- yield level dynamic test and three high intensity dynamic tests (Dynamic Test 1, 2 and 3). It is observed from the tests that flexure-shear-critical column B1 (first story, column axis B) suffered significant loss of lateral resistant force and gravity load support in Dynamic Test 1. In Dynamic Test 2, Column B1 continued to lose lateral resistant force. Non-ductile column A1 experienced shear failure as well, but without too much axial load degradation. In Dynamic Test 3, column A1 gradually lost the axial load support and the whole frame was partially collapsed on the east side. A shear failure model was also proposed by relating the shear failure to the column rotation measured in region of plastic hinge at the ends of column (assuming the plastic hinge length equal to the column section width).  38  D3  C3  B3  A3  D2  C2  B2  A2  C1  B1  A1  D1  Figure 2.16  (a) Ghannoum 2006-A1  60  40  Shear (kN)  20  (b) Ghannoum 2006-B1  60  Test Data 1 Test Data 2 Test Data 3 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  40  Shear (kN)  Information of specimen of Ghannoum 2006 Tests  0  20  0  -20  -20  -40  -40  -60  -60 -6  -4  -2  0  Hori Drift (%)  2  4  6  -6  -4  -2  0  2  4  6  Hori Drift (%)  Note: The column is named by the form “Test name- column name”. For example, the column “Ghannoum 2006-A1” refers to the column A1 in Ghannoum 2006 Tests. Column A1 is the first story column in axis A. Figure 2.17  Hysteretic response of non-ductile columns in Ghannoum 2006 Tests 39  2.4.7  Shin 2005 Tests Shin (2006) conducted shaking table tests on twelve single-story, single-bay reinforced  concrete planar frames that consist of two one-third scale columns connected by a rigid steel beam. There are three kinds of experimental setups, of which specimens with Setup I has two ductile columns, specimens with Setup II includes one ductile on the east side and nonductile column on the west, and specimens with Setup III consists of two non-ductile columns. The general experimental setup is shown in Figure 2.18. Four identical specimens were built for each experimental setup and subjected to a pulse-like ground motion with large velocity pulses (Kobe Earthquake) or long-duration motions with low velocity (Chile Earthquake) shown in Figure 2.19. Experimental results show that frame behaviors was strongly affected by column properties, axial stress level and the input ground motion. Specimens with Setup I exhibited ductile response when subjected to Kobe and Chile input motions for both axial load cases and no shear strength degradation was observed. The non-ductile columns in Setup II specimens experienced shear failure and shear degradation with residual shear strength. Specimens with Setup III collapsed shortly after shear failure of shear-critical columns. Higher axial load ratio also resulted in more rapid shear failure of the columns and the collapse of specimens at lower column drifts. The columns subjected to Chile input motions tended to experience cyclic shear degradation (lateral resistance decreased from one cycle to the next without negative slope in lateral force-displacement relations), whereas specimens subjected to Kobe input motions more likely had in-cycle shear degradation (lateral resistant force degrades within one cycle, associated with negative slope in the lateral force-displacement relations).  40  EastC  WestC  (a) Elevation view of the specimens  (b) Cross-section for non-ductile columns  (c) Cross-section for ductile columns Source: Shin (2007)  Figure 2.18  Information of specimens of Shin 2005 Tests  41  GM Acceleration (g)  1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0  10  20  30  Time (Sec)  40  50  60  70  (a) Chile input ground motion 1.0 0.5 0.0 -0.5 -1.0 -1.5 0  5  10  15  20  Time (Sec)  25  30  (b) Kobe input ground motion Figure 2.19  Input ground motions of Shin 2005 Tests  Shin 2005 Tests  5  Pseudo-Acceleration S(a) (g)  GM Acceleration (g)  1.5  T1=0.210 T2=0.200  4 3  Setup I Setup II Setup III Chile Earthquake Kobe Earthquake  2 1 T3=0.207  0 0  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec)  Figure 2.20  2  Response spectra for Shin 2005 Tests  42  (a) Shin 2005-II-Test 1-Chile-0.1-WestC  (b) Shin 2005-II-Test 2-Kobe-0.1-WestC 50  50 Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  25  Shear (kN)  Shear (kN)  25  0  0  -25  -25  -50  -50 -8  -4  0  4  8  -8  -4  0  Hori Drift (%)  8  (d) Shin 2005-II-Test 8-Kobe-0.24-WestC  50  50  25  25  Shear (kN)  Shear (kN)  (c) Shin 2005-II-Test 7-Chile-0.24-WestC  0  0  -25  -25  -50  -50 -8  -4  0  4  8  -8  -4  0  Hori Drift (%)  4  8  Hori Drift (%)  (e) Shin 2005-III-Test 4-Chile-0.1-EastC  (f) Shin 2005-III-Test 4-Chile-0.1-WestC  50  50  25  25  Shear (kN)  Shear (kN)  4  Hori Drift (%)  0  -25  0  -25  -50  -50 -8  -4  0  Hori Drift (%)  4  -8  -4  0  4  Hori Drift (%)  43  (h) Shin 2005-III-Test 6-Kobe-0.1-WestC 50  25  25  Shear (kN)  Shear (kN)  (g) Shin 2005-III-Test 6-Kobe-0.1-EastC 50  0  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  -25  0  -25  -50  -50 -4  -2  0  2  4  6  8  -4  -2  0  Hori Drift (%) (i) Shin 2005-III-Test 10-Chile-0.24-EastC  4  6  8  (j) Shin 2005-III-Test 10-Chile-0.24-WestC  50  50  25  25  Shear (kN)  Shear (kN)  2  Hori Drift (%)  0  -25  0  -25  -50  -50 -4  -2  0  2  Hori Drift (%)  4  -4  -2  0  2  4  Hori Drift (%)  Note: The column is named by the form “Test name-Setup-Test series-Input ground motion-Designed axial load ratio- Column name”. For example, the column “Shin 2005-III-Test 10-Chile-0.24-EastC” refers to the eastern column in specimen with type II experimental setup and designed to experience axial load ratio equal to 0.24. It is tested in series Test 10, with Chile earthquake as the input ground motion. Figure 2.21  Hysteretic response of non-ductile columns in Shin 2005 Tests  44  2.4.8  Elwood 2002 Tests Elwood (2002) performed shaking table tests on two one-story, two-bay reinforced  concrete frames S1 and S2 to investigate the shear failure and axial load failure of columns that leads to the collapse of frame system during earthquake shaking. Specimens S1 and S2 have identical experimental setups consisting of a non-ductile column in the middle and two circular ductile outside columns, as shown in Figure 2.23. Specimen S2 was subjected to higher axial load ( P Ag f c'  0.2 ) compared with that of S1 ( P Ag f c'  0.1 ). Only center  columns of S1 and S2 are included in the Database since the outside columns are circular ductile columns and should be discussed separately. Results from specimens S1 and S2 showed that the behavior of frame structures and the column drift capacity are closely related to the axial stress to which the column was subjected. For specimen S1 with moderate axial load on the center column, the column sustained most of its initial axial load after the center column experienced shear failure. For specimen S2 that was subjected to higher axial load, shear failure occurred at a lower drift and was followed shortly by axial-load failure of the center column.  Elwood 2002 Tests  Pseudo-Acceleration S(a) (g)  4  3.5  T2 =0.21s  3  2% Damping 5% Damping Specimen S1 Specimen S2  2.5 2  1.5 1 T1=0.22s  0.5 0 0  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (Sec)  Figure 2.22  2  Response spectra for Elwood 2002 Tests 45  CenC  (a) Experimental setup of the specimens S1 and S2  (b) Cross-section for non-ductile columns  (c) Cross section for ductile columns Source: Elwood (2002)  Figure 2.23  (a) Elwood 2002-S1-CenC  100  60  80 60 40  Shear (kN)  40  (b) Elwood 2002-S2-CenC  100  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  80  Shear (kN)  Information of specimens of Elwood 2002 Tests  20 0  20 0  -20  -20  -40  -40  -60  -60  -80  -80  -100  -100 -8  -4  0  4  Hori Drift (%)  Figure 2.24  8  -8  -4  0  4  8  Hori Drift (%)  Hysteretic response of non-ductile columns in Elwood 2002 Tests  46  2.5 2.5.1  Test Results Interpretation P-Delta Correction The data series “shear” ( V ) for each column provided in the Database is the column  lateral force directly recorded by the force transducers or other instruments during the experiments. As the column lateral deformation  increased, axial load P would cause additional moment. Due to this so called P-delta effect, the actual column lateral force, referred to as effective lateral force ( Veff ), is larger than the recorded shear force. Throughout this research, effective lateral force Veff instead of recorded shear V will be adopted to construct lateral load-deformation relations and evaluate available models, unless specified otherwise. The column effective force could be derived by correcting the P-Delta effect as shown in Figure 2.25.  V  Veff  V   P L  L  Source: PEER Structural Performance Database User‟s Manual Figure 2.25  Illustration for correcting P-Delta effect  For most columns included in The Database, there is not much difference in lateral force before and after the P-Delta correction, since the axial load imposed on the columns in The Database is relatively low and the drift achieved by columns, especially non-ductile  47  columns were very limited. For example, the hysteretic response of recorded shear and effective lateral force for column Elwood 2002-S2-CenC is shown in Figure 2.26(a). (a) Elwood 2002-S1-CenC  (b) NCREE 2005-P2-C3 60  100  Effective Force Veff 80  Recorded Shear V 40  Lateral Force (kN)  60  Lateral Force (kN)  40 20 0 -20  20  0  -40 -20  -60  Effective Force Veff  -80  Recorded Shear V -40  -100 -6  -4  -2  0  2  Hori Drift (%)  Figure 2.26  4  6  8  -5  0  5  10  15  Hori Drift (%)  P-Delta correction for columns in The Database  However, some ductile columns in concrete frames consisting of multiple columns may experience very high axial load due to the redistribution of gravity load from the non-ductile columns that suffered shear failure. P-Delta effect may greatly influence the lateral force for those ductile columns. For example as shown in Figure 2.26 (b), before the P-Delta correction the ductile column C3 in specimen P2 of NCREE 2005 tests seemed to experience severe shear degradation after the non-ductile center column C2 failed; while after correcting the P-Delta effect, it is found that C3 actually exhibited very ductile behavior and probably went through strain hardening as shown by the increase of shear strength in the last few cycles. Herein unless otherwise specified, the data of column lateral force will adopt effective lateral force obtained by correcting P-delta effect of recorded shear.  48  2.5.2  Column Section Analysis Based on geometric information and material properties of column section, moment-  curvature analysis are carried out to obtain the moment-curvature relationships for columns and the key response measures such as first yield point and yield point. The first yield point is defined as the point that the outermost concrete compressive strain reaches 0.002 or the longitudinal reinforcement yields, whichever comes the first. The corresponding moment and later force are the first yield moment M FirstYield and first yield force FFirstYield . The yield point is defined as the point that outermost concrete compressive strain of column reaches 0.004. The moment and lateral force at the yield point are named as ideal moment M 0.004 and ideal force F0.004 , respectively. Simplified method of applying equivalent concrete block is used to calculate the moment-curvature relationship. It assumes that concrete stress could be represented by the equivalent concrete stress block using α1 and β1, parameters that can be calculated by Equation 2.1(NBCC, 2005).     11   t'    t'   c   c      1   4   t'      c   2  3     t  6  2  '     c    Equation 2.1  Where  t = concrete strain at the top of the column section;  c' = concrete compressive strain. With different top concrete compressive strain  t , the corresponding moment and curvature could be calculated using the force and moment equilibrium principle.  49  This simplified method could predict the yield point and the ideal moment fairly accurately, yet the data for first yield point is unreliable. For example for the center columns in specimen S1 and S2 of Elwood 2002 Tests shown in Table 2.6, the key points calculated by simplified method are compared with the results from a numerical software such as UC Fyber (Elwood. et al., 2002). Table 2.6  Results of the section analysis for columns in Elwood 2002 Tests  Section analysis (UC Fyber) MFirstYield φFirstYield  Simplified Method  comparison  M0.004  φ0.004 MFirstYield φFirstYield M0.004  φ0.004 MFirstYield φFirstYield M0.004  φ0.004  kN·m  1/km  kN·m  1/km  kN·m  1/km  kN·m  1/km  kN·m rads/km kN·m  1/km  S1  49.7  21  57.9  55.4  46.4  22.8  53.8  50.7  6.6%  8.6%  7.1%  8.5%  S2  59.2  24.7  61.4  45.4  45.4  19.9  56.4  40.7  23.3%  19.4%  8.1%  10.4%  2.6  The Number of Cycles at Key Points  2.6.1  Procedure to Count the Number of Cycles at Key Points For columns subjected to seismic load, the number of lateral-force cycles that a column  experiences is a good indicator of the energy dissipation and column damage. The number of cycles is counted through an automatic algorithm developed in MATLAB. Using column Shin 2005-II-Test 1-Chile-0.1-WestC (west column in specimen setup II with Chile input ground motion and design axial load ratio as 0.1 in Shin 2005 Tests) as an example, the procedure is described as follows:   Define the semi-cycles by selecting the semi-cycle points at which the lateral force changes direction, as shown in Figure 2.27. Record the position of each semi-cycle point in the data series. For example, the position of the first semi-cycle point (SemiCycleID=1) is 259 (SemiCyclePos=259). The algorithm starts to select the points when effective force exceeds 30% of peak lateral force for the first time. 50  50 Semi-Cycle points  25 0  -25 -50 6  8  Figure 2.27    10  12  14  16  Time (Sec)  18  20  22  Semi-cycle points for column Shin 2005-II-Test 1-Chile-0.1-WestC  Define the cycles. Three adjacent semi-cycle points are grouped as the key points of one cycle. For example, the first semi-cycle point is considered as the start point of cycle 1 (CycleID=1) and the third semi-cycle point is the end point of cycle 1; The third semi-cycle point is also the start point of cycle 2 (CycleID=2) and the fifth semicycle point is the end point of cycle 2, and so on. Group the data between the start point and end point of each cycle and record the maximum drift and lateral force in both positive and negative directions for each cycle, as shown in Figure 2.28. Shin 2005-II-Test 1-Chile-WestC 45 Envelope Test Data Max Eff Force Max Hori Drift  30  Lateral Force (kN)  Effective Force (kN)  Eff force  15  0  -15  -30  -45  -6  -4  -2  0  2  4  6  Hori Drift (%)  Figure 2.28  The maximum lateral force and drift points in each cycle  51    Find the position of the key points and count the number of the cycles that columns have been through. For example, it is found that the key point “80% Max Veff” occurred in cycle 23 (CycleID=23).    Criteria for counting the number of cycles. To exclude the effect of the small oscillation of lateral force and better reflect the energy dissipation and column damage, insignificant cycles whose maximum lateral force is less than 30% Max Veff are neglected. Thus columns have been through 15 cycles before reaching the key point “80% Max Veff”. Figure 2.29 plots the relations between total number of cycles that column went  through before the axial load failure with the maximum column axial load ratio measured during the tests. Results show that while the axial load varied greatly, the number of cycles did not change much and was not directly correlated with the axial load ratio. .  Total No. of Cycles  60  NCREE 2009-MCFS  50  NCREE 2009-HCFS  40  NCREE 2009-MUFS NCREE 2005-P2  30  NCREE 2005-L 20  Ghannoum 2006  10  Elwood 2002 Shin 2005-Chile-0.1  0 0.0  0.1  0.2  0.3  0.4  Shin 2005-Chile-0.24  Max Axial load Ratio (P max/fc'Ag) Figure 2.29  Number of cycles with column axial load ratio  52  2.6.2  Number of Cycles for Columns in Shin 2005 Tests Table 2.7 summarizes number of cycles that columns in Shin 2005 Tests went through  up to the first yield point (column achieved 75% of maximum lateral force), the point column reached maximum lateral force, shear failure point and the axial load failure point. Table 2.7  Number of cycles for columns in Shin 2005 Tests  (a) Columns in Shin 2005 Tests subjected to Chile earthquake load Test Shin 2005 Tests (Chile Earthquake) Ductile columns Non-ductile columns Column Type 47 54 55 51 58 59 Column ID 38 39 42 43 46 50 I I I I II II II III III II III III Setup EastC WestC EastC WestC EastC EastC WestC EastC WestC WestC EastC WestC Column Ground Motion Chile Chile Chile Chile Chile Chile Chile Chile Chile Chile Chile Chile Designed Axial 0.085 0.083 0.201 0.201 0.090 0.201 0.086 0.087 0.092 0.202 0.192 0.198 Load Ratio 7 7 3 4 13 2 13 15 18 9 9 8 0-Δyield  19 0-Max Veff 0-80%Max Veff N.A. 0-axial failure 81  27 N.A. 83  24 N.A. 86  6 N.A. 84  32 8 N.A. N.A. 55 70  15 81 30  19 25 34  21 22 36  16 16 16  10 16 29  9 12 15  (b) Columns in Shin 2005 Tests subjected to Kobe earthquake load Test Shin 2005 Tests (Kobe Earthquake) Column Type Ductile columns Non-ductile columns Column ID 40 41 44 45 48 52 49 53 56 57 I I I I II II II II III III Setup EastC WestC EastC WestC EastC EastC WestC WestC EastC WestC Column Ground Motion Kobe Kobe Kobe Kobe Kobe Kobe Kobe Kobe Kobe Kobe Initial axial 0.189 0.191 0.083 0.087 0.086 0.085 0.092 0.088 0.192 0.198 load ratio 1 1 1 1 1 1 1 1 1 1 0-Δyield  2 0-Max Veff 0-80%Max Veff N.A. 0-axial failure 16  2 N.A. 18  1 N.A. 16  2 N.A. 17  2 1 N.A. N.A. 17 16  2 2 2  2 2 2  2 2 2  2 2 2  From Table 2.7 (a) for columns subjected to the Chile earthquake load, it is observed that increasing axial load could greatly reduce the numbers of cycles that non-ductile columns went through for each key point, by comparing the results of column 51, 58 and 59 53  with column 47, 54 and 55; while for ductile columns, no significant effect of axial load could be found, except that increasing axial load may cause the column to achieve maximum effective force in fewer cycles. Results in Table 2.7 (b) show that columns subjected to Kobe earthquake achieved almost all the key points in first two cycles. The ductile columns went through several cycles, yet the number was significantly lower than the identical columns that were subjected to Chile earthquake. It is observed during the test with Kobe ground motion that the lateral force of the non-ductile columns was relatively small (less than 20% of maximum lateral force) initially. Then a sudden increase in lateral force occurred within one cycle and was followed by the column shear failure shortly. Figure 2.30 compared the number of cycles that non-ductile columns went through with column stiffness keff and column drift capacities at shear failure and axial load failure. Non-ductile columns in Shin 2005 Tests were tested in specimens with Setup II (had one ductile and one non-ductile column connected by a rigid beam) and III (consisting two non-ductile columns). One should be cautious when comparing the number of cycles for columns in different specimens, since the cycles might be affected by the structural properties of the specimens, such as frame period and stiffness. The data of non-ductile columns in specimens with setup II was highlighted in a black circle.  54  (a) Cycles up to first yield  No. of Cycles  20  Shin 2005-Chile-0.1  16  Shin 2005-Chile-0.24  12 8 4  0 0  1000  2000  3000  4000  Column Stiffness keff (kN/m)  (b) Cycles up to 80% Max Veff  30 25  25  No. of Cycles  No. of Cycles  (c) Cycle up to 80% Max Veff  30  20 15 10 5  20 15 10 5  0  0 0  2  4  6  0  8  0.5  40 35 30 25 20 15 10 5 0  (d) Cycles up to axial load failure  0  2  4  6  Measured drift at axial load failure  Figure 2.30  2.7  1  1.5  2  δMeas/δCalc (Shear failure)  No. of Cycles  No. of Cycles  Measured drift at 80% Max Veff  8  (e) Cycles up to axial load failure  40 35 30 25 20 15 10 5 0 0  0.5  1  1.5  2  δMeas/δCalc (Axial load failure)  Number of cycles for columns in Shin 2005 Tests  Experimental Envelope of Lateral Force-Drift Relations To assess the generalized load-deformation relationships recommended by ASCE/SEI  41 and better understand the column response, an experimental envelope is developed by picking the outermost data from the cycles in hysteretic response of lateral force for each column in The Database. The information about each cycle for columns in the Database could be found by following the procedures discussed in section 2.6.1. An example of the experimental envelope is shown in Figure 2.31 for the column Shin 2005-II-Test 1-Chile-0.1-  55  WestC, with (a) The experimental envelope with hysteretic response of column lateral force; (b) The normalized envelope curve. 45  1.2 Experimental  Normalized Envelope  Test Data  0.8  Normalized Force (V/Vmax)  30  Lateral Force (kN)  15  0  -15  0.4  0.0  -0.4  -30  -0.8  -45  -1.2 -6  -4  -2  0  2  4  6  Hori Drift (%)  (a) Experimental envelope with test data Figure 2.31  -4  -2  0  2  4  Ductility (μ=Δ/Δyield)  (b) Normalized envelope  Experimental envelope for column Shin 2005-II-Test 1-Chile-0.1-WestC  A normalized envelope is also constructed by dividing the horizontal drift with the yield drift  y (defined in section 3.2.5) and the lateral force by the maximum lateral load achieved in each direction. For example, the normalized envelope for Shin 2005-II-Test 1Chile-0.1-WestC is shown in Figure 2.31 (b). In the normalized experimental envelopes for columns, the x-coordinate is the ductility demand and the y-coordinate is the non-dimensional effective force ratio related to the maximum lateral force achieved in each direction during the tests. Thus comparing the normalized experimental envelope for different columns will provide a better understanding of column behavior during the dynamic tests. The observations from the experimental envelope are discussed in section 6.5.1.  56  Chapter 3: Column Effective Stiffness 3.1  Introduction In this chapter, two methods to interpret the measured column effective stiffness EI eff  are presented, namely the “PEER Method” and “TEST Method”. With the obtained measured effective stiffness, performances of the method proposed by ASCE/SEI 41 and the ThreeComponent model are evaluated and the effect of number of cycles is investigated. Equation Chapter 3 Section 1 3.2 3.2.1  Measured Column Effective Stiffness Introduction For majority structural models predicting the behavior of reinforced concrete frames  subjected to earthquake shaking, lateral force-deformation relations need to be established for column components. The column effective stiffness dominates the column behavior in the initial stage and casts a significant influence on the yield displacement and ductility demand of structures. In linear analysis, column effective stiffness controls the prediction of structural period and lateral load distribution. The assumptions made about the column effective stiffness significantly affect the accuracy of structural models. For most columns in the single-storey specimens, the top connecting beam and the column footings are relatively stiff and the column could be considered as fixed against rotation at both ends. Assuming a linear variation of curvature over the height of the column, the measured effective modulus of rigidity EI Meas , herein referred to as effective stiffness for simplicity, could be derived for columns with double curvature:  EI Meas   2a 3 2a 3 Vy 2a 2 Vy keff   3 3 y 3 y  Equation 3.1  57  Where Vy = lateral force at yielding point, which could be taken as the measured maximum effective force VMax or the ideal force F0.004 from section analysis discussed in section 2.5.2;  y = measured yield displacement and will be discussed further in following chapters;  y = measured yield drift; a = column shear span or equivalent cantilever length, which could be taken as half of the column clear height for fixed-fixed columns; keff = column lateral stiffness, the slope of the initial linear line from the lateral force-deformation relations, as shown in Figure 3.1. Ideally the area S1 should equal to S2. Shear V  Vmax S2  V FirstYield  (S1 ≈ S2) S1 keff Displacement Δ  Δy Figure 3.1  Column lateral stiffness keff from lateral load-deformation relations  The measured effective stiffness is often expressed as a fraction of the gross-section stiffness EI g . The modulus of elasticity E is estimated by E  4500 f c' (MPa) and I g is the moment of inertia (mm4). The ratio of EI eff / EI g will be referred to as effective stiffness ratio herein. In order to assess the performance of available effective stiffness models with the column data from the Database, consistent methods need to be defined to interpret the lateral  58  force-deformation data. There are two methods that have been widely used in past research, namely PEER method (Camerillo, 2003) and TEST method (Yavari, 2011). Both methods are applied to the test data from first test series if columns are subjected to more than one test, so there is no prior cracking from last dynamic test and could represent the initial behavior of the column.  3.2.2  Effect of Column Axial Load Since the columns included in The Database were subjected to seismic loads, the axial  load varied greatly in the positive and negative direction, especially for outside columns in multi-column frame subjected to overturning moment. For example, from the hysteretic response of lateral load and axial load shown in Figure 3.2 for outside non-ductile column C1 in specimen HCFS of NCREE 2009 Tests, it is observed that the column axial load varied from 150kN in negative direction to 420kN in positive direction and column C1 was much stiffer in positive direction due to the higher axial load. NCREE 2009-HCFS-C1 100  500 Test Data  75  Test Data 400  TEST  Axial Load (kN)  50  Lateral Force (kN)  1st Yield  PEER  25 0  -25  300  200  -50  100 -75  0  -100  -3  -2  -1  0  Hori Drift (%)  Figure 3.2  1  2  3  -3  -2  -1  0  1  2  3  Hori Drift (%)  Influence of varying column axial load on column effective stiffness 59  The greatly varying axial load, and the resulting distinct column stiffness, indicates that the measured column effective stiffness should be interpreted separately in positive and negative direction for each column in The Database, instead of averaging the results in both directions. As a result, there will be 59  2  118 data points with regard to the measured of column effective stiffness.  3.2.3  PEER Method Camarillo (2003) proposed to determine the column effective stiffness from lateral  force-deformation relations based on the effective force at first yield point and the yield point (obtained from the section analysis defined in section 2.5.2). This method is adopted to obtain the yield displacement and initial column stiffness for the columns included in PEER database and thus referred to here as “PEER” method. From the column lateral force-deformation response, the initial stiffness kPEER is determined by linear straight line for the origin to the point at which the lateral force equal to first yield force FFirstYield , as shown in Figure 3.3 (a). The line with slope equal to initial stiffness kPEER is extended to the point that column effective force reaches the ideal force  kPEER and the corresponding drift is considered yield drift  y  PEER . As shown in Figure 3.3(b), for columns whose strength did not exceed the ideal force (not at least 7% higher), for instance the shear critical columns that experienced shear failure before the flexural yielding, the above method was no longer valid and the first yield point is adjusted as the point at which the lateral force reaches 95% of maximum lateral force ( V  FFirstYield  0.95VMax ).  60  (a) if FMax  1.07 FFirstYield  (b) if FMax  1.07 FFirstYield Source: Camarillo (2003)  Figure 3.3  Interpretation of column effective stiffness by PEER method  From the initial stiffness kPEER interpreted by PEER method, the column effective stiffness ratio could be calculated by Equation 3.2  EI PEER  2a 3 EI g  kPEER / EI g 3  Equation 3.2  Results of column effective stiffness ratio EI PEER EI g interpreted by PEER method are plotted in Figure 3.4 and the data was grouped by the shaking table tests they belong to. 0.6  NCREE 2009 NCREE 2007 NCREE 2005 NCREE 2004 Ghannoum 2006 Elwood 2002 Shin 2005-I Shin 2005-II Shin 2005-III ASCE 41  0.5  EI PEER / EIg  0.4 0.3 0.2 0.1 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial axial load ratio (P ini /Agfc')  Figure 3.4  Column effective stiffness ratio of PEER method  61  Most columns included in The Database were subjected to low initial axial load ratio ( Pini Ag f c'  0.25 ) and the effective stiffness ratio EI PEER EI g for most columns were below 0.3. The column effective stiffness increases slightly with the increase of axial load ratio.  3.2.4  TEST Method Another method commonly used by researchers (e.g. Yavari, 2011) to interpret the  column effective stiffness is to define the initial stiffness as shown in Figure 3.5(a) based on 75% of the maximum lateral force (considering the point of 75%VMax as first yield point) and corresponding drift on the measured lateral force-deformation relation,. This method is named as “TEST Method” since it is based on the lateral force-deformation test data. Figure 3.5 shows an example of interoperating column stiffness by TEST Method. From initial stiffness kTEST , column effective stiffness ratio is calculated by Equation 3.3.  EITEST EI g   2a 3 kTEST EI g 3  Equation 3.3  Shin 2005-II-Test 1-Chile-0.1-WestC 50 Test Data PEER TEST  Lateral Force (kN)  25  Δ  Δ0.75MaxV ΔMaxV  0  -25  -50  -6  -4  -2  0  2  4  6  Hori Drift (%)  (a) Test Method  (b) Effective stiffness for column in Shin 2005 Tests Source: (a) is adapted from Yavari (2011)  Figure 3.5  Interpretation of column effective stiffness by TEST method 62  Similar to PEER method, Fig 3.6 shows the effective stiffness ratio derived by TEST method. Most columns included in the Database were subjected to low initial axial load ratio ( Pini Ag f c'  0.4 ) and the effective stiffness ratio EITEST EI g for most columns were below 0.3. The column effective stiffness increases slightly with the increase of axial load ratio. 0.6  NCREE 2009 NCREE 2007 NCREE 2005 NCREE 2004 Ghannoum 2006 Elwood 2002 Shin 2005-I Shin 2005-II Shin 2005-III ASCE 41  EITEST / EIg  0.5  0.4 0.3 0.2 0.1 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial axial load ratio (P ini /Agfc')  Figure 3.6  3.2.5  Column effective stiffness ratio from TEST method  Measured Column Effective Stiffness The effective stiffness interpreted by PEER method and TEST method is compared in  Figure 3.7. Results show that for most columns, there is not much difference in the effective stiffness from these two methods. The effective stiffness EITEST from TEST method seems to represent the mean value of the EI PEER from PEER method. 2  NCREE 2009  EI TEST / EIPEER  NCREE 2007  1.5  NCREE 2005 NCREE 2004  Ghannoum 2006  1  Elwood 2002  Shin 2005-I  0.5  Shin 2005-II  Shin 2005-III Mean  0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial axial load ratio (Pini /Agfc')  Figure 3.7  Comparison of column effective stiffness from PEER method and TEST method  63  The effective stiffness interpreted by both methods for the column C1 in NCREE 2009 Tests and Elwood 2002 Tests is very similar as shown in Figure 3.8. NCREE 2009-MCFS-C1  Elwood 2002-S1-CenC  100  100 Test Data  Test Data  80  PEER  80  60  TEST  60  TEST  40  Lateral Force (kN)  Lateral Force (kN)  40  PEER  20 0 -20  20 0 -20  -40  -40  -60  -60  -80  -80  -100  -100 -4  -2  0  Hori Drift (%)  Figure 3.8  2  4  -4  -2  0  2  4  6  Hori Drift (%)  Examples of column effective stiffness from PEER method and TEST method  However, for some columns there are large discrepancies in the results of effective stiffness from these two methods, especially in the case for columns had maximum lateral force VMax less than the first yield force FFirstYield obtained from the moment curvature analysis. According to PEER method, the first yield point should be taken as the point that achieved 95% of VMax instead. For example, the maximum lateral force of the shear critical column NCREE 2007-S4-C3 in Figure 3.9 (a) was less than the calculated first yield force and due to the effect that the lateral force increased suddenly in the failure cycle, the first yield point could only be selected in the failure cycle and was quite close to the point with maximum force in negative direction, resulted in a much lower stiffness kPEER than that from TEST method.  64  Similarly, the column NCREE 2005-L-C4 in Figure 3.9(b) had relatively low lateral force and the maximum lateral force VMax in first few cycles happened to be slightly lower than 0.95VMax . Thus according to PEER method, the first yield point had to be selected among the data after the column had clearly yielded, resulting in an erroneous interpretation of column initial response. NCREE 2007-S4-C3  NCREE 2005-L-C4  200  40  150  Test Data  Test Data  PEER  PEER  TEST 100  TEST 20  ThreeComp  ThreeComp ASCE  Lateral Force (kN)  Lateral Force (kN)  ASCE  50 0 -50 -100  0  -20  -150 -200  -40 -1.5  -1.0  -0.5  0.0  Hori Drift (%)  Figure 3.9  0.5  1.0  1.5  -6  -3  0  3  6  Hori Drift (%)  Large discrepancies in effective stiffness from PEER method and TEST method  By interpolating the lateral stiffness based on the first yield point from section analysis, the PEER method seems to provide the estimation of column effective stiffness with certain physical meaning. However, the PEER Method can be unreliable and erroneous due to several reasons. Firstly, either initial axial load or maximum axial load was used to carry out the moment curvature analysis for simplicity, while some of the columns were actually subjected to greatly varying axial load during the dynamic tests. Thus it was difficult to get the “true” first yield point of columns and the corresponding lateral force. Secondly, for most columns that had maximum lateral force less than the idea force F0.004 , the column usually achieved 0.95Vmax at rather later stage and even quite closed to the failure point. The effective 65  stiffness based on the selected first yield point failed to represent the initial behavior of the columns. On the other hand, the TEST method generally represented the column behavior fairly well, since the initial response of columns up to 75%VMax was relatively linear for most columns. This method could also be considered as a simplified way to achieve equal energy up to the maximum lateral effective force in most cases ( S1  S2 as shown in Figure 3.1). In conclusion, the measured effective stiffness for columns included in The Database will be obtained by following TEST method. Notice that the column stiffness in positive and negative directions is discussed separately in this chapter, while in the following chapters, the measured stiffness and yield drift for columns is obtained by averaging the results in both directions, i.e. EI eff   EITEST  Post  EITEST  Nega  2 , keff   kTEST  Post  kTEST  Nega  2 and   y   y TEST  Post   y TEST  Nega  2 .  3.3  Effect of Key Parameters on Measured Column Effective Stiffness The measured column effective stiffness ratio EI Meas EI g is plotted with key  parameters in Figure 3.10 for the columns included in The Database. The key parameters considered include: (a) Initial axial load ratio Pini Ag f c' ; (b) Longitudinal reinforcement ratio l ; (c) Aspect ratio a / D ; (d) Ratio of longitudinal reinforcement diameter to the depth of column section db / D ; (e) Concrete compressive strength on test day f c' ; (f) Ratio of yield stress of longitudinal reinforcement to concrete compressive strength f yl f c' ;  66  In this chapter, the test data is grouped by the column failure type classified by ASCE/SEI 41(discussed in section 6.3). The data series “Shear” in orange triangular marker represents the shear critical columns in The Database (corresponding to the Condition iii in ASCE/SEI 41). The data series “Flex-Shear” (corresponding to Condition ii) in red diamond marker and data series “Flexure” (corresponding to Condition i) in green round marker refer to the flexure-shear critical columns and flexure critical columns respectively. These two data series are from columns that are connected with a rigid beam and hence lateral drifts can be attributed entirely to column deformations. The data series “Flex-Shear (Beam)” in purple diamond marker and data series “Flexure (Beam)” in blue round marker represent the flexure-shear critical columns and flexure critical columns that were connected with flexible beam that may introduce additional flexibility to the total response, and hence, are expected to be softer than that estimated based on column stiffness alone. Results in Figure 3.10 show that effective stiffness ratio tends to increase consistently with axial load ratio. Increasing ratio of yield strength of longitudinal reinforcement to concrete compressive strength f yl f c' also results in a slight increase in column effective stiffness ratio. Other parameters, such as longitudinal reinforcement ratio l and concrete compressive strength f c' , are not strongly correlated with column effective stiffness ratio.  67  4  Shear Flex-Shear Flex-Shear (Beam) Flexure Flexure (Beam)  0.3  Long. Rein. Ratio ρl (%)  Initial Axial Load Ratio P/Agfc' (MPa)  0.4  0.2  0.1  0  2  1  0 0  0.1  0.2  0.3  EI Meas /EI g  0.4  0.5  0  4 3 2 1  0.2  0.3  EI Meas /EI g  0.4  0.5  0.4  0.5  0.4  0.5  0.08 0.06 0.04 0.02  0  0 0  0.1  0.2  0.3  EI Meas /EI g  0.4  0.5  0  0.1  0.2  0.3  EI Meas /EI g  0.4 Long. strength/Concrete Strength fyl/fc'  40  Concrete strength fc' (MPa)  0.1  0.1  Long. diameter/Section Depth db /D  5  Aspect Ratio a/D  3  35  30  25  20  0.3  0.2  0.1  0 0  0.1  0.2  0.3  EI Meas /EI g Figure 3.10  0.4  0.5  0  0.1  0.2  0.3  EI Meas /EI g  The measured effective stiffness with key parameters  68  3.4  Evaluation of ASCE/SEI 41 Procedure The current seismic rehabilitation standard ASCE/SEI 41 recommends to estimate the  effective stiffness ratio based on the column axial load level. Similar procedures are adopted in the other code and standards, such as American Concrete code, ACI 318-08 and commentary to the New Zealand concrete code, NZS 3101-06(NZS 2006). According to ASCE/SEI 41 procedure shown in Figure 3.11, the effective stiffness equals to 0.3EI g when columns are subjected to axial load ratio less than 0.1, while for columns have axial load ratio higher than 0.5, the effective stiffness shall be taken as 0.7 EI g . Linear interpolation of the effective stiffness ratio is required for columns subjected to axial load ratio in between ( 0.1  P Ag f c'  0.7 ).  Effective Stiffness Ratio (EIeff/EIg)  0.8 0.7 0.6 0.5 0.4 0.3  0.2 0.1 0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  Axial Load Ratio Figure 3.11  Column effective stiffness ratio estimated by ASCE/SEI 41 procedure  Figure 3.12 plots the ratio of measured column effective stiffness EI Meas to calculated results EI ASCE from ASCE/SEI 41 procedures.  69  2  Shear Flex-Shear Flex-Shear (Beam) Flexure Flexure (Beam) Mean  EIMeas/EIASCE  1.5  1  0.5  0  0  0.1  0.2  0.3  0.4  Initial Axial Load Ratio Pini /Agfc'  Figure 3.12  Evaluation of column effective stiffness estimated by ASCE/SEI 41 procedure  It is observed in Figure 3.12 that the majority data of EI Meas EI ASCE falls below unity representing the case that estimated effective stiffness matches exactly with the measured column stiffness. ASCE/SEI 41 procedure generally overestimates the column effective stiffness for all types of columns.  3.5  Evaluation of Three-component Model Elwood and Eberhard (2009) proposed a three-component model to estimate the  column yield deformation by explicitly taking into account the deformation due to the flexure behavior  flex , shear response  shear and bar-slip effect  slip , by Equation 3.4.  y ThreeComp   flex   shear   slip 2a 2 y 3 M  0.004 Av Geff   flex   shear   slip   Equation 3.4  adb f s y 8u  Where  y =the curvature at the yield point (the extreme concrete compressive strain reaches 0.004) obtained from the moment curvature analysis.  70  Av = effective shear area, usually taken as 5/6 of the gross-section area for rectangular 1 1 E columns; Geff = G   c , the effective shear modulus is reduced to be half of the elastic 2 2 2.4  shear modulus G due to crack in concrete blocks.  d b = nominal diameter of longitudinal reinforcement. f s =stress in the tension rebar at the yield point of the column, usually taken as the yield strength of longitudinal reinforcement f yl . The bond stress u for elastic analysis is ranged from u  0.5 f c ' to  u  1.0 f c ' (MPa) and here taken as 0.8 f c ' (MPa). Figure 3.13 plots the ratio of measured contributions from each of the deformation components (  flex ,  shear and  slip ) for the columns included in The Database, for (a) flexureshear critical columns, (b) flexure critical columns and (c) shear critical columns. It is observed from Figure 3.13 that for most columns, around half of the column deformations up to the yielding point are attributed to the column flexure behavior. The deformations due to bar slip effect could account for more than 50% of total yield displacement. While the shear response caused less than 5% deformations for shear critical columns, the contribution of shear component was negligible for flexure-shear critical columns and flexure critical columns.  71  Flexure  Deformation Component (%)  100  Shear  Bar slip  80 60  40 20  0 13 15 20 21 24 25 28 29 30 31 47 49 51 53 54 55 56 57 58 59 1  2  3  4  5  6  7  8  9 32 33 36 37  Column ID  (a) Flexure-shear critical columns Flexure  Deformation Component (%)  100  Shear  Bar slip  80 60 40  20 0 22 23 26 27 38 39 40 41 42 43 44 45 46 48 50 52 10 11 12 34 35  Column ID  (b) Flexure critical columns Flexure  Deformation Component (%)  100  Shear  Bar slip  80 60 40  20 0 14  16  17  18  19  Column ID  (c) Shear critical columns Figure 3.13  Deformation components from three-component model  72  With the column yield drift calculated by three-component model, the column effective stiffness could be estimated by Equation 3.5.  EIThreeComp  2 F0.004 a3  3 y _ ThreeComp  Equation 3.5  The ratio of measured column effective stiffness EI Meas to calculated effective stiffness EIThreeComp from Three-component model is plotted in Figure 3.14.  EIMeas/EIThreeComp  2 1.5 1  Shear Flex-Shear Flex-Shear (Beam) Flexure Flexure (Beam) Mean  0.5 0  0  0.1  0.2  0.3  0.4  Initial Axial Load Ratio Pini /Agfc'  Figure 3.14  Evaluation of column effective stiffness estimated by three-component model  In order to facilitate engineers to estimate the column effective stiffness in preliminary design stage, Elwood and Eberhard (2009) recommended a simplified model to calculate the effective stiffness ratio based on the Three-component model.  0.2   EI SimpThreeComp EI g  0.45  2.5P / Ag f c'   1.0  db  D  1  110      D  a   Equation 3.6  The calculated effective stiffness EI SimpThreeComp and the results of from three-component model, EIThreeComp , are compared in Figure 3.16. Results show that other than shear-critical columns in the database that had small aspect ratio a / d , effective stiffness EI Simp ThreeComp tends to be slightly larger than EIThreeComp from three-component model for most columns.  73  Since there is no significant difference between the calculated EIThreeComp and EI SimpThreeComp , it is appropriate to estimate column effective stiffness by applying the simplified three-component model in preliminary analysis.  EIMeas/EISimp-ThreeComp  2  Shear Flex-Shear Flex-Shear (Beam) Flexure Flexure (Beam) Mean  1.5 1  0.5 0  0  Figure 3.15  0.1  0.2 Initial Axial Load Ratio Pini /Agfc'  0.3  0.4  Evaluation of column effective stiffness estimated by simplified three-component model  EISimp-ThreeComp/EIThreeComp  2 1.5 1 Shear Flex-Shear Flex-Shear (Beam) Flexure (Beam) Flexure  0.5 0  0  0.1  0.2  0.3  0.4  Initial Axial Load Ratio Pini /Agfc'  Figure 3.16  Comparison of effective stiffness from three-component model and simplified model  74  3.6  Comparison with the Static Cyclic Tests Data While the ASCE/SEI 41 procedure tends to predict stiffer column initial response than  that observed, both three-component model and its simplified version could either overestimate or underestimate the column effective stiffness. The calculated effective stiffness EIThreeComp (discussed in section 3.5) is more likely to represent the mean value of the measured effective stiffness. Table 3.1 summarizes the statistical results of the effective stiffness models for columns included in The Database. Table 3.1  Statistical results of column effective stiffness ratio for columns in The Database  Max  Mean  Median  Standard Deviation  COV (%)  (a) Flexure-critical columns EITEST EI ASCE 0.17  0.98  0.57  0.57  0.15  26.9  EITEST EIThreeComp  0.36  1.74  0.97  0.95  0.22  22.6  EITEST EI SimpThreeComp  0.30  1.24  0.74  0.73  0.20  27.1  (b) Flexure -shear-critical columns EITEST EI ASCE 0.28 1.21  0.70  0.66  0.17  24.9  EITEST EIThreeComp  0.45  1.79  1.15  1.10  0.26  22.4  EITEST EI SimpThreeComp  0.45  1.34  0.89  0.87  0.20  22.1  (c) Shear-critical columns EITEST EI ASCE 0.65  1.27  0.80  0.74  0.18  22.6  EITEST EIThreeComp  0.97  2.19  1.27  1.18  0.35  27.4  EITEST EI SimpThreeComp  1.25  2.66  1.59  1.44  0.42  26.1  Min  Elwood and Eberhard (2009) selected 221 rectangular columns from the PEER database and use the static cyclic test data for those columns to evaluate the aforementioned effective stiffness models. The main properties of the dataset compiled by Elwood and Eberhard (2009) are summarized in Table 3.2.  75  Table 3.2  Properties of the dataset in Elwood and Eberhard (2009)  Parameters  Symbol  Aspect Ratio  a/d  Concrete compressive strength  f c'  Long. Rein. Yield Stress  f yl  Unit  Min  Max  Mean  Median  1.5  7.6  3.6  3.2  MPa  21.0  118.0  52.3  36.5  MPa  318  587  456  453  Long. Rein. Ratio Ratio of depth of section to long. rein. diameter  l  0.010  0.060  0.024  0.021  D / db  12  32  18  16  Axial Load Ratio  P f c' Ag  0.00  0.63  0.23  0.2  f c'  0.09  0.71  0.32  0.3  Normalized Shear stress  v  Table 3.3 compared the evaluation of the effective stiffness models by the static cyclic test data and dynamic test data. Table 3.3  Evaluation of column effective stiffness models with dynamic and static cyclic test data  Min (a) Columns in The Database EITEST EI ASCE 0.17  Max  Mean  Median  COV (%)  1.27  0.66  0.63  27.5  EITEST EIThreeComp  0.36  2.19  1.09  1.07  24.8  EITEST EI SimpThreeComp  0.30  2.66  0.89  0.83  35.4  (b) Columns in the static cyclic test dataset EITEST EI ASCE 0.27 1.95 0.82 EITEST EIThreeComp 0.45 1.84 0.97  0.83  36.0  0.92  26.6  EITEST EI SimpThreeComp  0.94  25.5  0.46  1.63  0.95  76  3.7  Effects of Number of Cycles Figure 3.17 plots the number of cycles that columns go through before achieving 75%  of maximum lateral effective force (the first yield point specified in TEST method discussed in section 3.2.4) with the measured column effective stiffness keff for some non-ductile columns included in The Database.  Cycles up to first yield point  25  NCREE 2009-MCFS NCREE 2009-HCFS  No. of Cycles  20  NCREE 2009-MUFS NCREE 2005-P2  15  NCREE 2005-L  10  Ghannoum 2006  5  Elwood 2002  0  Shin 2005-Chile-0.1 0  1000  2000  3000  4000  5000  6000  7000  Shin 2005-Chile-0.24  Effective Stiffness K eff (kN/m) (a) Number of cycles with effective stiffness keff  Cycles up to first yield point  25  NCREE 2009-MCFS  No. of Cycles  NCREE 2009-HCFS 20  NCREE 2009-MUFS  15  NCREE 2005-P2 NCREE 2005-L  10  Ghannoum 2006 Elwood 2002  5  Shin 2005-Chile-0.1  0 0.0  0.5  1.0  1.5  2.0  Shin 2005-Chile-0.24  Effective Stiffness KTEST / KThreeComp (b) Number of cycles with normalized effective stiffness keff Figure 3.17  Number of cycles with column effective stiffness  keff  It is observed from Figure 3.17 (a) that keff seems to increase slightly as number of the cycles increase. However it is not appropriate to compare the results for all the columns  77  together, since the lateral stiffness keff determined by the section properties and column length varied greatly for columns in different test programs. If grouping the columns into different clusters as shown in Figure 3.17 (a) and only comparing the columns with similar properties experimental setups, the lateral stiffness keff increased slightly as columns went through less number of cycles for most columns. This  negative trend between the lateral stiffness and number of cycles is understandable, since less reversed cycles in the initial stage indicates less flexural cracks and column damage, making the columns stiffer. The negative correlation between the number of cycles and lateral column stiffness could be observed more obviously in Figure 3.17 (b), which plots the relations between the normalized effective stiffness ( kMeas kThreeComp ) and number of cycles up to the first yield point. Similar trend could be found in the relations between the number of cycles and effective stiffness ratio EI eff EI g as shown in Figure 3.18.  78  Cycles up to first yield point  25  NCREE 2009-MCFS NCREE 2009-HCFS  No. of Cycles  20  NCREE 2009-MUFS NCREE 2005-P2  15  NCREE 2005-L  10  Ghannoum 2006  5  Elwood 2002  0  Shin 2005-Chile-0.1 0.0  0.1  0.2  0.3  0.4  0.5  Shin 2005-Chile-0.24  Effective Stiffness Ratio (EIeff/EIg) (a) Number of cycles with effective stiffness ratio EI eff EI g  Cycles up to first yield point  25  NCREE 2009-MCFS  No. of Cycles  NCREE 2009-HCFS 20  NCREE 2009-MUFS  15  NCREE 2005-P2 NCREE 2005-L  10  Ghannoum 2006 Elwood 2002  5  Shin 2005-Chile-0.1  0 0.0  0.5  1.0  1.5  2.0  Shin 2005-Chile-0.24  Effective Stiffness EITEST / EIThreeComp (a) Number of cycles with normalized effective stiffness ratio EI eff EI g Figure 3.18  Number of cycles with effective stiffness ratio  EI eff EI g  79  Chapter 4: Column Drift at Shear Failure 4.1  Definition of Column Drift at Shear Failure Non-ductile columns with light transverse reinforcement are vulnerable to shear failure.  As the shear damaged plane develops, the column could lose support for gravity load, leading to the collapse of the whole structure. Several models have been developed to estimate the column drift capacity at shear failure. In order to evaluate the performance of available models with The Database, measured drift at shear failure for each column needs to be interpreted in a consistent way. Based on the observations of column damage during the experiment and the lateral force-deformation (drift) data, there are generally three methods to define the column drift at shear failure:Equation Chapter 4 Section 1   Drift at peak shear force. This method suggests that the shear failure occurred when column achieves the maximum shear force recorded during the experiments. Drift at peak shear force could be easily obtained from the data of shear force and horizontal inter-story drift and will be referred to as  s MaxV (drift at Max V ).    Drift at peak effective force. According to this definition, column experienced shear failure when it achieved maximum lateral effective force. Effective lateral force could be obtained by correcting the P-delta effect of recorded shear-deformation relations. Drift at peak effective force is usually the same as or slightly higher than the drift at Max Shear and will be referred to as  s MaxVeff (drift at Max Veff ).    Drift at 80% peak effective force. This method defines that shear failure occurs when column losses 20% of its peak lateral force. The corresponding drift will be referred to as  s 80% MaxVeff (drift at 80% Max Veff ). Most drift capacity models are developed  80  to provide estimations of the column drift at 80% Max Veff. Compared with the column drift  s MaxVeff that was very sensitive to the experimental conditions (input ground motion, column axial load ratio etc.),  s 80% MaxVeff was more stable and better reflected the strength degradation as well as the drift capacity of columns. The drift obtained from this method is usually much larger than the drift from previous definitions, especially for flexure-critical and flexure-shear-critical columns.  Some columns may experience axial-load failure before loss of 20% peak lateral strength, for example the column C1 in NCREE 2009 Tests shown in Figure 4.1. In this case the  s 80% MaxVeff will instead be taken as the maximum column drift achieved before axialload failure.  NCREE 2009-MCFS-C1  120  60 30  0.2  Vert Drift (%)  90  Lateral Load (kN)  0.3  1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  0 -30  0.1  0.0  -60 -0.1 -90 -0.2  -120 -4  -2  0  Hori Drift (%)  2  4  -4  -2  0  2  4  Hori Drift (%)  Note: The definition of the axial-load failure could be found in section 5.1.4 Figure 4.1  Axial-load failure occurred before 20% loss of its lateral load  81  4.2  Effect of key Parameters on Measured Drift at Shear Failure The measured column drift  s MaxVeff and  s 80% MaxVeff for non-ductile columns are  plotted with key parameters in Figure 4.2 and Figure 4.3 respectively. The column plastic drift at 80% Max Veff with key parameters are plotted in Figure 4.4. The column plastic drift could be obtained by subtracting the yield drift measured during the tests  y (obtained from TEST method in section 3.2.5) from the  s 80% MaxVeff . The key parameters considered include: (a) maximum normalized shear stress v  f c ' ; (b) maximum  axial load ratio Pmax Ag f c ' , where Pmax is the initial axial load that column is subjected to, Ag is the area of column gross section; (c) transverse reinforcement ratio t ; (d) transverse  reinforcement indicator t f yt / f c' ; (e) ratio of transverse reinforcement spacing to depth s / d , where s is spacing of transverse reinforcement, d is effective depth of column cross-section; (f) aspect ratio a / d , where a is column shear span and could be taken as half of column clear height L . The measured column drift  s MaxVeff and  s 80% MaxVeff are plotted with key dynamic parameters in Figure 4.5. The dynamic parameters considered include: (a) the number of cycles that the column went through and the method to count the cycles could be found in section 2.6.1; (b) the relative strength factor R  S (a) W Vy , where Sa (T1 ) = pseudo acceleration for the fundamental structural period T1 calculated from the response spectrum (g), the data of the fundamental period T1 are available for most test frames in The Database; W = the weight of structure and could be calculated by adding the initial axial load of all columns in the  82  frame and V y = the yield strength of the specimen and could be calculated by adding the plastic shear demand of all columns. (c) the bracket duration of input ground motions tb (sec), defined as time between when the input motion exceeds the certain acceleration threshold for the first and last time. The acceleration threshold is selected as 0.1g in this research. The parameter bracket duration tb represents one of the main characteristics of the input motions. The number of cycles and relative strength factor R describes the interactions of the structures and input motions, which are unique in dynamic tests. In this chapter, the test data is grouped by the column failure type classified by ASCE/SEI 41 (discussed in section 6.3). The data series “Shear” (corresponding to the Condition iii in ASCE/SEI 41) in green triangular marker represents the shear critical columns included in The Database. The data series “Flex-Shear” (corresponding to the Condition ii in ASCE/SEI 41) in red diamond marker represents the flexure-shear critical columns that were connected to a rigid beam on the upper end. The data series “Flex-Shear (Beam)” in purple diamond marker represents the flexure-shear-critical columns in the frame connected with flexure beam on the upper end and the data of measured drift may be higher than the actual column drift capacity due to the additional flexibility contributed by the beam. The “Flex-Shear (Lower)” in hollow red diamond marker represents the flexure-shear critical columns that did not fail during the tests and the drift data could be considered as the lower bound of the column drift capacity. The “Flex-Shear (Flex)” in blue diamond marker represents the columns that experienced flexure failure during the test, yet were misclassified by ASCE/SEI 41 as flexure-shear-critical columns.  83  0.4 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.7 0.6  0.5  Maximum Axial Load Ratio Pmax /Agfc'  Normalized Shear v/√(fc') (√(MPa)  0.8  0.4  0.3 0.2 0.1  0 0  1  2  3  4  5  0.35 0.3  0.25 0.2  0.15 0.1 0.05  0  6  0  1  Column drift at Max Veff (%)  2  3  4  5  6  Column drift at Max Veff (%) 0.1  Trans. rein. ratio ρt (×102 )  Trans. Rein. Indicator ρtfyt/fc'  0.4  0.3  0.2  0.1  0.08 0.06 0.04  0.02  0  0 0  1  2  3  4  5  6  0  1  Column drift at Max Veff (%)  2  3  4  5  6  Column drift at Max Veff (%) 5  0.8  4  Aspect ratio a/d  spacing to depth ratio s/d  1  0.6 0.4  0.2  3 2 1  0  0 0  1  2  3  4  5  Column drift at Max Veff (%)  6  0  1  2  3  4  5  6  Column drift at Max Veff (%)  Figure 4.2 Measure column drift at Max Veff with key parameters 84  Maximum Axial Load Ratio Pmax /Agfc'  Normalized Shear v/√(fc') (√(MPa)  0.8 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.7 0.6  0.5 0.4  0.3 0.2 0.1  0.4  0.35 0.3  0.25 0.2  0.15 0.1 0.05  0  0 0  1  2  3  4  5  6  0  Column drift at 80% Max Veff (%)  2  3  4  5  6  0.1  Trans. Rein. Indicator ρtfyt/fc'  Trans. rein. ratio ρt (×102 )  0.4  0.3  0.2  0.1  0.08 0.06 0.04  0.02  0  0 0  1  2  3  4  5  6  0  Column drift at 80% Max Veff (%)  1  2  3  4  5  6  Column drift at 80% Max Veff (%)  1  5  0.8  4  Aspect ratio a/d  spacing to depth ratio s/d  1  Column drift at 80% Max Veff (%)  0.6 0.4 0.2  3 2 1  0  0 0  1  2  3  4  5  Column drift at 80% Max Veff (%)  6  0  1  2  3  4  5  6  Column drift at 80% Max Veff (%)  Figure 4.3 Measured column drift at 80% Max Veff with key parameters 85  0.4 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.7 0.6  0.5  Maximum Axial Load Ratio Pmax/Agfc'  Normalized Shear v/√(fc') (√(MPa)  0.8  0.4  0.3 0.2 0.1  0 0  1  2  3  4  5  0.35 0.3  0.25 0.2  0.15 0.1 0.05  0  6  0  Plastic drift at 80% Max Veff (%)  2  3  4  5  6  0.1  Trans. Rein. Indicator ρtfyt/fc'  Trans. rein. ratio ρt (×102 )  0.4  0.3  0.2  0.1  0.08 0.06 0.04  0.02  0  0 0  1  2  3  4  5  6  0  Plastic drift at 80% Max Veff (%)  1  2  3  4  5  6  Plastic drift at 80% Max Veff (%)  1  5  0.8  4  Aspect ratio a/d  spacing to depth ratio s/d  1  Plastic drift at 80% Max Veff (%)  0.6 0.4 0.2  3 2 1  0  0 0  1  2  3  4  5  Plastic drift at 80% Max Veff (%)  6  0  1  2  3  4  5  6  Plastic drift at 80% Max Veff (%)  Figure 4.4 Measured column plastic drift at 80% Max Veff with key parameters 86  60  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  40  Number of Cycles up to 80% Max Veff  Number of Cycles up toMax Veff  60  20  0  0  1  2  3  4  5  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  40  20  0  6  0  1  Column drift at Max Veff (%)  3  4  5  6  10  Overstrength factor R=S(a)*W/Vy  Overstrength factor R=S(a)*W/Vy  10  8 6  4 2  0 0  1  2  3  4  5  8 6  4 2  0  6  0  Column drift at Max Veff (%)  1  2  3  4  5  6  Column drift at 80% Max Veff (%)  20  20  Duration of input motions (Sec)  Duration of input motions (Sec)  2  Column drift at 80% Max Veff (%)  15  10  5  0  15  10  5  0 0  1  2  3  4  Column drift at Max Veff (%)  5  6  0  1  2  3  4  5  6  Column drift at 80% Max Veff (%)  Figure 4.5 Measured column drift with key dynamic parameters 87  Figure 4.2 and Figure 4.3 show that column drifts at Max Veff and 80% Max Veff are inversely correlated with the maximum axial load ratio. The column drifts also seem to decrease with increasing transverse reinforcement indicator. Other parameters are not strongly related to the column drifts at shear failure. Since several columns included in The Database had very similar column detailing and experimental setups, the column drift data at shear failure is “banded” at certain parameters, such as transverse reinforcement ratio t and aspect ratio a / d . Therefore the column data from The Database could be used to evaluate the available drift capacity models, but one must be cautious in developing new models based on this limited data set. Results in Figure 4.2 show that column drifts at shear failure, especially the 80% Max Veff , tends to decrease with increasing relative strength factor R and more number of cycles.  The longer duration of the input motions tb also slightly decrease the column drift, yet the trend is not very obvious and the data is pretty “banded” (the columns in the same frame have same tb but different column drift at shear failure). None of the three dynamic parameters seems to be directly related to column drift at Max Veff , suggesting more randomness and uncertainties are existed and it is better to use column drift at 80% Max Veff as the indicator of the column shear failure.  4.3 4.3.1  Evaluation of Pujol et al. (1999) Model Procedure Based on test data of 92 columns in 15 cyclic test series, Pujol et al. (1999) observed  that the ratio of maximum column drift at shear failure to the column aspect ratio a / d  88  tended to be positively correlated with transverse reinforcement index t f yt / v . An empirical model was proposed to provide conservative estimate of the maximum column drift at shear failure by Equation 4.1.   s Max  a  1 t f yt a     d  100 v d   4   Equation 4.1  Where t = transverse reinforcement ratio; f yt = yield stress of transverse reinforcement (MPa); a = column shear span and for columns have fixed-fixed end conditions, a could be calculated as half of column clear height ( a  L 2 ) ; d = effective depth of column section and could be taken as 80% of column depth D ; v = maximum shear stress (MPa). The properties of the database used by Pujol et al. (1999) could be found in Table 4.1. Table 4.1  Properties of column database in Pujol et al. (1999)  Parameters Aspect Ratio Concrete compressive strength Long. Rein. Ratio  Symbol  Maximum shear stress  4.3.2  Max  Min  5  1.3  86.2  20.7  l  0.005  0.0051  t  0.0164  0.0  0.2  0.0  N.A.  0.167  a/d  f c'  Trans. Rein. Ratio Axial Load Ratio  Unit  MPa  ' c  P f Ag  v  f c'  MPa  Evaluation with The Database According to Pujol et al. (1999) model, the column drift at shear failure is inversely  related to the column shear stress v . For non-ductile columns vulnerable to shear failure, the calculated shear strength and the measured maximum lateral force should be ideally same.  89  Yet in fact the shear strength V0 ' calculated by ASCE/SEI 41 model (introduced in section 6.2.3) could either underestimate or overestimate the column shear strength. Both the calculated shear strength V0 ' and the measured maximum effective force VMax are considered in the assessment of the Pujol et al. (1999) model. The calculated column drift is plotted with the measured column drift  s MaxVeff and  s 80% MaxVeff in Figure 4.7 and Figure 4.6 respectively, with (a) calculated column drift based on shear strength V0 ' from ASCE/SEI 41; (b) calculated drift based on maximum effective force VMax . 6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Calculated column drift (%) (VMax)  Calculated column drift (%) (V 0 ')  6  2  0  4  2  0 0  2  4  Measured drift at 80% Max Veff (%)  (a) Based on shear strength V0 ' Figure 4.6  6  0  2  4  6  Measured drift at 80% Max Veff (%)  (b) Based on maximum effective force VMax .  Calculated drift by Pujol et al. (1999) model with measured drift at 80% Max Veff  Results in Figure 4.6 are consistent with the intent of the model to provide conservative estimate of “the maximum column drift at shear failure”, since the measured drift at 80% Max Veff exceeded the calculated drift capacity based on either V0 ' or VMax for all non-ductile columns in The Database.  90  As shown in Figure 4.7, Pujol et al. (1999) model also underestimate the column drift at Max Veff for most of the columns. 6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Calculated column drift (%) (VMax)  Calculated column drift (%) (V0')  6  2  4  2  0  0  0  2  4  Measured column drift at Max Veff (%)  (a) Based on shear strength V0 ' Figure 4.7  6  0  2  4  6  Measured column drift at Max Veff (%)  (b) Based on maximum effective force VMax  Calculated drift by Pujol et al. (1999) with measured drift at Max Veff  Figure 4.8 plots the ratio (  s 80% MaxVeff  Cal ) of measured drift at 80% Max Veff to the calculated column drifts. The horizontal line “Mean” shows the mean value of the data for flexure-shear-critical columns. It is observed that Pujol et al. (1999) model underestimated column drift capacity for flexure-shear-critical columns and tended to be more conservative for shear-critical columns with low aspect ratio. The statistical results of the evaluation are reported in Table 4.2.  91  (δ s)80%MaxVeff / δ Calc (V0')  4.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Inital Axial Load Ration (Pini /Agfc') (a) The ratio of measured drift at 80% Max Veff to calculated drift with V0 '  (δ s)80%MaxVeff / δ Calc (VMax)  4.0 3.5 3.0 2.5 2.0 1.5  1.0 0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc') (b) The ratio of measured drift at 80% Max Veff to calculated drift with VMax Figure 4.8  Evaluation of Pujol et al. (1999) model  Figure 4.8(b) ignores the test data for shear-critical columns NCREE 2007-S1-C2 and NCREE 2007-S3-C1, which had exceptional high ratio  s 80% MaxVeff  Cal equal to 4.25 and 5.94 respectively. The measured maximum lateral force VMax is almost two times of the calculated shear strength V0 ' for those two columns as shown in Figure 6.4. Thus the calculated column drift  Cal based on VMeas was significantly lower than the measured column drift and resulted in rather high ratio of  s 80% MaxVeff  Cal .  92  Table 4.2  Statistical results of Pujol et al. (1999) model  (δs)80%MaxVeff / δCal (V0') (δs)80%MaxVeff / δCal (VMax) (δs)MaxVeff / δCal (V0') (δs)MaxVeff / δCal (VMax)  4.3.3  Min  Max  Mean  Median  Standard Deviation  COV (%)  1.13  3.12  1.93  1.92  0.54  27.8  1.27  3.11  2.09  2.15  0.46  22.2  0.84  2.51  1.37  1.31  0.35  25.7  0.76  2.19  1.50  1.49  0.37  24.8  Comparison with the Evaluation with Sezen (2002) Database In Elwood and Moehle (2003), Pujol et al. (1999) model was evaluated with the cyclic  test data of columns included Sezen (2002) database. Figure 4.9 plots the calculated drift at 80% Max Veff by Pujol et al. (1999) model for (a) columns in the Database, (b) columns included in Sezen (2002) database.  Calculated column drift (%) (V Max)  6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  2  0 0  2  4  6  Measured drift at 80% Max Veff (%)  (a) Columns in the Database Figure 4.9  (b) Columns in Sezen (2002) database  Evaluation of Pujol et al. (1999) model with columns in The Database and Sezen (2002) database  93  While overestimating the drift capacity for six columns in Sezen (2002) database, Pujol et al. (1999) provided conservative estimate of the drift capacity at 80% Max Veff for all the columns subjected to the dynamic tests. The mean value and coefficient of variation of  s 80% MaxVeff  Calc for columns in Sezen (2002) database are 1.71 and 42%. The mean value of ratio  s 80% MaxVeff  Calc for columns in The Database was higher (2.09), suggesting that the dynamic nature of the shaking table tests could have resulted in an increased the measured column drift at shear failure. There was also less variability in the dynamic test data, with coefficient of variation less than 30%.  4.4 4.4.1  Evaluation of Kato and Ohnishi (2002) Model Procedure Kato and Ohnishi (2002) proposed a model to calculate the column plastic drift at loss  of 20% peak lateral force by taking into account the maximum edge strain in concrete core and axial load ratio. The total drift at 80% Max Veff could be estimated by summing drift at column yielding (  y / L ) and the calculated plastic drift (  P / L ) shown in Equation 4.2.  s  y P   L L L   m cp D    je P   L   m cp D  j   e     2      0  en  1 3  3en         2   1  en  2   3 3    3  5e  4      n  3             Equation 4.2    94  Where  cp =maximum edge strain in the core concrete. For non-ductile columns with large-spaced transverse reinforcement in the Database, the concrete core was considered to be unconfined and thus set  cp equal to 0.002; en =equivalent axial load ratio; D =depth of column cross-section; je =depth of concrete core and shall be taken as the distance between the tension and compression reinforcement.  m = coefficient selected to achieve good agreement between the proposed model and the test dada. Based on the data of 36 columns subjected to cyclic pseudo-static tests, Kato and Ohnishi (2002) recommended m  2.4 for estimating column drift at shear failure.  en = equivalent axial load ratio could be found in Figure 4.10 or calculated by Equation 4.3:   p  0   p  1 3  en   p 5  4 15   s   0.33  1 3   p  2 3 1      3  2  5  p   s   0.33  2 3 1      p  2   s  As y   je  je   s       3  Equation 4.3  where  =min. axial load/max. axial load;  p =maximum axial load ratio, PMax Ag f c' ;  As =area of longitudinal reinforcement located at the center of the section;  y =yield stress of longitudinal reinforcement;  s = stress of longitudinal reinforcement.  Source: Ghannoum (2007) Figure 4.10  Equivalent axial load ratio from Kato and Ohnishi (2002) Model 95  From the Figure 4.10, we can see that for columns subjected to low axial load ratio (less than 0.33), en equals to the maximum axial load ratio.  4.4.2  Evaluation with The Database The calculated column drift at shear failure are compared with the measured drift at  80% Max Veff in Figure 4.11, with (a) calculated column drift based on the measured yield drift  y ; (b) calculated column drift based on the yield drift  y ThreeComp calculated from three-component model (introduced in section 3.5). 6  Calculated column drift (%) δy-ThreeComp  Calculated column drift (%) δy  6  4  2  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0  4  2  0 0  2  4  6  0  2  4  6  Measured drift at 80% Max Veff (%)  Measured drift at 80% Max Veff(%)  (a) Based on the measured yield drift  y  (b) Based on the yield drift  y ThreeComp  Figure 4.11  Calculated column drift at 80% Max Veff by Kato and Ohnishi (2002) model  There is large scatter existing in the calculated column drift by Kato and Ohnishi (2002) model, especially when applying calculated yield drift from three-components model. The ratio of measured drift at Max Veff to calculated column drift by Kato and Ohnishi (2002) model is plotted in Figure 4.12. The horizontal line “Mean” shows the mean value of the data for flexure-shear-critical columns.  96  (δs) 80%MaxVeff/ δ Calc (δy)Meas  2.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  1.5 1.0 0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc') (a) The ratio of measured drift to calculated drift with measured yield drift (δs )80%MaxVeff / δCalc (δy-ThreeCoomp )  2.0 1.5  1.0 0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (P ini /Agfc') (b) The ratio of measured drift to calculated drift with yield drift from three-component model Figure 4.12  Evaluation of Kato and Ohnishi (2002) model  Results in Figure 4.12 show that Kato and Ohnishi (2002) model could either underestimate or overestimate the column drift at 80% Max Veff and seems to represent the mean value of the test results. The model is developed to estimate the column drift for flexure-shear-critical columns and turned out to be very unconservative for shear-critical columns. The statistical results of ratio  s 80% MaxVeff  Calc for Kato and Ohnishi (2002) model for flexure-shear-critical columns are summarized in Table 4.3.  97  Table 4.3  Statistical results of Kato and Ohnishi (2002) model  (δs)80%MaxVeff / δCalc (δy) (δs)80%MaxVeff / δCalc (δy-ThreeComp) 4.4.3  Standard COV (%) Deviation  Min  Max  Mean  Median  0.28  1.36  0.88  0.92  0.28  31.9  0.31  1.65  0.98  1.03  0.33  33.6  Comparison with the Evaluation with Sezen (2002) Database The accuracy of the Kato and Ohnishi (2002) model largely depends on the accuracy of  estimated column yield drift. To avoid introducing additional uncertainties and errors, the calculated drift at shear failure was based on the measured yield drift when evaluating the model with the Sezen (2002) database. Results are plotted in Figure 4.13 for (a) columns in the Database (with calculated drift based on measured yield drift  y ), and (b) columns included in Sezen (2002) database. The shaded data points in Figure 4.13 (b) denoted the column with high axial load ratio ( P Ag f c'  0.25 ).  Calculated column drift (%) ( δy)  8 7  6 5 4 3 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  2 1  0 0  1  2  3  4  5  6  7  8  Measured column drift (%)  (a) Columns in The Database Figure 4.13  (b) Columns in Sezen (2002) database  Evaluation of Kato and Ohnishi (2002) model with dynamic and cyclic test data  98  Compared with the columns in Sezen (2002) database, the data of columns subjected to seismic load seems to be shifted closer to the unit line, indicating that the model was less unconservative for most columns. However, it is noticed that most data beneath the unit line in Figure 4.13 (a) belong to the columns connected with flexible beam (in purple diamond markers). The actual column drift capacity might be lower than the measured drift for those columns and data points may as well scattered in the upper triangular area of the plot similar to that in Figure 4.13 (b). The ratio of  s 80% MaxVeff  Calc had slightly lower mean value and higher coefficient of variation (equal to 0.84 and 44% respectively) for columns in Sezen (2002) database.  4.5 4.5.1  Evaluation of Zhu et al. (2006) Model Procedure In Zhu et al. (2006), 125 columns with detailing and properties typically found in old  reinforced concrete structures were selected from the PEER database and used to develop a probabilistic model to estimate the median column drift at loss of 20% its peak lateral strength. The properties of the column database used are summarized in Table 4.4. According to Zhu et al. (2006), columns were first classified into two zones: Zone F (flexure-dominated columns) and Zone S (shear-dominated columns). Generally, the columns included in Zone F have adequate transverse reinforcements and are expected to experience column failure associated with flexure damage, while the columns in Zone S are non-ductile columns that are vulnerable to shear failure. The rules of classifying columns in Zone F and Zone S can be found in Zhu et al. (2006). For simplicity this research will  99  classify the flexure-critical columns (columns in Condition i according to ASCE/SEI 41) into Zone F and non-ductile columns (Columns in Condition ii and Condition iii) into Zone S. Then the median drift at shear failure for columns in each zone could be calculated as: s a P  0.013  0.031 d d Ag f c'  Zone S:   s median  2.02t  0.025  Zone F:   s median  0.049  0.716l  0.120  Table 4.4  t f yt f  ' c   0.042  Equation 4.4  s P  0.070 d Ag f c'  Properties of the column database used in Zhu et al. (2006)  Parameters  Symbol  Unit  Max  Min  Aspect Ratio  a/d  7  1.2  Hoop spacing to depth ratio  s/d  1.2  0.1  Concrete compressive strength  f  ' c  MPa  56.2  16  Long. Rein. Yield Stress  f yl  MPa  587  318  Long. Rein. Ratio  l  0.033  0.012  Trans. Rein. Yield Stress  f yt  616  249  Trans. Rein. Ratio  t  0.022  0.0006  0.8  0  3.1  0.2  Axial Load Ratio Normalized shear stress  4.5.2  v    ' c  Pini f Ag f c' 6   st f st  MPa    MPa  Evaluation with The Database The calculated column drift at shear failure are compared with the measured drift at  80% Max Veff in Figure 4.14, with (a) calculated column drift based on the initial axial load  Pini ; (b) calculated column drift based on the maximum axial load Pmax . The ratio of measured drift at Max Veff to calculated column drift by Zhu et al. (2006) model is plotted in Figure 4.15.  100  6  Calculated column drift (%) (Pini)  Calculated column drift (%) (Pmax)  6  4  2  0  ±σ Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower)  4  2  0 0  2  4  6  0  Measured column drift (%)  (a) Calculated drift based on Pini Figure 4.14  2  4  6  Measured column drift (%)  (b) Calculated drift based on Pmax  Calculated column drift by Zhu et al. (2006) model  (δs)80%MaxVeff / δ Calc (Pini)  3.0 2.5 2.0 1.5  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  1.0 0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  Initial Axial Load Ration (Pini /Agfc') (a) The ratio of measured drift to calculated drift with initial axial load Pini  (δs) 80%MaxVeff / δ Calc (Pmax)  3.0 2.5 2.0 1.5 1.0  0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  Initial Axial Load Ration (Pini /Agfc')  (b) The ratio of measured drift to calculated drift with maximum axial load Pmax Figure 4.15  Evaluation of Zhu et al. (2006) model 101  Zhu et al. (2006) model was intend to represent the median value of the column drift capacity at shear failure, yet the results in Figure 4.15 show that the calculated column drift generally underestimate the drift capacity at 80% Max Veff for most flexure-shear-critical columns. On the other hand, the model can be very unconservative for shear-critical columns. The statistical results of ratio  s 80% MaxVeff  Cal are reported in Table 4.5. Table 4.5  Statistical results of Zhu et al. (2006) model  (δs)80%MaxVeff / δCal (Pini) (δs)80%MaxVeff / δCal (Pmax)  4.5.3  Min  Max  Mean  Median  Standard Deviation  COV (%)  0.80  2.45  1.53  1.46  0.44  28.9  0.83  2.79  1.67  1.57  0.49  29.4  Comparison with the Evaluation with Cyclic Test Data The calculated column drift by Zhu et al. (2006) are plotted in Figure 4.16 for (a)  columns in The Database (with calculated drift based on maximum axial load), (b) columns in the database used to develop the probability model.  Calculated column drift (%) (Pmax)  6  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex) ±σ  4  2  0 0  2  4  Measured column drift (%) (a) Columns in The Database  Figure 4.16  6  (b) Columns subjected to static cyclic tests  Evaluation of Zhu et al. (2006) model with dynamic and cyclic tests data 102  Since Zhu et al. (2006) greatly underestimate the column drift capacity as discussed before, the mean value of the ratio  s 80% MaxVeff  Cal for flexure-shear-critical columns in The Database is more than 1.5 and the data in Figure 4.16(a) is greatly shifted towards the lower triangle area, while the data for the flexure-shear-critical columns (data series in the “*” marker) was more evenly distributed around the diagonal unit line.  4.6 4.6.1  Evaluation of Elwood and Moehle (2003, 2005) Model Procedure Based on the test data of 50 flexure-shear-critical columns in Sezen (2002) database,  Elwood and Moehle (2003) proposed an empirical linear model to estimate the column drift at 80% Max Veff :  s   1 1 v 1  5 t   ' 30 20 f c 100  Equation 4.5  Where t = transverse reinforcement ratio; f c' = concrete compressive stress on test day (MPa); v = maximum shear stress (MPa). Elwood and Moehle (2005) observed that the column drift at shear failure seems to be inversely related to the axial load ratio for columns with low transverse reinforcement ratio  t  0.004  . Therefore, a variation of the linear model taking into account the influence of column axial load ratio was proposed:  s   3 1 v 1 P 1  4 t    ' ' 100 40 f c 40 Ag f c 100  Equation 4.6  103  Where t =transverse reinforcement ratio; f c' =concrete compressive stress on test day (MPa); v = maximum shear stress (MPa); P =column axial load; Ag =area of column gross section.  4.6.2  Evaluation with The Database The calculated column drift at shear failure by the above linear models are compared  with the measured drift at 80% Max Veff in Figure 4.17, with: (a) calculated drift by Elwood and Moehle (2003) model based on ASCE/SEI 41 shear strength V0 ' ; (b) calculated drift by Elwood and Moehle (2003) model based on measured maximum effective force VMax ; (c) calculated drift by Elwood and Moehle (2005) model based on ASCE/SEI 41 shear strength  V0 ' and initial axial load Pini ; (d) calculated drift by Elwood and Moehle (2005) model based on the measured maximum effective force VMax and maximum axial load Pmax . It is noticed that the calculated column drift of both linear models is largely controlled by the reinforcement ratio t . Since most of the non-ductile columns in The Database had relatively low transverse reinforcement ratio ( t  0.0010 0.0016 ), the calculated column drift tends to fall between 1.5 and 3. As a result, the data of calculated drift in Figure 4.17 was concentrated at a flat banded area instead of being scattered around the diagonal unit line, indicating that some other critical factors might be missing or the importance of the key parameters included in the models need to be re-weighed for columns subjected to low axial load.  104  6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Calculated column drift (%) (VMax)  Calculated column drift (%) (V0')  6  2  0  4  2  0 0  2  4  6  0  Measured column drift (%)  Calculated column drift (%) (VMax,Pmax)  Calculated column drift (%) (V0 ', Pini)  6  6  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  4  (b) Elwood and Moehle (2003) with VMax  (a) Elwood and Moehle (2003) with V0 ' 6  2  Measured column drift (%)  2  0  4  2  0 0  2 4 Measured column drift (%)  6  (c) Elwood and Moehle (2005) with V0 ' and Pini Figure 4.17  0  2 4 Measured column drift (%)  6  (d) Elwood and Moehle (2005) with VMax and Pmax  Calculated column drift by Elwood and Moehle (2003, 2005) models  Comparing Figure 4.17 (a) with (b) and (c) with (d), it is observed that the calculated drift did not change much with the different options of column axial load ratio and shear stress applied in both linear models.  105  Figure 4.18 plots the ratio of measured column drift over calculated drift   s 80% MaxVeff Cal  with: (a) calculated drift from Elwood and Moehle (2003) model based on  V0' ; (b) calculated drift from Elwood and Moehle (2005) model based on V0' and Pini  (δs )80%MaxVeff / δ Calc (V0 ',Pini)  3.5  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  3.0 2.5 2.0 1.5 1.0 0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini/Agfc') (a) The ratio of measured drift to calculated drift for Elwood and Moehle (2003) model (δs )80%MaxVeff / δ Calc (V0 ',Pini)  3.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex)  2.5 2.0 1.5  1.0 0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  Axial Load Ration (P/Agfc')  (b) The ratio of measured drift to calculated drift for Elwood and Moehle (2005) model Figure 4.18  Evaluation of Elwood and Moehle (2003, 2005) models  The statistical results of the linear models for flexure-shear critical columns are reported in Table 4.6.  106  Table 4.6  Statistical results of Elwood and Moehle (2003, 2005) models  Min  Max  Mean Median  Standard Deviation  COV (%)  (a) Elwood and Moehle (2003) model  (δs)80%MaxVeff / δCal (V0') (δs)80%MaxVeff / δCal (VMax)  0.88 1.00  2.41  1.45  1.51  0.38  26.5  2.60  1.59  1.53  0.40  25.5  (b) Elwood and Moehle (2005) model  (δs)80%MaxVeff / δCal (V0',Pini) (δs)80%MaxVeff / δCal (VMax,Pmax)  4.6.3  0.88  2.16  1.45  1.48  0.36  24.7  0.96  2.54  1.61  1.69  0.40  24.7  Comparison with the Evaluation with Sezen (2002) Database Elwood and Moehle (2005) model was developed based on the cyclic test data of  columns in Sezen (2002) database. Figure 4.19 plots the calculated drift at 80% Max Veff by Elwood and Moehle (2005) model for (a) columns in the Database, (b) columns included in Sezen (2002) database.  Calculated column drift (%) (V0 ', Pini)  6  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  5  4 3 2 1 0 0  1  2 3 4 5 Measured column drift (%)  (a) Columns in The Database Figure 4.19  6  (b) Columns included in Sezen (2002) database  Evaluation of Elwood and Moehle (2005) model with The Database and Sezen (2002) database 107  Since the transverse reinforcement ratio t for most flexure-shear-critical columns in the Database is around 0.0010 to 0.0020, the dynamic column data in Figure 4.19 (a) should be compared with the data series in square makers in Figure 4.19 (b). It is observed that while the data from cyclic tests scattered around the diagonal unit line in Figure 4.19 (b), the dynamic test data is more flat and seemed to be shifted towards the lower triangular area, indicating that the models underestimate column drift capacity. The performance of shear strength models and drift capacity models is compared in Figure 4.20 for (a) columns in The Database and (b) columns in the Sezen (2002) database. Results showed that shear strength could be estimated more accurately than the column drift capacity at shear failure. 2  2  VMax /V0 '  VMax /V0 '  1.5  1  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.5  1  0  0  1  2  3  0 0  1  2 δMeas /δCalc  (a) Columns in The Database  3 δMeas /δCalc  (b) Columns in Sezen (2002) database  Figure 4.20 Measure shear strength versus measured drift normalized by the calculated shear strength and calculated column drift  The evenly distributed data around the point (1.0, 1.0) in Figure 4.20 (b) indicated that for columns subjected to cyclic lateral load, underestimating or overestimating of the shear strength did not necessary result in underestimation or overestimation the column drift 108  capacity at shear failure. On the other hand, the data in Figure 4.20 (a) is observed to be concentrated in the upper right part, suggesting current models tend to underestimate the column shear strength and the drift capacity at shear failure.  4.6.4  Column Plastic Drift at Shear Failure Figure 4.21 compares the plastic drift calculated from linear models with measured  plastic drift, which is obtained by deducting the measured yield drift from total drift. 6  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Calculated plastic drift (%) (V Meas )  Calculated plastic drift (%) (V0')  6  2  0  4  2  0  0  2  4  6  0  Measured column drift (%)  Calculated plastic drift (%) (Vmeas ,Pmax)  Calculated plastic drift (%) (V0 ',Pini)  6  6  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  4  (b) Elwood and Moehle (2003) with VMax  (a) Elwood and Moehle (2003) with V0 ' 6  2  Measured column drift (%)  2  0  4  2  0 0  2 4 Measured column plastic drift (%)  6  (c) Elwood and Moehle (2005) with V0 ' and Pini Figure 4.21  0  2 4 Measured column plastic drift (%)  6  (d) Elwood and Moehle (2005) with VMax and Pmax  Plastic drift calculated by Elwood and Moehle (2003, 2005) model 109  The data of calculated plastic drift for shear-critical columns in Figure 4.21 seems to scatter around a dashed line parallel to the unit line. The shift of the data indicates that compared with cyclic test data, the plastic drift capacity at shear failure in shaking table tests increased, possibly due to the dynamic effect. Recall that no obvious trend could be found for calculated total column drift from linear models in Figure 4.17. Given the trends seen in Fig 4.20 considering plastic drift rather than total, models for estimating of the plastic drift at shear failure probably should be considered instead. Plastic column drift model has several advantages over the total drift model: (a) The plastic drift is essential to develop plastic hinge models. Columns in existing reinforced concrete structures usually have different detailing and connections (e.g. splices) with the beam (or column base) at the upper end and the bottom end. Plastic hinge model could capture the column damage and shear failure more appropriately than the model based on the behaviour of entire column. (b) Plastic drift is a good indicator of column damage and could better reflect the column drift capacity at shear failure. For example, a relatively flexible column may experience extended deformation before the column yielding. Thus at the point that the total column drift reaches the calculated drift from linear models, the column plastic deformation might be very small and it would be inappropriate to define it as shear failure point.  Several columns in Shin 2005 Tests and Elwood 2002 Tests are subjected to input motions scaled from the ground motions recorded in a subduction earthquake (Chile 1985  110  earthquake). It is noticed in Figure 4.22 (a) that the plastic drifts calculated from Elwood and Moehle (2005) model for those columns (subjected to Chile ground motions) are not as conservative as that for other columns. If excluding the columns subjected to subduction earthquake and using least-square fit of the test data for those columns subjected to general type of ground motions (e.g. Chi-Chi Earthquake), a new empirical plastic drift model at 80% Max Veff for flexure-shear-critical columns is proposed based on maximum axial load Pmax and measured maximum effective force VMax . The plastic drift model for columns subjected to subduction earthquake is scaled from the proposed model to achieve best match of the test data, taking into account the ideas that: (a) as column goes through more significant cycles when subjected to subduction earthquake, the effectiveness of transverse reinforcement should be reduced due to the bond degradation; (b) the negative impact of shear stress should be amplified, since the column subjected to subduction earthquake is likely to experience high shear stress level more times; (c) the overall plastic drift is found to be lower than that for similar columns subjected to general type of ground motions. The proposed empirical plastic drift capacity models are presented in Equation 4.7 General Earthquake:  s Subduction  0.049  0.60t  0.05  P v  0.049 ' Ag f c f c'  Subduction Earthquake:  s General  0.045  0.50t  0.05  Equation 4.7  P v  0.062 ' Ag f c f c'  The calculated plastic drifts at 80% Max Veff from the proposed empirical models are plotted in Figure 4.22 (b). The statistical results of the proposed models and Elwood and Moehle (2005) model are summarized in Table 4.7.  111  5  5  4  Calculated plastic drift (%) (Proposed)  Calculated plastic drift (%) (Elwood et al., 2005)  Subduction  General  3  2  1  0 0  1  2  3  4  Subduction 4  General  3  2  1  0  0  5  Measured column plastic drift (%) Figure 4.22  1  2  3  4  5  Measured column plastic drift (%)  Proposed empirical plastic drift model for flexure-shear-critical columns  Results in Figure 4.22 show that the proposed empirical models improves the estimation of column plastic drift capacities compared with Elwood and Moehle (2005) model and could reflect the trends of the column plastic drift capacity at shear failure fairly well. Table 4.7 shows that the model tends to represent the mean value of the column drift capacity, but large varieties still existed in the data. Table 4.7  Statistical results of proposed plastic drift models  Min  Max  (a) Proposed plastic drift model 1.94 (δs)Meas / (δs)Subduction 0.38 0.48 1.67 (δs)Meas / (δs)General 0.38 1.94 (δs)Meas / (δs)Cal (b) Elwood and Moehle (2005) model 1.65 (δs)Meas / (δs)Subduction 0.40 0.00 1.87 (δs)Meas / (δs)General 0.00 1.87 (δs)Meas / (δs)Cal  Mean  Median  Standard Deviation  COV (%)  1.03  1.13  0.54  52  1.02  0.97  0.34  33  1.02  1.03  0.41  40  0.84  0.78  0.41  49  0.75  0.75  0.57  76  0.80  0.78  0.51  63  The large variation (coefficient of variation) existed in the data of the ratio of measured plastic drift to calculated plastic drift from proposed model could be greatly reduced if the 112  coefficient of the axial load ratio term is positive, yet this contradict to the fact that higher axial load tends to reduce the column drift capacity. The quantity of the columns subjected to subduction earthquake included in The Database is also very limited. It is noted that most flexure-shear-critical columns used for developing this model achieved greatly varied plastic drift at 80% Max Veff , yet had very similar transverse reinforcement ratio ( t is around 0.0016) and were subjected to relatively low axial load ratio ( P Ag f c' is between 0.08 and 0.25) and shear stress ( v  f c' is around 0.3).  Some other parameters that could better represent the characteristics of subduction input ground motions might be missing. With more shaking table tests with subduction earthquakes and other types of earthquakes as input ground motions conducted on columns covering larger property range, a more reliable plastic drift capacity model could be developed.  4.7  Effect of Number of Cycles Figure 4.23 plots the number of cycles that columns went through up to the shear  failure point with the column drift at 80% Max Veff . The ordinate of Figure 4.23(b) is the normalized column drifts obtained from dividing the measured column drift by the calculated drift. The calculated drift at shear failure was obtained from Elwood and Moehle (2005) model (discussed in section 4.6.1) based on the maximum axial load PMax and measured maximum effective force VMax . Using the normalized column drifts could roughly take into account the effect of axial load ratio and shear stress  113  level and gain a better idea of the relations between number of cycles and column drift capacity. Results in Figure 4.23 (a) show that there is no strong correlation between the number of cycles that columns went through and the drift capacity at 80% Max Veff .  Cycles up to 80% Max Veff  60  NCREE 2009-MCFS NCREE 2009-HCFS  No. of Cycles  50  NCREE 2009-MUFS  40  NCREE 2005-P2  30  NCREE 2005-L  20  Ghannoum 2006  10  Elwood 2002 Shin 2005-Chile-0.1  0 0  0.5  1  1.5  2  2.5  3  3.5  4  Shin 2005-Chile-0.24  δMeas/δCalc (a) Number of cycles with normalized column drift at 80% Max Veff  Cycles up to 80% Max Veff  60  NCREE 2009-MCFS NCREE 2009-HCFS  No. of Cycles  50  NCREE 2009-MUFS  40  NCREE 2005-P2  30  NCREE 2005-L  20  Ghannoum 2006  10  Elwood 2002 Shin 2005-Chile-0.1  0 0  1  2  3  4  5  6  Shin 2005-Chile-0.24  Measured drift at 80% Max V eff δMeas (%) (b) Number of cycles with column drift at 80% Max Veff Figure 4.23  Number of cycles with normalized column drift at shear failure  114  Chapter 5: Column Drift Capacity at Axial-Load Failure 5.1  Definition of Axial-Load Failure The behaviour of ductile columns with adequate transverse reinforcement is dominated  by flexure response. The failure mechanisms of these flexure-critical columns are often assumed to be either reaching the ultimate curvature that the column section could withstand or suffering structural instability due to P-Delta effect.Equation Chapter 5 Section 1 Most available models estimating ultimate column drift capacity are based on the ultimate concrete and steel strains with consideration of P-Delta effect. These models are not applicable to shear-dominated columns that have light transverse reinforcement. As the shear failure plane develops, non-ductile columns may experience axial-load failure and lose gravity load support after shear failure, potentially leading to the collapse of the structures. There is lack of consensus on the definition of column axial-load failure, since loss of axial load support is closely related to many factors, such as column axial load response, lateral strength degradation and the column lengthening/shortening. In order to evaluate the performance of models proposed to estimate column drift capacity at axial-load failure, a consistent method needs to be adopted to define the axial-load failure point and the measured column drift at axial-load failure.  5.1.1  Definition Based on Hysteretic Response of Axial Load Since axial-load failure occurs when columns lose support for gravity load, it is  intuitive to associate axial-load failure with the sudden change of column axial load response. Using the west column in Shin 2005 Tests as an example (Figure 5.1), axial-load failure  115  could be defined as the point at which axial load suddenly dropped (point in blue diamond marker). It is observed that lateral force degraded to almost zero at this point too. Shin 2005-III-Chile-0.24-WestC  50  120  90  Axial Load (kN)  25  Lateral Force (kN)  150  Test Data 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift AF based on axial load  0  60  -25 30  0  -50 -4  -2  0  Hori Drift (%)  Figure 5.1  2  -4  4  -2  0  2  4  Hori Drift (%)  Axial-load failure based on axial load response for column in Shin 2005 Tests  The sudden change of the column axial load may also be accompanied by the abrupt change in slope of the column lengthening/shortening-horizontal drift response, as shown in Figure 5.2 for column C3 in NCREE 2007 Tests.  NCREE 2007-S3-C3  400  column lengthening/shortening (%)  300  Axial Load (kN)  0.3  Test Data 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift  200  100  0.2  0.1  0.0  -0.1  -0.2  -0.3 0 -4  -2  0  Hori Drift (%)  Figure 5.2  2  4  -0.4 -4  -2  0  2  4  Hori Drift (%)  Axial-load failure based on axial load response for column in NCREE 2007 Tests 116  Elwood 2002-S1-CenC  Ghannoum 2006-A1 120  200  90  150  Axial Load (kN)  250  Axial Load (kN)  150  60  30  Test Data 1 Test Data 2 Test Data 3 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift  0  -30 -6  -4  -2  0  2  4  Hori Drift (%)  Figure 5.3  Test Data 1st Yield Max VEff 810% Max VEff Axial Load Failure Max Achieved Drift  100  50  0  -50 6  -8  -4  0  4  8  Hori Drift (%)  Axial-load failure definitions for columns without sudden change in axial load response  However, for some columns that experienced axial-load failure, there was no obvious change in the axial load response. For example, the column A1 in Ghannoum 2006 Tests and the center columns in specimen S1 of Elwood 2002 Tests lost the gravity load support in the later stage of the test (axial-load failure could be considered occurred at the point shown in blue round marker), yet there seemed to be no significant change in the axial load response in Figure 5.3. It would be difficult to identify the axial-load failure point solely based on axial load hysteretic response for these columns. The axial load response is also affected by the type of the frames. For columns in specimens with multiple columns, such as specimens in NCREE 2005 Tests, axial load redistribution within the frame is allowed after the column axial-load failure; while in specimens with only two columns, such as specimens in Shin 2005 Tests, no axial load redistribution is allowed and the specimen tends to collapse after the column failed.  117  5.1.2  Definition Based on Degradation of Lateral Resistance Past studies (Kabeyasawa. et al. 2002) show that columns tend to lose support for  gravity load when column lateral resistance degrades to zero. Using column C1 in specimen P2 of NCREE 2005 Tests shown in Figure 5.4 as an example, the axial-load failure point could be defined based on the degradation of lateral resistance.  NCREE 2005-P2-C1  40  1.0  0.5  Vert Drift (%)  Lateral Force (kN)  20  0  -20  -1.5  -2.0  -60 -6  -4  -2  0  2  4  -0.5  -1.0  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift AF based on Shear Deg  -40  0.0  6  Hori Drift (%)  Figure 5.4  -6  -4  -2  0  2  4  6  Hori Drift (%)  Axial-load failure based on shear degradation  However for some non-ductile columns, the lateral strength did not suffer severe shear degradation (the residual shear is not less than 10% of its peak strength) prior to the loss of axial load support. For example as shown in Figure 5.5(a), it would be inappropriate to apply this definition for column A1 in specimen MCFS of NCREE 2009 Tests whose lateral resistance did not degrade to zero during the tests. What‟s more, in the case of column C3 of NCREE 2007 Tests shown in Figure 5.5 (b), the lateral force degraded to almost zero at the very end of the test, yet the column was severely damaged before this point and the horizontal drift data was not reliable (the upper  118  part of the columns slid away along the shear damaged plane). Drift at the point with zero lateral resistance was exceptional high for those columns and defining the axial-load failure at this point might result in a very unconservative interpretation of the column drift capacity at loss of axial load support.  NCREE 2009-MCFS-A1  90  NCREE 2007-S3-C3  200  Test Data 1st Yield Max Veff 80% Max Veff Axial Load Failure Max Achieved Drift  150 60  30  Lateral Force (kN)  Lateral Force (kN)  100  0  -30  1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  -60  -2  0  2  4  Hori Drift (%)  Figure 5.5  5.1.3  0 -50 -100 -150 -200  -90 -4  50  6  -4  -2  0  2  4  Hori Drift (%)  Inappropriate to define the column axial-load failure based on shear degradation  Definition Based on Column Shortening/Lengthening The axial-load failure of non-ductile columns with light transverse reinforcement is  usually induced by the development of a diagonal failure plane after column shear failure. The column may fail to transfer axial load with the sliding of columns along the critical diagonal crack, indicating that the change in vertical deformation response might be a good indicator of column axial-load failure. Yavari (2011) defined the onset of column axial-load failure as “the point at which the peak vertical displacement due to column lengthening is recorded followed by a sudden shortening of the column” (Yavari, 2011, p. 59). The axial-load failure points for columns in 119  NCREE 2009 Tests and NCREE 2007 Tests interpreted by this definition are shown in Figure 5.6.  NCREE 2007-S4-C3  NCREE 2009-HCFS-B1  0.3  0.2 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  0.2  Vert Drift (%)  Vert Drift (%)  0.1  Test Data 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift  0.0  0.1  0.0  -0.1 -0.1  -0.2  -0.2 -4  -2  0  2  4  -2  Hori Drift (%)  Figure 5.6  5.1.4  -1  0  1  2  Hori Drift (%)  Axial-load failure based on column lengthening/shortening  Measured Column Drift at Axial-Load Failure For a majority of non-ductile columns in The Database, the point with sudden change  in the sign of the slope of the column shortening/lengthening-horizontal drift relations is in accordance with the change of axial load. This point is also able to reflect the shear degradation fairly well. Therefore the column axial-load failure point in this research will be defined based on the column shortening/lengthening data (derived from the vertical deformation). However based on this definition, the column drift at the axial-load failure point might be very small for some columns; for example, column C3 in specimen MCFS in NCREE 2009 Tests as shown in Figure 5.7.  120  Since this chapter focuses on estimating the drift capacity up to the point that column losses gravity load support, the measured column drift at axial-load failure  a shall be taken as the maximum drift that the column achieved before the axial-load failure point as defined above. The measured plastic drift at axial-load failure could be calculated by subtracting the measured yield drift  y (obtained from TEST method discussed in section 3.2.4) from the total column drift at axial-load failure.  NCREE 2009-HCFS-C1  120  0.10  90 0.05  Vert Drift (%)  Lateral Load (kN)  60 30 0 1st Yield Max VEff 80% Max VEff Axial Load Failure Max Achieved Drift Test Data 1 Test Data 2  -30 -60 -90  -2  -1  0  1  2  3  4  -0.05  -0.10  -0.15  -120 -3  0.00  5  -0.20 -3  Hori Drift (%)  Figure 5.7  5.2  -2  -1  0  1  2  3  4  5  Hori Drift (%)  Axial-load failure at low lateral drift for column in NCREE 2009 Tests  Effect of Key Parameters on Measured Drift at Axial-Load Failure Figure 5.8 plots the measured column drifts at axial-load failure for non-ductile with  several key parameters. The plastic drift at axial-load failure is plotted in Figure 5.9. The key parameters considered and the columns that each data series represents are similar to that for column drift at shear failure in section 4.2 of this thesis. It is observed from Figure 5.8 that the column drift tends to decrease with increasing axial load ratio and normalized shear stress. Increasing transverse reinforcement indicator 121  also appears to increase the column drift at axial-load failure. Other parameters are not strongly correlated with either the column drift or plastic drift capacities. Results in Figure 5.10 show that column drift at axial-load failure tends to decrease with increasing relative strength factor R , possibly could be explained by the fact that weaker structure system is more likely to incur dynamic instability and the resulted dynamic excitations (or the inertia forces) caused the column to experience axial-load failure at lower drift. Going through more number of cycles slightly increase the column drift capacity for most non-ductile columns. The relation between the bracket duration and the column drift is not very obvious and the data is pretty “banded” (the columns in the same frame have same tb but different column drift at axial-load failure).  122  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.7  0.6 0.5  Maximum axial load ratio Pmax /Agfc'  Normalized Shear v/√(fc') (√(MPa)  0.8  0.4  0.3 0.2 0.1  0.4 0.35  0.3 0.25 0.2  0.15 0.1 0.05  0  0 0  2  4  6  8  10  0  Column drift at axial-load failure (%)  2  4  6  8  10  Column drift at axial-load failure (%) 0.1  Trans. rein. ratio ρt (×102 )  Trans. Rein. Indicator ρtfyt/fc'  0.4  0.3  0.2  0.1  0.08 0.06 0.04 0.02  0  0  0  2  4  6  8  10  0  Column drift at axial-load failure (%)  2  4  6  8  10  Column drift axial-load failure (%) 5  0.8  4  Aspect ratio a/d  spacing to depth ratio s/d  1  0.6 0.4 0.2  3 2 1  0  0  0  2  4  6  8  Column drift at axial-load failure (%)  10  0  2  4  6  8  10  Column drift axial-load failure (%)  Figure 5.8 Measured drift at axial-load failure with key parameters 123  0.4  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  0.7  0.6 0.5  Maximum Axial Load Ratio Pmax /Agfc'  Normalized Shear v/√(fc') (√(MPa)  0.8  0.4  0.3 0.2 0.1  0 0  2  4  6  8  0.35  0.3 0.25 0.2  0.15 0.1 0.05  0 0  10  Plastic drift at axial-load failure (%)  2  4  6  8  10  Plastic drift at axial-load failure (%) 0.1  Trans. rein. ratio ρt (×102 )  Trans. Rein. Indicator ρtfyt/fc'  0.4  0.3  0.2  0.1  0.08 0.06 0.04 0.02 0  0  0  2  4  6  8  0  10  Plastic drift at axial-load failure (%)  0.8  4  Aspect ratio a/d  spacing to depth ratio s/d  5  0.6 0.4  0  0  4  6  8  8  10  2 1  2  6  3  0.2  Plastic drift at axial-load failure(%)  4  Plastic drift at axial-load failure (%)  1  0  2  10  0  2  4  6  8  10  Plastic drift at axial-load failure (%)  Figure 5.9 Measured plastic drift at axial-load failure with key parameters 124  70  50  Overstrength factor S(a)*W/V y  60  Number of Cycles  10  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  40 30  20 10  0  8  6 4 2  0 0  2  4  6  8  10  Column drift at axial-load failure (%)  0  2  4  6  8  10  Column drift at axial-load failure (%)  20  Bracket duration t b (sec)  15  10  5  0 0  2  4  6  8  10  Column drift at axial-load failure (%) Figure 5.10 Measured drift at axial-load failure with dynamic parameters  125  5.3 5.3.1  Evaluation of Shear-Friction Model Procedure According to Elwood and Moehle (2003), axial load has to be transferred across the  shear failure plane to be supported after column suffers shear failure. This mechanism is known as shear friction. The classic shear-friction model related the shear Vsf transferred across the shear-failure plane (i.e. the main crack) with the normal stress N (perpendicular to the cracking surface) by an effective coefficient of friction  . Vsf   N  Equation 5.1  By drawing a free-body diagram across the shear damaged plane and ignoring the contribution from longitudinal reinforcement, the effective coefficient of friction  for nonductile columns with light transverse reinforcement could be expressed as:  P 1 Ast f st d c s  P 1  tan  Ast f st d c s tan   Equation 5.2  Where P = column axial load; Ast = area of transverse reinforcement; f st = yield strength of transverse reinforcement; d c = distance between the center line of longitudinal reinforcement in tension and compression;  = critical crack angle and is assumed to be 65 . Elwood and Moehle (2003) plotted the calculated coefficient of friction  with the measured drift at loss of gravity load support for 12 full scale flexure-shear-critical columns tested by Lynn (2001) and Sezen (2002). A trend was observed and approximated by a linear line:    tan    100  Axial  0 4  Equation 5.3  126  Combining the aforementioned two equations, Elwood and Moehle (2003) proposed a Shear-Friction model to estimate the column drift capacity at axial load failure for nonductile columns: 1   tan   4  P 1 100 tan   Ast f st d c s tan  2   Axial  5.3.2  Equation 5.4  Evaluation of Shear-Friction Model with The Database The calculated column drift by Shear-Friction model are compared with the measured  column drift at axial-load failure in Figure 5.11. The ratio of calculated column drift to measured drift  a  Cal is plotted in Figure 5.12, with (a) calculated drift based on initial axial load Pini ; (b) calculated drift based on maximum axial load Pmax . 8  Calculated column drift (%) (Pmax)  Calculated column drift (%) (Pini)  8  6  4  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  2  0  6  4  2  0 0  2 4 6 Measured column drift (%)  (a) Calculated drift based on Pini Figure 5.11  8  0  2 4 6 Measured column drift (%)  8  (b) Calculated drift based on Pmax  Calculated column drift at axial-load failure of shear-friction model  127  3.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  δ a / δ Cal (Pini)  2.5 2.0 1.5 1.0  0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (P ini/Agfc') (a) Calculated column drift at axial-load failure with initial axial load Pini 3.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  δ a / δ Cal (Pmax)  2.5 2.0 1.5  1.0 0.5 0.0  0  0.05  0.1 0.15 0.2 0.25 Initial Axial Load Ration (Pini /Agfc')  0.3  0.35  0.4  (b) Calculated column drift at axial-load failure with maximum axial load Pmax Figure 5.12  Evaluation of shear-friction model  Results in Figure 5.12(a) show that calculated drift based on initial axial load Pini tends to overestimate the column drift capacity at axial-load failure for most flexure-shear-critical columns. With more accurate estimation of the column axial load level (the maximum axial load in this case), the calculated drift seems to represent the mean value of the drift capacity at axial-load failure, though with large variability existed in the data. Table 5.1  Statistical results of shear-friction model  δa / δCal (Pini) δa / δCal (Pmax)  Min  Max  Mean  Median  Standard Deviation  COV (%)  0.42  2.03  0.97  0.88  0.39  40.4  0.51  2.20  1.23  1.01  0.48  39.2  128  5.3.3  Comparison with the Cyclic Test Data Elwood and Moehle (2003) suggested that effective coefficient of friction  calculated  based on axial load P and critical crack angle  could be related to the column drift at axialload failure. Figure 5.13 plots the calculated coefficient of friction  with the measured drift at axial-load failure for columns in The Database, with (a) coefficient  calculated based on initial axial load Pini ; (b) coefficient  calculated based on maximum axial load Pmax . The purple solid line represents the Equation 5.3 recommended by Elwood and Moehle (2003). 2.5  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  2  Coefficient of friction μ (Pmax)  Coefficient of friction μ (Pini)  2.5  1.5  2  1.5  1  0.5  1  0.5  0  0 0  2 4 6 8 Measured column drift (%)  (a) Calculated drift based on Pini Figure 5.13  10  0  2 4 6 8 Measured column drift (%)  10  (b) Calculated drift based on Pmax  Relation between effective coefficient of friction   and column drift at axial-load failure  Results in Figure 5.13 show that if calculating the coefficient of friction  based on maximum axial load Pmax for shear-critical columns, the measured drift is evenly distributed around the solid purple line. If calculating the  based on initial axial load Pini , the data is likely to be shifted towards left and become more scattered. While columns were subjected to varying axial load during shaking table tests, it is believed that maximum axial load 129  Pmax represented the column axial load stress level at axial-load failure better than the  designed axial load Pini . Results in Figure 5.13(b) indicate that the column drift at axial-load failure during the dynamic tests follows a similar trend to that observed from the cyclic test data. For shear-critical columns, no such trend was observed and the coefficient of friction  seemed to be positively correlated to the column drift at axial-load failure. However, no  reliable conclusion could be drawn due to the limited number of test data. Figure 5.14 plots the Shear-Friction model with the measured drift at axial-load failure with (a) columns in the Database; and (b) columns from Lynn (2001) tests and Sezen (2002) Tests. 40  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex) Shear-Friction model  Pmax/(Astfstdc/s)  30  20  10  0 0  2 4 6 Measured column drift (%)  (a) Columns in The Database Figure 5.14  8  (b) Columns in static cyclic tests  Evaluation of shear-friction model by dynamic and cyclic test data  The concave curve in Figure 5.14 indicated that the Shear-Friction model is very sensitive to the axial load factor and transverse reinforcement details. Especially for columns subjected to relatively low axial load, a small change in axial load level may result in a significant difference in the predicted column drift capacity. 130  Calculated column drift (%) (Pmax)  8  6  4  2  0 0  2 4 6 Measured column drift (%)  8  (a) Columns in The Database Figure 5.15  5.4 5.4.1  (b) Columns in static cyclic tests  Evaluation of shear-friction model by dynamic and cyclic tests data  Evaluation of Kato and Ohnishi (2002) Model Procedure The drift capacity model proposed by Kato and Ohnishi (2002) discussed in section 4.4  could also be applied to estimate the column drift at axial load failure  a by using m  3.6 :  a  y  P   L L L   m cp D    je P   L   m cp D  j   e     2      0  en  1 3  3en         2  1 2   en    3 3 4     3 5e      n 3            Equation 5.5    131  5.4.2  Evaluation with The Database The calculated plastic column drift is compared with the measured plastic drift at axial-  load failure in Figure 5.16. Results show that Kato and Ohnishi (2002) model generally overestimate the plastic column drift at axial-load failure.  Calculated column plastic drift (%)  8  6  4 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  2  0 0  2  4  6  8  Measured column plastic drift (%)  Figure 5.16  Calculated column plastic drift at axial-load failure by Kato and Ohnishi (2002) model  The accuracy of the Kato and Ohnishi (2002) model largely depends on the accuracy of estimated column yield drift. The calculated column drift at shear failure are compared with the measured drift at axial-load failure in Figure 5.17. The test data seems to be scattered around a dashed line parallel to the diagonal unit line in Figure 5.17, indicating the coefficient m selected as 3.6 based on the cyclic test data might be too large when estimating the drift capacity at axial-load failure for columns subjected to earthquake load. The ratio of measured drift at axial-load failure to calculated column drift is plotted in Figure 5.18. Results show that Kato and Ohnishi (2002) model overestimates the column drift capacity for most of the flexure-shear-critical columns.  132  8  Calculated column drift (%) (δy-ThreeComp )  Calculated column drift (%) ( δy )  8  6  4 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  2  0 0  2  4  6  6  4  2  0  8  0  Measured column drift (%)  (a) Calculated drift based on  y Figure 5.17  2  4  (b) Calculated drift based on  8   y ThreeComp  Calculated column drift at axial-load failure by Kato and Ohnishi (2002) model  2.0  δa / δ Calc (Measured δYield)  6  Measured column drift (%)  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  1.5 1.0  0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (P ini /Agfc') (a) The ratio of measured drift to calculated drift with measured yield drift  y δa / δ Calc (Three-Comp δYield)  2.0  Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  1.5 1.0 0.5 0.0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  Initial Axial Load Ration (Pini /Agfc') (b) The ratio of measured drift to calculated drift with yield drift from three-components Figure 5.18  0.4   y ThreeComp  Evaluation of Kato and Ohnishi (2002) model 133  Table 5.2  Statistical results of Kato and Ohnishi (2002) model  δa / δCal (δy-Meas) δa / δCal (δy-ThreeComp)  5.5 5.5.1  Min  Max  Mean  Median  Standard Deviation  COV (%)  0.36  1.64  0.80  0.82  0.25  31.0  0.40  1.60  0.87  0.85  0.27  30.8  Evaluation of Zhu et al. (2006) Model Procedure Based on a database consisting 28 columns that experienced axial load loss during  unidirectional pseudo-static tests, Zhu et al. (2006) proposed a probability model to give the median prediction of the column drift at axial-load failure for flexure-shear-critical columns that experience yielding before shear failure. According to the classic shear-friction model proposed by Elwood and Mohele (2005), the coefficient of friction  could be expressed by Equation 5.2:  P 1 Ast f st d c s  P 1  tan  Ast f st d c s tan  Subsequently, Zhu et al. (2006) related the coefficient  with the drift capacity at axialload failure for the 28 columns and proposed to estimate the median drift capacity by:   Axial median  0.184exp  1.45   5.5.2  Equation 5.6  Evaluation of Zhu et al. (2006) Model Figure 5.19 plots the calculated column drift by Zhu et al. (2006) model with the  measured drift at axial-load failure with (a) calculated column drift based initial axial load Pini ; (b) calculated column drift based maximum axial load Pmax . 134  8  Calculated column drift (%) (P max)  Calculated column drift (%) (P ini)  8  6  4  2  0  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  6  4  2  0  0  2 4 6 Measured column drift (%)  (a) Calculated drift based on Pini Figure 5.19  8  0  2 4 6 Measured column drift (%)  8  (b) Calculated drift based on Pmax  Calculated column drift at axial-load failure by Zhu et al. (2006) model  Notice that this probability model is a deterministic model developed independent of the shear capacity of the columns. Therefore, if the calculated drift at axial-load failure is lower than the column drift at shear failure by Zhu et al. (2006) model introduced in section 4.5, the axial-load failure should be considered occurred simultaneously with the shear failure and the drift capacity at shear failure should be used instead. Considering the above assumptions, the calculated column drift by Zhu et al. (2006) model is adjusted and plotted in Figure 5.20.  135  8  Calculated column drift (%) (P max)  Calculated column drift (%) (P ini)  8  6  4  2  0  2 4 6 Measured column drift (%)  8  4  2  0  Figure 5.20  2 4 6 Measured column drift (%)  8  (b) Calculated drift based on Pmax  (a) Calculated drift based on Pini  Adjusted calculated column drift at axial-load failure by Zhu et al. (2006) model Shear Flex-Shear Flex-shear (Lower) Flex-Shear (Beam) Flex-Shear (Flex) Mean  2.0  δ a / δ Calc (Pini)  6  0  0  2.5  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  1.5 1.0  0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  Initial Axial Load Ration (Pini /Agfc')  0.3  0.35  0.4  (a) Calculated column drift at axial load failure with initial axial load Pini  δ a / δ Calc (Pmax)  2.5  2.0 1.5 1.0  0.5 0.0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini/Agfc')  (b) Calculated column drift at axial load failure with maximum axial load Pmax Figure 5.21  Evaluation of Zhu et al. (2006) model 136  The ratio of measured column drift to calculated drift  a  Cal is plotted in Figure 5.21. Results in Figure 5.20(a) and Figure 5.21(a) show that the calculated column drift based on initial axial load Pini tends to overestimate the column capacity at axial-load failure for most flexure-shear-critical columns. The data in Figure 5.20 (b) is distributed more evenly along the diagonal unit line, indicating that with relatively accurate estimation of column axial load level (maximum axial load Pmax in this case), Zhu et al. (2006) model tends to represent the mean value of the column drift capacity fairly well. Table 5.3  Statistical results of Zhu et al. (2006) model  δa / δCal (Pini) δa / δCal (Pmax)  5.5.3  Min  Max  Mean  Median  Standard Deviation  COV (%)  0.32  1.62  0.87  0.88  0.34  39.4  0.52  2.05  1.15  1.12  0.34  29.6  Comparison with Static Cyclic Test Data The Zhu et al. (2006) probability model was developed based on the test data of 28  columns subjected to cyclic lateral load. The calculated column drift for these columns is plotted in Figure 5.22 (b). Results in Figure 5.22 show that Zhu et al. (2006) could capture the mean value of column drift at axial-load failure for columns in both the dynamic tests and cyclic tests fairly well.  137  Calculated column drift (%) (Pmax)  12 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  10 8 6 4 2 0  0  2  4  6  8  10  12  Measured column drift (%)  (a) Columns in The Database Figure 5.22  5.6  (b) Columns in static cyclic tests  Evaluation of Zhu et al. (2006) model with the dynamic and cyclic test data  Effect of Number of Cycles Figure 4.23 plots the number of cycles that columns went through up to the axial-load  failure with the normalized measured column drift. The ordinate of Figure 4.23 is the normalized column drifts obtained from dividing the measured column drift by the calculated drift. The calculated drift at axial-load failure was obtained from Shear-Friction model (discussed in section 5.3.1) based on the maximum axial load ratio and measured shear strength. Using the normalized column drifts could roughly take into account the effect of axial load ratio and shear stress level and gain a better idea of the relations between number of cycles and column drift capacity. In Figure 5.23 (a), there is a positive correlation between the number of cycles and normalized column drift at axial-load failure. This could be explained by the fact that the cracks in one direction would be closed when column deformed to the opposite direction and  138  the column damage was more evenly distributed as the column went through more reverse cycles.  Cycles up to axial-load failure  No. of Cycles  60  NCREE 2009-MCFS  50  NCREE 2009-HCFS  40  NCREE 2009-MUFS  30  NCREE 2005-P2  20  Ghannoum 2006  10  Elwood 2002 Shin 2005-Chile-0.1  0 0  0.5  1  1.5  2  2.5  3  3.5  4  Shin 2005-Chile-0.24  δMeas/δCalc (a) Number of cycles with normalized column drift at axial-load failure  Cycles up to axial-load failure  No. of Cycles  60  NCREE 2009-MCFS  50  NCREE 2009-HCFS  40  NCREE 2009-MUFS  30  NCREE 2005-P2  20  Ghannoum 2006  10  Elwood 2002 Shin 2005-Chile-0.1  0 0  1  2  3  4  5  6  7  Shin 2005-Chile-0.24  Measured drift at axial-load failure δMeas (%) (b) Number of cycles with column drift at axial-load failure Figure 5.23  Number of cycles with column drift capacity at axial-load failure  139  Chapter 6: Evaluation of ACSE/SEI 41-06 Supplement 6.1  Introduction This chapter focuses on using The Database to evaluate the provisions of the current  seismic rehabilitation standard ASCE/SEI 41 regarding reinforced concrete columns, such as column shear strength, column classification and level of conservertism for modeling parameters. Some refinements and changes of this standard are provided based on the test data from The Database.Equation Chapter 6 Section 1  6.2 6.2.1  Column Shear Strength Evaluation of Shear Strength Ratio VMax/Vn For columns with inadequate transverse reinforcement, ASCE/SEI 41 adopted the shear  strength model proposed by Sezen (2004) .According to ASCE/SEI 41, column shear strength Vn could be calculated by Equation 6.1.  Vn  kV0  k Vc  Vt   0.5 f '  Nu c  0.8 Ag Vc    1 '  M / Vd  0.5 f A c g    Av f y d    s / d   0.5    s  Av f y d   Vt  0.5  0.5   s / d   1.0  s   0   s / d   1.0       Equation 6.1  The definition of each parameter could be found in Sezen (2004) model introduced in section 1.2.3 of this thesis. It is recommended by ASCE/SEI 41 that if spacing of column transverse reinforcement is larger than half of the effective depth of column section  140  ( s  d 2 ) in the direction of applied shear force, the transverse reinforcement should be taken as no more than 50% effective when resisting shear or torsion, and thus Vs  0.5 Av f y d / s .Moreover, if the spacing is larger than the effective depth of the column  section ( s  d ), the transverse reinforcement shall be considered ineffective and set Vs  0 . In order to evaluate the accuracy of column shear strengths estimated by ASCE/SEI 41, the shear strength ratio is calculated by dividing measured maximum lateral force VMax by calculated shear strength Vn . The results are plotted in Figure 6.1 and grouped by the failure types of columns (discussed in section 1.1.3). The dashed line represents the mean value of the shear strength ratio for flexure-shear-critical columns. 2.5  Shear Flexure-Shear Flexure Mean  V Max / V n  2  1.5 1 0.5 0  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ratio (Pini/Agfc') Figure 6.1  Evaluation of shear strength ratio (VMax/Vn)  For flexure-shear-critical and shear-critical columns, the shear strength ratio data are well above the horizontal line representing the case that measured shear strength exactly matches the calculated value, indicating the ASCE/SEI 41 generally underestimates the shear strength for non-ductile columns. The mean value of VMax Vn is 1.45 for flexure-shear columns with a coefficient of variation of 18.3%.  141  Since the ASCE/SEI 41 shear strength model is mainly proposed to calculate the shear strength of non-ductile columns, it provides a poor estimation for shear resistance of ductile columns. This may be attributed to the fact that the plastic demand was much smaller than the shear strength for flexure-critical columns and those ductile columns encountered flexure yielding before achieving the shear strength.  6.2.2  Evaluation of Shear Strength Ratio VMax/V0 When estimating column shear strength by ASCE/SEI 41, modifier factor k is applied  to account for the reduction of column drift capacity due to high ductility demand. If the modifier factor k is removed, shear strength could be estimated as V0  Vn / k  Vc  Vs . 2.5  V Max / V 0  2  1.5 1  Shear Flexure-Shear Flexure Mean  0.5  0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini/Agfc') Figure 6.2  Evaluation of shear strength ratio (VMax/V0)  Figure 6.2 plots the shear strength ratio VMax V0 . For flexure-shear-critical columns, ignoring the reduction due to ductility demand brings the data closer to the unit horizontal line. The statistical results in Table 6.1 also indicate that the calculated V0 provides a better estimation of column shear strength (the mean value decreased from 1.45 to 1.38). V0 still underestimated the shear strength for shear critical columns.  142  Evaluation of Shear Strength Ratio VMax/V0’  6.2.3  Further, if reduction on transverse reinforcement contribution is ignored for columns with widely spaced ties ( s / d  0.5 or s / d  1.0 ), the shear strength could be calculated as V0 ' by Equation 6.2. V0 '  Vn '/ k  Vc  Vs '  0.5 f ' Nu c Vc =  1  M / Vd 0.5 f c' Ag  A f d Vs '  v y s    0.8 Ag    Equation 6.2  The resulting shear strength ratio VMax V0 ' is evaluated in Figure 6.3. 3  Shear Flexure-Shear Flexure Mean  V Max / V 0'  2.5 2 1.5  1 0.5 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ratio (P ini/Agfc') Figure 6.3  Evaluation of shear strength ratio (VMax/V0’)  Results show that for flexure-shear-critical columns, calculated shear strength V0 ' tends to represent the mean value of column shear strength. The resulted shear strength ratio has a coefficient of variation of 17.0% and mean value of 1.09. It is observed that V0 ' underestimates the shear strength of shear-critical columns compared to that of flexure-shear-critical columns. It is observed that a sudden increase in lateral force occurred in the cycle of shear failure for shear-critical columns, as shown in Figure 6.4. 143  The shear-critical columns in NCREE 2007 Tests have low aspect ratio a d (around 1.5 for columns in specimen S3 and 2. 0 for columns in S4) that exceeded the property range of the Sezen (2002) database ( 2< a d  4 ) that the shear strength model was developed based on. The columns have relative heavy longitudinal reinforcement ( l  3.24% ) and very light transverse reinforcement ( t  0.067% ). The behaviour of 90-degree hooks of the transverse reinforcement could not be estimated reliably. Scaled from full size columns, some shear-critical columns may experience higher strain rate during the dynamic tests, which contributed to the increase in column lateral resistances.  144  (a) NCREE 2007-S3-C1  250  (b) NCREE 2007-S3-C3  250  Test Data  200  Max Veff  150  ASCE  200 150 100  50  Shear (kN)  Shear (kN)  100  0  -50  50 0 -50  -100  -100  -150  -150  -200  -200  -250  -250 -4  -2  0  Hori Drift (%)  2  4  -2  (c) NCREE 2007-S4-C1  2  Test Data  Test Data  200  Max Shear  Max Shear  150  ASCE  100  50  50  Shear (kN)  100  Shear (kN)  1  (d) NCREE 2007-S4-C3  250  150  0  Hori Drift (%)  250 200  -1  0  -50  ASCE  0 -50  -100  -100  -150  -150  -200  -200 -250  -250 -2  -1  0  Hori Drift (%)  Figure 6.4  1  2  -2  -1  0  1  2  Hori Drift (%)  Sudden increase in lateral resistance for columns in NCREE 2007 Tests  145  6.2.4  Influence of Axial Load Ratio In the above evaluation of column shear strength, initial axial load ratio was used to  estimate the shear strength. However, columns subjected to dynamic loads would experience varying axial loads due to the overturning moment. Other axial loads measured during the tests, such as maximum axial load Pmax and the axial load PMaxVeff at the time that columns achieved maximum effective force, could also be applied in the shear strength model. The corresponding shear strength ratio is plotted in Figure 6.5. Results show that shear strength ratio is not very sensitive to the selected axial load ratio. Shear strength V0 ' calculated by following ASCE/SEI 41 procedure could represent the mean value of column shear strengths fairly well and is not very sensitive to the applied axial load ratio. 2.5 Initial P Max P P@Vmax  V Meas / V 0 '  2 1.5 1 0.5  0 0  0.05  Figure 6.5  0.1 0.15 0.2 Initial Axial Load Ratio (Pini/Agfc')  0.25  0.3  Shear strength ratio calculated with different axial load ratio  Table 6.1 summarizes the statistical characteristics for the aforementioned methods of estimating column shear strength.  146  Table 6.1  Evaluation of ASCE/SEI 41 shear strength model  Method  Min  Max  Mean  Median  Standard Deviation  COV (%)  (a)flexure-shear-critical columns  Vn  kV0  k Vc  Vs   0.78  1.85  1.45  1.48  0.26  18.3  Vn '  kV0 '  k Vc  Vs '  0.78  1.85  1.38  1.44  0.29  20.7  V0 '  Vc  Vs '  0.70  1.47  1.09  1.13  0.18  16.9  V0 '  Vc  Vs ' (w. Pmax)  0.63  1.22  1.00  1.07  0.17  17.2  V0 '  Vc  Vs ' (w. P@VMax)  0.64  1.24  1.03  1.08  0.17  16.9  Vn  kV0  k Vc  Vs   1.18  2.22  1.82  1.82  0.40  21.7  Vn '  kV0 '  k Vc  Vs '  1.14  2.22  1.77  1.72  0.39  22.3  V0 '  Vc  Vs '  1.04  1.94  1.58  1.61  0.33  21.1  V0 '  Vc  Vs ' (w. Pmax)  0.92  1.88  1.45  1.48  0.34  23.5  V0 '  Vc  Vs ' (w. P@VMax)  0.97  2.14  1.65  1.80  0.45  27.0  (b)shear-critical columns  6.3 6.3.1  Column Classification ASCE/SEI 41 Column Classification According to ASCE/SEI 41, columns are classified into three conditions based on shear  strength Vn , plastic shear capacity V p and transverse reinforcement detailing. As illustrated in Figure 1.3 in section 1.2.4, columns in Condition i roughly correspond to flexure-critical columns, while columns in Condition ii and iii generally referred to flexure-shear-critical columns and shear-critical columns respectively. The accuracy of the ASCE/SEI 41 classification method could be evaluated by the matrix shown in Table 6.2. The diagonal data in the blue zone indicates the number of columns that are classified properly, for example, “5” indicates that there were 5 columns failed in shear during the tests and were also classified as shear-critical columns by ASCE/SEI 41. The column data that falls in the green zone provides a “safer” estimation of  147  the column failure type in terms of column drift capacity, for example “23” indicates that there were 23 columns that are classified as shear-critical columns, yet were observed to have experienced flexure-shear failure during the tests. On the other hand, we would prefer to avoid classifying the column into the red zone. When evaluating the performance of column classification method recommended by ASCE/SEI 41, only columns that were observed to experience failure during the tests are included in the Table 6.2, ensuring sure that the “observed failure type” is fairly reliable. Table 6.2  Evaluation of ASCE/SEI 41 column classification method 1.0  V p V0  0.6  Flexure  Flexure (Condition i) 0  Flexure-Shear  0  4  23  27  Shear  0  0  5  5  Total  0  6  28  34   Flexure  Shear  Observed Failure Type  ASCE/SEI 41 Classification Flexure-Shear Shear (Condition ii) (condition iii) 2 0  Total 2  Table 6.2 shows that five columns that were observed to experience pure shear failure are classified properly by ASCE/SEI 41. In NCREE 2007 tests, two ductile outside columns C1 and C3 were observed to experience flexure failure, yet were classified as condition ii columns. ASCE/SEI 41 adopts  Vp  0.6V0 as the criteria for the condition i and those two flexural-critical columns had ratio of Vp / V0 slightly above 0.6 (0.66 and 0.68 respectively). This conservatism is appropriate since ASCE/SEI 41 intends to avoid the case that flexure-shear-critical columns are unconservatively classified as flexure-critical columns in condition i. Most flexure-shear-critical columns were misclassified under condition iii. According to ASCE/SEI 41, shear failure occurs at zero plastic rotation for columns in condition iii, 148  which is way too conservative for most non-ductile columns since these columns usually developed certain deformation prior to the shear failure. Misclassifying flexure-shear-critical columns into condition iii would greatly underestimate the drift capacities of columns. As discussed in section 6.2.3, column shear strength Vn calculated by ASCE/SEI 41 is generally much lower than measured column shear capacity. It is intuitive that column classification procedure should use the „real‟ shear strength to classify columns, instead of being conservative as is commonly done in strength based assessments. Therefore, it is better to ignore the reduction due to large-spacing transverse reinforcement as specified in ASCE/SEI 41 and use V0'  Vn' / k  Vc  Vs' when estimating shear strength. Table 6.3 summarized the results of this variation of the ASCE/SEI 41 classification method. Most of the flexure-shear-critical columns are classified properly. Table 6.3  Evaluation of variation of ASCE/SEI 41 classification method '  1.0  V p V0  0.6  Flexure  Flexure (Condition i) 0  Flexure-Shear  0  14  13  27  Shear  0  0  5  5  Total  0  16  18  34   Flexure  Shear  Observed Failure Type  ASCE/SEI 41 Classification Flexure-Shear Shear (Condition ii) (Condition iii) 2 0  Total 2  It is noticed that for most flexure-shear-critical columns that are misclassified as Condition iii, the ratio of plastic shear capacity V p over shear strength Vn' k  is just slightly above 1.0 and it is too conservative to classify these columns as shear-critical columns.  149  If the criterion for flexure-shear columns is changed to 1.1  Vp Vn' k   0.6 instead of 1.0  Vp Vn' k   0.6 , the results of modified ASCE/SEI 41 classification method are  summarized in Table 6.4. Table 6.4  Evaluation of modified ASCE/SEI 41 classification method  1.1  V p  Vn' / k   0.6  Observed Failure Type   Flexure  Shear  ASCE/SEI 41 Classification Flexure (Condition i)  Flexure-Shear Shear (Condition ii) (Condition iii)  Total  Flexure  0  2  0  2  Flexure-Shear  0  27  0  27  Shear  0  0  5  5  Total  0  29  5  34  Results in Table 6.4 shows that by using non-reduced column shear strength ( V0 '  Vn ' k ) and the modified classification criteria, all flexure-shear-critical columns are classified properly by failure type and thus the resulting load-deformation relation will be more accurate and result in a more economical retrofit solution. In the following sections, columns are classified based on the modified ASCE/SEI classification method.  6.4 6.4.1  Modeling Parameters of Generalized Load-Deformation Relation Measured Plastic Rotations During seismic response of older concrete frames, plastic rotations are generally  concentrated at the ends of concrete columns. In Ghaunnoum 2006 tests, column rotations are measured over a plastic hinge with length equal to the depth of column section. However, there is lack of consensus on the plastic hinge length. Also it is very difficult to measure column rotations at the ends directly for most shaking table tests. Therefore, column rotation  150  is taken as the inter-story drift ratio, which assumes that the plastic hinge length equals to column clear height L. The yield rotation  y equals to the column drift at column yielding and plastic rotation at shear failure and axial-load failure can be calculated by Equation 6.3.  y   y L    a Mp 3 EI ASCE  Equation 6.3  Where L = the column clear height, a = the shear span that equals to L/2 for fixed-fixed columns. M p = the moment strength obtained from section analysis; EI ASCE = the effective column stiffness recommended by ASCE/SEI 41 (discussed in section 3.4). According to ASCE/SEI 41, plastic rotation at shear failure and axial-load failure is calculated by subtracting the yield rotation  y from total rotation. The total column rotation  l and  a is taken as the measured column drift at shear failure and axial-load failure. The method to obtain the plastic rotation is consistent with the definition used to construct the table of recommended modeling parameters in Elwood et al. (2007). For the purpose of evaluating the modeling parameters a and b , the plastic rotation ratio are defined as the ratio of measured plastic rotation to the corresponding modeling parameters. Plastic Rotation Ratio (PRR) at shear failure = l   y  a  Equation 6.4  Plastic Rotation Ratio (PRR) at axial-load failure =  a   y  b  Equation 6.5  Where  l = the total rotation at shear failure;  a = the total rotation at axial-load failure;  y = the yield rotation.  151  6.4.2  Influence of Varying Column Axial Load The recommended values of modeling parameters a and b for columns can be found in  ASCE/SEI 41 (Table 6-8 Modeling parameters and numerical acceptance criteria for nonlinear procedures-reinforced concrete columns). For columns in condition i and iii, plastic rotations a and b are based on axial load ratio P / Ag f c' and transverse reinforcement ratio t  Av / bw s ; linear interpolation between values is required to achieve best estimation. For condition ii, modeling parameters a and b are obtained based on axial load ratio P / Ag f c' , transverse reinforcement ratio t and normalized shear stress V / bw d f c' ; linear interpolation among these three variables can be done in any order. Due to the overturning moment demands during the dynamic tests, outside columns in frame system are usually subjected to axial load (e.g. the maximum axial load Pmax ) larger than the initial axial load Pini . Some middle columns in the frame were subjected to additional axial load imposed by prestressed force and the movement of the actuator during the shaking table tests caused varying axial load on columns. Also the calculated shear strength V0 ' for non-ductile columns may be different from the measured shear strength VMax . Since the modeling parameters a and b represent the plastic rotations at significant loss of lateral strength and loss of axial load support, a and b are compared with the measured plastic rotation at shear and axial-load failure for The Database in Figure 6.6 and Figure 6.7, respectively. Similar to the evaluation in Chapter 4 and Chapter 5, the test data is grouped by the column failure type that generally corresponds to the ASCE/SEI 41 classification. The data series “Shear” (corresponding to the Condition iii in ASCE/SEI 41) in green triangular  152  marker represents the shear-critical columns included in the Database. The data series “FlexShear” (corresponding to the Condition ii in ASCE/SEI 41) in red diamond marker represents the flexure-shear-critical columns that were connected to a rigid beam on the upper end. The data series “Flex-Shear (Beam)” in purple diamond marker represents the flexure-shearcritical columns in the frame connected with flexure beam on the upper end and the data of measured drift may be higher than the actual column drift capacity due to the additional flexibility contributed by the beam. The “Flex-Shear (Lower)” in hollow red diamond marker represents the flexure-shear-critical columns that did not fail during the tests and the drift data could be considered as the lower bound of the column drift capacity. The “Flex-Shear (Flex)” in blue diamond marker represents the columns that experienced flexure failure during the test, yet were misclassified by ASCE/SEI 41 as flexure-shear-critical columns. 6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Parameter a (×10-2 ) (Pmax ,VMax)  Parameter a (×10-2 ) (Pini , V0 ')  6  2  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  2  0  0  0  2  4  6  Measured drift at 80% Max Veff (%)  (a) Parameter a based on Pini and V0 ' Figure 6.6  0  2  4  6  Measured drift at 80% Max Veff (%)  (b) Parameter a based on Pmax and VMax  Evaluation of modeling parameter a for non-ductile columns  153  6 Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  Parameter b (×10-2) (Pmax , VMax)  Parameter b (×10-2 ) (Pini , V0 ')  6  2  0  Shear Flex-Shear Flex-Shear (Beam) Flex-Shear (Lower) Flex-Shear (Flex)  4  2  0 0  2  4  6  Measured drift at axial-load failure (%)  (a) Parameter b based on Pini and V0 ' Figure 6.7  0  2  4  6  Measured drift at axial-load failure (%)  (b) Parameter b based on Pmax and VMax  Evaluation of modeling parameter b for non-ductile columns  Results in Figure 6.6 and Figure 6.7 show that the modeling parameters a and b based on initial axial load Pini and calculated shear strength V0 ' are slightly higher than the results interpolated based on maximum axial load ratio Pmax and measured shear strength Vmeas . Yet the difference between the modeling parameter is less than 10% for most columns based on different column axial load and shear strength. Also in structural design stage, engineer could only estimate the column plastic rotation at loss of lateral load based on the calculated shear strength and designed axial load level which could be represented by initial axial load ratio. Therefore for the assessment of level of conservatism provided by ASCE/SEI 41, modeling parameters a and b are interpolated based on initial axial load Pini and calculated shear strength V0 ' in the following sections.  154  6.4.3  Assessment of ASCE/SEI 41 Modeling Parameter a  6.4.3.1  Plastic Rotation Ratio at Loss of Lateral Resistance  As shown in Figure 1.1, modeling parameter a represents the column plastic rotation at significant loss of lateral resistant force. It is generally assumed that column shear failure occurred when the lateral force degrades to 80% of its peak strength and the lateral resistance would degrade rapidly afterwards. Figure 6.8 plots results of the assessment of parameter a for columns in the database. The data in solid markers in Figure 6.8 is PRR (plastic rotation ratio) for columns that actually failed during the test, while the hollow marker represents the PRR based on maximum drift ratio for columns that did not fail in tests and could be considered as lower bound of PRR. The red solid diamond marker (Condition ii) represents the PRR for flexure-shear critical columns that were connected by rigid beam in reinforced concrete frame, while the PRR of non-ductile columns that were connected by a flexible beam are represented by purple solid diamond marker, namely Condition ii (Beam). Influenced by the deflection of the flexible beam, the measured column rotation may include significant rigid body rotation at the ends of columns. The PRR represented by purple solid diamond marker might be higher than the actual value and should be evaluated with caution. The horizontal line at 1.0 represents the case where test results exactly match the plastic rotation limit in ASCE/SEI 41. Two ductile outside columns C1 and C3 in specimen S1 of NCREE 2007 tests were observed to experience flexure failure, yet classified as columns in condition ii. The yellow diamond marker, namely Condition ii (Flex-shear), represents the PRR for those two columns by following the procedure of interpolating table value a for columns in Condition  155  ii. Since these two columns demonstrated ductile response and experienced extended deformation, the PRR is much greater than 1 as expected. If the columns are considered to be in Condition i, as shown in yellow circular marker, the PRR is much closer to horizontal unit line and better estimations of plastic rotation are achieved.  PPR ( Plastic Rotation Ratio)  5  Condition iii  4  Condition ii  3  Condition ii (Beam) Condition ii (lower)  2  Condition ii (FlexShear) Condition ii (Flex)  1  0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc')  Figure 6.8  Plastic rotation ratio at significant loss of lateral resistant force  For columns in condition iii that are expected to experience shear failure prior to flexure yielding, ASCE/SEI 41 suggests that no plastic rotation could be relied upon the loss of lateral resistance force. Therefore, the plastic rotation ratio has to be replaced by total rotation ratio l /  y in Figure 6.8, represented by triangular markers. Results indicated that ignoring all the plastic rotation is conservative for those shear-critical columns. However, the plastic rotation l   y at the time of loss of significant lateral resistance for 6 shear-critical columns included in The Database is less than 1%, justifying the conservatism of recommendation of ASCE 41 for the sake of safety. PRR for most flexure-critical columns is larger than 1, indicating ASCE/SEI 41 tends to provide conservative estimation of the plastic rotations at significant loss of lateral resistance for columns in condition ii. This conservatism is appropriate since axial capacity degradation is closely associated with the development of shear-failure plane. Almost all 156  columns in The Database were subjected to axial load ratio less than 0.25; hence caution should be taken when applying ASCE/SEI 41 to columns with higher axial load ratio.  6.4.3.2  Probability of Failure  From Figure 6.8, it is shown that significant scatter existed among the data of plastic rotation ratio, emphasizing the need to determine the level of safety provided by ASCE/SEI 41 provisions, rather than to analyze the mean value and coefficient variation of the data itself. Results of plastic rotation ratio could be represented by a lognormal cumulative distribution, commonly referred to as a fragility curve, as shown in Figure 6.9. The fragility curve is constructed based on the test data of flexure-shear-critical columns that experienced the shear failure during the tests (the data series in red diamond marker in Figure 6.8). 1 0.9  Probability of Failure (Pf)  0.8 0.7 0.6  0.5 0.4 0.3 0.2  Flexure-Shear-critical  0.1  Fragility curve  0 0  Figure 6.9  1  2 3 Plastic rotation ratio  4  5  Lognomal fragility curve for plastic rotation at significant loss of lateral resistance  Based on Figure 6.9, there is 11% probability of failure for columns in “Condition ii” when experiencing plastic rotation demand equal to parameter a specified in ASCE/SEI 41.  157  The “failure” here refers to the point at which the column has lost more than 20% of its lateral resistance. According to ASCE/SEI 41, applying the parameter a specified in the table shall achieve the probability of failure Pf less than 15% for columns vulnerable to shear failure, i.e. columns in condition ii and iii, while for flexure critical columns in condition i, the probability of failure Pf up to 35% is acceptable since flexure failure is less sudden and brittle. The resulting probability of failure at loss of significant lateral resistance for flexureshear-critical columns in The Database is consistent with ASCE/SEI 41 targets.  6.4.4  Assessment of ASCE/SEI 41 Modeling Parameter b  6.4.4.1  Plastic Rotation Ratio at Loss of Axial Load Capacity  Modeling parameter b in ASCE/SEI 41 refers to plastic rotation at the time that column loses gravity load support. The ratio of measured plastic rotation at axial-load failure over the table value is plotted in Figure 6.10. The type of columns corresponding to each data  PPR ( Plastic Rotation Ratio)  series can be found in section 6.4.3.1 of this thesis. 5  Condition iii  4  Condition ii  3  Condition ii(Beam) Condition ii(lower)  2  Condition ii(FlexShear) Condition ii(Flex)  1  0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc')  Figure 6.10  Plastic rotation ratio at loss of axial load support  158  Results show that PRR for most shear-critical columns is close to or below the unit horizontal line, indicating that the plastic rotation estimated by ASCE/SEI 41 can be unconservative for columns in Condition iii. It is noticed that unlike other shear-critical columns, PPR is greater than 4 for column C2 in specimen S1 in NCREE 2007 Tests. It might be attributed to the fact that unlike other shear-critical columns in The Database that were part of portal frames, column C2 is the middle column in the two-bay planar frame and axial load could be redistributed to outside ductile columns after column axial-load failure, resulting in larger apparent deformation capacity (the column NCREE 2007-S1-C2 was subjected to tension instead of axial compression in the later stage of the tests). Interaction with other two ductile outside columns may greatly increase the drift capacity at axial-load failure for column C2. ASCE/SEI 41 tends to provide conservative estimation of plastic rotation at loss of gravity load support for most of columns in “Condition ii”. Two non-ductile columns (in Condition ii) C1 and C2 in specimen L of NCREE 2005 tests have exceptional high plastic rotation ratio equal to 3.7 and 4.2 respectively. With extended deformations in the positive direction were observed from the hysteretic response, yet those two columns experienced substantial damage after the failure of lap-sliced longitudinal reinforcements. The failure modes of the columns may be changed to the flexural failure, since the column could not transfer the moment and shear resistance after the failure of lap-spliced longitudinal reinforcement. The frame was weaker but the column would not experience shear failure.  159  6.4.4.2  Probability of Failure  Results of plastic rotation ratio could be represented by a lognormal cumulative distribution, commonly referred to as a fragility curve, as shown in Figure 6.11. Since loss of gravity load support of certain columns could result in the collapse of structural system, higher level of safety should be achieved. Elwood et al. (2007) suggested that the probability of failure Pf should be less than 15% if columns experience a plastic rotation demand equal to the parameter b specified in ASCE/SEI 41. “Failure” here refers to the point that the column loses axial load support. The fragility curve is constructed based on the test data of flexure-shear-critical columns that experienced axial-load failure during the tests (the data series in red diamond marker in Figure 6.10). 1 0.9  Probability of Failure (Pf)  0.8 0.7 0.6 0.5 0.4 0.3  0.2  Flexure-Shear-critical  0.1  Fragility curve  0 0  1  2  3  4  5  plastic rotation ratio Figure 6.11  Lognomal fragility curve for plastic rotation at loss of axial-load support  Based on Figure 6.11, there is a 7.5% probability of failure for columns in “Condition ii” experiencing a plastic rotation demand equal to b specified in ASCE/SEI 41. 160  6.4.4.3  Comparison of Plastic Rotation at Shear Failure and Axial-Load Failure  Figure 6.12 compares the plastic rotation at loss of axial load capacity and at significant loss of lateral resistant force. Result of  p   Axial    p   Shear  in Figure 6.12(a) is the drift “gap”  that column exhibited between shear failure and the axial-load failure. The data of parameters a  b in purple markers corresponds to the  p   Axial    p   Shear  specified by ASCE/SEI 41  was found to be around 0.006 for most non-ductile columns. Because that the columns in The Database had similar transverse reinforcement ratio, axial load ratio as well as the shear stress level, the resulting parameters a and b are very close to each other. Measured plastic rotation in green markers is more scattered and many columns had  p   Axial    p   Shear  close  to zero, since these column lost 20% of its peak lateral strength at a point quite close to or even after axial-load failure. Results in Figure 6.12(b) also show that the measured  p   Axial     p Shear  tends to be  slightly higher than 1.0, while parameter b a for most flexure-shear-critical columns was around 1.5.  161  Meas-Condition iii Meas-Condition ii Meas-Condition ii (Beam) ASCE-Condition iii ASCE-Condition ii ASCE-Condition ii (Beam)  1.5  (×10-2)  θ p(Axial) - θp(Shear)  2  1  0.5  0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc')  (a) Plastic rotation at axial-load failure minus the plastic rotation at shear failure 2.5 Meas-Condition ii Meas-Condition ii (Beam) ASCE-Condition ii ASCE-Condition ii (Beam)  θp (Axial) / θp (Shear)  2 1.5  1 0.5 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ratio (Pini /Agfc')  (b) Ratio of plastic rotation at axial-load failure over the plastic rotation at shear failure Figure 6.12  Comparison of plastic rotation at axial-load failure and shear failure  It is observed that non-ductile columns in The Database experienced 20% of shear degradation at relatively large drift and many of them achieved same column drift at shear failure and axial-load failure. Since the shaking table tests usually last for a relatively short time, columns could only go through limited number of cycles during the tests and might not dissipate as much energy as the columns subjected to reversed-cyclic tests. Therefore gap between the measured drift at shear failure and at axial-load failure is quite small for columns subjected to shaking table tests. This also helps to explain the fact that while the drift models could capture the column drift capacity at axial-load failure fairly well, the measured drifts at shear failure tend to greatly exceeds the calculated drifts from available models. 162  6.4.5  Modeling Parameters for Columns in Condition i Unlike the static cyclic tests conducted on single columns in past research, shaking  table tests are often terminated at the collapse of whole frames or the end of strong input ground motion. Due to the early failure of non-ductile columns in the frames, the collapse of the structure induced the P-Delta failure of the flexure-critical columns in Condition i. The maximum plastic rotation measured during the tests is compared with the modeling parameter b in Figure 6.13 (in solid blue round marker). For flexure-critical columns belong to the frames that did not collapse are marked in blue hollow circle and the maximum plastic rotation could be considered as the lower bound of the drift capacity at axial-load failure. 10  10  Condition i Parameter b (×10-2) (Pmax)  Parameter b (×10-2) (Pini)  Condition i Condition i (Lower)  8  6  4  Condition i (Lower)  8  6  4  2  2  0  0  0  2  4  6  8  10  0  2  4  6  8  Measured Max plastic rotation (%)  Measured Max plastic rotation (%)  (a) Parameter b based on Pini  (b) Parameter b based on Pmax  Figure 6.13  10  Evaluation of modeling parameter b for columns in Condition i  The modeling parameters for columns in Condition i is obtained by linearly interpolating the values specified in ASCE/SEI 41 based on the transverse reinforcement ratio and axial load ratio, thus the parameter b in Figure 6.13 (a) and (b) is relatively more sensitive to the axial load ratio than that for columns in other conditions.  163  Plastic rotation ratio (Pini)  2.5 Condition i  2  Condition i (Lower)  1.5 1 0.5 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc') (a) PRR based on initial axial load Pini Plastic rotation ratio (Pmax)  2.5 Condition i  2  Condition i (Lower)  1.5 1 0.5 0 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Initial Axial Load Ration (Pini /Agfc') (a) PRR based on maximum axial load Pmax Figure 6.14  Plastic rotation ratio for columns in Condition i  The PRR (plastic rotation ratio) based on initial axial load Pini and maximum axial load  Pmax is plotted in Figure 6.14. Results show that modeling parameter b based on initial axial load Pini in Figure 6.14 (a) appears to represent the mean value of the plastic rotation at axialload failure for flexure-critical columns and is pretty conservative for most columns in Condition i that experienced P-Delta failure during the tests. The modeling parameter b based on Pini is unconservative for the ductile columns C3 in Specimen S1 of NCREE 2007 Tests (PRR=0.73), ductile columns C1 (PRR=0.83) and D1 (PRR=0.84) in Ghannoum 2006 Tests. After the axial-load failure of non-ductile columns in the frame, flexure-critical column C1 in Ghannoum 2006 Tests and column C3 in Specimen  164  S1 of NCREE 2007 Tests picked up significant amount of axial load redistributed from the axially collapsed non-ductile columns. The modeling parameter b based on maximum axial load Pmax in Figure 6.14 (b) is indeed a better reflection of the plastic drift capacity for those two columns (PPR equals to 1.51 and 1.16 based on Pmax ) respectively. The PPR based on Pmax for column D1 in Ghannoum 2006 is slightly lower than 1.0 (PRR=0.92). It is noticed that the test data for several columns are truncated, such as column D1 in Ghannoum 2006 Tests, since the column drift and shear force measured from the instruments are not reliable beyond certain point. The maximum drift obtained from the hysteretic response of columns may be lower than the actual column plastic drift.  6.5 6.5.1  Column Lateral Load-Deformation Backbones Experimental Envelope of Effective Lateral Force To assess the generalized load-deformation relationships recommended by ASCE/SEI  41 and better understand the column response, experimental envelopes and normalized envelopes are developed for non-ductile columns in The Database by following the procedure discussed in section 2.7 of this research. Figure 6.15 to Figure 6.19 present the normalized experimental envelopes for nonductile columns in several test programs (i.e. NCREE 2009 Tests, NCREE 2007 Tests, Elwood 2002 Tests and Shin 2005 Tests). The dashed line in the figures shows the column response and shear degradation beyond the axial-load failure point (defined in section 5.1.3). The numbers in the bracket are the maximum axial load ratio measured during the tests for certain columns.  165  1.2  1.0  1.0  0.8  0.8  0.6  (0.23)  0.4  (0.28)  0.2  (0.43)  0.0  NCREE NCREE NCREE NCREE NCREE NCREE NCREE NCREE NCREE NCREE  -0.2  -0.4 -0.6 -0.8 -1.0  (0.26) 2009-HCFS-A1 2009-HCFS-B1 2009-HCFS-C1 2009-MCFS-A1 2009-MCFS-B1 2009-MCFS-C1 2009-MUFS-A1 2009-MUFS-B1 2009-MUFS-B1 2009-MUFS-C1  Normalized Force (V/Vmax)  Normalized Force (V/Vmax)  1.2  0.6  0.4 0.2 0.0 -0.2  NCREE NCREE NCREE NCREE NCREE NCREE NCREE NCREE NCREE  -0.4 -0.6 -0.8 -1.0  -1.2  2009-HCFS-A1 2009-HCFS-B1 2009-HCFS-C1 2009-MCFS-A1 2009-MCFS-B1 2009-MCFS-C1 2009-MUFS-A1 2009-MUFS-B1 2009-MUFS-C1  -1.2 -4  -2  0  2  4  6  -3  -2  -1  Ductility (μ=Δ/Δyield)  0  1  2  3  4  5  Hori Drift (%)  Note: the number in the bracket is the maximum axial load ratio measured during the test for each column; the dashed line is the column behavior beyond the axial-load failure point. Figure 6.15  Normalized experimental envelopes for columns in NCREE 2009 Tests  1.2  1.2  1.0  1.0  (0.12)  Normalized Force (V/Vmax)  Normalized Force (V/Vmax)  0.8 0.6 0.4  (0.25)  0.2 0.0 -0.2 -0.4  0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6  -0.6  -0.8  -0.8 Elwood 2002-S1-CenC Elwood 2002-S2-CenC  -1.0  Elwood 2002-S1-CenC Elwood 2002-S2-CenC  -1.0 -1.2  -1.2 -4  -2  0  2  Ductility (μ=Δ/Δyield)  4  6  -4  -2  0  2  4  6  Hori Drift (%)  Note: the number in the bracket is the maximum axial load ratio measured during the test for each column; the dashed line is the column behavior beyond the axial-load failure point. Figure 6.16  Normalized experimental envelope for Elwood 2002 Tests  166  Shin 2005-II-Test 1-Chile-0.1-WestC Shin 2005-II-Test 7-Chile-0.24-WestC Shin 2005-III-Test 4-Chile-0.1-EastC Shin 2005-III-Test 4-Chile-0.1-WestC Shin 2005-III-Test 10-Chile-0.24-EastC Shin 2005-III-Test 10-Chile-0.24-WestC  Shin 2005-II-Test 1-Chile-0.1-WestC Shin 2005-II-Test 7-Chile-0.24-WestC Shin 2005-III-Test 4-Chile-0.1-EastC Shin 2005-III-Test 4-Chile-0.1-WestC Shin 2005-III-Test 10-Chile-0.24-EastC Shin 2005-III-Test 10-Chile-0.24-WestC  1.2  1.2  1.0  0.8  Normalized Force (V/Vmax)  Normalized Force (V/Vmax)  (0.24)  (0.13)  (0.23)  0.4  0.0  -0.4  (0.13)  0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6  (0.24)  -0.8  -0.8  (0.13)  -1.0 -1.2  -1.2 -3  -2  -1  0  1  2  -4  3  -2  Ductility (μ=Δ/Δyield)  Figure 6.17  2  4  Normalized envelopes for columns subjected to Chile Earthquake in Shin 2005 Tests Shin 2005-II-Test 2-Kobe-0.1-WestC Shin 2005-II-Test 8-Kobe-0.24-WestC Shin 2005-III-Test 6-Kobe-0.1-EastC Shin 2005-III-Test 6-Kobe-0.1-WestC  Shin 2005-II-Test 2-Kobe-0.1-WestC Shin 2005-II-Test 8-Kobe-0.24-WestC Shin 2005-III-Test 6-Kobe-0.1-EastC Shin 2005-III-Test 6-Kobe-0.1-WestC  1.2  1.2  1.0  1.0  0.8 0.6  0.8  (0.12)  (0.14)  Normalized Force (V/Vmax)  Normalized Force (V/Vmax)  0  Hori Drift (%)  0.4  (0.23)  (0.12)  0.2 0.0 -0.2  -0.4  0.6 0.4  0.2 0.0 -0.2 -0.4  -0.6  -0.6  -0.8  -0.8  -1.0  -1.0 -1.2  -1.2 -2  0  2  Ductility (μ=Δ/Δyield)  Figure 6.18  4  6  -2  0  2  4  6  Hori Drift (%)  Normalized envelopes for columns subjected to Kobe Earthquake in Shin 2005 Tests  167  1.2  1.2 1.0  Normalized Force (V/Vmax)  0.8 0.6  2007-S1-C2 2007-S3-C1 2007-S3-C3 2007-S4-C1 2007-S4-C3  1.0  Normalized Force (V/Vmax)  NCREE NCREE NCREE NCREE NCREE  (0.16)  0.4 0.2  (0.04)  0.0 -0.2  (0.12)  (0.09)  -0.4  0.8 0.6 0.4  0.2 0.0 -0.2  -0.4  -0.6  -0.6  -0.8  -0.8  -1.0  -1.0  NCREE NCREE NCREE NCREE NCREE  2007-S1-C2 2007-S3-C1 2007-S3-C3 2007-S4-C1 2007-S4-C3  -1.2  -1.2 -4  -2  0  Ductility (μ=Δ/Δyield)  2  4  -4  -2  0  2  4  Hori Drift (%)  Note: the number in the bracket is the maximum axial load ratio measured during the test for each column; the dashed line is the column behavior beyond the axial-load failure point. Figure 6.19  Normalized envelopes for shear-critical columns in NCREE 2007 Tests  Results in Figure 6.15 show that the columns in specimen HCFS subjected to higher axial load ratio demonstrated less deformation capacity than columns in specimen MCFS in the direction of column failure. In the other direction, columns showed very similar behavior. Results in Figure 6.16 show that the column subjected to higher axial load (center column in S2) experienced more rapid and sever shear degradation than the column with lower axial load (the center column in S1). This trend is consistent with the response for flexure-shear critical columns shown in Figure 6.15 to Figure 6.19. The absolute value of the negative degrading slope increased as the column axial load ratio increased, indicating the shear degradation is more rapid and severe. As shown in Figure 6.19, the middle column NCREE 2007-S1-C2 had relatively low maximum axial load ratio equal to 0.04 and was subjected to axial tension force instead of  168  compression at the later stage of the tests, possibly due to the axial load redistribution within the system after shear failure occurred. This shear-critical column lost lateral resistance very quickly and experienced sudden axial-load failure, suggesting that columns subjected to tension force are even more vulnerable to shear degradation and loss of axial load support. From the normalized envelopes for non-ductile columns in The Database, it is observed that most non-ductile columns subjected to earthquake load achieved the maximum lateral force when ductility demand μ reached 1.5 and would suffer shear failure and axial-load failure before ductility demand μ reached 4.0. A simplified normalized lateral load-deformation response for non-ductile columns could be constructed. With calculated shear strength and yield drift estimated by available models, a simplified lateral load-deformation relation could be established in preliminary structural analysis. Figure 6.20 plots the normalized envelopes for non-ductile columns in The Database with the proposed simplified normalized backbone curve. The key points used to construct the simplified normalized lateral load-deformation for flexure-shear-critical columns connected with rigid beam are noted in Figure 6.20 (a). As shown in Figure 6.20 (b), the load-drift responses of flexure-shear-critical columns connected with relatively flexible beam are found to exhibit more deformation capacity due to the flexibility contributed by the beam compared with the proposed backbone which captures only the deformation capacity of the column. On the other hand, the load-drift responses of shear-critical columns shown in Figure 6.20 (c) exhibit lower deformation capacity compared with the simplified backbone for flexure-shear columns, demonstrating the limited deformation capacity of columns failing in pure shear.  169  1.2  (2.0 , 1.0)  (1.0 , 1.0)  1.0  Shin 2005-II-Test 1-Chile-0.1-WestC Shin 2005-II-Test 7-Chile-0.24-WestC  Normalized Force (V/Vmax)  0.8 0.6  Shin 2005-III-Test 4-Chile-0.1-EastC  0.4  Shin 2005-III-Test 4-Chile-0.1-WestC  (2.5 , 0.2)  0.2  Shin 2005-III-Test 10-Chile-0.24-EastC  (0 , 0)  0.0  Shin 2005-III-Test 10-Chile-0.24-WestC  (-2.5 , -0.2)  -0.2  Shin 2005-II-Test 2-Kobe-0.1-WestC  -0.4  Shin 2005-II-Test 8-Kobe-0.24-WestC  -0.6  Shin 2005-III-Test 6-Kobe-0.1-EastC -0.8 Shin 2005-III-Test 6-Kobe-0.1-WestC -1.0  (-2.0 , -1.0)  (-1.0 , -1.0)  Simplified Backbone  -1.2 -6  -5  -4  -3  -2  -1  0  1  2  3  4  5  6  Ductility (μ=Δ/Δyield) (a) Flexure-shear-critical columns connected with rigid beam 1.2  (1.0 , 1.0)  1.0  Elwood 2002-S1-CenC  (2.0 , 1.0)  Elwood 2002-S2-CenC  Normalized Force (V/Vmax)  NCREE 2005-P2-C1 0.8  NCREE 2005-P2-C2  0.6  NCREE 2005-L-C1  NCREE 2005-L-C2 0.4  Ghannoum 2006-A1 Ghannoum 2006-B1  0.2  (2.5 , 0.2)  NCREE 2009-HCFS-A1  (0 , 0)  0.0  (-2.5 , -0.2)  NCREE 2009-HCFS-B1 NCREE 2009-HCFS-C1  -0.2  NCREE 2009-MCFS-A1 NCREE 2009-MCFS-B1  -0.4  NCREE 2009-MCFS-C1 -0.6  NCREE 2009-MUFS-A1  -0.8  NCREE 2009-MUFS-B1 NCREE 2009-MUFS-B1  -1.0  (-2.0 , -1.0) -1.2 -6  -5  -4  -3  -2  NCREE 2009-MUFS-C1  (-1.0 , -1.0) -1  0  1  Simplified Backbone 2  3  4  5  6  Ductility (μ=Δ/Δyield) (b) Flexure-shear-critical columns connected with relatively flexible beam  170  1.2  (1.0 , 1.0)  1.0  (2.0 , 1.0)  NCREE 2007-S1-C2 NCREE 2007-S1-C2  Normalized Force (V/Vmax)  0.8  NCREE 2007-S3-C1  0.6  NCREE 2007-S3-C1  0.4  NCREE 2007-S3-C3  (2.5 , 0.2)  0.2  NCREE 2007-S3-C3  (0 , 0)  0.0  NCREE 2007-S4-C1  (-2.5 , -0.2)  -0.2  NCREE 2007-S4-C1  -0.4  NCREE 2007-S4-C3  -0.6  NCREE 2007-S4-C3  -0.8  Simplified Envelope  -1.0  (-2.0 , -1.0)  (-1.0 , -1.0)  -1.2 -6  -5  -4  -3  -2  -1  0  1  2  3  4  5  6  Ductility (μ=Δ/Δyield) (c) Shear-critical columns Figure 6.20  Normalized envelopes for non-ductile columns with proposed simplified envelopes  Figure 6.21 plots the normalized experimental envelope for non-ductile columns with column horizontal drift. It is observed in Figure 6.21 (a) that most flexure-shear critical columns achieve maximum lateral resistance when column drifts reach 1.2% and start to degrade when column drifts equal to 2.5%. A simplified backbone curve directly related to column drift is constructed in Figure 6.21 (a). The proposed backbone curve with the experimental envelopes for flexure-shear-critical columns connected with flexible beam and shear-critical columns are proposed in Figure 6.21 (b) and (c) respectively. Similar to the results shown in Figure 6.20 (b), columns connected with flexible beam exhibit more deformation due to the flexibility contributed by the beam. The shear-critical columns in Figure 6.21 (c) demonstrate very limited deformation compared with the simplified backbone curve.  171  The simplified envelope did not capture the load-drift relations for column NCREE 2005-L-C1 very well as shown in Figure 6.21 (b) (test series in the magenta line), possibly because the column response was changed to be dominated by flexural behaviour due to the failure of lap-spliced longitudinal reinforcement. 1.2  Normalized Force (V/Vmax)  Shin 2005-II-Test 1-Chile-0.1-WestC  (2.5 , 1.0)  (1.2 , 1.0)  1.0  Shin 2005-II-Test 7-Chile-0.24-WestC  0.8  Shin 2005-III-Test 4-Chile-0.1-EastC  0.6  Shin 2005-III-Test 4-Chile-0.1-WestC Shin 2005-III-Test 10-Chile-0.24-EastC  0.4  Shin 2005-III-Test 10-Chile-0.24-WestC  0.2  (3.0 , 0.2)  0.0  Shin 2005-II-Test 2-Kobe-0.1-WestC  (0 , 0)  Shin 2005-II-Test 8-Kobe-0.24-WestC  (-3.0 , -0.2)  -0.2  Shin 2005-III-Test 6-Kobe-0.1-EastC Shin 2005-III-Test 6-Kobe-0.1-WestC  -0.4  NCREE 2004-S2-C1  -0.6  NCREE 2004-S2-C2  -0.8  NCREE 2004-S3-C1  -1.0  NCREE 2004-S3-C2  (-2.5 , -1.0)  -1.2 -6  -5  -4  -3  -2  (-1.2 , -1.0) -1  0  1  Simplified Backbone  2  3  4  5  6  Horizontal Drift (%) (a) Flexure-shear-critical columns connected with rigid beam  172  1.2  NCREE 2009-HCFS-A1  (2.5 , 1.0)  (1.2 , 1.0)  1.0  NCREE 2009-HCFS-B1 NCREE 2009-HCFS-C1  0.8  Normalized Force (V/Vmax)  NCREE 2009-MCFS-A1 0.6  NCREE 2009-MCFS-B1  0.4  NCREE 2009-MCFS-C1 NCREE 2009-MUFS-A1  (3.0 , 0.2)  0.2  NCREE 2009-MUFS-B1  NCREE 2009-MUFS-C1  (0 , 0)  0.0  Elwood 2002-S1-CenC  (-3.0 , -0.2)  -0.2  Elwood 2002-S2-CenC NCREE 2005-P2-C1  -0.4  NCREE 2005-P2-C2  -0.6  NCREE 2005-L-C1  -0.8  NCREE 2005-L-C2  Ghannoum 2006-A1  -1.0  (-2.5 , -1.0)  -1.2  Ghannoum 2006-B1  (-1.2 , -1.0)  Simplified Backbone  -6  -5  -4  -3  -2  -1  0  1  2  3  4  5  6  Horizontal Drift (%) (b) Flexure-shear-critical columns connected with relatively flexible beam 1.2  (2.5 , 1.0)  (1.2 , 1.0)  1.0  NCREE 2007-S1-C2  NCREE 2007-S1-C2  Normalized Force (V/Vmax)  0.8 0.6  NCREE 2007-S3-C1  0.4  NCREE 2007-S3-C1  0.2  (3.0 , 0.2)  0.0  NCREE 2007-S3-C3  (0 , 0)  NCREE 2007-S3-C3  (-3.0 , -0.2)  -0.2  NCREE 2007-S4-C1  -0.4  NCREE 2007-S4-C1  -0.6 NCREE 2007-S4-C3  -0.8 NCREE 2007-S4-C3  -1.0  (-2.5, -1.0)  -1.2 -6  -5  -4  -3  (-1.2 , -1.0) -2  -1  0  1  Simplified Backbone  2  Horizontal Drift (%)  3  4  5  6  (b) Shear-critical columns Figure 6.21  Envelope for non-ductile columns with normalized force and column horizontal drift  173  Figure 6.22 plots the experimental envelopes for non-ductile columns with normalized force and column plastic rotation. The column plastic rotation is calculated by subtracting the measured yield drift  y (discussed in section 3.2.5) from the total column drift. As shown in Figure 6.22 (a), the proposed simplified backbone could represent the plastic rotation for columns connected with rigid beam fairly well, only slightly underestimating the plastic drift for two columns (column Shin 2005-III-Test 4-Chile-0.1WestC and Shin 2005-III-Chile-0.24-WestC). The plastic rotation for columns connected with flexible beams shown in Figure 6.22 (b) is generally lager than that of columns connected with rigid beam. Columns connected with relatively flexible beams in The Database are usually from multiple-bay multiple-storey frames and thus the column interactions and axial load redistribution mechanism may allow the columns to exhibit more extended plastic deformation after the column yielding.  174  1.2  Normalized Force (V/Vmax)  1.0  (0, 1.0)  Shin 2005-II-Test 1-Chile-0.1-WestC Shin 2005-II-Test 7-Chile-0.24-WestC Shin 2005-III-Test 4-Chile-0.1-EastC Shin 2005-III-Test 4-Chile-0.1-WestC Shin 2005-III-Test 10-Chile-0.24-EastC Shin 2005-III-Test 10-Chile-0.24-WestC Shin 2005-II-Test 2-Kobe-0.1-WestC Shin 2005-II-Test 8-Kobe-0.24-WestC Shin 2005-III-Test 6-Kobe-0.1-EastC Shin 2005-III-Test 6-Kobe-0.1-WestC NCREE 2004-S2-C1 NCREE 2004-S2-C2 NCREE 2004-S3-C1 NCREE 2004-S3-C2 Simplified Backbone  (1.3 , 1.0)  0.8  0.6 0.4 0.2  (1.8 , 0.2)  0.0 0  1  2  3  4  Column Plastic Drift (%)  (a) Flexure-shear-critical columns connected with rigid beam 1.2  NCREE 2009-HCFS-A1 NCREE 2009-HCFS-B1 NCREE 2009-HCFS-C1 NCREE 2009-MCFS-A1 NCREE 2009-MCFS-B1 NCREE 2009-MCFS-C1 NCREE 2009-MUFS-A1 NCREE 2009-MUFS-B1 NCREE 2009-MUFS-C1 Elwood 2002-S1-CenC Elwood 2002-S2-CenC NCREE 2005-P2-C1 NCREE 2005-P2-C2 NCREE 2005-L-C1 NCREE 2005-L-C2 Ghannoum 2006-A1 Ghannoum 2006-B1 Simplified Backbone  (1.3, 1.0)  (0, 1.0)  Normalized Force (V/Vmax)  1.0  0.8 0.6 0.4 0.2  (1.8 , 0.2)  0.0 0  1  2  3  4  Column Plastic Drift (%)  5  6  (b) Flexure-shear-critical connected with relatively flexible beam 1.2  NCREE 2007-S1-C2  Normalized Force (V/Vmax)  (0, 1.0) 1.0  (1.3 , 1.0)  NCREE 2007-S1-C2 NCREE 2007-S3-C1 NCREE 2007-S3-C1  0.8  NCREE 2007-S3-C3 NCREE 2007-S3-C3  0.6  NCREE 2007-S4-C1 NCREE 2007-S4-C1  0.4  NCREE 2007-S4-C3 NCREE 2007-S4-C3  0.2  (1.8 , 0.2)  Simplified Backbone  0.0 0  1  2  3  4  5  6  Column Plastic Drift (%)  (c) Shear-critical columns Figure 6.22  Envelope with normalized force and column plastic drift  175  6.5.2  ASCE/SEI Backbone Curve The ASCE/SEI 41 backbone curves for non-ductile columns in The Database are  constructed based on the plastic shear demand, measured maximum effective force as well as the effective stiffness ratio and modeling parameter a , b and c recommended by ASEC/SEI 41. For example the ASCE/SEI backbone curves for column Shin 2005-II-Test 1-Chile-0.1 are plotted in Figure 6.23.  45 Experimental  B  ASCE (w. Calc)  30  C  ASCE (w. Meas)  Lateral Force (kN)  15  D  A  0  E  E  D  -15  -30  C  B  -45 -6  Figure 6.23  -4  -2  0  Hori Drift (%)  2  4  6  ASCE backbone cures for column Shin 2005-II-Test 1-Chile-0.1-WestC  The backbone curve “ASCE (w. Calc)” is constructed by following procedure: The slope for the column initial response from A to B (the initial stiffness keff ) in Figure 1.1 is derived from the ratio of modulus of gross section ( EI eff EI g ) recommended by ASCE/SEI 41, assuming that the column has fixed-fixed end restraints. 1. The ordinate of the point B (the maximum lateral force in the backbone curve) for flexure-shear-critical column is the plastic shear demand V p obtained from the moment-curvature analysis; for shear-critical column, the calculated shear strength 176  V0 ' is used instead since the plastic shear demand V p is expected to be larger than V0 ' . 2. The column sustains the maximum lateral force up to the point C. The drift gap between the point B and C is the modeling parameter a obtained from the table in ASCE/SEI 41 based on the initial axial load ratio and calculated shear strength V0 ' . 3. The lateral force then degrades to the point D, the residual force could be calculated by modeling parameter c in ASCE/SEI 41 based on the initial axial load ratio and calculated shear strength. 4. The column response is terminated at point E. The drift gap between point B and E is the modeling parameter b based on the initial axial load ratio and calculated shear strength. The backbone curve “ASCE (w. Meas)” is constructed by following the similar procedure, except that: 1. The coordinate B is the measured yield drift from the TEST method discussed in section 3.2.4 and the ordinate is the measured maximum effective force in both positive and negative directions. 2. The modeling parameters a , b and c are obtained from the ASCE/SEI 41 based on the measured maximum axial load ratio and measured shear strength VMax .  Figure 6.24 to Figure 6.30 plot the ASCE backbone curve with the experimental envelope for the non-ductile columns included in The Database. The dashed line represented the column behaviour beyond the axial-load failure point (defined in section 5.1.4).  177  (a) NCREE 2009-HCFS-A1  (b) NCREE 2009-HCFS-B1  120  120 Experimental ASCE (w. Calc) ASCE (w. Meas)  Experimental 90  ASCE (w. Calc) ASCE (w. Meas)  60  60  30  Shear (kN)  Shear (kN)  90  0  -30  30 0  -30  -60  -60  -90  -90  -120  -4  -2  0  2  4  -120 -4  Hori Drift (%)  (c) NCREE 2009-HCFS-C1  120  ASCE (w. Meas)  Shear (kN)  Shear (kN)  0  ASCE (w. Meas)  30 0  -30  -30  -60  -60  -90  -90 -120  -120 -4  -2  0  2  Hori Drift (%)  -4  4  (e) NCREE 2009-MCFS-B1  0  Hori Drift (%)  2  4  Experimental  90  ASCE (w. Calc) ASCE (w. Meas)  60  -2  (f) NCREE 2009-MCFS-C1  120  Experimental  90  ASCE (w. Calc) ASCE (w. Meas)  60  30  Shear (kN)  Shear (kN)  4  ASCE (w. Calc)  60  30  120  2  Experimental  90  ASCE (w. Calc)  60  0  Hori Drift (%)  (d) NCREE 2009-MCFS-A1  120  Experimental  90  -2  0  30 0  -30  -30  -60  -60  -90  -90  -120  -120 -4  -2  0  Hori Drift (%)  2  4  -4  -2  0  Hori Drift (%)  2  4  178  100  (h) NCREE 2009-MUFS-B1  100  Experimental  Experimental  80  80  ASCE (w. Calc)  ASCE (w. Calc)  60  60  ASCE (w. Meas)  Lateral Force (kN)  Shear (kN)  40 20 0 -20 -40  20 0 -20 -40 -60  -80  -80  -100  -100  -4  -2  0  Hori Drift (%)  2  -4  4  -2  0  2  4  Hori Drift (%)  (i) NCREE 2009-MUFS-C1  100  Experimental  80  ASCE (w. Calc)  60  Lateral Force (kN)  ASCE (w. Meas)  40  -60  UFS-B1  )  (g) NCREE 2009-MUFS-A1  ASCE (w. Meas)  40 20 0 -20 -40 -60 -80  -100 2  4  -4  -2  0  2  4  Hori Drift (%)  Note: The dashed line represents the column behavior beyond the axial-load failure point Figure 6.24  ASCE/SEI 41 backbone curves for columns in NCREE 2009 Tests  179  (a) NCREE 2005-P2-C1  40  (b) NCREE 2005-P2-C2  40  Experimental ASCE (w. Calc)  20  ASCE (w. Meas)  0  Shear (kN)  Shear (kN)  20  -20  -20  Experimental  -40  0  ASCE (w. Calc) ASCE (w. Meas)  -60 -6  -4  -2  0  2  4  6  -40 -6  Hori Drift (%)  2  4  6  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  40  ASCE (w. Meas)  ASCE (w. Meas)  20  Shear (kN)  20  Shear (kN)  0  (d) NCREE 2005-L-C2  60  Experimental  40  -2  Hori Drift (%)  (c) NCREE 2005-L-C1  60  -4  0  0  -20  -20  -40  -40  -60  -60 -10  -5  0  5  Hori Drift (%)  Figure 6.25  10  -10  -5  0  5  10  Hori Drift (%)  ASCE/SEI 41 backbone curves for columns in NCREE 2005 Tests  180  (a) Ghannoum 2006-A1  60  (b) Ghannoum 2006-B1  60  Experimental  Experimental  ASCE (w. Calc)  40  ASCE (w. Calc)  40  ASCE (w. Meas)  ASCE (w. Meas)  20  Shear (kN)  Shear (kN)  20  0  -20  0  -20  -40  -40  -60  -60 -6  -4  -2  0  2  4  6  -6  -4  Hori Drift (%)  Figure 6.26  2  4  6  (b) Elwood 2002-S2-CenC  100  80  80  60  60  40  40  20  20  Shear (kN)  Shear (kN)  0  ASCE/SEI 41 backbone curves for columns in Ghannoum 2006 Tests  (a) Elwood 2002-S1-CenC  100  -2  Hori Drift (%)  0 -20 -40  0 -20 -40  Experimental  -60 -80  Experimental  -60  ASCE (w. Calc)  ASCE (w. Calc)  -80  ASCE (w. Meas)  ASCE (w. Meas)  -100  -100 -6  -4  -2  0  Hori Drift (%)  Figure 6.27  2  4  6  8  -6  -4  -2  0  2  4  6  8  Hori Drift (%)  ASCE/SEI 41 backbone curves for columns in Elwood 2002 Tests  181  (a) NCREE 2004-S2-C1  40  (b) NCREE 2004-S2-C2  60  40  20  Shear (kN)  Shear (kN)  20 0  -20  0  -20 Experimental  Experimental  -40  -40  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  ASCE (w. Meas)  -60  -60 -8  -4  0  4  Hori Drift (%)  -8  8  8  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  40  ASCE (w. Meas)  ASCE (w. Meas)  20  Shear (kN)  20  Shear (kN)  4  (d) NCREE 2004-S3-C2  60  Experimental  40  0  Hori Drift (%)  (c) NCREE 2004-S3-C1  60  -4  0  0  -20  -20  -40  -40  -60  -60 -6  -4  -2  0  Hori Drift (%)  Figure 6.28  2  4  6  -6  -4  -2  0  2  4  6  Hori Drift (%)  ASCE/SEI 41 backbone curves for columns in NCREE 2004 Tests  182  (a) Shin 2005-II-Chile-0.1-WestC  50  Experimental  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  0  -25  0  -25  -50  -50 -8  -4  0  4  8  -8  Hori Drift (%) 50  (c) Shin 2005-II-Chile-0.24-WestC Experimental  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  4  8  ASCE (w. Meas)  25  Shear (kN)  Shear (kN)  0  Hori Drift (%)  (d) Shin 2005-II-Kobe-0.24-WestC  0  -25  0  -25  -50  -50 -8  -4  0  4  Hori Drift (%)  8  -8  (e) Shin 2005-III-Chile-0.1-EastC  -4  0  4  Hori Drift (%)  8  (f) Shin 2005-III-Chile-0.1-WestC 50  Experimental  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  ASCE (w. Meas)  25  Shear (kN)  25  Shear (kN)  -4  50  25  50  ASCE (w. Meas)  25  Shear (kN)  25  Shear (kN)  (b) Shin 2005-II-Kobe-0.1-WestC 50  0  -25  0  -25  -50  -50 -6  -4  -2  0  2  Hori Drift (%)  4  6  -6  -4  -2  0  2  4  6  Hori Drift (%)  183  (g) Shin-2005-III-Kobe-0.1-EastC  (h) Shin 2005-III-Kobe-0.1-WestC  50  50 Experimental  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  0  -25  0  -25  -50  -50 -6  -4  -2  0  2  4  6  -6  Hori Drift (%)  -2  0  2  4  6  50  Experimental  Experimental  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas)  ASCE (w. Meas)  25  Shear (kN)  25  -4  Hori Drift (%) (j) Shin 2005-III-Chile-0.24-WestC  (i) Shin 2005-III-Chile-0.24-EastC  50  Shear (kN)  ASCE (w. Meas)  25  Shear (kN)  25  Shear (kN)  Experimental  0  -25  0  -25  -50  -50 -4  -2  0  2  Hori Drift (%)  4  -4  -2  0  2  4  Hori Drift (%)  Note: The dashed line represents the column behavior beyond the axial-load failure point Figure 6.29  ASCE/SEI 41 backbone curves for columns in Shin 2005 Tests  184  (a) NCREE 2007-S3-C1  250  Experimental  200  ASCE (w. Calc)  150  ASCE (w. Meas)  100  100  50  50  Shear (kN)  Shear (kN)  Experimental  200  ASCE (w. Calc)  150  (b) NCREE 2007-S3-C3  250  0 -50  0 -50  -100  -100  -150  -150  -200  -200  -250  ASCE (w. Meas)  -250 -2  -1  0  1  2  -2  Hori Drift (%)  (c) NCREE 2007-S4-C1  200  1  2  Experimental 150  ASCE (w. Calc)  ASCE (w. Calc)  ASCE (w. Meas) 100  100  50  50  Shear (kN)  Shear (kN)  0  Hori Drift (%)  (d) NCREE 2007-S4-C3  200  Experimental 150  -1  0 -50  0 -50  -100  -100  -150  -150  -200  ASCE (w. Meas)  -200 -2  -1  0  1  2  -2  -1  Hori Drift (%)  (e) NCREE 2007-S1-C2  200  2  Experimental  200  ASCE (w. Calc)  100  1  (a) NCREE 2007-S3-C1  250  Experimental  150  0  Hori Drift (%)  ASCE (w. Calc)  150  ASCE (w. Meas)  ASCE (w. Meas)  100  Shear (kN)  Shear (kN)  50 0 -50  50 0 -50  -100 -100 -150 -150  -200  -200  -250 -4  -2  0  Hori Drift (%)  Figure 6.30  2  4  -2  -1  0  1  2  Hori Drift (%)  ASCE/SEI 41 backbone curves for columns in NCREE 2007 Tests 185  The ASEC/SEI 41 generally predicts stiffer initial response of non-ductile columns. In NCREE 2009 Tests, outside columns A1 in the negative direction and columns C1 in the positive direction are subjected to higher axial load due to the overturning moment. It is observed in Figure 6.24 (a), (c), (d) and (f) that the maximum measured shear forces VMax are very close to the V p in the positive direction for columns A1 and in the negative direction for columns C1; while in the opposite direction, the higher axial load increased the maximum shear VMax that exceeded the plastic shear demand V p based on the initial axial load Pini . For columns in specimen MCFS shown in Figure 6.24 (a), (b) and (c), the modeling parameters predict the plastic rotation at shear failure fairly well but are a little bit unconservative for the plastic rotation at axial load failure. For columns in specimen HCFS shown in Figure 6.24 (d), (e) and (f), the modeling parameters a and b seem to be very conservative. The beam to column strength ratio was relatively low for specimen HCFS of which columns were subjected to high axial load ratio and the extra flexibility contributed by the beam caused the total inter-story drift to be larger than the actual column deformation. Yavari (2011) suggested using the chord rotation (definition is introduced in section 2.4.2) instead to describe the column deformation. The chord rotation  t at the top of the columns in NCREE 2009 Tests could be estimated as 64% of the total rotation (inter-story drift ratio) based on the flexure rigidity of the columns and beams (Yavari, 2011).  t  0.64   L   0.64total  Equation 6.6  If using the chord rotation instead, better agreement between the column behaviour and ASCE/SEI 41 modeling parameters could be achieved for columns in NCREE 2009 Tests.  186  Similarly, the deformation of columns connected with flexure beams shown in Figure 6.25, Figure 6.26 and Figure 6.27 is also greater than the actual column drift and modelling parameters appeared to be very conservative. Cautious should be taken when evaluating the drift capacities of these columns and the chord rotations or plastic rotation at column ends probably should be used instead.  187  Chapter 7: Conclusions and Future Work 7.1  Summaries The Shaking Table Tests Column Database comprised of 59 reinforced concrete  columns from seven test programs is compiled in this research. Key geometrical information, material properties and test data for each column are provided to facilitate the researchers to investigate the behavior of columns subjected to seismic load. Based on the test data of 36 non-ductile columns with light transverse reinforcement ( t  0.007 ), this research evaluates several available structural models for non-ductile columns that are typically used in old reinforced concrete structures, such as column effective stiffness models and the drift capacities models at shear failure and axial-load failure. The provisions of current seismic rehabilitation standard ASCE/SEI 41 Supplement 1 regarding the concrete columns are also evaluated with The Database. The standard generally overestimates the column effective stiffness and could either underestimate or overestimate the shear strength. The modeling parameters provide fairly conservative estimate of the column drift capacities and satisfied the targeted probability of failure. Refinements of the shear strength model and criteria for column classification are suggested. Columns are subjected to varying axial load during the shaking table tests. Especially for outside columns in a frame with multiple bays, the difference between axial load in the positive and negative direction could be very significant due to the overturning moment. When evaluating the models estimating column shear strength and drift capacities, the maximum axial load ratio is believed to better represent the column axial load level; while in estimation of column effective stiffness, it is appropriate use the initial axial load ratio.  188  It is noticed that the non-ductile columns included in The Database had very similar experimental setups and column detailing, such as transverse reinforcement ratio t , aspect ratio a / d and the ratio of transverse reinforcement spacing to column depth s / d . Most columns are subjected to axial load ratio P / Ag f c' less than 0.25. Therefore the column data from The Database could be used to evaluate the available drift capacity models, but one should take cautious when developing new models based on this dataset.  7.2 7.2.1  Conclusions Column Effective Stiffness The measured column lateral stiffness keff is interpreted from lateral force-deformation  relations by following TEST method, which regards the point that column achieves 75% of maximum lateral resistance VMax as the first yield point. The PEER method is not selected, because it is difficult to identify the accurate first yield point from the section analysis with varying axial load and the lateral stiffness based on the selected first yield point for some columns (e.g. the first yield point with lateral resistance equal to 0.95VMax ) could not reflect the column initial response well. Most columns in The Database have fixed-fixed ends and the column effective stiffness EI eff could be calculated from lateral stiffness keff . ASCE/SEI 41 recommends estimating the effective stiffness based on the axial load ratio and it is found that ASCE/SEI 41 tends to greatly overestimate the column stiffness. The ratio of measured stiffness to calculated stiffness has a mean value equal to 0.66 and coefficient of variation equal to 27.5%.  189  Elwood and Eberhard (2009) proposed a three-component model to calculate the column yield displacement by explicitly taking into account the deformation due to the flexure behavior, shear response and bar slip effects. Results show that around half of deformation are caused by flexure behavior and the deformation due to the bar slip effects could count for 50% of the total deformation. The shear response contributed less than 5% for shear-critical columns and was negligible for flexure and flexure-shear critical columns. The ratio of measured effective stiffness EITEST to the stiffness EIThreeComp estimated by three-component model has a mean value of 1.09 and coefficient of variation of 24.8%. The simplified three-component model calibrated by Elwood and Eberhard (2009) is evaluated with The Database. The resulted stiffness ratio is close to that of three-component model for most columns, with a mean value of 0.89 and coefficient of variation of 35.4%.  7.2.2  Drift Capacities at Shear Failure The column drift at shear failure could be defined as the column drift at maximum  shear force (Max V ), maximum effective lateral force (Max Veff ) and 80% of maximum effective lateral force (80% Max Veff ). Most models are developed to estimate the column drift capacity at significant loss of lateral resistance, i.e. drift at 80% Max Veff . Compared with the cyclic test data for flexure-shear-critical columns that have similar axial load ratio and shear stress level, the measured drift at 80% Max Veff generally exceeds the calculated drift from most available models, possibly due to the fact that the dynamic effect and the column interactions within the frame could increase the column drift capacity.  190  Pujol et al. (1999) model provided a conservative estimation of the column drift at 80% Max Veff for flexure-shear critical columns. The performance of Kato and Ohnishi (2002) model proposed to estimate the column plastic drift at 80% Max Veff largely depends on the accuracy of estimated yield drift. When applying the measured yield drift, Kato and Ohnishi (2002) model could either overestimate or underestimate the column drift and there is larger scatter existing in the data. With calculated yield drift by three-component model, this model is likely to underestimate the column drift capacity for flexure-shear critical columns. The calculated column drift from the probability model proposed by Zhu et al. (2006) and the empirical linear models developed by Elwood and Moehle (2005) generally fell in a flat banded region, because most non-ductile columns have similar transverse reinforcement ratio t and relatively low axial load ratio. Both models tend to greatly underestimate the column drift capacity. Comparing the calculated plastic drift from Elwood and Moehle (2005) with the measured plastic drift, it is found that the flexure-shear-critical columns subjected to subduction earthquake (e.g. Chile earthquake) tend to have lower plastic drift capacity. An empirical plastic drift model is proposed to estimate the plastic drift at 80% Max Veff for columns subjected to general type of earthquake and subduction earthquake. The proposed plastic drift model could reflect the trend of the measured plastic drift pretty well, yet large variability existed in the data. All four models greatly overestimate the drift capacities of six short columns in NCREE 2007 Tests, indicating the shear failure mechanism of shear-critical columns may be  191  essentially different from flexure-shear-critical columns and the drift capacity should be calculated separately.  7.2.3  Drift Capacities at Axial-Load Failure This research suggests defining the axial-load failure based on the column lengthening  shortening, since it is closely related to the column damage (i.e. the sliding along the shear damaged plane). Though axial-load failure point represented the initiation of the condition that column lost support for gravity load, it is difficult to define the failure point directly based on axial load response. Past research also shows that column tends to loss gravity support when lateral force degraded to almost zero, yet using the shear degradation to define the axial load failure can be either impossible or erroneous. Elwood and Moehle (2005) proposed a shear-friction model by relating the effective coefficient of friction  with the measured column drift. The shear-friction model based on the maximum axial load ratio represented the mean value of the measured column drift at axial-load failure for the flexure-shear critical columns and described the variability in the data fairly well. If based on the initial axial load ratio, the model tends to overestimate the column drift capacity and the data is also more scattered. Kato and Ohnishi (2002) model greatly overestimate the column drift capacity at axialload failure and large variability existed in the data. The ratio of measure column drift to calculated drift by Zhu et al. (2006) probability model which was developed based on shear-friction model had a mean value of 1.25 and coefficient of variation around 30%, consistent with the intent of the model to provide a median estimation of column drift capacity at axial-load failure.  192  The shear-friction model and Zhu et al. (2006) probability model are developed based on the limited amount static cyclic test data of non-ductile columns that have been tested up to the axial-load failure. However, both models are capable of representing the mean value of the measured column drift at axial-load failure for flexure-shear-critical columns included in The Database, indicating that the dynamic tests did not increase much of the ultimate column drift capacity and the shear-friction models could capture the failure mechanism properly. Yet the performance of the models depends on the estimation of the column axial load level and underestimation of column axial load may result in unconservative calculated column drift capacity.  7.2.4  Evaluation of ASCE/SEI 41 The current seismic rehabilitation standard ASCE/SEI 41 recommends calculating the  column shear strength based on the Sezen (2004) model and reducing the contribution from concrete blocks for non-ductile columns with transverse reinforcement spacing larger than half of the cross-section depth ( s  d 2 ). Results show the reduction due to the ductility demand and large spacing transverse reinforcement could lead to a very conservative estimation of column strength. When following the ASCE/SEI 41 method to classify columns, the reduction on the shear strength should be ignored. It is also found that some flexure-shear critical columns were misclassified into Condition iii with the ratio of VP V0' slightly higher than 1.0. Thus this research suggests modifying the criteria of columns in Condition ii as 0.6  VP V0'  1.1 instead of 0.6  VP V0'  1.0 .  193  The modeling parameters of a and b for columns in Condition ii and Condition iii are evaluated with The Database. It is found that parameter a tends to underestimate the column plastic drift at significant loss of lateral strength and there is a11% probability of failure for columns in “Condition ii” subjected to plastic rotation demand equal to parameter a specified in ASCE/SEI 41. Parameter b also greatly underestimates the plastic drift capacity at axial load failure, with the probability of failure around 7.5%. The maximum measured plastic drifts of flexure-critical columns are used to evaluate the modeling parameter b for columns in Condition i. Results show that the axial load ratio greatly influences the column plastic drift capacity and parameter b specified in ASCE/SEI 41 generally provides conservative estimate of the plastic drift capacity at axial-load failure.  7.3  Future Research Inter-story drift ratio has been commonly used as the indicator of the column drift  capacity in engineering practice. However, the inter-story column drift included the deformation contributed from the joint and beam response. The deformation measured over certain length (plastic hinge length) at column ends or the chord rotation should be reordered in future shaking table tests. Evaluation of the available drift capacity models showed that large variability existed in the total column drift. The models based on the plastic drift or the plastic rotation at column ends should be developed to better describe the column drift capacity. The drift capacity and lateral stiffness of columns in shaking table tests are likely to be higher than that of columns subjected to cyclic tests. High strain rate in longitudinal reinforcement is measured during the dynamic test and strain rate effect should be further  194  studied. Most strain gages installed in the specimen were damaged and not much reliable strain rate data was available. In future tests, the way that strain gages are installed and protected shall be improved. The input ground motion casts great influence on the column behaviour, such as the number of cycles and drift capacity. Future dynamic tests with greater range of input ground motion could be conducted on the frame specimens and the effect of input ground motions on the structural behaviour should be further studied. Parameters representing the characteristics of ground motions, such as the PGA (peak ground acceleration), duration and number of cycles, probably could be introduced in estimation of the column drift capacity. Most flexure-shear critical columns in The Database had similar geometrical properties, transverse confinements and axial load level, yet the drift capacity at shear failure and axialload failure varied significantly. The displacement histories (the number of cycles that column went through), strain rate effect, energy dissipation, column interactions within the frame and the mechanism of axial load redistribution may also cast great influence on the column behaviour and drift capacities. Further research on the moment frame rehabilitation should pay more attention on those issues. While Shear-Friction model represented the failure mechanism and column drift capacity at axial load failure fairly well, the shear critical columns expected to suffer pure shear failure may lose the gravity load support suddenly and faster. More tests should be done on shear critical columns to investigate the column behaviour and failure mechanism.  195  References  ASCE 2008. Seismic Rehabilitation of Existing Buildings. ASCE/SEI 41, Supplement 1. Reston,VA: American Society of Civil Engineers. Berry, M. Parrish, M. & Eberhard, M. 2004. PEER Structural Performance Database User‟s Manual, Pacific Earthquake Engineering Research Center, University of California, Berkeley. [available at: www.ce.washington.edu/~peera1]. Camarillo, H. 2003. Evaluation of Shear Strength Methodologies for Reinforced Concrete Columns. Master, University of Washington, Seattle. Elwood, K. 2003. Shaking Table Tests and Analytical Studies on the Gravity Load Collapse of Reinforced Concrete Frames. Ph.D. Dissertation, University of California, Berkeley. Elwood, K., Eeri, M., Matamoros, A., Wallace, J., Lehman, D., Heintz, J., Mitchell, A., Moore, M., Valley, M. & Lowes, L. 2007. Update to ASCE/SEI 41 Concrete Provisions. Earthquake Spectra, 23, 493. Elwood, K. J. & Eberhard, M. O. 2009. Effective Stiffness of Reinforced Concrete Columns. ACI Structural Journal, 106 (4), 476-484. Elwood, K. J. & Moehle, J. P. 2005.Drift Capacity of Reinforced Concrete Columns with Light Transverse Reinforcement. Earthquake Spectra, 21, 71-89. Ghannoum, W. M. 2007. Experimental and Analytical Dynamic Collapse Study of a Reinforced Concrete Frame with Light Transverse Reinforcement. Ph.D. Dissertation, University of California, Berkeley.  196  Kato, D. & Ohnishi, K. 2002. Axial Load Carrying Capacity of R/C Columns Under Lateral Load Reversals. Third US-Japan Workshop on Performance Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, Seattle, WA, PEER Report. 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Institute for Research in construction, Ottawa, Ontario, Canada. Pujol, S., Ramirez, J. A. & Sozen, M. A. 1999. Drift Capacity of Reinforced Concrete Columns Subjected to Cyclic Shear Reversals. American Concrete Institute, Farmington Hills, Michigan, SP-187, 255-274.  197  Sezen, H. 2002. Seismic response and modeling of reinforced concrete building columns, Ph.D. Dissertation, Department of Civil and Environmental Engineering, University of California, Berkeley. Sezen, H. & Moehle, J. 2004. Shear strength model for lightly reinforced concrete columns. ASCE Journal of Structural Engineering, 130, 1692. Shin, Y. B. 2007. Dynamic Response of Ductile and Non-Ductile Reinforced Concrete Columns. Ph.D. Dissertation, University of California, Berkeley. Su, R.S. 2007. Shake table tests on reinforced concrete short columns failed in shear. Master thesis, Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. Wu C.L., Yang Y.S. & Loh C.H. 2006. Dynamic Gravity Load Collapse of Non-ductile RC Frames I : Experimental Approach. 8th US National Conference on Earthquake Engineering - the 100th Anniversary Earthquake Conference Commemorating the 1906 San Francisco Earthquake, San Francisco, Moscone Center, April 18-22. Wu, C.L., Loh, C.H. & Yang, Y.S. 2006. Shaking table tests on gravity load collapse of lowductility RC frames under near-fault earthquake excitation. Advances in Experimental Structural Engineering, Itoh and Aoki, Eds: 725-732. Wu, C. L., Hwang, S. J., Yang, Y. S. & Su, R. S. 2007. Shake Table Tests on Reinforced Concrete Short Columns Failing in Shear. National Center for Research on Earthquake Engineering. Wu, C. L., Kuo, W. W., Hwang, S. J., Yang, Y. S. & Loh, C. H. 2009. Study on the Collapse Behavior of Nonductile Reinforced Concrete Frames Subjected to Earthquake Loadings. National Center for Research on Earthquake Engineering.  198  Wu, C. L., Kuo, W. W., Yang, Y. S., Hwang, S. J., Elwood, K. J., Loh, C. H. & Moehle, J. P. 2009. Collapse of a Non-Ductile Concrete Frame: Shaking Table Tests. Earthquake Engineering & Structural Dynamics, 38, 205-244. Yang Y.S., Wu C.L., Loh C.H., Lin C.H. & Chao S.H. 2006. Dynamic Gravity Load Collapse of Non-ductile RC Frames II : Computational Approach. 8th US National Conference on Earthquake Engineering - the 100th Anniversary Earthquake Conference Commemorating the 1906 San Francisco Earthquake, San Francisco, Moscone Center, April 18-22. Yavari, S., Elwood, K. J. & Wu, C. L. 2009. Collapse of a non-ductile concrete frame: Evaluation of analytical models. Earthquake Engineering & Structural Dynamics, 38, 225-241.  199  Appendices Appendix A Properties of the Shaking Table Tests Column Database Column Information Specimen Column Scale  Geometry D  b  L  Long. Rein. f c'  dl  c  (mm) (mm) (mm) (Mpa) (mm) NCREE 2009 Tests 1 MCFS A1 1/2.25 2 MCFS B1 1/2.25 3 MCFS C1 1/2.25 4 HCFS A1 1/2.25 5 HCFS B1 1/2.25 6 HCFS C1 1/2.25 7 MUF A1 1/2.25 8 MUF B1 1/2.25 9 MUF C1 1/2.25 10 MUFS A1 1/2.25 11 MUFS B1 1/2.25 12 MUFS C1 1/2.25 NCREE 2007 Tests 13 S1 C1 1/2 14 S1 C2 1/2 15 S1 C3 1/2 16 S3 C1 1/2  1400 1400 1400 1400 1400 1400 1400 1400 1400 1400 1400 1400  34.0 34.0 34.0 34.4 34.4 34.4 35.8 35.8 35.8 36.5 36.5 36.5  12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7  fyl  l  (Mpa)  dt  Trans. Rein. Tie s fyt Type  (mm)  t  (mm) (Mpa) (%)  200 200 200 200 200 200 200 200 200 200 200 200  200 200 200 200 200 200 200 200 200 200 200 200  17 17 17 17 17 17 17 17 17 17 17 17  439.0 439.0 439.0 439.0 439.0 439.0 467.0 467.0 467.0 467.0 467.0 467.0  2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53% 2.53%  5 5 5 5 5 5 5 5 5 5 5 5  r90 r90 r90 r90 r90 r90 r135 r135 r135 r90 r90 r90  120 120 120 120 120 120 40 40 40 120 120 120  469.0 469.0 469.0 469.0 469.0 469.0 475.0 475.0 475.0 475.0 475.0 475.0  0.16% 0.16% 0.16% 0.16% 0.16% 0.16% 0.49% 0.49% 0.49% 0.16% 0.16% 0.16%  250 250 250 250  250 1000 29.9 12.7 16 250 1000 29.9 12.7 16 250 1000 29.9 12.7 16 250 750 29.9 12.7 16  436.4 436.4 436.4 436.4  1.62% 3.24% 1.62% 3.24%  4 4 4 4  r135 50 643.3 0.30% r90 150 643.3 0.07% r135 50 643.3 0.30% r90 150 643.3 0.07%  Classification Failure Vp V0' Type (kN)  (kN)  61.91 68.63 61.34 68.57 72.71 68.23 60.58 69.17 60.68 60.08 69.38 60.40  59.79 66.95 59.28 66.82 78.23 75.78 119.45 118.03 120.85 54.05 72.83 69.42  FS  114.93 189.68 170.74 88.21 116.60 190.99 246.86 136.80  FS  FS FS FS F F F FS FS FS FS S  S FS S  200  Column Information Specimen Column Scale  Geometry D  b  L  Long. Rein. f c'  dl  c  (mm) (mm) (mm) (Mpa) (mm) 17 18 19  S3 C2 1/2 S4 C1 1/2 S4 C2 1/2 NCREE 2005 Tests 20 P2 C1 1/3 21 P2 C2 1/3 22 P2 C3 1/3 23 P2 C4 1/3 24 L C1 1/3 25 L C2 1/3 26 L C3 1/3 27 L C4 1/3 NCREE 2004 Tests 28 S2 C1 1/2 29 S2 C2 1/2 30 S3 C1 1/2 31 S3 C2 1/2 Ghannoum 2006 Tests 32 S1 A1 1/3 33 S1 B1 1/3 34 S1 C1 1/3 35 S1 D1 1/3  fyl  l  (Mpa)  dt  Trans. Rein. Tie s fyt Type  (mm)  250 250 750 29.9 12.7 16 436.4 3.24% 250 250 1000 29.9 12.7 16 436.4 3.24% 250 250 1000 29.9 12.7 16 436.4 3.24%  t  (mm) (Mpa) (%)  Classification Failure Vp V0' Type (kN)  (kN)  4 4 4  r90 r90 r90  150 643.3 0.07% 249.90 140.29 150 643.3 0.07% 177.36 99.69 150 643.3 0.07% 183.18 106.21  S  F  S S  150 150 150 150 150 150 150 150  150 150 150 150 150 150 150 150  1000 1000 1000 1000 1000 1000 1000 1000  33.8 33.8 33.8 33.8 32.3 32.3 32.3 32.3  9.53 9.53 6 6 9.53 9.53 6 6  14 14 12 12 14 14 12 12  470.9 470.9 231.7 231.7 470.9 470.9 231.7 231.7  2.53% 2.53% 1.39% 1.39% 2.53% 2.53% 1.39% 1.39%  3.2 3.2 5 5 3.2 3.2 5 5  r90 r90 r135 r135 r90 r90 r135 r135  100 100 33 33 100 100 33 33  548.1 548.1 661.9 661.9 548.1 548.1 661.9 661.9  0.16% 0.16% 1.19% 1.19% 0.16% 0.16% 1.19% 1.19%  33.85 31.98 21.16 13.90 33.64 39.15 16.18 15.00  36.33 32.44 176.12 171.04 34.06 38.84 172.81 171.97  200 200 200 200  200 200 200 200  1730 1730 1730 1730  31.8 31.8 31.8 31.8  9.53 9.53 9.53 9.53  9.5 9.5 9.5 9.5  385.9 385.9 385.9 385.9  1.43% 1.43% 1.43% 1.43%  4 4 4 4  r90 r90 r90 r90  125 125 125 125  385.9 385.9 385.9 385.9  0.10% 0.10% 0.10% 0.10%  30.62 30.66 30.88 30.88  39.21 39.24 39.41 39.40  FS  152 152 152 152  152 152 152 152  991 991 991 991  24.6 24.6 24.6 24.6  9.53 9.53 6.35 6.35  17 17 17 17  441.3 441.3 482.6 482.6  2.45% 2.45% 1.09% 1.09%  3.18 3.18 4.76 4.76  r90 r90 r135 r135  102 102 31.8 31.8  655.0 655.0 556.3 556.3  0.15% 0.15% 1.10% 1.10%  32.51 34.88 23.04 19.19  36.35 39.93 143.63 140.03  F  F FS FS F F FS FS  FS FS FS  F FS FS  201  Column Information Specimen Column Scale  Geometry D  b  L  Long. Rein. f c'  dl  (mm) (mm) (mm) (Mpa) (mm) Elwood 2002 Tests 36 S1 CenC 37 S2 CenC Shin 2005 Tests 38 Setup I EastC 39 Setup I WestC 40 Setup I EastC 41 Setup I WestC 42 Setup I EastC 43 Setup I WestC 44 Setup I EastC 45 Setup I WestC 46 Setup II EastC 47 Setup II WestC 48 Setup II EastC 49 Setup II WestC 50 Setup II EastC 51 Setup II WestC 52 Setup II EastC 53 Setup II WestC 54 Setup III EastC 55 Setup III WestC  c  fyl  l  (Mpa)  dt  Trans. Rein. Tie s fyt Type  (mm)  t  (mm) (Mpa) (%)  Classification Failure Vp V0' Type (kN)  (kN)  1/2 1/2  229 229 1473 24.6 12.7 25 479.3 2.48% 4.76 d90 152 689.7 0.18% 73.07 91.30 229 229 1473 23.9 12.7 25 479.3 2.48% 4.76 d90 152 689.7 0.18% 77.66 104.04  F  1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3  152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152  F  152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152  991 991 991 991 991 991 991 991 991 991 991 991 991 991 991 991 991 991  25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.6  9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53  12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 12.2 444.1 13.8 444.1 12.2 444.1 13.8 444.1 12.2 444.1 13.8 444.1 12.2 444.1 13.8 444.1 13.8 444.1 13.8 444.1  2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46% 2.46%  4.76 4.76 4.76 4.76 4.76 4.76 4.76 4.76 4.76 3.17 4.76 3.17 4.76 3.17 4.76 3.17 3.17 3.17  r135 r135 r135 r135 r135 r135 r135 r135 r135 r90 r135 r90 r135 r90 r135 r90 r90 r90  38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 102 38.1 102 38.1 102 38.1 102 102 102  549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 554.5 549.0 554.5 549.0 554.5 549.0 554.5 554.5 554.5  0.92% 0.92% 0.92% 0.92% 0.92% 0.92% 0.92% 0.92% 0.92% 0.15% 0.92% 0.15% 0.92% 0.15% 0.92% 0.15% 0.15% 0.15%  30.93 30.87 30.98 31.03 33.57 33.37 33.33 33.37 31.54 31.76 30.88 31.73 33.59 34.50 33.53 34.43 31.81 32.00  113.13 113.03 113.23 113.32 117.62 117.29 117.22 117.29 113.19 32.99 113.04 32.96 117.65 37.56 117.55 37.45 33.05 33.29  F  F F F F F F FS F FS F FS F FS FS FS FS FS  202  Column Information Specimen Column Scale  Geometry D  b  L  Long. Rein. f c'  dl  (mm) (mm) (mm) (Mpa) (mm) 56 57 58 59  Setup III Setup III Setup III Setup III  EastC WestC EastC WestC  1/3 1/3 1/3 1/3  152 152 152 152  152 152 152 152  991 991 991 991  25.6 25.6 25.6 25.6  9.53 9.53 9.53 9.53  c  fyl  l  (Mpa) 13.8 444.1 13.8 444.1 13.8 444.1 13.8 444.1  dt  Trans. Rein. Tie s fyt Type  (mm) 2.46% 2.46% 2.46% 2.46%  3.17 3.17 3.17 3.17  t  (mm) (Mpa) (%) r90 r90 r90 r90  102 102 102 102  554.5 554.5 554.5 554.5  0.15% 0.15% 0.15% 0.15%  Classification Failure Vp V0' Type (kN)  (kN)  31.98 31.86 34.29 34.29  33.25 33.12 37.22 37.42  FS FS  FS FS  Note: D = height of column section; b = width of column section; L= clear height of the column; fc‟= concrete compressive stress at test day; dl = the diameter of longitudinal reinforcement; c = clear cover (measured from the outer face of the concrete to outer face of longitudinal reinforcement); fyt = yield stress of longitudinal reinforcement; ρl = longitudinal reinforcement ratio; dt = the diameter of transverse reinforcement; Tie Type = the type of the transverse reinforcement, “r135” means rectangular-shaped hoops with 135-degree hooks and “d90” means diamond-shaped transverse reinforcement with 90-degree hooks; s = spacing of transverse reinforcement; fyl = yield stress of transverse reinforcement; ρt = volumetric ratio of transverse reinforcement; Vp = plastic shear demand obtained from section analysis; V0‟= the calculated shear strength from ASCE/SEI 41(discussed in section 6.2.3).  203  Appendix B Information of Test Series in Shaking Table Tests Column Database  ID  Column Name  Prestressed Initial Test Series Force* axial load Pini  Initial axial load ratio  Max Min axial load axial load ratio ratio  Pini/Agfc'  Pmax/Agfc' Pmin/Agfc'  (kN)  1  NCREE 2009-MCFS-A1  N  2  NCREE 2009-MCFS-B1  N  3  NCREE 2009-MCFS-C1  N  4  NCREE 2009-HCFS-A1  Y  5  NCREE 2009-HCFS-B1  Y  6  NCREE 2009-HCFS-C1  Y  7  NCREE 2009-MUFS-A1  N  8  NCREE 2009-MUFS-B1  N  9  NCREE 2009-MUFS-C1  N  10  NCREE 2009-MUF-A1  N  Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2  138.4 138.3 266.1 266.1 142.4 142.3 257.3 257.3 499.0 498.9 255.9 256.0 130.0 130.2 259.5 259.5 132.3 132.3 137.4 137.3  0.102 0.102 0.196 0.196 0.105 0.105 0.187 0.187 0.363 0.363 0.186 0.186 0.089 0.089 0.178 0.178 0.091 0.091 0.096 0.096  0.229 0.215 0.277 0.266 0.234 0.259 0.308 0.345 0.401 0.429 0.304 0.323 0.162 0.158 0.247 0.233 0.202 0.190 0.207 0.199  0.021 0.003 0.005 0.169 0.027 -0.005 0.098 0.080 0.342 -0.020 0.105 0.069 0.028 0.051 0.136 -0.020 0.016 0.021 0.039 0.046  PGA  Shear Max demand Hori. Drift Vmax  δ  max  (g)  (kN)  (%)  0.82 0.88 0.88 0.82 0.82 0.88 0.86 0.88 0.86 0.88 0.86 0.88 0.82 1.02 0.82 1.02 0.82 1.02 0.76 0.96  70.3 68.5 86.7 86.7 85.1 94.8 85.0 83.0 105.0 100.6 78.9 94.2 54.2 43.5 93.0 69.3 73.1 61.5 62.2 53.2  2.26 5.50 5.38 2.15 2.15 5.38 2.19 3.87 2.18 3.61 2.26 4.10 2.96 2.92 2.75 2.62 2.80 2.71 2.95 2.97  204  ID  Column Name  Prestressed Initial Test Series Force* axial load N  Pini  N  (kN)  11  NCREE 2009-MUF-B1  N  12  NCREE 2009-MUF-C1  N  13  NCREE 2007-S1-C1  N  14  NCREE 2007-S1-C2  N  15  NCREE 2007-S1-C3  N  16 17 18 19 20 21 22 23 24 25 26  NCREE 2007-S3-C1 NCREE 2007-S3-C3 NCREE 2007-S4-C1 NCREE 2007-S4-C3 NCREE 2005-P2-C1 NCREE 2005-P2-C2 NCREE 2005-P2-C3 NCREE 2005-P2-C4 NCREE 2005-L-C1 NCREE 2005-L-C2 NCREE 2005-L-C3  N N N N N N N N N N N  Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2  270.4 270.4 137.5 137.7 156.6 345.2 52.7 -319.8 166.8 360.0 170.2 191.7 102.7 152.1 68.3 19.7 110.7 39.8 36.8 102.9 60.6  Initial axial load ratio  Max Min axial load axial load ratio ratio  Pini/Agfc'  Pmax/Agfc' Pmin/Agfc'  0.189 0.189 0.096 0.096 0.084 0.185 0.028 -0.171 0.089 0.193 0.091 0.103 0.055 0.081 0.090 0.026 0.146 0.052 0.051 0.142 0.083  0.220 0.254 0.226 0.208 0.313 0.303 0.039 -0.168 0.322 0.296 0.150 0.165 0.091 0.118 0.150 0.111 0.428 0.132 0.111 0.161 0.135  0.148 0.166 0.037 0.036 0.042 0.158 -0.311 -0.319 0.035 0.174 0.045 0.018 0.007 0.005 0.020 -0.035 0.051 -0.111 0.028 0.034 0.051  PGA  Shear Max demand Hori. Drift Vmax  δ  max  (g)  (kN)  (%)  0.76 0.96 0.76 0.96 0.95 0.58 0.95 0.56 0.95 0.56 1.20 1.20 1.15 1.15 1.60 1.53 1.60 1.60 1.66 1.66 1.66  77.6 75.7 74.4 64.7 150.8 143.0 162.9 19.1 149.3 119.8 203.0 126.5 147.1 154.2 48.3 37.2 41.1 28.3 43.2 41.7 25.8  2.78 2.80 2.79 2.82 3.38 5.93 3.35 4.18 3.37 3.27 1.28 1.44 0.91 1.22 4.98 4.98 11.00 11.00 8.50 8.50 8.50  205  ID  Column Name  Prestressed Initial Test Series Force* axial load Pini  Initial axial load ratio  Max Min axial load axial load ratio ratio  Pini/Agfc'  Pmax/Agfc' Pmin/Agfc'  (kN)  27 28 29 30 31  NCREE 2005-L-C4 NCREE 2004-S2-C1 NCREE 2004-S2-C2 NCREE 2004-S3-C1 NCREE 2004-S3-C2  N N N N N  32  Ghannoum 2006-A1  N  33  Ghannoum 2006-B1  N  34  Ghannoum 2006-C1  N  35  Ghannoum 2006-D1  N  DT 1 DT 2 DT 3 HY DT 1 DT 2 DT 3 HY DT 1 DT 2 DT 3 HY DT 1 DT 2 DT 3 HY  49.7 142.4 142.4 145.9 145.9 45.3 63.7 75.3 46.3 97.1 50.3 26.1 96.9 90.0 130.2 148.5 90.6 41.4 30.3 27.1 41.5  0.068 0.112 0.112 0.115 0.115 0.079 0.111 0.132 0.081 0.170 0.088 0.046 0.169 0.157 0.228 0.260 0.158 0.072 0.053 0.047 0.073  0.127 0.148 0.150 0.180 0.145 0.209 0.224 0.168 0.128 0.246 0.128 0.055 0.182 0.265 0.352 0.402 0.168 0.148 0.137 0.091 0.114  0.035 0.062 0.064 0.089 0.084 -0.036 0.047 0.090 0.038 0.034 -0.016 0.012 0.154 0.098 0.166 0.198 0.143 -0.025 -0.030 -0.036 0.034  PGA  Shear Max demand Hori. Drift Vmax  δ  max  (g)  (kN)  (%)  1.66 0.17 0.17 0.63 0.63 N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A.  29.9 41.4 45.6 38.4 37.7 45.5 35.5 14.2 18.8 47.4 11.4 9.3 19.6 33.7 31.1 31.0 17.4 30.9 19.0 14.1 14.6  8.50 4.31 4.31 4.49 4.49 4.66 4.17 1.88 0.58 5.11 4.46 2.19 0.59 5.41 4.72 5.05 0.62 5.58 5.09 4.28 0.51  206  ID  Column Name  Prestressed Initial Test Series Force* axial load Pini  Initial axial load ratio  Max Min axial load axial load ratio ratio  Pini/Agfc'  Pmax/Agfc' Pmin/Agfc'  (kN)  36 37 38 39 40 41 42 43 44 45 46 47 48  Elwood 2002-S1-CenC Elwood 2002-S2-CenC Shin 2005-I-Test 3-Chile-0.1EastC Shin 2005-I-Test 3-Chile-0.1WestC Shin 2005-I-Test 5-Kobe-0.1EastC Shin 2005-I-Test 5-Kobe-0.1WestC Shin 2005-I-Test 9-Chile0.24-EastC Shin 2005-I-Test 9-Chile0.24-WestC Shin 2005-I-Test 11-Kobe0.24-EastC Shin 2005-I-Test 11-Kobe0.24-WestC Shin 2005-II-Test 1-Chile0.1-EastC Shin 2005-II-Test 1-Chile0.1-WestC Shin 2005-II-Test 2-Kobe0.1-EastC  Y Y  PGA  Shear Max demand Hori. Drift Vmax  δ  max  (g)  (kN)  (%)  127.9 298.4  0.100 0.239  0.122 0.251  -0.006 0.033  0.79 0.73  83.2 93.6  6.07 7.24  N  Test 3  50.8  0.085  0.134  0.021  1.13  42.1  5.39  N  Test 3  49.5  0.083  0.133  0.037  1.13  39.7  5.39  N  Test 5  52.2  0.088  0.141  0.063  0.90  43.3  8.07  N  Test 5  53.4  0.090  0.122  0.046  0.90  37.7  8.07  Y  Test 9  119.4  0.201  0.248  0.148  1.06  44.9  4.84  Y  Test 9  119.4  0.201  0.253  0.148  1.06  42.7  4.84  Y  Test 11  112.5  0.189  0.247  0.151  0.87  47.4  5.95  Y  Test 11  113.7  0.191  0.227  0.153  0.87  43.4  5.95  N  Test 1  51.6  0.087  0.131  0.044  1.22  41.9  6.94  N  Test 1  50.9  0.086  0.126  0.044  1.01  39.1  4.26  N  Test 2  49.5  0.083  0.132  0.046  0.89  43.9  8.93  207  ID  Column Name  Prestressed Initial Test Series Force* axial load Pini (kN)  Initial axial load ratio  Max Min axial load axial load ratio ratio  Pini/Agfc'  Pmax/Agfc' Pmin/Agfc'  PGA  Shear Max demand Hori. Drift Vmax  (g)  (kN)  δ  max  (%)  Shin 2005-II-Test 2-KobeN Test 2 50.5 0.085 0.120 0.034 0.89 37.1 4.36 0.1-WestC Shin 2005-II-Test 7-Chile50 Y Test 7 119.8 0.202 0.353 0.071 1.18 47.0 6.93 0.24-EastC Shin 2005-II-Test 7-Chile51 Y Test 7 119.9 0.202 0.242 0.137 0.83 43.7 3.40 0.24-WestC Shin 2005-II-Test 8-Kobe52 Y Test 8 118.2 0.199 0.375 0.027 0.96 46.9 6.42 0.24-EastC Shin 2005-II-Test 8-Kobe53 Y Test 8 118.0 0.199 0.228 0.144 0.96 42.6 3.41 0.24-WestC Shin 2005-III-Test 4-Chile54 N Test 4 51.7 0.087 0.132 0.046 0.81 38.2 3.83 0.1-EastC Shin 2005-III-Test 4-Chile55 N Test 4 54.9 0.092 0.132 0.053 0.81 37.6 3.83 0.1-WestC Shin 2005-III-Test 6-Kobe56 N Test 6 54.4 0.092 0.139 0.061 1.00 41.4 3.35 0.1-EastC Shin 2005-III-Test 6-Kobe57 N Test 6 52.5 0.088 0.118 0.033 1.00 37.2 4.28 0.1-WestC Shin 2005-III-Test 10-Chile58 Y Test 10 114.2 0.192 0.238 0.148 0.87 43.2 3.16 0.24-EastC Shin 2005-III-Test 10-Chile59 Y Test 10 117.6 0.198 0.234 0.155 0.87 40.9 2.48 0.24-WestC *In order to achieve high column axial load level in shaking table tests, prestressed force (e.g. by using pneumatic jack) might be applied to increase the column axial load on top of the lead weight packets. “Y” means that prestressed force is applied during the tests and “N” means the opposite. 49  208  

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